{
  "$comment_zh": "AMRAL 符號全域對照表：符號、定義、出處，純文字/JSON，不依賴 SEDB。",
  "$comment_en": "AMRAL global symbol reference: symbol, definition, source. Plain JSON, no SEDB dependency to read or use.",
  "generated_by": "見證, 2026-08-24",
  "count": 1845,
  "symbols": [
    {
      "id": "ns.framework.u",
      "latex": "u(x,t)",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "速度場",
      "label_en": "velocity field",
      "definition_zh": "Navier–Stokes 方程的未知速度場，全系列每一輪的分析對象。",
      "definition_en": "The unknown Navier-Stokes velocity field; the object of analysis across every round of the program."
    },
    {
      "id": "ns.framework.p",
      "latex": "p",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "壓力",
      "label_en": "pressure",
      "definition_zh": "Navier–Stokes 壓力場，透過 Biot–Savart / Calderón–Zygmund 型核由速度場非局部決定。",
      "definition_en": "The Navier-Stokes pressure field, determined nonlocally from the velocity field via a Biot-Savart / Calderon-Zygmund type kernel."
    },
    {
      "id": "ns.framework.Tstar",
      "latex": "T^\\ast",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "假設的有限爆炸時刻",
      "label_en": "hypothetical finite blow-up time",
      "definition_zh": "整個 C1-C6 論證都在 blow-up 假設 t\\uparrow T^\\ast 下逐步排除或收緊候選機制。",
      "definition_en": "The whole C1-C6 argument works under the hypothesis of blow-up as t approaches T* from below, progressively excluding or narrowing candidate mechanisms."
    },
    {
      "id": "ns.framework.nu",
      "latex": "\\nu",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "運動黏滯係數",
      "label_en": "kinematic viscosity",
      "definition_zh": "Navier–Stokes 方程中的黏滯係數，出現在拋物光滑化與能量耗散估計中。",
      "definition_en": "The viscosity coefficient in the Navier-Stokes equations; appears in parabolic smoothing and energy dissipation estimates."
    },
    {
      "id": "ns.framework.S",
      "latex": "S = \\nabla_{\\rm sym}u",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "應變張量",
      "label_en": "strain tensor",
      "definition_zh": "速度場的對稱梯度，GP（幾何-壓力）與 TS 標籤幾何分析的核心對象。",
      "definition_en": "The symmetric gradient of the velocity field; the central object of the GP (geometry-pressure) and TS geometric analysis.",
      "defining_relation": "S = \\nabla_{\\rm sym}u"
    },
    {
      "id": "ns.framework.omega",
      "latex": "\\omega = \\nabla\\times u",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "渦度",
      "label_en": "vorticity",
      "definition_zh": "速度場的旋度，與應變張量共同構成 Miller 應變-渦度恆等式等外部結果的輸入。",
      "definition_en": "The curl of the velocity field; feeds, alongside the strain tensor, into external results such as Miller's strain-vorticity identity.",
      "defining_relation": "\\omega = \\nabla\\times u"
    },
    {
      "id": "ns.framework.Q",
      "latex": "Q",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "應變-渦度二次不變量",
      "label_en": "strain/vorticity quadratic invariant",
      "definition_zh": "應變與渦度構成的二次張量不變量，從 C6-D 起是 GP 標籤幾何質量的主要載體。",
      "definition_en": "A quadratic tensor invariant built from strain and vorticity; from C6-D onward it is the main carrier of GP-label geometric mass."
    },
    {
      "id": "ns.framework.L3",
      "latex": "L^3",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "臨界 Lebesgue 空間",
      "label_en": "critical Lebesgue space",
      "definition_zh": "Navier–Stokes 的尺度臨界 Lebesgue 空間；‖u‖_3 是 C6-J 起 backward Leray 分析中不變的臨界範數之一。",
      "definition_en": "The scale-critical Lebesgue space for Navier-Stokes; the norm |u|_3 is one of the two scale-invariant critical norms used from C6-J onward."
    },
    {
      "id": "ns.framework.Hhalf",
      "latex": "\\dot H^{1/2}",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "臨界齊次 Sobolev 空間",
      "label_en": "critical homogeneous Sobolev space",
      "definition_zh": "另一個尺度臨界空間，C6-J 起與 L^3 並列作為 blow-up 序列必須發散的臨界拓撲，C6-M 起也是頻譜可見度通道的宿主空間。",
      "definition_en": "A second scale-critical space; from C6-J onward, used alongside L^3 as a critical topology in which the blow-up sequence must diverge, and from C6-M onward also hosts the spectral visibility channel."
    },
    {
      "id": "ns.framework.alphabet6",
      "latex": "\\{A,T,G,P,H,F\\}",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "六元殘餘缺陷字母表",
      "label_en": "six-class residual defect alphabet",
      "definition_zh": "C5-M 封階時把所有復發倖存者狀態壓成的六元有限字母表；C6 的起點是稽核其中三個候選復發循環。",
      "definition_en": "The six-letter finite alphabet C5-M compressed every recurrent survivor state into at its phase closure; C6 opens by auditing three candidate recurrent cycles drawn from it.",
      "notes": "C6-A audits the three candidates T, G<->P, H<->F drawn from this alphabet."
    },
    {
      "id": "ns.framework.TS",
      "latex": "TS",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "時空共享來源遺傳型缺陷類",
      "label_en": "temporal-spatial shared-source hereditary defect class",
      "definition_zh": "C6-E/F 精煉後留下的三個 minimal-survivor 候選之一，追蹤共享中間/算子來源密度的時空遺傳性。",
      "definition_en": "One of the three minimal-survivor candidates surviving C6-E/F's refinement; tracks the spacetime heredity of shared middle/operator source densities.",
      "notes": "Detailed definition: ns.c6.c6e.ts_state (C6-E), refined in ns.c6.c6g.ts_node (C6-G)."
    },
    {
      "id": "ns.framework.GP",
      "latex": "GP",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "幾何-壓力遺傳型缺陷類",
      "label_en": "geometry-pressure hereditary defect class",
      "definition_zh": "三個 minimal-survivor 候選之一，追蹤應變/Q 幾何與壓力簽名的遺傳耦合。",
      "definition_en": "One of the three minimal-survivor candidates; tracks the hereditary coupling between strain/Q geometry and pressure signature.",
      "notes": "Detailed definition: ns.c6.c6g.gp_node (C6-G)."
    },
    {
      "id": "ns.framework.HF",
      "latex": "HF",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "高階強迫非線性相干缺陷類",
      "label_en": "high-order forcing nonlinear-coherent defect class",
      "definition_zh": "三個 minimal-survivor 候選之一，追蹤高階導數 sign-thick 核心與非線性 Duhamel 相干性。",
      "definition_en": "One of the three minimal-survivor candidates; tracks high-order derivative sign-thick cores and nonlinear Duhamel coherence.",
      "notes": "Detailed definition: ns.c6.c6g.hf_node (C6-G)."
    },
    {
      "id": "ns.framework.Kij",
      "latex": "K_{ij}(x)\\sim|x|^{-3}",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "壓力 Biot–Savart 型核",
      "label_en": "pressure Biot-Savart-type kernel",
      "definition_zh": "全空間壓力核，p=K_{ij}*(u_iu_j)；其二階導數核 |\\nabla^2K_{ij}(x)|\\lesssim|x|^{-5} 是 C6-O/P/Q 壓力尾端估計的關鍵可積性來源。",
      "definition_en": "The whole-space pressure kernel, p = K_ij * (u_i u_j); its second derivative |grad^2 K_ij(x)| <~ |x|^-5 is the key integrable-decay fact behind the pressure-tail closure results of C6-O/P/Q."
    },
    {
      "id": "ns.c6.Un",
      "latex": "U_n",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "臨界場 blow-up 序列",
      "label_en": "critical blow-up field sequence",
      "definition_zh": "backward-Leray 重整化下的臨界場序列，C6-K 起所有奇異載體分析的起點物件。",
      "definition_en": "The critical field sequence under backward-Leray renormalization; the starting object for all singular-carrier analysis from C6-K onward."
    },
    {
      "id": "ns.c6.mu_n",
      "latex": "\\mu_n",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "正規化臨界 L^3 機率測度",
      "label_en": "normalized critical L^3 probability measure",
      "definition_zh": "把發散的絕對質量 M_n 與其分布形狀分離：d\\mu_n = |U_n|^3/\\|U_n\\|_3^3\\,dx。",
      "definition_en": "Separates diverging absolute mass M_n from its distributional shape: dmu_n = |U_n|^3 / |U_n|_3^3 dx.",
      "defining_relation": "d\\mu_n = |U_n|^3/\\|U_n\\|_3^3\\,dx",
      "notes": "Renamed mu_{3,n} in C6-L onward when written in original (non-rescaled) physical variables u(t_n)."
    },
    {
      "id": "ns.c6.Mn",
      "latex": "M_n",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "絕對發散臨界質量",
      "label_en": "absolute diverging critical mass",
      "definition_zh": "即 \\|U_n\\|_3^3，隨 blow-up 假設發散的絕對質量部分。",
      "definition_en": "Equal to |U_n|_3^3; the diverging absolute-mass part under the blow-up hypothesis."
    },
    {
      "id": "ns.c6.chi_def_n",
      "latex": "\\chi_n^{def}(R)",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "缺陷可見度",
      "label_en": "defect visibility",
      "definition_zh": "已追蹤的缺陷核心在半徑 R 內看見的臨界質量比例；C6-K.4 Defect-Visibility Dichotomy 的主角。",
      "definition_en": "The fraction of critical mass the tracked defect core sees within radius R; the central object of C6-K.4's Defect-Visibility Dichotomy."
    },
    {
      "id": "ns.c6.rho_n_concentration",
      "latex": "\\rho_n",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "集中半徑",
      "label_en": "concentration radius",
      "definition_zh": "C6-K.2 Concentration-Radius Trichotomy 的核心參數：可趨零（次尺度）、趨有限正值（同尺度）、或趨無窮（多重/擴散）。",
      "definition_en": "The core parameter of the C6-K.2 Concentration-Radius Trichotomy: can shrink to zero (secondary-scale), converge to a finite positive value (same-scale), or diverge (multiplicity/diffusion)."
    },
    {
      "id": "ns.c6.Rfiber",
      "latex": "\\mathfrak R_{fiber}(K)",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "臨界纖維半徑",
      "label_en": "critical fiber radius",
      "definition_zh": "緊緻缺陷集合 K 的臨界纖維半徑；C6-J 的 Critical Fiber Escape Theorem 證明它對任何被無窮次造訪的緊緻集合必為無窮。",
      "definition_en": "The critical fiber radius of a compact defect set K; C6-J's Critical Fiber Escape Theorem proves it must be infinite for any compact set visited infinitely often."
    },
    {
      "id": "ns.c6.phi_j_profiles",
      "latex": "\\phi_j",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "正交尺度/核心剖面",
      "label_en": "orthogonal scale/core profiles",
      "definition_zh": "在振幅正規化後套用 Gallagher–Koch–Planchon 剖面分解得到的正交剖面；C6-K.10 明確警告這些只是形狀分類器，不是 N–S 對稱下的子解。",
      "definition_en": "Orthogonal profiles obtained by applying Gallagher-Koch-Planchon profile decomposition after amplitude normalization; C6-K.10 explicitly warns these are shape classifiers only, not N-S-symmetric daughter solutions."
    },
    {
      "id": "ns.c6.Vn_amplitude",
      "latex": "V_n = U_n/L_n",
      "series": "NS",
      "first_appearance": "C6-K",
      "label_zh": "振幅正規化輔助形狀序列",
      "label_en": "amplitude-normalized auxiliary shape sequence",
      "definition_zh": "除以發散振幅後得到 \\|V_n\\|_3=1 的序列，是合法套用剖面分解的前提，但除法本身不是 N–S 對稱。",
      "definition_en": "The sequence normalized by the diverging amplitude so that |V_n|_3 = 1, a precondition for legally applying profile decomposition -- though the division itself is not an N-S symmetry.",
      "defining_relation": "V_n = U_n/L_n,\\ \\|V_n\\|_3=1"
    },
    {
      "id": "ns.c6.eta_n_carrier",
      "latex": "\\eta_n",
      "series": "NS",
      "first_appearance": "C6-L",
      "label_zh": "缺陷載體機率",
      "label_en": "defect carrier probability",
      "definition_zh": "每個標籤（TS/GP/HF）自己的正規化載體機率測度，TS 用共享來源密度、GP 用 Q-加權幾何、HF 用 sign 高集合構造。",
      "definition_en": "Each label's (TS/GP/HF) own normalized carrier probability measure, built from the shared-source density for TS, Q-weighted geometry for GP, and the component/sign high-set indicator for HF."
    },
    {
      "id": "ns.c6.Omega_D3",
      "latex": "\\Omega_{D3,n}",
      "series": "NS",
      "first_appearance": "C6-L",
      "label_zh": "缺陷/奇異質量可見度重疊係數",
      "label_en": "defect/singular-mass visibility overlap coefficient",
      "definition_zh": "\\Omega_{D3,n}=1-d_{TV}(\\mu_n,\\eta_n)\\in[0,1]；全系列第一個把奇異臨界質量與缺陷標籤放進同一個測度論物件的量。",
      "definition_en": "Omega_D3,n = 1 - d_TV(mu_n, eta_n) in [0,1]; the first quantity in the series to put singular critical mass and defect labels into one measure-theoretic object.",
      "defining_relation": "\\Omega_{D3,n}=1-d_{TV}(\\mu_n,\\eta_n)"
    },
    {
      "id": "ns.c6.xi_n_joint",
      "latex": "\\xi_n = (\\mu_n\\wedge\\eta_n)/\\Omega_{D3,n}",
      "series": "NS",
      "first_appearance": "C6-L",
      "label_zh": "聯合可見載體測度",
      "label_en": "joint visible carrier measure",
      "definition_zh": "當 \\Omega_{D3,n}\\ge\\omega_0>0 時可建出的聯合測度，使任何被它看見的集合同時被 \\mu_n 和 \\eta_n 看見。",
      "definition_en": "The joint measure extractable when Omega_D3,n >= omega_0 > 0, such that any set it sees is simultaneously seen by both mu_n and eta_n."
    },
    {
      "id": "ns.c6.An_separating",
      "latex": "A_n",
      "series": "NS",
      "first_appearance": "C6-L",
      "label_zh": "漸近測度分離集",
      "label_en": "asymptotic measure separating set",
      "definition_zh": "Spectator Separation Theorem 抽出的集合，使 \\mu_n(A_n)\\to1 同時 \\eta_n(A_n)\\to0：可見度趨零是真正的漸近測度分離，不只是重疊偏低。",
      "definition_en": "The set extracted by the Spectator Separation Theorem such that mu_n(A_n) -> 1 while eta_n(A_n) -> 0: vanishing visibility is genuine asymptotic measure separation, not merely weak correlation."
    },
    {
      "id": "ns.c6.ell_n",
      "latex": "\\ell_n",
      "series": "NS",
      "first_appearance": "C6-L",
      "label_zh": "質量載體尺度 / 內尺度",
      "label_en": "critical-mass carrier scale / inner rescaling length",
      "definition_zh": "C6-L 用於次尺度物理重縮放的內尺度；C6-N 起同一符號改指相對主導質量核心的球半徑，兩者概念相通但輪次間不完全同一物件，見 notes。",
      "definition_en": "The inner scale used for C6-L's physical secondary-scale rescaling; from C6-N onward the same symbol denotes the radius of a relative-dominant mass core -- related but not literally the identical object across rounds, see notes.",
      "notes": "Track this symbol's exact referent per-round when cross-referencing; it is reused with closely related but not always identical meaning from C6-L through C6-Q."
    },
    {
      "id": "ns.c6.Hn_plus",
      "latex": "H_n^+ = \\rho_n^{-2}",
      "series": "NS",
      "first_appearance": "C6-L",
      "label_zh": "次尺度內層未來視界",
      "label_en": "secondary-scale inner future horizon",
      "definition_zh": "C6-L.8 Secondary-Scale Horizon Theorem 的核心量：內層核心越深，H_n^+\\to\\infty，即原本的終結時刻在其自身拋物鐘裡變得無窮遠。",
      "definition_en": "The central quantity of C6-L.8's Secondary-Scale Horizon Theorem: as the inner core deepens, H_n^+ -> infinity, i.e. the original terminal time recedes to infinite distance on the inner core's own parabolic clock.",
      "defining_relation": "H_n^+ = \\rho_n^{-2}"
    },
    {
      "id": "ns.c6.Sigma_n_LP",
      "latex": "\\Sigma_n(q,x)",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "LP 臨界頻譜相空間機率測度",
      "label_en": "Littlewood-Paley critical spectral phase-space probability measure",
      "definition_zh": "d\\Sigma_n(q,x)=2^q|\\Delta_qU_n|^2/\\sum_j2^j\\|\\Delta_jU_n\\|_2^2\\,dx；把可見度的第一個通道從 L^3 擴展到頻譜/\\dot H^{1/2}。",
      "definition_en": "dSigma_n(q,x) = 2^q |Delta_q U_n|^2 / sum_j 2^j |Delta_j U_n|_2^2 dx; extends the visibility framework's first channel from L^3 to the spectral/H-dot^{1/2} domain.",
      "defining_relation": "d\\Sigma_n(q,x)=\\dfrac{2^q|\\Delta_qU_n|^2}{\\sum_j2^j\\|\\Delta_jU_n\\|_2^2}dx"
    },
    {
      "id": "ns.c6.sigma_n_spatial",
      "latex": "\\sigma_n(x)",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "頻譜可見度空間邊際",
      "label_en": "spectral visibility spatial marginal",
      "definition_zh": "\\Sigma_n 的空間邊際，是 C6-L 質量測度 \\mu_n 的 \\dot H^{1/2} 版本。",
      "definition_en": "The spatial marginal of Sigma_n; the H-dot^{1/2} analogue of C6-L's mass measure mu_n."
    },
    {
      "id": "ns.c6.Omega_DH",
      "latex": "\\Omega_{DH,n}",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "頻譜缺陷可見度",
      "label_en": "spectral defect visibility",
      "definition_zh": "\\Omega_{DH,n}=1-d_{TV}(\\sigma_n,\\eta_n)；證明 L^3-spectator 剖面依然可能是 \\dot H^{1/2}-visible carrier 的核心量。",
      "definition_en": "Omega_DH,n = 1 - d_TV(sigma_n, eta_n); the central quantity proving an L^3-spectator profile can still be a Dot-H^{1/2}-visible carrier.",
      "defining_relation": "\\Omega_{DH,n}=1-d_{TV}(\\sigma_n,\\eta_n)"
    },
    {
      "id": "ns.c6.CP_capacity",
      "latex": "C_P",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "遠場壓力容量",
      "label_en": "far-pressure capacity",
      "definition_zh": "C_P=\\int|a_P|，把壓力升格成第三個 carrier channel 的第一步；C6-M.5 給出分離剖面下的定量上界 C_P^{far}\\lesssim d^{-5}\\|v\\|_2^2。",
      "definition_en": "C_P = integral of |a_P|; the first step promoting pressure to a third carrier channel. C6-M.5 gives the quantitative bound C_P^far <~ d^-5 |v|_2^2 for separated profiles.",
      "defining_relation": "C_P=\\int|a_P|"
    },
    {
      "id": "ns.c6.GammaP_coherence",
      "latex": "\\Gamma_P = R_P/C_P",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "遠場壓力相干度",
      "label_en": "far-pressure coherence",
      "definition_zh": "定向遠場壓力來源的相干度，與容量 C_P 一起構成 Pressure Alignment Identity。",
      "definition_en": "The coherence of the oriented far-pressure source, forming the Pressure Alignment Identity together with the capacity C_P.",
      "defining_relation": "\\Gamma_P = R_P/C_P"
    },
    {
      "id": "ns.c6.piP_plus",
      "latex": "\\pi_P^+",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "同向壓力來源機率",
      "label_en": "aligned same-sign pressure-source probability",
      "definition_zh": "與奇異質量測度 \\mu_3 取重疊 \\Omega_{3P}^+，用於 Pressure-Coherent Singular-Carrier Theorem。",
      "definition_en": "Overlapped with the singular-mass measure mu_3 to give Omega_{3P}^+, feeding the Pressure-Coherent Singular-Carrier Theorem."
    },
    {
      "id": "ns.c6.aj_retention",
      "latex": "a_j = \\mu(C_{j+1})/\\mu(C_j)",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "逐層巢狀載體保留比",
      "label_en": "per-level nested carrier retention ratio",
      "definition_zh": "巢狀重綁定每一層的質量保留比例；C6-N.12 之後證明真正無限深的固定比例巢狀鏈在單一光滑切片上不可能，須理解成跨世代漸近現象。",
      "definition_en": "The mass retention ratio at each level of nested rebinding; C6-N.12 later proves a truly infinite fixed-fraction nested chain is impossible within one smooth slice, and must be read as a cross-generation asymptotic phenomenon instead.",
      "defining_relation": "a_j = \\mu(C_{j+1})/\\mu(C_j)"
    },
    {
      "id": "ns.c6.beta_m_product",
      "latex": "\\beta_m = \\beta_0\\prod_{j<m}a_j",
      "series": "NS",
      "first_appearance": "C6-M",
      "label_zh": "巢狀保留乘積",
      "label_en": "nested retention product",
      "definition_zh": "深度 m 層後的保留比例；Finite Loss-Count Theorem 與 Asymptotically Lossless Nesting Principle 的核心量。",
      "definition_en": "The retention fraction after m nesting levels; the central quantity of the Finite Loss-Count Theorem and the Asymptotically Lossless Nesting Principle.",
      "defining_relation": "\\beta_m = \\beta_0\\prod_{j<m}a_j"
    },
    {
      "id": "ns.c6.m3abs",
      "latex": "m_{3,n}^{abs} = \\int_{C_n}|u|^3dx",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "絕對臨界載體負載",
      "label_en": "absolute critical carrier load",
      "definition_zh": "在固定核心上的臨界質量積分，N-S 重縮放下無因次；C6-N 提出的「正確一般奇異載體判準」就建立在這個量上，而非相對比例。",
      "definition_en": "The critical mass integral over a fixed core, dimensionless under N-S rescaling; C6-N's proposed correct general singular-carrier criterion is built on this quantity rather than on relative fraction.",
      "defining_relation": "m_{3,n}^{abs} = \\int_{C_n}|u(x,t_n)|^3dx"
    },
    {
      "id": "ns.c6.chi3rel",
      "latex": "\\chi_{3,n}^{rel} = \\mu_{3,n}(C_n)",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "相對臨界質量比例",
      "label_en": "relative critical mass fraction",
      "definition_zh": "核心承擔的全域相對 L^3 比例；C6-N.1 Carrier Visibility Hierarchy 證明 \\chi_{3,n}^{rel}\\to0 不蘊含「非真正奇異載體」——這是對 C6-L/M 語義的永久性修正。",
      "definition_en": "The core's share of the global relative L^3 norm; C6-N.1's Carrier Visibility Hierarchy proves chi_{3,n}^rel -> 0 does NOT imply 'not a genuine singular carrier' -- a permanent correction to C6-L/M's semantics.",
      "defining_relation": "\\chi_{3,n}^{rel} = \\mu_{3,n}(C_n)"
    },
    {
      "id": "ns.c6.An_record",
      "latex": "A_n = \\|u(t_n)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "記錄峰值振幅",
      "label_en": "record peak amplitude",
      "definition_zh": "在 t\\le t_n 上的最大值記錄振幅，peak-amplitude rescaling 的正規化基準；C6-P 起要求它嚴格取歷史最大值（record time）以取得逆向有界性。",
      "definition_en": "The running-maximum amplitude on t <= t_n, the normalization base for peak-amplitude rescaling; from C6-P onward it is required to be a strict historical record (record time) to obtain backward boundedness.",
      "defining_relation": "A_n = \\|u(t_n)\\|_\\infty = \\max_{s\\le t_n}\\|u(s)\\|_\\infty"
    },
    {
      "id": "ns.c6.an_peakscale",
      "latex": "a_n = A_n^{-1}",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "峰值振幅尺度",
      "label_en": "peak amplitude scale",
      "definition_zh": "record 振幅的倒數，peak-amplitude N–S 重縮放的空間/時間標度單位；C6-N.6 證明相對主導質量載體必藏一個遠小於 \\ell_n 的 a_n。",
      "definition_en": "The reciprocal of the record amplitude; the spatial/temporal scale unit of peak-amplitude N-S rescaling. C6-N.6 proves a relative-dominant mass carrier necessarily hides an a_n far smaller than ell_n.",
      "defining_relation": "a_n = A_n^{-1}"
    },
    {
      "id": "ns.c6.AnC_local",
      "latex": "A_n^C",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "局部載體振幅",
      "label_en": "local carrier amplitude",
      "definition_zh": "\\|u(t_n)\\|_{L^\\infty(C_n)}，相對主導核心自身內部的最大振幅。",
      "definition_en": "The local L-infinity norm of u(t_n) restricted to the relative-dominant carrier C_n itself."
    },
    {
      "id": "ns.c6.TnC_typeII",
      "latex": "\\mathfrak T_n^C = \\ell_nA_n^C",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "載體尺度 Type-II 升級參數",
      "label_en": "carrier-scale Type-II escalation parameter",
      "definition_zh": "C6-N.2 Relative Carrier Amplitude Theorem 證明相對主導必迫使此參數發散，因此相對主導載體必屬 Type-II。",
      "definition_en": "C6-N.2's Relative Carrier Amplitude Theorem proves relative dominance forces this parameter to diverge, so a relative-dominant carrier must be Type-II.",
      "defining_relation": "\\mathfrak T_n^C = \\ell_nA_n^C \\to \\infty"
    },
    {
      "id": "ns.c6.vn_peakrescale",
      "latex": "v_n(z,\\tau)",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "峰值振幅 N–S 重縮放場",
      "label_en": "peak-amplitude N-S rescaled field",
      "definition_zh": "v_n(z,\\tau)=A_n^{-1}u(x_n+z/A_n,\\,t_n+\\tau/A_n^2)；record 性質給出 \\|v_n(\\tau)\\|_\\infty\\le1 對所有 \\tau\\le0。",
      "definition_en": "v_n(z,tau) = A_n^{-1} u(x_n + z/A_n, t_n + tau/A_n^2); the record property gives |v_n(tau)|_infty <= 1 for all tau <= 0.",
      "defining_relation": "v_n(z,\\tau) = A_n^{-1}u\\!\\left(x_n+\\dfrac{z}{A_n},\\,t_n+\\dfrac{\\tau}{A_n^2}\\right)"
    },
    {
      "id": "ns.c6.vinfty_ancient",
      "latex": "v_\\infty",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "有界 ancient N–S 極限解",
      "label_en": "bounded ancient N-S limit profile",
      "definition_zh": "v_n 沿子序列的局部極限，滿足 |v_\\infty(0,0)|=1，是非平凡有界 ancient 解；C6-P 證明其為 FLAT（恆定常數）或攜帶真正的 GP/HF/TS ancient 缺陷。",
      "definition_en": "The local subsequential limit of v_n, satisfying |v_infty(0,0)|=1, a nontrivial bounded ancient solution; C6-P proves it is either FLAT (a constant) or carries a genuine GP/HF/TS ancient defect."
    },
    {
      "id": "ns.c6.Hn_minus",
      "latex": "H_n^- = t_nA_n^2",
      "series": "NS",
      "first_appearance": "C6-N",
      "label_zh": "峰值重縮放的倒向壽命",
      "label_en": "backward lifetime of the peak rescaling",
      "definition_zh": "隨 t_n\\to T^\\ast、A_n\\to\\infty 而趨於無窮，保證每個固定 [-T,0] 最終落在重縮放定義域內。",
      "definition_en": "Tends to infinity as t_n -> T* and A_n -> infinity, guaranteeing every fixed interval [-T,0] eventually lies inside the rescaled domain.",
      "defining_relation": "H_n^- = t_nA_n^2 \\to \\infty"
    },
    {
      "id": "ns.c6.RnMP_ratio",
      "latex": "\\mathfrak R_n^{MP} = A_n\\ell_n",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "質量／峰值尺度比",
      "label_en": "mass/peak scale ratio",
      "definition_zh": "= \\ell_n/a_n；對 Type-II 相對主導載體發散到無窮，是質量核心在峰值框架中變成擴張球的半徑。",
      "definition_en": "Equal to ell_n/a_n; diverges to infinity for a Type-II relative-dominant carrier, and is the radius the mass core expands to in the peak frame.",
      "defining_relation": "\\mathfrak R_n^{MP} = \\dfrac{\\ell_n}{a_n} = A_n\\ell_n"
    },
    {
      "id": "ns.c6.mu3npeak",
      "latex": "\\mu_{3,n}^{peak}",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "峰值框架全域臨界質量機率",
      "label_en": "peak-frame global critical-mass probability",
      "definition_zh": "在峰值變數 z 下的正規化全域相對 L^3 質量測度；C6-O.1 證明它對每個固定球的測度趨零。",
      "definition_en": "The normalized global relative L^3 mass measure in peak variables z; C6-O.1 proves its measure on every fixed ball tends to zero."
    },
    {
      "id": "ns.c6.Theta3peak",
      "latex": "\\Theta_3^{peak}",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "全域質量峰值緊緻係數",
      "label_en": "peak tightness coefficient for global mass",
      "definition_zh": "\\Theta_3^{peak}=\\lim_{R\\to\\infty}\\liminf_n\\mu_{3,n}^{peak}(B_R)；C6-O.1 證明此係數恆為 0，即全域相對質量在峰值框架中完全不緊。",
      "definition_en": "Theta_3^peak = lim_{R->infty} liminf_n mu_{3,n}^peak(B_R); C6-O.1 proves this coefficient is always 0 -- global relative mass is fully non-tight in the peak frame.",
      "defining_relation": "\\Theta_3^{peak}=\\lim_{R\\to\\infty}\\liminf_n\\mu_{3,n}^{peak}(B_R) = 0"
    },
    {
      "id": "ns.c6.etaDpeak",
      "latex": "\\eta_n^{D,peak}",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "峰值框架缺陷載體推前測度",
      "label_en": "peak-frame pushforward defect carrier probability",
      "definition_zh": "把 \\eta_n^D 沿峰值座標映射 T_n^{peak}(z)=x_n+a_nz 推前到峰值框架的結果。",
      "definition_en": "The result of pushing eta_n^D forward along the peak-frame coordinate map T_n^peak(z) = x_n + a_n z."
    },
    {
      "id": "ns.c6.ThetaDpeak",
      "latex": "\\Theta_D^{peak}",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "缺陷峰值緊緻係數",
      "label_en": "defect peak tightness coefficient",
      "definition_zh": "缺陷載體在峰值框架中的緊緻度；三種狀態 O-DT(=1 峰值緊緻)、O-DP(部分)、O-DE(=0 峰值逃逸)。C6-O.2 證明若 =1 則此標籤必是全域相對 L^3 旁觀者。",
      "definition_en": "How tight the defect carrier is in the peak frame; three regimes O-DT (=1, peak-tight), O-DP (partial), O-DE (=0, peak-escape). C6-O.2 proves that if it equals 1, the label is necessarily a global relative L^3 spectator."
    },
    {
      "id": "ns.c6.Akpeak",
      "latex": "\\widehat A_{k,n}^{peak} = A_{k,n}/A_{0,n}^{k+1}",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "峰值正規化 k 階導數振幅",
      "label_en": "peak-normalized k-th derivative amplitude",
      "definition_zh": "無因次的 k 階導數峰值比；C6-O 的 Derivative-to-Peak Trichotomy 分成 flattening(→0)、peak-scale(→有限)、subpeak escape(→∞)三支，但 C6-P.2 證明對固定 k，第三支在 record peak 不可能發生。",
      "definition_en": "The dimensionless k-th-derivative-to-peak ratio; C6-O's Derivative-to-Peak Trichotomy splits into flattening (->0), peak-scale (->finite), and subpeak escape (->infinity), but C6-P.2 proves the third branch is impossible at record peaks for any fixed k.",
      "defining_relation": "\\widehat A_{k,n}^{peak} = \\dfrac{A_{k,n}}{A_{0,n}^{k+1}} = \\|D^kv_n(0)\\|_\\infty"
    },
    {
      "id": "ns.c6.bkn_subpeak",
      "latex": "b_{k,n} = A_{k,n}^{-1/(k+1)}",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "次峰值導數尺度",
      "label_en": "subpeak derivative scale",
      "definition_zh": "C6-O 提出的、疑似比峰值更小的導數尺度；C6-P.3 證明對固定 k，b_{k,n}/a_n 有一致正下界，此尺度不存在。",
      "definition_en": "The derivative scale C6-O proposed as possibly smaller than the peak; C6-P.3 proves for fixed k that b_{k,n}/a_n has a uniform positive lower bound, so this scale does not exist.",
      "defining_relation": "b_{k,n} = A_{k,n}^{-1/(k+1)}",
      "notes": "Superseded by C6-P.3 for every fixed k -- kept for historical/cross-reference reasons, not as a live candidate scale."
    },
    {
      "id": "ns.c6.Thetatnpeak",
      "latex": "\\Theta_{t,n}^{peak} = A_n^2\\Delta t_n",
      "series": "NS",
      "first_appearance": "C6-O",
      "label_zh": "峰值時間窗比",
      "label_en": "peak time-window ratio",
      "definition_zh": "把原始事件持續時間換算成峰值 ancient 時間 \\tau 的尺度；C6-O.7 的 Peak Time-Window Trichotomy 分 O-T0/O-T1/O-T∞ 三支。",
      "definition_en": "Converts the original event duration into peak ancient-time (tau) scale; C6-O.7's Peak Time-Window Trichotomy splits into O-T0/O-T1/O-T-infinity.",
      "defining_relation": "\\Theta_{t,n}^{peak} = A_n^2\\Delta t_n"
    },
    {
      "id": "ns.c6.Ck_bound",
      "latex": "C_k(\\nu)",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "固定階數的一致峰值導數界",
      "label_en": "uniform fixed-order record-peak derivative bound",
      "definition_zh": "C6-P.1 Fixed-Order Record-Peak Derivative Rigidity 由 bounded mild N–S 拋物光滑化（外部定理）給出，僅依賴 k、\\nu：\\|D^kv_n(0)\\|_\\infty\\le C_k。",
      "definition_en": "Given by C6-P.1's Fixed-Order Record-Peak Derivative Rigidity, using bounded-mild-solution parabolic smoothing (external theorem); depends only on k and nu: |D^k v_n(0)|_infty <= C_k.",
      "defining_relation": "\\|D^kv_n(0)\\|_\\infty \\le C_k(\\nu)"
    },
    {
      "id": "ns.c6.rho_an",
      "latex": "\\rho_{an}",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "一致正空間解析半徑",
      "label_en": "uniform positive spatial analytic radius",
      "definition_zh": "record peak 有界性經固定正光滑化延遲後給出的一致解析半徑，證明高階原始導數的階乘成長只是普通解析基線。",
      "definition_en": "The uniform analytic radius obtained after a fixed positive smoothing delay from record-peak boundedness; shows raw high-order derivative factorial growth is the ordinary analytic baseline, not a new concentration scale."
    },
    {
      "id": "ns.c6.Akan_normalized",
      "latex": "\\mathfrak A_{k,n}^{an}",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "解析正規化導數根",
      "label_en": "analytic-normalized derivative root",
      "definition_zh": "\\|D^kv_n(0)\\|_\\infty^{1/(k+1)}/(k!)^{1/(k+1)}；一致有界於 n,k，是高階逃逸必須改用的正確座標。",
      "definition_en": "|D^k v_n(0)|_infty^{1/(k+1)} / (k!)^{1/(k+1)}; uniformly bounded in n and k, and the correct coordinate a genuine high-order escape must instead use.",
      "defining_relation": "\\mathfrak A_{k,n}^{an} = \\dfrac{\\|D^kv_n(0)\\|_\\infty^{1/(k+1)}}{(k!)^{1/(k+1)}}"
    },
    {
      "id": "ns.c6.Rkpeak_root",
      "latex": "\\mathcal R_{k,n}^{peak}",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "峰值正規化 Grujić–Xu 導數根",
      "label_en": "peak-normalized Grujić-Xu derivative root",
      "definition_zh": "沿用 Grujić–Xu 判準已內建的階乘正規化根形式；峰值框架下對 n,k 一致有界，說明「order root blow-up」與固定正解析半徑不相容。",
      "definition_en": "Reuses the factorial-normalization form already built into the Grujic-Xu criterion; uniformly bounded in n and k in the peak frame, showing 'order root blow-up' is incompatible with a fixed positive analytic radius."
    },
    {
      "id": "ns.c6.b_flat_constant",
      "latex": "b",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "FLAT ancient 解的非零常數向量",
      "label_en": "nonzero constant vector of the FLAT ancient profile",
      "definition_zh": "若一階峰值比 \\widehat A_{1,n}^{peak}\\to0，則 v_\\infty(x,0)\\equiv b；backward uniqueness（外部定理）進一步強迫 v_\\infty(x,\\tau)\\equiv b 對所有 \\tau\\le0，這個分支不攜帶任何 TS/GP/HF 缺陷。",
      "definition_en": "If the first-order peak ratio tends to 0, v_infty(x,0) = b; backward uniqueness (external theorem) further forces v_infty(x,tau) = b for all tau <= 0, and this branch carries no TS/GP/HF defect at all.",
      "defining_relation": "v_\\infty(x,\\tau)\\equiv b,\\ |b|=1"
    },
    {
      "id": "ns.c6.Gammakloc",
      "latex": "\\Gamma_{k,n}^{loc}(R)",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "局部固定階導數捕獲比",
      "label_en": "local fixed-order derivative capture ratio",
      "definition_zh": "固定階導數在有界峰值球內的局部最大值與全域峰值最大值之比；C6-P.7 Fixed-Order Carrier Dichotomy 用它區分局部捕獲(P-KVIS)與 Derivative-Carrier Translation Escape(P-KESC)。",
      "definition_en": "The ratio of the local maximum of a fixed-order derivative within a bounded peak ball to the global peak maximum; used by C6-P.7's Fixed-Order Carrier Dichotomy to distinguish local capture (P-KVIS) from Derivative-Carrier Translation Escape (P-KESC)."
    },
    {
      "id": "ns.c6.ThetaGPanc",
      "latex": "\\Theta_{GP}^{anc}",
      "series": "NS",
      "first_appearance": "C6-P",
      "label_zh": "ancient GP 元資料元組",
      "label_en": "ancient GP metadata tuple",
      "definition_zh": "封裝 Q 負載、幾何錐、壓力 Hessian、有限半徑來源與簽章的 ancient GP 局部狀態；跨世代舊來源身分與核心圖不自動繼承。",
      "definition_en": "The ancient GP local-state tuple packaging Q load, geometric cone, pressure Hessian, finite-radius provenance, and signature; cross-generation old provenance/core-graph identity does not transfer automatically."
    },
    {
      "id": "ns.c6.E1_strain_energy",
      "latex": "E_1(t) = \\tfrac12\\|S(t)\\|_{\\dot H^1}^2",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "H^1 應變能量",
      "label_en": "H^1 strain energy",
      "definition_zh": "TS 算子通道真正使用的時間成長帳本；C6-Q.1 對它的成長率 E_1' 給出完全局部的精確空間密度代表。",
      "definition_en": "The temporal growth ledger the TS operator channel actually uses; C6-Q.1 gives its growth rate E_1' a fully local exact spatial density.",
      "defining_relation": "E_1(t) = \\tfrac12\\|S(t)\\|_{\\dot H^1}^2 = \\tfrac12\\|\\nabla S(t)\\|_2^2"
    },
    {
      "id": "ns.c6.gOloc",
      "latex": "g_O^{loc}",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "E_1' 的完全局部精確空間密度",
      "label_en": "fully local exact spatial density for E_1'",
      "definition_zh": "不含 P_{st}、不含壓力奇異積分的成長率密度；C6-Q.1 Projection-Free H^1 Growth Identity 的核心產物，之後成為偏好的 TS 成長載體提升(Local Growth Representative Lift)。",
      "definition_en": "The growth-rate density containing no P_st and no pressure singular integral; the core product of C6-Q.1's Projection-Free H^1 Growth Identity, later adopted as the preferred TS growth carrier (Local Growth Representative Lift).",
      "defining_relation": "g_O^{loc} = -[(u\\cdot\\nabla)S+S^2]\\!:\\!(-\\Delta S) - \\nu|\\Delta S|^2"
    },
    {
      "id": "ns.c6.gOproj",
      "latex": "g_O^{proj}",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "投影代表提升密度",
      "label_en": "projected representative lift density",
      "definition_zh": "C6-E 原始使用的成長密度代表，依賴 Q_{SV}=P_{st}(\\cdots)；C6-Q.2 指出它與 g_O^{loc} 同積分卻不必同空間分布，二者是同一時間邊際的不同載體模型，都需標記來源(lift provenance)。",
      "definition_en": "The growth density representative C6-E originally used, depending on Q_SV = P_st(...); C6-Q.2 points out it integrates to the same value as g_O^loc but need not share its spatial distribution -- both are distinct carrier models of the same temporal marginal, and each must record its lift provenance.",
      "notes": "In its original C6-E/F appearance this quantity was simply called g_O (see ns.c6.c6e.g_o, ns.c6.c6f.g_o); C6-Q introduces the g_O^{proj} name to distinguish it from the new projection-free g_O^{loc}."
    },
    {
      "id": "ns.c6.Pst_projection",
      "latex": "P_{st}",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "應變約束空間正交投影",
      "label_en": "orthogonal projection onto the strain constraint space",
      "definition_zh": "投影到 L^2_{st} 的算子，自 C6-E 起用於 TS 算子密度；C6-Q.4 從 Miller–Sawyer 等距同構推出顯式公式 P_{st}M=-2\\nabla_{\\rm sym}(-\\Delta)^{-1}P_{df}\\operatorname{div}M，證明它是零階 Calderón–Zygmund 矩陣算子。",
      "definition_en": "The orthogonal projection onto L^2_st, used in the TS operator density from C6-E onward; C6-Q.4 derives the explicit formula P_st M = -2 grad_sym (-Delta)^-1 P_df div M from the Miller-Sawyer isometry, proving it is a degree-zero Calderon-Zygmund matrix operator.",
      "defining_relation": "P_{st}M = -2\\nabla_{\\rm sym}(-\\Delta)^{-1}P_{df}\\operatorname{div}M",
      "notes": "First introduced in earlier rounds (C6-E/F) as an abstract projection; its explicit closed form is a C6-Q result, so this entry is filed under C6-Q's detailed pass even though the symbol itself predates it."
    },
    {
      "id": "ns.c6.Kst_kernel",
      "latex": "K_{st}(x) = |x|^{-3}\\Omega(x/|x|)",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "P_st 的顯式零階 CZ 核",
      "label_en": "explicit degree-zero Calderon-Zygmund kernel of P_st",
      "definition_zh": "|\\nabla K_{st}(x)|\\le C|x|^{-4}；對有界來源，遠場尾端振盪只有 O(R_0/R)，故 P_{st} 遠場非局部性被壓成有限維常數模式加消失振盪。",
      "definition_en": "|grad K_st(x)| <= C|x|^-4; for a bounded source, the far-tail oscillation is only O(R_0/R), so P_st's far-field nonlocality compresses to a finite-dimensional constant mode plus vanishing oscillation."
    },
    {
      "id": "ns.c6.CR_constant_mode",
      "latex": "C_R(\\tau)",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "P_st 遠端尾端的常數 STF 背景模式",
      "label_en": "constant STF background mode of the P_st far tail",
      "definition_zh": "投影到零跡對稱矩陣空間(五個純量自由度)的常數背景，是遠場非局部性壓縮後僅剩的有限維不確定性。",
      "definition_en": "The constant background projected into the space of trace-free symmetric matrices (five scalar degrees of freedom); the only finite-dimensional uncertainty left after compressing the far-field nonlocality."
    },
    {
      "id": "ns.c6.ER_oscillation",
      "latex": "E_R^F(x)",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "消失遠端尾端振盪餘項",
      "label_en": "vanishing far-tail oscillation remainder",
      "definition_zh": "\\|E_R^F\\|_{L^\\infty(B_{R_0})}\\le CR_0/R\\to0；Local-Constant Tail Reduction 的核心估計。",
      "definition_en": "|E_R^F|_{L-infty(B_{R_0})} <= C R_0/R -> 0; the central estimate of the Local-Constant Tail Reduction.",
      "defining_relation": "\\|E_R^F\\|_{L^\\infty(B_{R_0})} \\le C\\,R_0/R"
    },
    {
      "id": "ns.c6.QSV_criterion",
      "latex": "Q_{SV}",
      "series": "NS",
      "first_appearance": "framework",
      "label_zh": "Miller 投影算子範數判準場",
      "label_en": "Miller's projected operator-norm criterion field",
      "definition_zh": "Q_{SV}=P_{st}((u\\cdot\\nabla)S+S^2+\\tfrac34\\omega\\otimes\\omega)；C6-Q 明確區分 growth carrier(g_O^{loc}) 與 operator-norm carrier(Q_{SV}) 是不同物件，後者仍genuinely 需要 P_{st}。",
      "definition_en": "Q_SV = P_st((u.grad)S + S^2 + 3/4 omega tensor omega); C6-Q explicitly separates the growth carrier (g_O^loc) from the operator-norm carrier (Q_SV) as distinct objects -- the latter still genuinely needs P_st.",
      "defining_relation": "Q_{SV} = P_{st}\\!\\left((u\\cdot\\nabla)S+S^2+\\tfrac34\\omega\\otimes\\omega\\right)",
      "notes": "Originates in the TS/GP/HF sub-thread at ns.c6.c6e.q_sv (C6-E) and ns.c6.c6f.q_sv (C6-F), before this C6-Q entry's explicit growth-carrier/operator-norm-carrier distinction."
    },
    {
      "id": "ns.c6.wn_recenter",
      "latex": "w_n(y,\\tau) = v_n(y+z_n,\\tau)",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "載體重定心平移",
      "label_en": "carrier recentering translation",
      "definition_zh": "利用 N–S 平移對稱重新定心逃逸中的缺陷載體；record 有界性與固定階導數界在平移後保持一致。",
      "definition_en": "Uses the N-S translation symmetry to recenter an escaping defect carrier; record boundedness and fixed-order derivative bounds remain uniform after translation.",
      "defining_relation": "w_n(y,\\tau) = v_n(y+z_n,\\tau)"
    },
    {
      "id": "ns.c6.winfty_satellite",
      "latex": "w_\\infty",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "衛星 ancient 缺陷剖面",
      "label_en": "Satellite Ancient Defect Profile",
      "definition_zh": "w_n 沿子序列的局部極限，攜帶與原始逃逸載體相同的非零局部絕對缺陷負載；C6-Q.9 Satellite Ancient Defect Extraction 的產物，證明空間載體逃逸不是終點。",
      "definition_en": "The local subsequential limit of w_n, carrying the same nonzero local absolute defect load as the original escaping carrier; the product of C6-Q.9's Satellite Ancient Defect Extraction, proving spatial carrier escape is not terminal."
    },
    {
      "id": "ns.c6.Xn_physical_center",
      "latex": "X_n = x_n + a_nz_n",
      "series": "NS",
      "first_appearance": "C6-Q",
      "label_zh": "物理衛星中心",
      "label_en": "physical satellite center",
      "definition_zh": "把峰值變數 z_n 換回原始物理座標得到的衛星中心；C6-Q.10 Satellite Physical-Center Trichotomy 依 d_n^{phys}=a_n|z_n| 分三支。",
      "definition_en": "The satellite center in original physical coordinates, obtained by converting the peak-variable center z_n back; C6-Q.10's Satellite Physical-Center Trichotomy splits by d_n^phys = a_n |z_n|.",
      "defining_relation": "X_n = x_n + a_nz_n"
    },
    {
      "id": "ns.c6.c6a.mathfrak_d_c5",
      "latex": "\\mathfrak D_{C5}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "C5 殘餘狀態空間（六類）",
      "label_en": "C5 six-class residual state space",
      "definition_zh": "C5-M 將整個 C5 residual state space 壓縮後得到的六個殘餘缺陷類別集合 $\\{\\mathsf A,\\mathsf T,\\mathsf G,\\mathsf P,\\mathsf H,\\mathsf F\\}$，是 C6 全系列 cycle 分析的起點；C6-A 第2節隨即將 $\\mathsf A$ 移出 physical SCC 討論範圍，得到五類的 $V_6$。",
      "definition_en": "The six residual defect classes $\\{\\mathsf A,\\mathsf T,\\mathsf G,\\mathsf P,\\mathsf H,\\mathsf F\\}$ obtained by compressing the entire C5 residual state space in C5-M; this is the starting point for all C6 cycle analysis. C6-A section 2 immediately removes $\\mathsf A$ from physical-SCC consideration, yielding the five-class $V_6$.",
      "defining_relation": "\\mathfrak D_{C5} = \\{\\mathsf A, \\mathsf T, \\mathsf G, \\mathsf P, \\mathsf H, \\mathsf F\\}",
      "notes": "Uses sans-serif \\mathsf letters; compare with $V_6$, which restates the five physical classes in plain italic type (T,G,P,H,F) after excluding A."
    },
    {
      "id": "ns.c6.c6a.class_a",
      "latex": "\\mathsf A",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "A 類（ancestry／合法性）",
      "label_en": "Class A (ancestry / legality)",
      "definition_zh": "六個殘餘缺陷類別之一，代表 ancestry / legality / theorem setup 相關的缺陷；C6-A 第2節指出 $A$ 可能表示 proof/theorem-entry failure 而非實際物理奇異機制，因此暫不納入 physical SCC extraction，直到 theorem-entry legality 被證明為止（對應 X-Integration guard G-AOUT）。",
      "definition_en": "One of the six residual defect classes, representing ancestry / legality / theorem-setup defects. C6-A section 2 notes that $A$ may signal a proof/theorem-entry failure rather than an actual physical singularity mechanism, so it is kept out of physical SCC extraction until theorem-entry legality is proved (corresponding to guard G-AOUT).",
      "notes": "Kept outside $V_6$, the five-class physical state space used for the round's SCC/cycle analysis."
    },
    {
      "id": "ns.c6.c6a.class_t",
      "latex": "\\mathsf T",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "T 類（時間相位振盪）",
      "label_en": "Class T (temporal phase oscillation)",
      "definition_zh": "六個殘餘缺陷類別之一，代表 temporal phase oscillation / concentration 缺陷；第14節指出 $T$ 未被 C5-C 的 scalar-ledger no-go 排除，但僅止於 $T\\overset{N}{\\looparrowright}T$（non-exclusion），並非已證的 PDE self-cycle，故稱為 candidate trap。",
      "definition_en": "One of the six residual defect classes, representing temporal phase oscillation / concentration defects. Section 14 notes that $T$ is not eliminated by C5-C's scalar-ledger no-go, but this only amounts to $T\\overset{N}{\\looparrowright}T$ (non-exclusion), not a proved PDE self-cycle — hence $T$ is called a candidate trap rather than a certified cycle."
    },
    {
      "id": "ns.c6.c6a.class_g",
      "latex": "\\mathsf G",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "G 類（應變—渦度幾何）",
      "label_en": "Class G (strain-vorticity geometry)",
      "definition_zh": "六個殘餘缺陷類別之一，代表 strain-vorticity field geometry 缺陷；是 Miller middle-eigenvalue/strain-vorticity 工作的主要 anchor 之一，也是候選 $G\\leftrightarrow P$ cycle（C6-A.3，第16節）的一端。",
      "definition_en": "One of the six residual defect classes, representing strain-vorticity field geometry defects. It is one of the main anchors of Miller's middle-eigenvalue / strain-vorticity work, and forms one endpoint of the candidate $G\\leftrightarrow P$ cycle (C6-A.3, section 16)."
    },
    {
      "id": "ns.c6.c6a.class_p",
      "latex": "\\mathsf P",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "P 類（壓力補償／溯源）",
      "label_en": "Class P (pressure compensation / provenance)",
      "definition_zh": "六個殘餘缺陷類別之一，代表 pressure compensation / provenance 缺陷；第1.4節指出因 Bradshaw–Tsai 的 local pressure expansion 提供 rigorous provenance，$P$ 並非純粹現象學式的 graph node，也是候選 $G\\leftrightarrow P$ cycle 的另一端。",
      "definition_en": "One of the six residual defect classes, representing pressure compensation / provenance defects. Section 1.4 notes that because Bradshaw–Tsai's local pressure expansion supplies rigorous provenance, $P$ is not a purely phenomenological graph node; it is also the other endpoint of the candidate $G\\leftrightarrow P$ cycle."
    },
    {
      "id": "ns.c6.c6a.class_h",
      "latex": "\\mathsf H",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "H 類（高階諧波／定理窗口缺陷）",
      "label_en": "Class H (high-order harmonic / theorem-window defect)",
      "definition_zh": "六個殘餘缺陷類別之一，代表 high-order harmonic / theorem-window defect；Grujić–Xu 2024 framework 提供其主要 external kill gate $\\mathsf H\\to\\mathrm{REG}$，也是候選 $H\\leftrightarrow F$ cycle（C6-A.4，第20節，本輪優先級最高的攻擊目標）的一端。",
      "definition_en": "One of the six residual defect classes, representing high-order harmonic / theorem-window defects. The Grujić–Xu 2024 framework supplies its main external kill gate $\\mathsf H\\to\\mathrm{REG}$; $H$ is also one endpoint of the candidate $H\\leftrightarrow F$ cycle (C6-A.4, section 20), the round's top-priority target."
    },
    {
      "id": "ns.c6.c6a.class_f",
      "latex": "\\mathsf F",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "F 類（強迫／階數變化債務）",
      "label_en": "Class F (forcing / order variation debt)",
      "definition_zh": "六個殘餘缺陷類別之一，代表 forcing / order variation debt；$H\\to F$ 方向已有 C5-J/L 的 turnover/roughness routing 支持並匯入 $\\text{VISC}\\vee\\text{PROJECTED-NL}$，但反向 $F\\to H$ 是否成立正是 C6-A 判定為尚未 certified 且列為 C6-B 首要任務的關鍵缺口。",
      "definition_en": "One of the six residual defect classes, representing forcing / order variation debt. The forward direction $H\\to F$ is already supported by C5-J/L's turnover/roughness routing into $\\text{VISC}\\vee\\text{PROJECTED-NL}$, but whether the reverse $F\\to H$ holds is exactly the gap C6-A judges not yet certified and assigns as C6-B's top priority."
    },
    {
      "id": "ns.c6.c6a.v6",
      "latex": "V_6",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "五類物理殘餘狀態空間",
      "label_en": "Five-class physical residual state space",
      "definition_zh": "排除 $A$ 之後，代表已進入 physical/legal regime 的五個 physical residual classes $\\{T,G,P,H,F\\}$；是本輪 projected may-graph（第3節）、SCC 分解（第4節）與投影 $\\pi:\\mathcal K\\to V_6$（C6-A.1）操作的底層頂點集合。",
      "definition_en": "The five physical residual classes $\\{T,G,P,H,F\\}$ obtained after excluding $A$, representing states that have already entered the physical/legal regime. This is the underlying vertex set for the round's projected may-graph (section 3), its SCC decomposition (section 4), and the projection $\\pi:\\mathcal K\\to V_6$ used in C6-A.1.",
      "defining_relation": "V_6 = \\{T,G,P,H,F\\}",
      "notes": "Here the five labels are typeset in plain italic (T,G,P,H,F), not the sans-serif \\mathsf used for the same classes in $\\mathfrak D_{C5}$; both notations refer to the same five physical classes."
    },
    {
      "id": "ns.c6.c6a.reg",
      "latex": "\\mathrm{REG}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "外部正則性終止態",
      "label_en": "External regularity sink state",
      "definition_zh": "代表已證明之正則性定理成立時所導向的外部 sink 狀態，不是 C5 六個殘餘類別之一；是 E 型 edge（external kill edge）$X\\overset{E}{\\longrightarrow}\\mathrm{REG}$ 的目標，也是 Grujić–Xu 2024 framework 作為 $\\mathsf H\\to\\mathrm{REG}$ 主要 kill gate 之落點。",
      "definition_en": "The external sink state reached when a published regularity theorem applies; it is not one of the six C5 residual classes. It is the target of type-E (external kill) edges $X\\overset{E}{\\longrightarrow}\\mathrm{REG}$, and is the landing point of the Grujić–Xu 2024 framework's main kill gate $\\mathsf H\\to\\mathrm{REG}$.",
      "notes": "A certified-composable cycle must avoid automatically triggering REG (C6-A.2 condition 4, section 12); the same requirement reappears as MSC-3 in the minimal survivor cycle certificate (section 23)."
    },
    {
      "id": "ns.c6.c6a.state_space_k",
      "latex": "\\pi:\\mathcal K\\to V_6",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "完整缺陷狀態空間與投影",
      "label_en": "Full defect state space and projection",
      "definition_zh": "$\\mathcal K$ 是承載完整（未粗化）缺陷狀態 $\\theta\\in\\mathcal K$ 的空間，$\\pi$ 是把 $\\theta$ 投影到粗略殘餘標籤 $V_6$ 的映射；對每個類別 $X$，$\\mathcal K_X$（如 $\\mathcal K_T,\\mathcal K_G,\\mathcal K_P,\\mathcal K_H,\\mathcal K_F$，第8節列出各自的 compact/compactified metadata，例如 $\\mathcal K_G$ 含 middle-gap coordinate、strain-direction measure 等）是 $\\pi$ 在 $X$ 上的纖維，用作 typed transition relation $R_e\\subset\\mathcal K_X\\times\\mathcal K_Y$ 的定義域/值域。C6-A.1 正是利用 $\\pi$ 的粗化性質證明 label-level SCC 不足以推出 PDE recurrent cycle。",
      "definition_en": "$\\mathcal K$ is the space of full (uncoarsened) defect states $\\theta\\in\\mathcal K$, and $\\pi$ is the map projecting $\\theta$ down to the coarse residual labels $V_6$. For each class $X$, $\\mathcal K_X$ (e.g. $\\mathcal K_T,\\mathcal K_G,\\mathcal K_P,\\mathcal K_H,\\mathcal K_F$, whose per-class compact/compactified metadata are listed in section 8 — e.g. $\\mathcal K_G$ includes middle-gap coordinate, strain-direction measure, etc.) is the fiber of $\\pi$ over $X$, and serves as the domain/codomain of typed transition relations $R_e\\subset\\mathcal K_X\\times\\mathcal K_Y$. C6-A.1 uses exactly this coarsening to prove that label-level SCCs cannot by themselves certify a PDE recurrent cycle.",
      "notes": "$\\mathcal K_X$ is introduced separately in section 8 as \"full defect state spaces,\" but is definitionally the fiber $\\pi^{-1}(X)\\subset\\mathcal K$ used throughout sections 9-24."
    },
    {
      "id": "ns.c6.c6a.edge_status_i",
      "latex": "X \\overset{I}{\\longrightarrow} Y",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "I 型：蘊含邊",
      "label_en": "Type I: implication edge",
      "definition_zh": "C6-A 定義的四種 typed edge status 之一，代表一條已證明的 routing theorem，具有明確的 source antecedent 與 target conclusion，不需額外條件即成立；是「certified composable」判準（C6-A.2）中允許的兩種合法邊類型之一。",
      "definition_en": "One of the four typed edge statuses C6-A defines. It means a proved routing theorem with an explicit source antecedent and target conclusion, holding unconditionally. It is one of the two edge statuses (with C) permitted in the \"certified composable\" criterion (C6-A.2).",
      "notes": "Single-letter status code local to the C6 typed-edge formalism; not to be confused with an identity map or index variable."
    },
    {
      "id": "ns.c6.c6a.edge_status_c",
      "latex": "X \\overset{C}{\\longrightarrow} Y",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "C 型：條件蘊含邊",
      "label_en": "Type C: conditional implication edge",
      "definition_zh": "四種 typed edge status 之一，代表該 implication 僅在額外 gate 成立時才被證明，例如 strong-middle cone、common far pressure、ancestry legality、theorem setup、bounded turnover 等；C6-A.2 要求同一 cycle 中所有 conditional antecedents 必須彼此相容才能構成 certified composable cycle。",
      "definition_en": "One of the four typed edge statuses. It means the implication is proved only under an extra gate — e.g. strong-middle cone, common far pressure, ancestry legality, theorem setup, bounded turnover. C6-A.2 requires that all conditional antecedents around one cycle be mutually compatible for the cycle to be certified composable.",
      "notes": "Not to be confused with a generic constant $C$; here it is one of the four edge-status codes $\\{I,C,N,E\\}$."
    },
    {
      "id": "ns.c6.c6a.edge_status_n",
      "latex": "X \\overset{N}{\\looparrowright} X",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "N 型：不可排除自環",
      "label_en": "Type N: non-exclusion self-loop",
      "definition_zh": "四種 typed edge status 之一，表示目前估計並未排除在 $X$ 中持續 recurrence 的可能性，但這不是一條 transition theorem——它既不保證存在對應的 Navier–Stokes orbit，也不保證一個 $X$ event 動態產生另一個；第7節明確指出 N-edges 是 survivor obligations，不是 certified dynamics，因此不能單獨用來定義 SCC。",
      "definition_en": "One of the four typed edge statuses. It states that current estimates have not ruled out recurrence within $X$, but this is not a transition theorem — it guarantees neither the existence of a Navier–Stokes orbit realizing recurrent $X$, nor that one $X$ event dynamically generates another. Section 7 states explicitly that N-edges are survivor obligations, not certified dynamics, and so cannot by themselves define an SCC.",
      "defining_relation": "X \\overset{N}{\\looparrowright} X",
      "notes": "Uses the loop arrow \\looparrowright specifically for self-recurrence; candidate class $T$'s status is exactly $T\\overset{N}{\\looparrowright}T$ (section 14)."
    },
    {
      "id": "ns.c6.c6a.edge_status_e",
      "latex": "X \\overset{E}{\\longrightarrow} \\mathrm{REG}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "E 型：外部終止邊",
      "label_en": "Type E: external kill edge",
      "definition_zh": "四種 typed edge status 之一，表示一條已發表的 regularity theorem 在其 antecedents 成立時關閉該路線，導向外部 sink $\\mathrm{REG}$；certified composable cycle（C6-A.2 條件4）要求 cycle 上沒有任何狀態會自動觸發此類 external REG kill gate。",
      "definition_en": "One of the four typed edge statuses. It means a published regularity theorem closes the route when its antecedents hold, directing flow to the external sink $\\mathrm{REG}$. The certified-composable criterion (C6-A.2, condition 4) requires that no state on the cycle automatically triggers such an external REG kill gate.",
      "notes": "Not to be confused with expectation notation $E[\\cdot]$ or Euler's number; its target is always $\\mathrm{REG}$."
    },
    {
      "id": "ns.c6.c6a.typed_transition_relation",
      "latex": "R_e \\subset \\mathcal K_X \\times \\mathcal K_Y",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "型別化轉移關係",
      "label_en": "Typed transition relation",
      "definition_zh": "C6-A 對「一條 edge」的精細化定義：不再只是粗略標籤箭頭 $X\\to Y$，而是完整狀態空間之間的關係 $R_e\\subset\\mathcal K_X\\times\\mathcal K_Y$，並附帶 source antecedent $\\mathcal A_e$、target constraint $\\mathcal B_e$、proof status、debt vector $d_e$ 與 provenance/scale/time metadata；是 cycle compatibility fiber product $\\mathfrak C(e_1,\\ldots,e_m)$（第11節）的基本 building block。",
      "definition_en": "C6-A's refinement of what \"an edge\" means: not merely a coarse label arrow $X\\to Y$, but a relation $R_e\\subset\\mathcal K_X\\times\\mathcal K_Y$ between full state spaces, carrying a source antecedent $\\mathcal A_e$, a target constraint $\\mathcal B_e$, a proof status, a debt vector $d_e$, and provenance/scale/time metadata. It is the basic building block of the cycle-compatibility fiber product $\\mathfrak C(e_1,\\ldots,e_m)$ (section 11).",
      "defining_relation": "R_e \\subset \\mathcal K_X \\times \\mathcal K_Y",
      "notes": "Proposition C6-A.1 (section 5) first argues this point using the per-pair notation $R_{XY}\\subset\\mathcal K_X\\times\\mathcal K_Y$ and $R_{YX}\\subset\\mathcal K_Y\\times\\mathcal K_X$, before section 9 formalizes the general per-edge notation $R_e$. Also, section 9's own listing of proof status gives only $I,C,E$ (omitting $N$), while section 43's edge ledger confirms the complete status alphabet is $\\sigma_e\\in\\{I,C,N,E\\}$."
    },
    {
      "id": "ns.c6.c6a.source_antecedent",
      "latex": "\\mathcal A_e \\subset \\mathcal K_X",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "邊的來源前提",
      "label_en": "Edge source antecedent",
      "definition_zh": "型別化轉移關係 $R_e$ 的組成之一，指該 edge 生效所需的來源狀態子集；C6-A.1 的 No-Go 論證核心即在於由 $R_{XY}$ 產生的 target metadata 未必落在 $R_{YX}$ 的來源前提 $\\mathcal A_e$ 之內，因此 $X\\leftrightarrow Y$ 不蘊含 composable two-cycle。也是第43節 edge proof-status ledger 中每條邊的固定欄位之一。",
      "definition_en": "One of the components of a typed transition relation $R_e$: the subset of source states required for the edge to fire. The core of the C6-A.1 No-Go argument is that target metadata produced by $R_{XY}$ need not lie inside the source antecedent $\\mathcal A_e$ required by $R_{YX}$, so $X\\leftrightarrow Y$ does not imply a composable two-cycle. It is also one of the fixed fields recorded for every edge in the section 43 proof-status ledger.",
      "defining_relation": "\\mathcal A_e \\subset \\mathcal K_X"
    },
    {
      "id": "ns.c6.c6a.edge_debt_vector",
      "latex": "d_e \\in [0,\\infty]^m",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "邊債務向量",
      "label_en": "Edge debt vector",
      "definition_zh": "第9節將每條 typed edge 附帶一個非負債務向量 $d_e\\in[0,\\infty]^m$；第24節將其具體展開為 $d(e)=(d_T,d_G,d_P,d_H,d_F,d_{clock},d_{time},\\ldots)$，座標可以是 integral toll、defect mass、pressure critical mass、derivative load、root variation、clock variation、time span 等；是計算循環總債務 $D(C)=\\sum_j d(e_j)$ 與後續 critical saturation、debt coercivity 判準的基本單位。",
      "definition_en": "Section 9 attaches a nonnegative debt vector $d_e\\in[0,\\infty]^m$ to every typed edge; section 24 expands it explicitly as $d(e)=(d_T,d_G,d_P,d_H,d_F,d_{clock},d_{time},\\ldots)$, with coordinates that may be integral toll, defect mass, pressure critical mass, derivative load, root variation, clock variation, or time span. It is the basic unit used to compute the total cycle debt $D(C)=\\sum_j d(e_j)$ and the later critical-saturation and debt-coercivity criteria.",
      "defining_relation": "d(e) = (d_T, d_G, d_P, d_H, d_F, d_{clock}, d_{time}, \\ldots)",
      "notes": "First introduced in compact form as $d_e\\in[0,\\infty]^m$ (section 9); section 24 gives the expanded coordinate form $d(e)$. Distinct from $D(C)$, the sum of these vectors over a whole cycle."
    },
    {
      "id": "ns.c6.c6a.disjunctive_hyperedge",
      "latex": "X \\longrightarrow \\{Y_1,\\ldots,Y_m\\}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "析取超邊",
      "label_en": "Disjunctive hyperedge",
      "definition_zh": "用來編碼 $X\\Rightarrow Y_1\\vee Y_2\\vee Y_3$ 型態 C5 陳述的記號：一條指向目標集合的析取超邊，而非三條同時成立的 mandatory ordinary edges；例如 root turnover 的 $\\text{TURNOVER}\\Rightarrow\\text{VISC}\\vee\\text{PROJECTED-NL}$。C6-A.7（第37節）證明將此類超邊逐一投影成 ordinary edges 只能得到 necessary candidate regions，不是充分的 recurrent 證書。",
      "definition_en": "The notation used to encode C5 statements of the form $X\\Rightarrow Y_1\\vee Y_2\\vee Y_3$: a single disjunctive hyperedge pointing at a target set, rather than three simultaneously mandatory ordinary edges — e.g. root turnover's $\\text{TURNOVER}\\Rightarrow\\text{VISC}\\vee\\text{PROJECTED-NL}$. C6-A.7 (section 37) proves that projecting such hyperedges into ordinary edges one at a time yields only necessary candidate regions, not sufficient recurrent certificates.",
      "defining_relation": "X \\longrightarrow \\{Y_1,\\ldots,Y_m\\}",
      "notes": "Feeds directly into the candidate-trap viability condition (section 36): a trap $S$ must intersect every mandatory disjunctive target set, $S\\cap\\{Y_1,\\ldots,Y_m\\}\\ne\\varnothing$."
    },
    {
      "id": "ns.c6.c6a.coarse_bidirectional_edge",
      "latex": "X \\leftrightarrow Y",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "粗化圖雙向邊",
      "label_en": "Coarse-graph bidirectional edge",
      "definition_zh": "表示投影後粗圖中 $X$、$Y$ 兩類之間同時存在正反向 typed transition relations（$R_{XY}$ 與 $R_{YX}$ 皆非空）；C6-A.1（Projected-SCC No-Go，第5節）的核心結論是：此記號成立並不蘊含存在一對相容狀態 $\\theta_X,\\theta_Y$ 使得 $(\\theta_X,\\theta_Y)\\in R_{XY}$ 且 $(\\theta_Y,\\theta_X)\\in R_{YX}$，即不蘊含 composable two-cycle。",
      "definition_en": "Denotes that, in the projected coarse graph, both forward and reverse typed transition relations exist between classes $X$ and $Y$ (both $R_{XY}$ and $R_{YX}$ nonempty). The central conclusion of C6-A.1 (Projected-SCC No-Go, section 5) is that this notation holds without implying the existence of a compatible pair $\\theta_X,\\theta_Y$ with $(\\theta_X,\\theta_Y)\\in R_{XY}$ and $(\\theta_Y,\\theta_X)\\in R_{YX}$ — i.e. it does not imply a composable two-cycle.",
      "notes": "Applied concretely to the round's two main candidates $G\\leftrightarrow P$ (C6-A.3) and $H\\leftrightarrow F$ (C6-A.4), both judged not yet certified."
    },
    {
      "id": "ns.c6.c6a.cycle_compatibility_set",
      "latex": "\\mathfrak C(e_1,\\ldots,e_m)",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "循環相容性纖維積",
      "label_en": "Cycle compatibility fiber product",
      "definition_zh": "對一組首尾相接的 typed edges $e_1:X_1\\to X_2,\\ldots,e_m:X_m\\to X_1$，定義 $\\mathfrak C(e_1,\\ldots,e_m)=R_{e_1}\\times_{\\mathcal K_{X_2}}R_{e_2}\\times_{\\mathcal K_{X_3}}\\cdots\\times_{\\mathcal K_{X_1}}R_{e_m}$，要求每條邊的 target metadata 都滿足下一條邊的 source antecedent；此集合非空是 C6-A.2「certified composable」判準的條件3，也是判定 $G\\leftrightarrow P$、$H\\leftrightarrow F$ 尚未 certified 的關鍵物件。",
      "definition_en": "For a chain of typed edges closing into a cycle, $e_1:X_1\\to X_2,\\ldots,e_m:X_m\\to X_1$, defines $\\mathfrak C(e_1,\\ldots,e_m)=R_{e_1}\\times_{\\mathcal K_{X_2}}R_{e_2}\\times_{\\mathcal K_{X_3}}\\cdots\\times_{\\mathcal K_{X_1}}R_{e_m}$, requiring that every edge's target metadata satisfy the next edge's source antecedent. Its nonemptiness is condition 3 of the C6-A.2 \"certified composable\" criterion, and it is the key object used to judge $G\\leftrightarrow P$ and $H\\leftrightarrow F$ as not yet certified.",
      "defining_relation": "\\mathfrak C(e_1,\\ldots,e_m) = R_{e_1} \\times_{\\mathcal K_{X_2}} R_{e_2} \\times_{\\mathcal K_{X_3}} \\cdots \\times_{\\mathcal K_{X_1}} R_{e_m}",
      "notes": "Distinct from the later cycle-state tuple $\\mathfrak C^{C6}$ (section 46), which packages this fiber product together with a recurrence map, $D(C)$, and budget coercivity into one record — same fraktur C, different object."
    },
    {
      "id": "ns.c6.c6a.certified_composable",
      "latex": "\\textbf{certified composable}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "「已認證可組合」判準",
      "label_en": "\"Certified composable\" criterion",
      "definition_zh": "C6-A.2（第12節）為投影標籤循環 $X_1\\to\\cdots\\to X_m\\to X_1$ 定義的性質：唯有同時滿足（1）每條邊具 implication/conditional 證明狀態、（2）所有 conditional antecedents 互相相容、（3）$\\mathfrak C(e_1,\\ldots,e_m)\\ne\\varnothing$、（4）沒有目標狀態自動觸發 external REG kill gate、（5）遞迴迭代保持 legality/scale/time metadata，才能稱為 certified composable；其推論是 label SCC 只是 certified cycle structure 的 over-approximation。",
      "definition_en": "The property C6-A.2 (section 12) defines for a projected label cycle $X_1\\to\\cdots\\to X_m\\to X_1$: it is certified composable only if (1) every edge has implication/conditional proof status, (2) all conditional antecedents are mutually compatible, (3) $\\mathfrak C(e_1,\\ldots,e_m)\\ne\\varnothing$, (4) no target state automatically triggers an external REG kill gate, and (5) recurrent iteration preserves legality/scale/time metadata. Its consequence is that a label SCC is only an over-approximation of certified cycle structure."
    },
    {
      "id": "ns.c6.c6a.recurrent_subset",
      "latex": "\\mathcal R_C \\subset \\mathfrak C(e_1,\\ldots,e_m)",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "循環的遞迴不變子集",
      "label_en": "Recurrent invariant subset of a cycle",
      "definition_zh": "第13節指出，即使 $\\mathfrak C(e_1,\\ldots,e_m)\\ne\\varnothing$ 也只證明存在一次相容的迴圈；要得到無窮 recurrent cycling，需要一個不變或遞迴子集 $\\mathcal R_C\\subset\\mathfrak C(e_1,\\ldots,e_m)$，其一次循環後的像會與自身相交。此物件把「projected cycle／composable cycle／recurrent cycle」三個層級中最強的一級形式化，C6 明確要求不可混淆這三層。",
      "definition_en": "Section 13 notes that even $\\mathfrak C(e_1,\\ldots,e_m)\\ne\\varnothing$ only certifies one compatible loop; obtaining infinite recurrent cycling requires an invariant or recurrent subset $\\mathcal R_C\\subset\\mathfrak C(e_1,\\ldots,e_m)$ whose image after one cycle intersects itself. This formalizes the strongest of the paper's three levels — projected cycle / composable cycle / recurrent cycle — which C6 explicitly warns must not be conflated.",
      "defining_relation": "\\mathcal R_C \\subset \\mathfrak C(e_1,\\ldots,e_m)"
    },
    {
      "id": "ns.c6.c6a.g_may",
      "latex": "\\mathcal G^{may}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "可能性圖（may-graph）",
      "label_en": "May-graph",
      "definition_zh": "使用所有 non-excluded/conditional 的 projected edges 構成的圖（第22節）；其 SCC 分解為 $\\mathrm{SCC}(\\mathcal G^{may})=\\{\\{T\\},\\{G,P,H,F\\}\\}$，與第4節的 projected may-SCC 結果一致，代表僅由粗化標籤層級可得的 over-approximation，而非 certified 的循環結構。",
      "definition_en": "The graph built from all non-excluded/conditional projected edges (section 22); its SCC decomposition is $\\mathrm{SCC}(\\mathcal G^{may})=\\{\\{T\\},\\{G,P,H,F\\}\\}$, matching the projected may-SCC result of section 4. It represents the coarse-label-level over-approximation, not a certified cycle structure.",
      "defining_relation": "\\mathrm{SCC}(\\mathcal G^{may}) = \\{\\{T\\},\\{G,P,H,F\\}\\}",
      "notes": "Section 47's formal status table abbreviates this same SCC value as $\\mathrm{SCC}_{may}$."
    },
    {
      "id": "ns.c6.c6a.g_cert",
      "latex": "\\mathcal G^{cert}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "已認證圖（cert-graph）",
      "label_en": "Certified graph",
      "definition_zh": "僅使用「fully composable implication transitions with verified endpoint compatibility」構成的圖（第22節），與 $\\mathcal G^{may}$ 相對；本輪 audit 的結論是目前 $\\mathcal G^{cert}$ 尚沒有任何非平凡 recurrent SCC 被 certified，這不代表 cycle 不可能，而是 cycle existence 本身變成一個 proof obligation。",
      "definition_en": "The graph built using only fully composable implication transitions with verified endpoint compatibility (section 22), contrasted with $\\mathcal G^{may}$. The round's audit concludes that $\\mathcal G^{cert}$ currently contains no certified nontrivial recurrent SCC — not because cycles are impossible, but because cycle existence itself has become a proof obligation.",
      "notes": "A recurrent SCC of $\\mathcal G^{cert}$ is effectively what section 35 calls a \"certified recurrent SCC,\" as opposed to a mere candidate trap."
    },
    {
      "id": "ns.c6.c6a.candidate_trap",
      "latex": "S \\subset V_6",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "候選陷阱集合",
      "label_en": "Candidate (survivor) trap",
      "definition_zh": "第35–36節定義的較弱概念：一個子集 $S\\subset V_6$ 即使不是 certified implications 的 SCC，仍可能「未被排除為 survivor trap」，只代表目前理論不強迫其離開；對於強制性析取路由 $\\theta\\Rightarrow Y_1\\vee\\cdots\\vee Y_m$ 的狀態，candidate trap 必須滿足 $S\\cap\\{Y_1,\\ldots,Y_m\\}\\ne\\varnothing$。C6-A 強調 candidate trap 與 certified recurrent SCC 不可混為一談，$T$、$G/P$、$H/F$ 目前都只是 candidate traps/cycles。",
      "definition_en": "The weaker notion defined in sections 35-36: a subset $S\\subset V_6$ can be \"not excluded as a survivor trap\" even when it is not an SCC of certified implications — it only means current theory does not force an exit. For a state with mandatory disjunctive routing $\\theta\\Rightarrow Y_1\\vee\\cdots\\vee Y_m$, a candidate trap must satisfy $S\\cap\\{Y_1,\\ldots,Y_m\\}\\ne\\varnothing$. C6-A stresses that a candidate trap must not be conflated with a certified recurrent SCC; $T$, $G/P$, and $H/F$ are all currently only candidate traps/cycles.",
      "defining_relation": "S\\cap\\{Y_1,\\ldots,Y_m\\}\\ne\\varnothing"
    },
    {
      "id": "ns.c6.c6a.t_star",
      "latex": "T^\\ast",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "假設性最大存在時間",
      "label_en": "Hypothetical maximal existence time",
      "definition_zh": "MSC-5（第23節，minimal survivor cycle certificate 的 time viability 條件）中出現的記號，指假設性的（有限時間 blow-up 情境下的）時間上界；一個 minimal survivor cycle 必須使無窮次迭代仍能容納於 $T^\\ast$ 之下才算滿足 time viability。",
      "definition_en": "The symbol appearing in MSC-5 (section 23, the time-viability condition of the minimal survivor cycle certificate), denoting the hypothetical time bound (under a finite-time blow-up scenario). A minimal survivor cycle must fit infinite iteration below $T^\\ast$ to satisfy time viability.",
      "notes": "Only lightly specified in this round — introduced as a hypothetical bound without further formula. Same symbol as ns.framework.Tstar; this entry documents its specific role in C6-A's MSC-5 time-viability condition rather than introducing a new meaning."
    },
    {
      "id": "ns.c6.c6a.total_cycle_debt",
      "latex": "D(C)",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "循環總債務",
      "label_en": "Total cycle debt",
      "definition_zh": "對有限循環 $C=(e_1,\\ldots,e_m)$，定義 $D(C)=\\sum_{j=1}^{m}d(e_j)$，即該循環上所有邊債務向量之和（第24節）；是 Cycle Critical Saturation（第26節，當 $D(C_n)\\to\\partial\\mathcal D$）與 debt coercivity（第28節）判準所依賴的核心量，也是 cycle-state tuple $\\mathfrak C^{C6}$ 的組成之一。",
      "definition_en": "For a finite cycle $C=(e_1,\\ldots,e_m)$, defines $D(C)=\\sum_{j=1}^{m}d(e_j)$, the sum of the edge debt vectors around the cycle (section 24). It is the core quantity used by Cycle Critical Saturation (section 26, when $D(C_n)\\to\\partial\\mathcal D$) and the debt-coercivity criterion (section 28), and is also a component of the cycle-state tuple $\\mathfrak C^{C6}$.",
      "defining_relation": "D(C) = \\sum_{j=1}^{m} d(e_j)"
    },
    {
      "id": "ns.c6.c6a.finite_budget_lemma_quantities",
      "latex": "b_n,\\ B_0,\\ b_0",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "有限預算循環排除引理之量",
      "label_en": "Finite-budget cycle exclusion lemma quantities",
      "definition_zh": "C6-A.5（第25節，Finite-Budget Cycle Exclusion Lemma）所用的量：$b_n\\ge0$ 為第 $n$ 代 recurrent cycle 所付的純量債務，$B_0$ 為其總和上界（$\\sum_{n=1}^{\\infty}b_n\\le B_0<\\infty$），$b_0$ 為其一致下界（$\\inf_nb_n\\ge b_0>0$）；引理證明若兩條件同時成立，則只有有限多代循環可能發生，證明利用 $Nb_0\\le\\sum_{n=1}^{N}b_n\\le B_0$ 得出 $N\\le B_0/b_0$。",
      "definition_en": "The quantities used in C6-A.5 (section 25, the Finite-Budget Cycle Exclusion Lemma): $b_n\\ge0$ is the scalar debt paid by the $n$-th recurrent-cycle generation, $B_0$ is an upper bound on their total ($\\sum_{n=1}^{\\infty}b_n\\le B_0<\\infty$), and $b_0$ is a uniform lower bound ($\\inf_nb_n\\ge b_0>0$). The lemma proves that if both hold, only finitely many cycle generations can occur, via $Nb_0\\le\\sum_{n=1}^{N}b_n\\le B_0$, giving $N\\le B_0/b_0$.",
      "defining_relation": "\\sum_{n=1}^{\\infty}b_n \\le B_0<\\infty, \\quad \\inf_n b_n \\ge b_0>0 \\ \\Rightarrow\\ N\\le B_0/b_0",
      "notes": "Section 26 identifies the obstruction to this lemma as exactly $b_n\\downarrow0$, motivating the \"Cycle Critical Saturation\" definition."
    },
    {
      "id": "ns.c6.c6a.cycle_critical_saturation",
      "latex": "\\textbf{Cycle Critical Saturation}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "循環臨界飽和",
      "label_en": "Cycle critical saturation",
      "definition_zh": "第26節定義的名詞，描述 C6-A.5 有限預算排除引理失效的障礙情形：當每代循環債務 $D(C_n)\\to\\partial\\mathcal D$（趨向債務空間邊界 $\\partial\\mathcal D$），亦即在所有 globally finite-budget 座標中都趨於零時，循環可以在總預算有限的情況下仍然存活；是 temporal load concentration、harmonic critical saturation、chain-clock critical saturation 等現象在 cycle 層級的類比。",
      "definition_en": "The term section 26 defines to describe the obstruction case where the C6-A.5 finite-budget exclusion lemma fails: when each generation's cycle debt $D(C_n)\\to\\partial\\mathcal D$ (tends to the boundary $\\partial\\mathcal D$ of the debt space), i.e. tends to zero in every globally finite-budget coordinate, a recurrent cycle can survive under a finite total budget. It is the cycle-level analogue of temporal load concentration, harmonic critical saturation, chain-clock critical saturation, and UV scale-weighted Zeno costs.",
      "defining_relation": "D(C_n) \\to \\partial\\mathcal D",
      "notes": "$\\partial\\mathcal D$ (boundary of the debt space) is introduced only within this definition and not elaborated further in this round."
    },
    {
      "id": "ns.c6.c6a.debt_coercivity",
      "latex": "d_B(C) \\ge \\epsilon_B > 0",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "循環強制性（債務強制性判準）",
      "label_en": "Cycle-coercive debt functional",
      "definition_zh": "第28節定義：對循環族 $\\mathcal C$，全域預算泛函 $B$ 若對每個循環實例 $C\\in\\mathcal C$ 都滿足 $d_B(C)\\ge\\epsilon_B>0$，稱為 cycle-coercive；此時若 $B_{total}<\\infty$，則 $\\mathcal C$ 中不存在無窮 recurrent cycle。這是本輪提出的、通往 cycle elimination 最直接的一般化路線，與 C6-A.5 的具體引理互補。",
      "definition_en": "Defined in section 28: for a cycle family $\\mathcal C$, a global budget functional $B$ is cycle-coercive if $d_B(C)\\ge\\epsilon_B>0$ for every cycle instance $C\\in\\mathcal C$. If $B_{total}<\\infty$, then no infinite recurrent cycle in $\\mathcal C$ exists. This is presented as the strongest simple route to cycle elimination, generalizing the concrete C6-A.5 lemma.",
      "defining_relation": "d_B(C) \\ge \\epsilon_B > 0"
    },
    {
      "id": "ns.c6.c6a.edge_ledger_tuple",
      "latex": "e = (X, Y, \\mathcal A_e, R_e, d_e, \\sigma_e)",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "邊證明狀態紀錄",
      "label_en": "Edge proof-status ledger record",
      "definition_zh": "第43節定義 C6 儲存每條邊所用的六元組：$X,Y$ 為所屬類別，$\\mathcal A_e$ 為前提，$R_e$ 為轉移關係，$d_e$ 為債務，$\\sigma_e\\in\\{I,C,N,E\\}$ 為證明狀態；此設計是為了防止未來把一條僅是可能性的邊誤晉升為定理級的邊。",
      "definition_en": "Section 43 defines the six-tuple C6 uses to store every edge: $X,Y$ are the classes, $\\mathcal A_e$ the antecedent, $R_e$ the transition relation, $d_e$ the debt, and $\\sigma_e\\in\\{I,C,N,E\\}$ the proof status. This bookkeeping is designed to prevent a possibility edge from later being accidentally promoted into a theorem edge.",
      "defining_relation": "e = (X, Y, \\mathcal A_e, R_e, d_e, \\sigma_e)",
      "notes": "Compare with $\\mathfrak E^{C6}$ (section 46), a parallel six-component \"True ETN\" edge-state record (source state, antecedent, target relation, proof status, debt, kill gates) serving a similar bookkeeping role within the broader True ETN framework."
    },
    {
      "id": "ns.c6.c6a.true_etn_edge_state",
      "latex": "\\mathfrak E^{C6}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "True ETN 邊狀態（C6 版本）",
      "label_en": "True ETN C6 edge state",
      "definition_zh": "第46節將 True ETN（無限維張力場）框架更新為 C6 版本的邊狀態記法：$\\mathfrak E^{C6}=(\\text{source state},\\text{antecedent},\\text{target relation},\\text{proof status},\\text{debt},\\text{kill gates})$，是本輪 typed edge 概念（$X,Y,\\mathcal A_e,R_e,d_e,\\sigma_e$ 等）與更早期跨輪 True ETN 基礎設施的整合介面。",
      "definition_en": "Section 46 updates the True ETN (infinite-dimensional tension field) framework's edge-state notation for C6: $\\mathfrak E^{C6}=(\\text{source state},\\text{antecedent},\\text{target relation},\\text{proof status},\\text{debt},\\text{kill gates})$. It is the integration point between this round's typed-edge concepts ($X,Y,\\mathcal A_e,R_e,d_e,\\sigma_e$, etc.) and the earlier cross-round True ETN infrastructure.",
      "defining_relation": "\\mathfrak E^{C6} = (\\text{source state}, \\text{antecedent}, \\text{target relation}, \\text{proof status}, \\text{debt}, \\text{kill gates})",
      "notes": "Parallels but is not identical to the section 43 edge ledger tuple $e=(X,Y,\\mathcal A_e,R_e,d_e,\\sigma_e)$; this is the True-ETN-facing formalization."
    },
    {
      "id": "ns.c6.c6a.true_etn_cycle_state",
      "latex": "\\mathfrak C^{C6}",
      "series": "NS",
      "first_appearance": "C6-A",
      "label_zh": "True ETN 循環狀態（C6 版本）",
      "label_en": "True ETN C6 cycle state",
      "definition_zh": "第46節定義的循環狀態記法：$\\mathfrak C^{C6}=(e_1,\\ldots,e_m,\\text{fiber compatibility},\\text{recurrence map},D(C),\\text{budget coercivity})$，將一條循環的所有邊、fiber compatibility、recurrence map、總債務 $D(C)$ 與 budget coercivity 打包成單一 True ETN 記錄。",
      "definition_en": "The cycle-state notation section 46 defines: $\\mathfrak C^{C6}=(e_1,\\ldots,e_m,\\text{fiber compatibility},\\text{recurrence map},D(C),\\text{budget coercivity})$, packaging a cycle's edges together with its fiber compatibility, recurrence map, total debt $D(C)$, and budget coercivity into a single True ETN record.",
      "defining_relation": "\\mathfrak C^{C6} = (e_1,\\ldots,e_m, \\text{fiber compatibility}, \\text{recurrence map}, D(C), \\text{budget coercivity})",
      "notes": "Distinct from $\\mathfrak C(e_1,\\ldots,e_m)$ (section 11), which is specifically the fiber-product SET itself; $\\mathfrak C^{C6}$ is the broader True-ETN record that includes that fiber compatibility as one component alongside $D(C)$ and budget coercivity. Same fraktur C letter — high risk of confusion between the two."
    },
    {
      "id": "ns.c6.c6b.f_visc_down",
      "latex": "F_{\\rm visc}^{\\downarrow}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "黏性（衰減側）強迫類",
      "label_en": "viscous (decay-side) forcing class",
      "definition_zh": "C5 粗糙強迫類拆分後的第一支，對應黏性／平滑側的 turnover（衰減型貢獻）。C6-B.1 證明黏性項在帶號導數極大點恆非正（$\\Delta f(x_\\ast,t)\\le0$），故 $F_{\\rm visc}^{\\downarrow}$ 不可能作為正向尖峰再生引擎；C6-B.2（Viscous Half-Cycle Elimination）據此將其從 $H/F$ candidate cycle 的正向 re-entry 路徑中永久剔除。",
      "definition_en": "The first branch produced by splitting C5's coarse forcing class: the viscous/smoothing-side turnover (decay-type contribution). Since Theorem C6-B.1 shows the viscous term is never positive at the signed derivative maximum ($\\Delta f(x_\\ast,t)\\le0$), $F_{\\rm visc}^{\\downarrow}$ cannot serve as a positive peak-regeneration engine; C6-B.2 (Viscous Half-Cycle Elimination) accordingly removes it permanently from the positive re-entry branch of the $H/F$ candidate cycle.",
      "notes": "Together with $F_{\\rm NL}^{\\pm}$ this is C6-B's core split of C5-L's undifferentiated forcing class $\\mathfrak V_k^{visc}+\\mathfrak V_k^{NL}$."
    },
    {
      "id": "ns.c6.c6b.f_nl_pm",
      "latex": "F_{\\rm NL}^{\\pm}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "投影非線性強迫類（可正可負）",
      "label_en": "projected nonlinear forcing class (signed)",
      "definition_zh": "C5 粗糙強迫類拆分後的第二支：投影非線性強迫，其對所選尖峰分量的作用可能是穩定（負向）也可能是增長（正向）。與 $F_{\\rm visc}^{\\downarrow}$ 不同，唯有此支具備成為正向 $H$ re-entry 引擎的可能性，因而是本輪後續全部 alignment／coherence 分析的對象。",
      "definition_en": "The second branch from splitting C5's coarse forcing class: the projected nonlinear forcing, whose effect on the selected peak component may be either stabilizing (negative) or growing (positive). Unlike $F_{\\rm visc}^{\\downarrow}$, only this branch can possibly serve as a positive $H$ re-entry engine, making it the object of all subsequent alignment/coherence analysis in this round."
    },
    {
      "id": "ns.c6.c6b.a_k",
      "latex": "A_k(t)=\\max_{|\\zeta|=k,\\ i}\\|D^\\zeta u_i(t)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "k 階導數尖峰振幅",
      "label_en": "order-k derivative peak amplitude",
      "definition_zh": "在時間 $t$，取所有 $k$ 階空間導數分量（對多重指標 $\\zeta$ 與向量分量 $i$ 取最大）的 $L^\\infty$ 範數之最大值，即該階導數場的整體尖峰振幅。C6-B 圍繞其正向變化（peak regeneration）建立全輪論證，是 Theorem C6-B.1、正向變分帳本（§6）與其後所有 re-entry lemma 的基礎物件。",
      "definition_en": "At time $t$, the maximum $L^\\infty$ norm over all order-$k$ spatial derivative components (maximized over multi-index $\\zeta$ and vector component $i$) — the overall peak amplitude of the order-$k$ derivative field. C6-B's entire argument (Theorem C6-B.1, the positive-variation ledger of §6, and every subsequent re-entry lemma) is organized around when and how this quantity can grow.",
      "defining_relation": "A_k(t)=\\max_{|\\zeta|=k,\\ i}\\|D^\\zeta u_i(t)\\|_\\infty"
    },
    {
      "id": "ns.c6.c6b.gamma_ell_duh",
      "latex": "\\Gamma_\\ell^{Duh}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "Duhamel 相干係數",
      "label_en": "Duhamel coherence coefficient",
      "definition_zh": "實際非線性響應 $\\|Z_\\ell\\|_\\infty$ 與其容量上界 $\\mathfrak C_\\ell^{Duh}$ 之比值，取值於 $[0,1]$。$\\Gamma^{Duh}\\approx1$ 表示時間／空間／分量貢獻在響應範數中相干疊加；$\\Gamma^{Duh}\\ll1$ 表示大量 Duhamel 抵消；$\\Gamma^{Duh}=0$ 表示儘管強迫容量為正，響應仍完全抵消。C6-B.3 證明沒有通用下界 $\\Gamma_\\ell^{Duh}\\ge c>0$ 能單由容量推出，這是全篇（乃至提議中 C6-C）最核心的 typed edge coordinate。",
      "definition_en": "The ratio of the actual nonlinear response $\\|Z_\\ell\\|_\\infty$ to its capacity upper bound $\\mathfrak C_\\ell^{Duh}$, valued in $[0,1]$. $\\Gamma^{Duh}\\approx1$ means time/space/component contributions add coherently in the response norm; $\\Gamma^{Duh}\\ll1$ means substantial Duhamel cancellation; $\\Gamma^{Duh}=0$ means complete cancellation despite positive forcing capacity. C6-B.3 proves no universal lower bound $\\Gamma_\\ell^{Duh}\\ge c>0$ follows from capacity alone — arguably the single most central typed edge coordinate in this round (and the proposed C6-C).",
      "defining_relation": "\\Gamma_\\ell^{Duh}=\\frac{\\|Z_\\ell\\|_\\infty}{\\mathfrak C_\\ell^{Duh}}\\in[0,1]",
      "notes": "§0 and §28 use the unsubscripted $\\Gamma^{Duh}$ for the same concept, as one coordinate of the re-entry coherence vector $\\Gamma^{re}$."
    },
    {
      "id": "ns.c6.c6b.f_nl_coh",
      "latex": "F_{\\rm NL}^{coh}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "相干非線性強迫",
      "label_en": "coherent nonlinear forcing",
      "definition_zh": "同時滿足 Duhamel 相干性、響應對繼承熱核的優勢、分量選擇相干性、鏈尺度符號厚度、定理適用合法性，以及整個定理窗口時間持續性等全部前提條件的非線性強迫事件。Theorem C6-B.7（Conditional Nonlinear Re-entry Theorem）證明唯有 $F_{\\rm NL}^{coh}\\to H$ 條件成立，而非粗糙的 $F_{\\rm NL}\\to H$。",
      "definition_en": "A projected-nonlinear forcing event satisfying all of: Duhamel response coherence, response dominance over inherited heat, component-selection coherence, chain-scale sign thickness, theorem setup legality, and whole-window temporal persistence. Theorem C6-B.7 (Conditional Nonlinear Re-entry Theorem) proves only the conditional $F_{\\rm NL}^{coh}\\to H$ — not the coarse $F_{\\rm NL}\\to H$.",
      "defining_relation": "F_{\\rm NL}^{coh}\\to H"
    },
    {
      "id": "ns.c6.c6b.h_force",
      "latex": "H_{\\rm force}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "產生強迫的 H 子型",
      "label_en": "forcing-producing subtype of H",
      "definition_zh": "持久壞窗類 $H$ 的一個子型，其後續演化會經由黏性或投影非線性 turnover congestion 產生新的強迫（$H\\Rightarrow H_{\\rm compact}\\vee F_{\\rm visc}\\vee F_{\\rm NL}$，§36）。由於 $H\\not\\Rightarrow F$ 並非無條件成立，唯有落在 $H_{\\rm force}$ 子型中的 $H$ 才可能啟動 $H/F$ candidate cycle 的下一世代，是 Theorem C6-B.9（Typed $H/F$ Cycle Reduction）中循環兩端的物件。",
      "definition_en": "A subtype of the persistent-bad-window class $H$: the branch whose subsequent evolution produces new forcing via viscous or projected-nonlinear turnover congestion ($H\\Rightarrow H_{\\rm compact}\\vee F_{\\rm visc}\\vee F_{\\rm NL}$, §36). Since $H\\not\\Rightarrow F$ does not hold unconditionally, only $H$ lying in the $H_{\\rm force}$ subtype can possibly initiate the next generation of the $H/F$ candidate cycle; it is the object at both ends of the cycle in Theorem C6-B.9 (Typed $H/F$ Cycle Reduction).",
      "defining_relation": "H\\Rightarrow H_{\\rm compact}\\vee F_{\\rm visc}\\vee F_{\\rm NL}"
    },
    {
      "id": "ns.c6.c6b.f_signed_component",
      "latex": "f(x,t)=\\sigma D^\\zeta u_i(x,t)",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "帶號極大化導數分量場",
      "label_en": "signed maximizing derivative component field",
      "definition_zh": "在 $A_k$ 的極大化分量／符號被達到的時空點，選取符號 $\\sigma\\in\\{\\pm1\\}$ 與對應導數分量 $D^\\zeta u_i$，構造純量場 $f=\\sigma D^\\zeta u_i$，使其在空間極大點 $x_\\ast$ 處等於正值 $A_k(t)$。此構造把 $L^\\infty$ 極大值問題化為單一純量拋物方程，是 Theorem C6-B.1（黏性不能再生 $D^k$ 尖峰）證明所依賴的最大值原理工具。",
      "definition_en": "At a space-time point where the maximizing component/sign of $A_k$ is attained, one chooses a sign $\\sigma\\in\\{\\pm1\\}$ and the corresponding derivative component $D^\\zeta u_i$ to form the scalar field $f=\\sigma D^\\zeta u_i$, equal to the positive value $A_k(t)$ at the spatial maximum $x_\\ast$. This reduces the $L^\\infty$-peak problem to a single scalar parabolic equation, the maximum-principle device behind the proof of Theorem C6-B.1 (viscosity cannot regenerate the $D^k$ peak).",
      "defining_relation": "f(x,t)=\\sigma D^\\zeta u_i(x,t),\\qquad \\sigma\\in\\{\\pm1\\},\\qquad f(x_\\ast,t)=A_k(t)>0"
    },
    {
      "id": "ns.c6.c6b.n_k_proj",
      "latex": "\\mathcal N_k^{proj}(t)",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "投影非線性強迫範數",
      "label_en": "projected nonlinear forcing norm",
      "definition_zh": "Leray 投影非線性項 $\\mathbb P((u\\cdot\\nabla)u)$ 之 $k$ 階導數，在所有分量／符號中取最大的 $L^\\infty$ 範數。這是本輪唯一能正向驅動 $A_k$ 尖峰增長的來源項，貫穿 Theorem C6-B.1（$D^+A_k\\le\\mathcal N_k^{proj}$）、正向變分帳本、對齊係數 $\\alpha_k^{peak}$、再生效率 $\\eta_k^{grow}$ 等全部後續論證。",
      "definition_en": "The order-$k$ derivative of the Leray-projected nonlinear term $\\mathbb P((u\\cdot\\nabla)u)$, maximized in $L^\\infty$ norm over all components/signs. This is the only source term in this round capable of positively driving growth of the $A_k$ peak, running through Theorem C6-B.1 ($D^+A_k\\le\\mathcal N_k^{proj}$), the positive-variation ledger, the alignment coefficient $\\alpha_k^{peak}$, and the regeneration efficiency $\\eta_k^{grow}$.",
      "defining_relation": "\\mathcal N_k^{proj}(t)=\\max_{|\\zeta|=k,i}\\left\\|D^\\zeta\\mathbb P((u\\cdot\\nabla)u)_i\\right\\|_\\infty"
    },
    {
      "id": "ns.c6.c6b.dini_derivative_a_k",
      "latex": "D^+A_k(t)",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "$A_k$ 的上 Dini 導數",
      "label_en": "upper Dini derivative of $A_k$",
      "definition_zh": "尖峰振幅 $A_k$ 在時間 $t$ 的上（右）Dini 導數，用以在 $A_k$ 未必處處可微時仍嚴謹描述其瞬時增長率。此為 Theorem C6-B.1 的核心量：因帶號空間極大點滿足 $\\Delta f(x_\\ast,t)\\le0$，故 $D^+A_k(t)\\le\\mathcal N_k^{proj}(t)$——黏性項對尖峰增長的貢獻恆非正，正向增長只能來自投影非線性項。",
      "definition_en": "The upper (right) Dini derivative of the peak amplitude $A_k$ at time $t$, used to rigorously bound its instantaneous growth rate even where $A_k$ need not be everywhere differentiable. This is the central object of Theorem C6-B.1: since $\\Delta f(x_\\ast,t)\\le0$ at the signed spatial maximum, $D^+A_k(t)\\le\\mathcal N_k^{proj}(t)$ — the viscous contribution to peak growth is never positive, so positive growth can only come from the projected nonlinear term.",
      "defining_relation": "D^+A_k(t)\\le\\mathcal N_k^{proj}(t)"
    },
    {
      "id": "ns.c6.c6b.var_i_plus_log_r_k",
      "latex": "\\operatorname{Var}_I^+\\log\\mathcal R_k",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "Grujić–Xu 根的正向變分",
      "label_en": "positive variation of the Grujić–Xu root",
      "definition_zh": "對 Grujić–Xu 正規化根 $\\mathcal R_k=A_k^{1/(k+1)}/(c^{k/(k+1)}(k!)^{1/(k+1)})$ 只計正向增量的變分（相對於 C5-L 原本計絕對變動的 $\\operatorname{Var}_I\\log\\mathcal R_k$）。C6-B 證明此正向變分僅受投影非線性強迫（經 $A_k$ 正規化後）的時間積分控制，即「正向根再生只受非線性源限制」——黏性項對其無貢獻。",
      "definition_en": "The \"positive-part-only\" variation of the Grujić–Xu normalized root $\\mathcal R_k=A_k^{1/(k+1)}/(c^{k/(k+1)}(k!)^{1/(k+1)})$, refining C5-L's original absolute-variation quantity $\\operatorname{Var}_I\\log\\mathcal R_k$. C6-B shows this positive variation is bounded solely by the time integral of the $A_k$-normalized projected nonlinear forcing — \"positive root regeneration is nonlinear-source limited,\" with no viscous contribution.",
      "defining_relation": "\\operatorname{Var}_I^+\\log\\mathcal R_k\\le\\frac1{k+1}\\int_I\\frac{\\mathcal N_k^{proj}(t)}{A_k(t)}dt",
      "notes": "$\\mathcal R_k$ itself (the explicit Grujić–Xu root formula) is inherited from C5-L / Grujić–Xu, not newly introduced by C6-B; the positive-part (\"+\") refinement and its nonlinear-only bound are C6-B's addition."
    },
    {
      "id": "ns.c6.c6b.f_nl_plus",
      "latex": "F_{\\rm NL}^{+}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "正向（增長支）非線性強迫",
      "label_en": "positive/growth branch of nonlinear forcing",
      "definition_zh": "$F_{\\rm NL}^{\\pm}$ 中僅取正向（增長）作用的分支，是 C6-B.2 論證後「$H/F$ candidate cycle 唯一還存活的正向回返分支」。它出現在本輪最終化簡的循環 $H_{\\rm force}\\to F_{\\rm NL}^{+}\\dashrightarrow H_{\\rm force}$（Theorem C6-B.9）中，連接 $H_{\\rm force}$ 與後續 coherence-gated re-entry。",
      "definition_en": "The positive/growth-only branch of $F_{\\rm NL}^{\\pm}$ — after C6-B.2's argument, \"the only surviving positive-return branch of the $H/F$ candidate cycle.\" It appears in this round's final reduced cycle $H_{\\rm force}\\to F_{\\rm NL}^{+}\\dashrightarrow H_{\\rm force}$ (Theorem C6-B.9), bridging $H_{\\rm force}$ to the subsequent coherence-gated re-entry."
    },
    {
      "id": "ns.c6.c6b.y_ell",
      "latex": "Y_\\ell",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "繼承熱核部分",
      "label_en": "inherited heat part",
      "definition_zh": "mild-form（Duhamel）表示中，$\\ell$ 階導數解 $D^\\ell u(t_1)=Y_\\ell+Z_\\ell$ 分解裡純由初始時刻 $t_0$ 資料經熱半群演化而來、不含非線性響應的一項。§18 的 One-Time Sign-Reentry Lemma 要求 $Y_\\ell$ 相對於非線性響應足夠小（$\\|Y_\\ell\\|_\\infty\\le\\epsilon A_Z$），生成的符號厚集合才會真正遺傳到實際導數場。",
      "definition_en": "In the mild-form (Duhamel) decomposition $D^\\ell u(t_1)=Y_\\ell+Z_\\ell$, the part coming purely from evolving the initial data at $t_0$ under the heat semigroup, with no nonlinear contribution. The One-Time Sign-Reentry Lemma (§18-19) requires $Y_\\ell$ to be sufficiently small relative to the nonlinear response ($\\|Y_\\ell\\|_\\infty\\le\\epsilon A_Z$) for the generated sign-thick set to actually be inherited by the real derivative field.",
      "defining_relation": "Y_\\ell=D^\\ell e^{\\nu(t_1-t_0)\\Delta}u(t_0)"
    },
    {
      "id": "ns.c6.c6b.z_ell",
      "latex": "Z_\\ell",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "非線性 Duhamel 響應",
      "label_en": "nonlinear Duhamel response",
      "definition_zh": "mild-form 分解 $D^\\ell u(t_1)=Y_\\ell+Z_\\ell$ 中由非線性項經 Duhamel 積分產生的部分。這是本輪判斷「強迫能否重新生成尖峰」的核心物件：其大小僅被 Duhamel capacity $\\mathfrak C_\\ell^{Duh}$ 上界，其相干程度由 $\\Gamma_\\ell^{Duh}$ 量化，其符號厚集合 $E_Z$ 決定能否觸發 one-time sign-reentry。",
      "definition_en": "In the mild-form decomposition $D^\\ell u(t_1)=Y_\\ell+Z_\\ell$, the part produced by the Duhamel integral of the nonlinear term. This is the central object for whether forcing can regenerate a peak in this round: its magnitude is only upper-bounded by the Duhamel capacity $\\mathfrak C_\\ell^{Duh}$, its coherence is quantified by $\\Gamma_\\ell^{Duh}$, and its sign-thick high set $E_Z$ determines whether it can trigger one-time sign-reentry.",
      "defining_relation": "Z_\\ell=-\\int_{t_0}^{t_1}D^\\ell e^{\\nu(t_1-s)\\Delta}\\mathbb P((u\\cdot\\nabla)u)(s)ds"
    },
    {
      "id": "ns.c6.c6b.c_ell_duh",
      "latex": "\\mathfrak C_\\ell^{Duh}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "Duhamel 響應容量",
      "label_en": "Duhamel (response) capacity",
      "definition_zh": "對非線性 Duhamel 積分核取範數後逐時間積分所得的上界量，由三角不等式滿足 $\\|Z_\\ell\\|_\\infty\\le\\mathfrak C_\\ell^{Duh}$。這只是容量上界，不是響應下界——正是 §9 到 §11（C6-B.3 Duhamel-Capacity No-Go）強調的核心區分：強迫容量大不代表實際響應大。",
      "definition_en": "The upper-bound quantity obtained by integrating the norm of the nonlinear Duhamel kernel in time, satisfying the triangle inequality $\\|Z_\\ell\\|_\\infty\\le\\mathfrak C_\\ell^{Duh}$. This is only a capacity upper bound, not a response lower bound — the distinction emphasized from §9 through §11 (C6-B.3, the Duhamel-Capacity No-Go): large forcing capacity does not imply large actual response.",
      "defining_relation": "\\mathfrak C_\\ell^{Duh}=\\int_{t_0}^{t_1}\\left\\|D^\\ell e^{\\nu(t_1-s)\\Delta}\\mathbb P((u\\cdot\\nabla)u)(s)\\right\\|_\\infty ds"
    },
    {
      "id": "ns.c6.c6b.alpha_k_peak",
      "latex": "\\alpha_k^{peak}(t)",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "尖峰源對齊係數",
      "label_en": "peak source-alignment coefficient",
      "definition_zh": "在帶號極大化導數點 $x_\\ast(t)$，投影非線性源實際朝向增大該尖峰方向的分量（正部分）與整體投影非線性強迫範數 $\\mathcal N_k^{proj}$ 之比值。§12 指出即使 $\\mathcal N_k^{proj}$ 很大，若 $\\alpha_k^{peak}\\approx0$（源與尖峰方向錯位），瞬時尖峰仍不會被正向驅動——強迫量值與對齊方向是兩個不同座標。",
      "definition_en": "At the signed maximizing derivative point $x_\\ast(t)$, the ratio of the positive part of the projected nonlinear source's component actually aligned toward increasing that peak, to the overall projected nonlinear forcing norm $\\mathcal N_k^{proj}$. §12 shows that even when $\\mathcal N_k^{proj}$ is large, if $\\alpha_k^{peak}\\approx0$ (source misaligned with the peak direction), the instantaneous peak is still not positively driven — forcing magnitude and alignment direction are distinct coordinates.",
      "defining_relation": "\\alpha_k^{peak}(t)=\\frac{\\left[-\\sigma D^\\zeta\\mathbb P((u\\cdot\\nabla)u)_i(x_\\ast,t)\\right]_+}{\\mathcal N_k^{proj}(t)}\\in[0,1]"
    },
    {
      "id": "ns.c6.c6b.eta_k_grow",
      "latex": "\\eta_k^{grow}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "再生（增長）效率",
      "label_en": "regeneration / growth efficiency",
      "definition_zh": "區間 $[t_0,t_1]$ 內尖峰振幅實際淨增量（正部分）與同區間投影非線性強迫累積供給量之比值，量化「花費的非線性容量」轉化為「實際尖峰再生」的效率。當某循環世代 $\\eta_k^{grow}\\to0$，代表花費越來越多非線性容量卻未相應再生尖峰——這是 cycle-break／critical-coherence 的訊號，而非 $H$ re-entry 的證據，也是提議中 C6-C 論文（proof obligation C3）要進一步研究的物件。",
      "definition_en": "The ratio of the actual net positive increase in peak amplitude over $[t_0,t_1]$ to the accumulated projected nonlinear forcing supplied over the same interval, quantifying how efficiently spent nonlinear capacity converts into actual peak regeneration. When $\\eta_k^{grow}\\to0$ along a recurrent cycle generation, increasing nonlinear capacity is spent without regenerating a comparable peak — a signal of cycle-break/critical-coherence, not evidence of $H$ re-entry, and flagged as a target for the proposed C6-C paper (proof obligation C3).",
      "defining_relation": "\\eta_k^{grow}=\\frac{[A_k(t_1)-A_k(t_0)]_+}{\\int_{t_0}^{t_1}\\mathcal N_k^{proj}(s)ds}\\in[0,1]"
    },
    {
      "id": "ns.c6.c6b.reentry_gate_chain",
      "latex": "F_{\\rm NL}\\to R_{\\rm amp}\\to R_{\\rm select}\\to R_{\\rm sign}\\to R_{\\rm setup}\\to R_{\\rm persist}\\to H",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "強迫再入五道中介閘門鏈",
      "label_en": "forcing-reentry five-gate chain",
      "definition_zh": "§16 將粗糙邊 $F_{\\rm NL}\\to H$ 拆解為五個必須依序通過的中介閘門：$R_{\\rm amp}$（未被抵消的 Duhamel 響應／尖峰再生）、$R_{\\rm select}$（生成分量在壞點仍是定理選中的分量／符號）、$R_{\\rm sign}$（鏈尺度空間符號厚度）、$R_{\\rm setup}$（Grujić–Xu 定理進入合法性）與 $R_{\\rm persist}$（整個定理窗口失敗）。這條鏈是本輪從「強迫很大」到「$H$」之間全部具體引理（C6-B.5、C6-B.6、C6-B.7 等）依序填補的骨架。",
      "definition_en": "§16 decomposes the coarse edge $F_{\\rm NL}\\to H$ into five intermediate gates that must be passed in sequence: $R_{\\rm amp}$ (noncancelled Duhamel response / peak regeneration), $R_{\\rm select}$ (the generated component remains the theorem-selected component/sign at the bad point), $R_{\\rm sign}$ (chain-scale spatial sign-thickness), $R_{\\rm setup}$ (Grujić–Xu theorem-entry legality), and $R_{\\rm persist}$ (entire theorem-window failure). This chain is the skeleton that all of this round's concrete lemmas (C6-B.5, C6-B.6, C6-B.7, etc.) fill in on the way from \"forcing is large\" to \"$H$.\""
    },
    {
      "id": "ns.c6.c6b.e_z",
      "latex": "E_Z",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "響應鏈尺度高值集合",
      "label_en": "response chain-scale high set",
      "definition_zh": "非線性 Duhamel 響應某分量／符號 $\\sigma Z_{\\ell,i}$ 超過閾值 $\\lambda_ZA_Z$（其中 $A_Z=\\|Z_\\ell\\|_\\infty$ 為響應峰值振幅）之空間點集合。C6-B.5（One-Time Sign-Reentry Lemma）要求此集合滿足 1D 鏈尺度符號厚度條件（對所有尺度 $[\\nu]$ 皆有 $b_{E_Z}(x_0,r,[\\nu])>\\delta$），才可能將 Duhamel 響應的「大」轉化為 Grujić–Xu 意義下的「壞幾何」，回應 C6-B.4 no-go 所指出「振幅大不蘊含符號厚集合」的缺口。",
      "definition_en": "The spatial set on which some component/sign $\\sigma Z_{\\ell,i}$ of the nonlinear Duhamel response exceeds the threshold $\\lambda_ZA_Z$ (where $A_Z=\\|Z_\\ell\\|_\\infty$ is the response peak amplitude). The One-Time Sign-Reentry Lemma (C6-B.5) requires this set to satisfy a 1D chain-scale sign-thickness condition ($b_{E_Z}(x_0,r,[\\nu])>\\delta$ for all scales $[\\nu]$) before the \"largeness\" of the Duhamel response can be converted into Grujić–Xu-type \"bad geometry,\" addressing the gap flagged by the C6-B.4 no-go (\"large amplitude does not imply a sign-thick set\").",
      "defining_relation": "E_Z=\\left\\{x:\\sigma Z_{\\ell,i}(x)>\\lambda_ZA_Z\\right\\},\\qquad b_{E_Z}(x_0,r,[\\nu])>\\delta\\ \\ \\forall[\\nu]",
      "notes": "$A_Z=\\|Z_\\ell\\|_\\infty$ (§17) is the local shorthand naming the response amplitude used to set $E_Z$'s threshold. The 1D chain-scale thickness function $b_E(\\cdot)$ itself is inherited notation from an earlier round (sign geometry / chain harmonic compatibility); C6-B applies it to the new set $E_Z$."
    },
    {
      "id": "ns.c6.c6b.epsilon_inherited_ratio",
      "latex": "\\epsilon",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "繼承場比例",
      "label_en": "inherited-field ratio",
      "definition_zh": "界定繼承熱核部分 $Y_\\ell$ 相對於響應峰值振幅 $A_Z$ 之「小」程度的比例常數，設定為 $\\|Y_\\ell\\|_\\infty\\le\\epsilon A_Z$。C6-B.5（One-Time Sign-Reentry Lemma）的門檻條件 $\\lambda_Z-\\epsilon>\\lambda(1+\\epsilon)$ 要求 $\\epsilon$ 充分小，繼承場才不會淹沒響應在符號厚集合 $E_Z$ 上造成的優勢，是一次性強迫再入憑證 $\\mathsf{REC}_1$ 的成分之一。",
      "definition_en": "The ratio constant bounding how \"small\" the inherited heat part $Y_\\ell$ is relative to the response peak amplitude $A_Z$, set via $\\|Y_\\ell\\|_\\infty\\le\\epsilon A_Z$. The threshold condition of the One-Time Sign-Reentry Lemma (C6-B.5), $\\lambda_Z-\\epsilon>\\lambda(1+\\epsilon)$, requires $\\epsilon$ sufficiently small so the inherited field does not swamp the dominance the response creates on $E_Z$; it is one component of the re-entry certificate $\\mathsf{REC}_1$.",
      "defining_relation": "\\|Y_\\ell\\|_\\infty\\le\\epsilon A_Z"
    },
    {
      "id": "ns.c6.c6b.m_sel",
      "latex": "m_{\\rm sel}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "分量選擇邊際",
      "label_en": "component-selection margin",
      "definition_zh": "在基準點 $x_0$，被生成／檢驗的導數分量／符號之值，減去該點所有其他分量／符號絕對值中的最大者。Grujić–Xu 空間測試是從實際導數張量在 $x_0$「被選中」的分量出發，故 C6-B.5 生成的符號厚集合唯有在 $m_{\\rm sel}>0$（清潔充分門檻）時才真正有效，否則定理可能選中另一個幾何不同的分量／符號。",
      "definition_en": "At the reference point $x_0$, the value of the generated/tested derivative component/sign minus the maximum absolute value among all other components/signs at that point. Since the Grujić–Xu spatial test operates on whichever component is actually \"selected\" from the derivative tensor at $x_0$, the sign-thick set generated by C6-B.5 is effective only when $m_{\\rm sel}>0$ (a clean sufficient gate) — otherwise the theorem may select a different component/sign with different geometry.",
      "defining_relation": "m_{\\rm sel}=\\sigma D^\\ell u_i(x_0)-\\max_{(\\zeta',j)\\ne(\\zeta,i)}|D^{\\zeta'}u_j(x_0)|"
    },
    {
      "id": "ns.c6.c6b.rec_1",
      "latex": "\\mathsf{REC}_{1}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "一次性強迫再入憑證",
      "label_en": "one-time forcing re-entry certificate",
      "definition_zh": "§21 定義的憑證物件，收集確立「一次性符號再入」（C6-B.5）所需的五個 typed 座標：Duhamel 相干係數 $\\Gamma_\\ell^{Duh}$（須非退化）、繼承場比例 $\\epsilon$（須低於閾值）、響應閾值分量 $\\lambda_Z$（須滿足 $\\lambda_Z-\\epsilon>\\lambda(1+\\epsilon)$）、選擇邊際 $m_{\\rm sel}$（須為正）與符號厚度 $\\beta_Z$（須大於 $\\delta$）。此憑證遠強於「$F_{\\rm NL}$ 很大」這一粗糙條件。",
      "definition_en": "The certificate object defined in §21, collecting the five typed coordinates needed to establish a \"one-time sign-reentry\" event (C6-B.5): the Duhamel coherence coefficient $\\Gamma_\\ell^{Duh}$ (must be nondegenerate), the inherited-field ratio $\\epsilon$ (must be below threshold), the response threshold fraction $\\lambda_Z$ (must satisfy $\\lambda_Z-\\epsilon>\\lambda(1+\\epsilon)$), the selection margin $m_{\\rm sel}$ (must be positive), and the sign-thickness $\\beta_Z$ (must exceed $\\delta$). This certificate is far stronger than the coarse condition \"$F_{\\rm NL}$ is large.\"",
      "defining_relation": "\\mathsf{REC}_{1}=\\left\\{\\Gamma_\\ell^{Duh},\\epsilon,\\lambda_Z,m_{\\rm sel},\\beta_Z\\right\\}"
    },
    {
      "id": "ns.c6.c6b.theta_ell",
      "latex": "\\Theta_\\ell(s,t_\\ast)",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "時間擾動量",
      "label_en": "temporal perturbation",
      "definition_zh": "$\\ell$ 階導數場在另一時刻 $s$ 與參考壞時刻 $t_\\ast$ 之間的 $L^\\infty$ 差異。C6-B.6（Sign-Thickness Persistence Lemma）證明只要 $(1+\\lambda)\\Theta_\\ell(s,t_\\ast)<m$，同一符號厚集合在時刻 $s$ 仍落在實際導數場的高值集合內，是把「一次性符號厚事件」延伸為「窗口持續性」（§22-26）論證鏈的核心控制量，並由 C5-L 的 $\\|\\partial_tD^\\ell u\\|_\\infty\\le C\\nu A_{\\ell+2}+\\mathcal N_\\ell^{proj}$ 估計其時間積分上界（§25）。",
      "definition_en": "The $L^\\infty$ difference of the order-$\\ell$ derivative field between another time $s$ and the reference bad time $t_\\ast$. The Sign-Thickness Persistence Lemma (C6-B.6) shows that whenever $(1+\\lambda)\\Theta_\\ell(s,t_\\ast)<m$, the same sign-thick set remains inside the actual derivative high set at time $s$ — the central control quantity extending a one-time sign-thick event into window persistence (§22-26), whose time integral is in turn bounded via C5-L's $\\|\\partial_tD^\\ell u\\|_\\infty\\le C\\nu A_{\\ell+2}+\\mathcal N_\\ell^{proj}$ (§25).",
      "defining_relation": "\\Theta_\\ell(s,t_\\ast)=\\|D^\\ell u(s)-D^\\ell u(t_\\ast)\\|_\\infty"
    },
    {
      "id": "ns.c6.c6b.m_threshold_margin",
      "latex": "m",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "符號厚度閾值優勢邊際",
      "label_en": "sign-thickness threshold dominance margin",
      "definition_zh": "在參考壞時刻 $t_\\ast$，壞集合 $E$ 上實際導數分量超出理論門檻 $\\lambda A_\\ell(t_\\ast)$ 的嚴格盈餘量（$m>0$）。C6-B.6（Sign-Thickness Persistence Lemma）證明只要時間擾動 $\\Theta_\\ell$ 累積量低於 $m/(1+\\lambda)$，此優勢邊際便足以撐過整個定理窗口。§28 的再入相干向量 $\\Gamma^{re}$ 中，同一概念以 $m_{\\rm thr}$（「實際閾值優勢邊際」）之名收錄。",
      "definition_en": "At the reference bad time $t_\\ast$, the strict surplus by which the actual derivative component on the bad set $E$ exceeds the theoretical threshold $\\lambda A_\\ell(t_\\ast)$ ($m>0$). The Sign-Thickness Persistence Lemma (C6-B.6) shows that as long as the accumulated temporal perturbation $\\Theta_\\ell$ stays below $m/(1+\\lambda)$, this dominance margin survives the entire theorem window. The re-entry coherence vector $\\Gamma^{re}$ (§28) records this same concept under the name $m_{\\rm thr}$ (\"actual threshold dominance margin\").",
      "defining_relation": "\\sigma D^\\ell u_i(x,t_\\ast)\\ge\\lambda A_\\ell(t_\\ast)+m\\quad(x\\in E),\\qquad m>0",
      "notes": "Bare single-letter \"$m$\" is the notation used at its point of definition (§22) and through the C6-B.6 proof (§23-25); §28's summary vector $\\Gamma^{re}$ relabels the same quantity $m_{\\rm thr}$."
    },
    {
      "id": "ns.c6.c6b.beta_z",
      "latex": "\\beta_Z",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "空間符號厚度",
      "label_en": "spatial sign-thickness",
      "definition_zh": "量化響應高值集合 $E_Z$ 之 1D 鏈尺度厚度的純量，須滿足 $\\beta_Z>\\delta$ 方能算作真正「符號厚」（即 §17 條件 $b_{E_Z}(x_0,r,[\\nu])>\\delta\\ \\forall[\\nu]$ 的簡記）。它是一次性強迫 re-entry 憑證 $\\mathsf{REC}_1$（§21）的成分之一，也是再入相干向量 $\\Gamma^{re}$（§28）中以邊際 $\\beta_Z-\\delta$ 形式出現的座標；當某循環世代 $\\beta_{Z,n}\\downarrow\\delta$（§29），代表符號厚度正朝臨界飽和退化。",
      "definition_en": "The scalar quantifying the 1D chain-scale thickness of the response high set $E_Z$; it must satisfy $\\beta_Z>\\delta$ to count as genuinely \"sign-thick\" (shorthand for the §17 condition $b_{E_Z}(x_0,r,[\\nu])>\\delta\\ \\forall[\\nu]$). It is one component of the one-time forcing re-entry certificate $\\mathsf{REC}_1$ (§21) and enters the re-entry coherence vector $\\Gamma^{re}$ (§28) as the margin coordinate $\\beta_Z-\\delta$; when a recurrent generation has $\\beta_{Z,n}\\downarrow\\delta$ (§29), sign-thickness is degenerating toward critical saturation.",
      "defining_relation": "\\beta_Z>\\delta"
    },
    {
      "id": "ns.c6.c6b.pi_time",
      "latex": "\\Pi_{\\rm time}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "持續性儲備",
      "label_en": "persistence reserve",
      "definition_zh": "再入相干向量 $\\Gamma^{re}$（§28）的分量之一，代表符號厚度優勢邊際 $m$ 能抵抗多少時間擾動累積（即 §25 給出的持續性充分條件 $(1+\\lambda)\\int(C\\nu A_{\\ell+2}+\\mathcal N_\\ell^{proj})dt<m$ 所反映的餘裕）而仍撐過整個定理窗口。當某循環世代 $\\Pi_{{\\rm time},n}\\downarrow0$（§29），代表持續性儲備正在耗盡，是 Forcing-Reentry Critical Saturation 的訊號之一。",
      "definition_en": "One component of the re-entry coherence vector $\\Gamma^{re}$ (§28), representing how much accumulated temporal perturbation the sign-thickness dominance margin $m$ can withstand (the \"slack\" reflected by the persistence sufficient condition $(1+\\lambda)\\int(C\\nu A_{\\ell+2}+\\mathcal N_\\ell^{proj})dt<m$ of §25) while still surviving the whole theorem window. When a recurrent generation has $\\Pi_{{\\rm time},n}\\downarrow0$ (§29), the persistence reserve is being depleted — one signal of Forcing-Reentry Critical Saturation."
    },
    {
      "id": "ns.c6.c6b.setup_flag",
      "latex": "\\mathsf{Setup}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "定理適用旗標",
      "label_en": "theorem-entry (setup) flag",
      "definition_zh": "再入相干向量 $\\Gamma^{re}$（§28）的分量之一，標記 Grujić–Xu 定理在給定 $(\\ell,t')$ 是否合法適用（theorem setup legal，見 §26 條件 1、§35 的 B-W2 setup failure 情形）。此旗標不成立時，即便其餘相干座標皆為正，符號厚事件仍不能計入 $H$。",
      "definition_en": "One component of the re-entry coherence vector $\\Gamma^{re}$ (§28), flagging whether the Grujić–Xu theorem is legally applicable at the given $(\\ell,t')$ (theorem setup legal; cf. §26 condition 1, and the B-W2 \"setup failure\" case of §35). If this flag fails, a sign-thick event cannot count toward $H$ regardless of how positive the other coherence coordinates are."
    },
    {
      "id": "ns.c6.c6b.gamma_re",
      "latex": "\\Gamma^{re}",
      "series": "NS",
      "first_appearance": "C6-B",
      "label_zh": "再入相干向量",
      "label_en": "re-entry coherence vector",
      "definition_zh": "C6-B 對內部 $F$ metadata 的擴充：一個七維座標向量，蒐羅 Duhamel 不抵消程度、非線性容量轉尖峰增長效率、分量選擇邊際、空間符號厚度邊際、閾值優勢邊際、持續性儲備與定理適用旗標。此為附掛在 $F_{\\rm NL}\\to H$ 邊上的 typed edge metadata；在 Theorem C6-B.9（Typed $H/F$ Cycle Reduction）中以「$\\Gamma^{re}$ gates」之名標註循環箭頭，並在 §49 被歸結為新的最小循環問題：這些座標能否在無限多次再生世代中同時保持非退化。",
      "definition_en": "C6-B's enrichment of internal $F$ metadata: a seven-coordinate vector collecting Duhamel noncancellation, nonlinear-capacity-to-peak-growth efficiency, component-selection margin, spatial sign-thickness margin, threshold dominance margin, persistence reserve, and the theorem-entry flag. This is typed edge metadata attached to the $F_{\\rm NL}\\to H$ edge; it labels the cycle arrow in Theorem C6-B.9 (Typed $H/F$ Cycle Reduction, as \"$\\Gamma^{re}$ gates\"), and §49 distills it into the new minimal cycle question: whether these coordinates can remain simultaneously nondegenerate over infinitely many re-entry generations.",
      "defining_relation": "\\Gamma^{re}=\\left(\\Gamma^{Duh},\\eta^{grow},m_{\\rm sel},\\beta_Z-\\delta,m_{\\rm thr},\\Pi_{\\rm time},\\mathsf{Setup}\\right)"
    },
    {
      "id": "ns.c6.c6c.y_ell",
      "latex": "Y_\\ell = D^\\ell e^{\\nu(t_1-t_0)\\Delta} u(t_0)",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "熱繼承項",
      "label_en": "Heat-inherited term",
      "definition_zh": "§2 定義為 Duhamel 分解 $D^\\ell u(t_1)=Y_\\ell+Z_\\ell$ 的線性部分：初始場 $u(t_0)$ 的 $\\ell$ 階導數經熱半群 $e^{\\nu(t_1-t_0)\\Delta}$ 演化到時刻 $t_1$ 的結果，代表「繼承自過去」而非由非線性 forcing 產生的部分，與 $Z_\\ell$ 互補。",
      "definition_en": "Defined in §2 as the linear (heat-propagated) summand of the Duhamel decomposition $D^\\ell u(t_1) = Y_\\ell + Z_\\ell$: the $\\ell$-th derivative of the initial field $u(t_0)$ evolved to time $t_1$ under the heat semigroup. It represents the part of the derivative inherited from the past rather than generated by nonlinear forcing, complementary to $Z_\\ell$.",
      "defining_relation": "Y_\\ell = D^\\ell e^{\\nu(t_1-t_0)\\Delta} u(t_0)",
      "notes": "Bounded via heat-contraction by A_ℓ(t_0) in §14; ratio ε = ||Y_ℓ||∞/||Z_ℓ||∞ (§29, inherited from C6-B) uses it."
    },
    {
      "id": "ns.c6.c6c.z_ell",
      "latex": "Z_\\ell = -\\int_{t_0}^{t_1} D^\\ell e^{\\nu(t_1-s)\\Delta} \\mathbb P((u\\cdot\\nabla)u)(s)ds",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "Duhamel 非線性響應",
      "label_en": "Duhamel nonlinear response",
      "definition_zh": "§2 定義的 Duhamel 分解中非線性部分：Navier–Stokes 非線性項 $(u\\cdot\\nabla)u$（經 Leray 投影 $\\mathbb P$）在 $[t_0,t_1]$ 上經熱半群傳播並積分而成。是本輪 coherence 理論的核心研究對象，其內部一致性結構（是否集中於單一未來目標、是否維持同號）貫穿全篇。",
      "definition_en": "The nonlinear summand of the Duhamel decomposition (§2): the accumulated effect of the Navier–Stokes nonlinear term (Leray-projected) propagated by the heat semigroup over $[t_0,t_1]$. Its internal coherence — whether it concentrates on one future target and maintains a consistent sign — is the central subject of C6-C.",
      "defining_relation": "Z_\\ell = -\\int_{t_0}^{t_1} D^\\ell e^{\\nu(t_1-s)\\Delta} \\mathbb P((u\\cdot\\nabla)u)(s)ds",
      "notes": "Equivalently Z_ℓ = ∫ q_ℓ(s)ds (§3)."
    },
    {
      "id": "ns.c6.c6c.q_ell",
      "latex": "q_\\ell(s) = - D^\\ell e^{\\nu(t_1-s)\\Delta} \\mathbb P((u\\cdot\\nabla)u)(s)",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "Duhamel 被積式",
      "label_en": "Duhamel integrand",
      "definition_zh": "§3 定義，為 $Z_\\ell$ 的被積函數，滿足 $Z_\\ell=\\int_{t_0}^{t_1}q_\\ell(s)ds$。其逐時刻 sup-norm $\\|q_\\ell(s)\\|_\\infty$ 用於定義全域容量 $\\mathfrak C_\\ell$ 與 coherence 機率測度 $\\mu_\\ell$，是全篇容量/一致性計算的基本建構元件。",
      "definition_en": "Defined in §3 as the integrand of $Z_\\ell$, satisfying $Z_\\ell = \\int_{t_0}^{t_1} q_\\ell(s)ds$. Its pointwise-in-time sup-norm underlies the global capacity $\\mathfrak C_\\ell$ and the coherence probability measure $\\mu_\\ell$, serving as the basic building block for all capacity/coherence computations in the round.",
      "defining_relation": "q_\\ell(s) = - D^\\ell e^{\\nu(t_1-s)\\Delta} \\mathbb P((u\\cdot\\nabla)u)(s)"
    },
    {
      "id": "ns.c6.c6c.capacity_ell",
      "latex": "\\mathfrak C_\\ell = \\int_{t_0}^{t_1} \\|q_\\ell(s)\\|_\\infty ds",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "全域 Duhamel 容量",
      "label_en": "Global Duhamel capacity",
      "definition_zh": "§3 定義，為 Duhamel 被積式 sup-norm 在時間窗上的積分，滿足三角不等式 $\\|Z_\\ell\\|_\\infty\\le\\mathfrak C_\\ell$。量測 forcing 的總可用容量而不論其在空間/時間上是否對齊，是 Duhamel coherence $\\Gamma_\\ell$ 的分母。",
      "definition_en": "Defined in §3 as the time-integral of $\\|q_\\ell(s)\\|_\\infty$ over $[t_0,t_1]$; satisfies $\\|Z_\\ell\\|_\\infty \\le \\mathfrak C_\\ell$. It measures total available forcing capacity irrespective of spatial or temporal alignment, and is the denominator of the Duhamel coherence ratio $\\Gamma_\\ell$.",
      "defining_relation": "\\mathfrak C_\\ell = \\int_{t_0}^{t_1} \\|q_\\ell(s)\\|_\\infty ds"
    },
    {
      "id": "ns.c6.c6c.gamma_duh",
      "latex": "\\Gamma_\\ell = \\Gamma_\\ell^{Duh} = \\frac{\\|Z_\\ell\\|_\\infty}{\\mathfrak C_\\ell} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "Duhamel 一致性",
      "label_en": "Duhamel coherence",
      "definition_zh": "§4 定義（$\\mathfrak C_\\ell>0$ 時），為實際響應 $\\|Z_\\ell\\|_\\infty$ 與全域容量 $\\mathfrak C_\\ell$ 之比。是本輪核心量：C6-C.1（§9）證明其精確分解為 $\\chi_\\ast^{target}\\gamma_\\ast^{time}$，並在 growth efficiency 上界（C6-C.3）、thick-target coherence 下界（C6-C.4）、reserve vector 首項中反覆作主控參數。全文亦省略下標記為 $\\Gamma^{Duh}$。C6-B 已證其不能僅由 $\\mathfrak C_\\ell$ 得正下界。",
      "definition_en": "Defined in §4 (when $\\mathfrak C_\\ell>0$) as the ratio of realized response to global capacity. This is the round's central quantity: C6-C.1 (§9) proves the exact factorization $\\Gamma_\\ell = \\chi_\\ast^{target}\\gamma_\\ast^{time}$, and it recurs as the controlling parameter in the growth-efficiency bound (C6-C.3), the thick-target coherence bound (C6-C.4), and as the first coordinate of the reserve vector. Written $\\Gamma^{Duh}$ with subscript suppressed elsewhere. C6-B already showed it admits no positive lower bound from $\\mathfrak C_\\ell$ alone.",
      "defining_relation": "\\Gamma_\\ell = \\Gamma_\\ell^{Duh} = \\frac{\\|Z_\\ell\\|_\\infty}{\\mathfrak C_\\ell}",
      "notes": "Threshold constant γ_0 (§10, \"Γ_ℓ≥γ_0>0\") is used as a standing nondegeneracy hypothesis. When Γ_{ℓ_n}→0 with comparable response, §26-27 (C6-C.5) shows 𝔠_{ℓ_n}/||Z_{ℓ_n}||∞ = Γ_{ℓ_n}^{-1}→∞, named \"Forcing-Capacity Inflation\" in §45 (boundary code CAPACITY)."
    },
    {
      "id": "ns.c6.c6c.response_peak",
      "latex": "(x_\\ast,i,\\sigma),\\ \\sigma\\in\\{\\pm1\\},\\ \\sigma Z_{\\ell,i}(x_\\ast)=\\|Z_\\ell\\|_\\infty",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "響應峰值位置/分量/符號",
      "label_en": "Response peak location/component/sign",
      "definition_zh": "§5 定義，取 Duhamel 響應 $Z_\\ell$ 達到 sup-norm 的最大化空間點 $x_\\ast$、分量指標 $i$ 與符號 $\\sigma\\in\\{\\pm1\\}$。此峰值三元組是後續單點量 $\\mathfrak C_\\ast$、$\\chi_\\ast^{target}$、$\\gamma_\\ast^{time}$ 的基準座標。",
      "definition_en": "Defined in §5 as the maximizing spatial point $x_\\ast$, component index $i$, and sign $\\sigma\\in\\{\\pm1\\}$ at which the Duhamel response attains its sup-norm. This peak triple is the reference coordinate for the pointwise quantities $\\mathfrak C_\\ast$, $\\chi_\\ast^{target}$, and $\\gamma_\\ast^{time}$ defined in §6-§8.",
      "defining_relation": "\\sigma Z_{\\ell,i}(x_\\ast) = \\|Z_\\ell\\|_\\infty",
      "notes": "All statements are said to admit a near-maximizer version if an exact maximizer fails to exist."
    },
    {
      "id": "ns.c6.c6c.capacity_star",
      "latex": "\\mathfrak C_\\ast = \\int_{t_0}^{t_1} |q_{\\ell,i}(s,x_\\ast)|ds",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "目標容量",
      "label_en": "Target capacity (at peak)",
      "definition_zh": "§6 定義，為實際送達最終峰值分量/位置 $(x_\\ast,i)$ 的容量，滿足 $0\\le\\mathfrak C_\\ast\\le\\mathfrak C_\\ell$，是 target concentration $\\chi_\\ast^{target}$ 的分子。",
      "definition_en": "Defined in §6 as the capacity actually delivered to the peak component/location $(x_\\ast,i)$, satisfying $0\\le\\mathfrak C_\\ast\\le\\mathfrak C_\\ell$; the numerator of the target concentration $\\chi_\\ast^{target}$.",
      "defining_relation": "\\mathfrak C_\\ast = \\int_{t_0}^{t_1} |q_{\\ell,i}(s,x_\\ast)|ds"
    },
    {
      "id": "ns.c6.c6c.chi_target_star",
      "latex": "\\chi_\\ast^{target} = \\frac{\\mathfrak C_\\ast}{\\mathfrak C_\\ell} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "未來目標集中度",
      "label_en": "Future-target concentration",
      "definition_zh": "§7 定義，量測全域 Duhamel 容量中實際「看得見」單一未來目標 $(x_\\ast,i)$ 的比例，是 C6-C.1（§9）精確分解 $\\Gamma_\\ell=\\chi_\\ast^{target}\\gamma_\\ast^{time}$ 的第一因子。若 $\\chi_{\\ast,n}^{target}\\to0$（§28 C-DIFF 分支），即 §47 boxed 命名的「Forcing Target Diffusion」（邊界代碼 DIFF）。",
      "definition_en": "Defined in §7, measuring the fraction of global Duhamel capacity actually visible to the single future target $(x_\\ast,i)$; the first factor in the exact factorization $\\Gamma_\\ell=\\chi_\\ast^{target}\\gamma_\\ast^{time}$ (C6-C.1, §9). If $\\chi_{\\ast,n}^{target}\\to0$ (branch C-DIFF, §28), this is named \"Forcing Target Diffusion\" in §47 (boundary code DIFF).",
      "defining_relation": "\\chi_\\ast^{target} = \\frac{\\mathfrak C_\\ast}{\\mathfrak C_\\ell}"
    },
    {
      "id": "ns.c6.c6c.gamma_time_star",
      "latex": "\\gamma_\\ast^{time} = \\frac{\\sigma\\int_{t_0}^{t_1} q_{\\ell,i}(s,x_\\ast)ds}{\\int_{t_0}^{t_1}|q_{\\ell,i}(s,x_\\ast)|ds} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "目標處時間符號一致性",
      "label_en": "Temporal sign coherence at the target",
      "definition_zh": "§8 定義（$\\mathfrak C_\\ast>0$ 時），量測被積式在峰值目標處於時間上維持同號（未被反號抵消）的程度，是 C6-C.1 分解的第二因子。若 $\\gamma_{\\ast,n}^{time}\\to0$（§28 C-CANCEL 分支），代表目標接收強烈時間反號抵消的 forcing（邊界代碼 CANCEL）。",
      "definition_en": "Defined in §8 (when $\\mathfrak C_\\ast>0$), measuring how much the integrand at the peak target maintains a consistent sign in time; the second factor in C6-C.1's factorization. If $\\gamma_{\\ast,n}^{time}\\to0$ (branch C-CANCEL, §28), the target receives strongly sign-cancelling forcing — \"temporal cancellation\" (boundary code CANCEL).",
      "defining_relation": "\\gamma_\\ast^{time} = \\frac{\\sigma\\int_{t_0}^{t_1} q_{\\ell,i}(s,x_\\ast)ds}{\\int_{t_0}^{t_1}|q_{\\ell,i}(s,x_\\ast)|ds}"
    },
    {
      "id": "ns.c6.c6c.mu_ell",
      "latex": "d\\mu_\\ell(s) = \\frac{\\|q_\\ell(s)\\|_\\infty}{\\mathfrak C_\\ell}ds",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "一致性機率測度",
      "label_en": "Coherence probability measure (on time)",
      "definition_zh": "§11 定義，將被積式 sup-norm 正規化為時間軸 $[t_0,t_1]$ 上的機率測度，用以將 $\\Gamma_\\ell$ 表為對齊標記 $a_\\ell(s)$ 之期望值 $\\Gamma_\\ell=\\int a_\\ell(s)\\,d\\mu_\\ell(s)$，為 §12 pushforward 測度 $\\nu_\\ell^{coh}$ 鋪路。",
      "definition_en": "Defined in §11 by normalizing the integrand's sup-norm into a probability measure on $[t_0,t_1]$. Lets $\\Gamma_\\ell$ be written as the expectation $\\Gamma_\\ell=\\int a_\\ell(s)\\,d\\mu_\\ell(s)$ of the alignment mark, and sets up the pushforward measure $\\nu_\\ell^{coh}$ of §12.",
      "defining_relation": "d\\mu_\\ell(s) = \\frac{\\|q_\\ell(s)\\|_\\infty}{\\mathfrak C_\\ell}ds"
    },
    {
      "id": "ns.c6.c6c.a_ell",
      "latex": "a_\\ell(s) = \\frac{\\sigma q_{\\ell,i}(s,x_\\ast)}{\\|q_\\ell(s)\\|_\\infty} \\in[-1,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "對齊標記",
      "label_en": "Alignment mark",
      "definition_zh": "§11 定義（分母非零時；否則設 $a_\\ell=0$），逐時刻量測被積式在峰值方向上的正規化投影：$a_\\ell(s)=1$ 表完全對齊，負值表反向抵消。滿足 $\\Gamma_\\ell=\\int a_\\ell\\,d\\mu_\\ell$，是 §12 Young 測度 $\\nu_\\ell^{coh}=(a_\\ell)_\\#\\mu_\\ell$ 的推前函數。",
      "definition_en": "Defined in §11 (denominator nonzero; else $a_\\ell=0$) as the pointwise-in-time normalized projection of the integrand onto the peak direction: $a_\\ell(s)=1$ means full alignment, negative values indicate cancelling. Satisfies $\\Gamma_\\ell=\\int a_\\ell\\,d\\mu_\\ell$, and is the map defining the Young measure $\\nu_\\ell^{coh}=(a_\\ell)_\\#\\mu_\\ell$ of §12.",
      "defining_relation": "a_\\ell(s) = \\frac{\\sigma q_{\\ell,i}(s,x_\\ast)}{\\|q_\\ell(s)\\|_\\infty}"
    },
    {
      "id": "ns.c6.c6c.nu_coh",
      "latex": "\\nu_\\ell^{coh} = (a_\\ell)_\\# \\mu_\\ell \\in \\mathcal P([-1,1])",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "一致性 Young 測度",
      "label_en": "Coherence Young measure/state",
      "definition_zh": "§12 定義，為對齊標記 $a_\\ell$ 在時間測度 $\\mu_\\ell$ 下的推前機率測度，滿足 $\\Gamma_\\ell=\\int_{-1}^1 a\\,d\\nu_\\ell^{coh}(a)$；因 $[-1,1]$ 緊致，遞迴 coherence 分佈存在弱收斂子序列。C6-C.2（§13，High-Coherence Concentration Lemma）用它證明：若 $\\Gamma_\\ell\\ge\\gamma_0$，低對齊尾集 $B_\\eta=\\{a\\le1-\\eta\\}$（$0<\\eta\\le2$）滿足 $\\nu_\\ell^{coh}(B_\\eta)\\le(1-\\gamma_0)/\\eta$，特例 $\\nu_\\ell^{coh}\\{a\\le0\\}\\le1-\\gamma_0$，即高 coherence 迫使多數容量對齊同一未來方向。",
      "definition_en": "Defined in §12 as the pushforward of $\\mu_\\ell$ under the alignment mark $a_\\ell$; satisfies $\\Gamma_\\ell=\\int_{-1}^1 a\\,d\\nu_\\ell^{coh}(a)$. Since $[-1,1]$ is compact, recurrent coherence profiles admit weakly convergent subsequences. C6-C.2 (§13, High-Coherence Concentration Lemma) uses it: if $\\Gamma_\\ell\\ge\\gamma_0$, the low-alignment tail set $B_\\eta=\\{a\\le1-\\eta\\}$ satisfies $\\nu_\\ell^{coh}(B_\\eta)\\le(1-\\gamma_0)/\\eta$, in particular $\\nu_\\ell^{coh}\\{a\\le0\\}\\le1-\\gamma_0$ — high coherence forces most capacity to align with one common future direction.",
      "defining_relation": "\\nu_\\ell^{coh} = (a_\\ell)_\\# \\mu_\\ell"
    },
    {
      "id": "ns.c6.c6c.derivative_amplitude_bound",
      "latex": "\\|Y_\\ell\\|_\\infty \\le A_\\ell(t_0)",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "導數振幅界",
      "label_en": "Derivative amplitude bound",
      "definition_zh": "§14 用於陳述熱半群 $L^\\infty$-收縮性質的既有量（承接自 C5/C6 系列，非本輪新定義）：$\\|Y_\\ell\\|_\\infty\\le A_\\ell(t_0)$，並得遞推界 $A_\\ell(t_1)\\le A_\\ell(t_0)+\\|Z_\\ell\\|_\\infty$，是 §15 peak-growth efficiency $\\eta_\\ell^{grow}$ 的構成要素。",
      "definition_en": "Used in §14 to state the heat semigroup's $L^\\infty$-contraction property, giving the recursive bound $A_\\ell(t_1)\\le A_\\ell(t_0)+\\|Z_\\ell\\|_\\infty$. Inherited notation from the C5/C6 series (not freshly defined here) for a derivative-amplitude bound at time $t$; a constituent of the peak-growth efficiency $\\eta_\\ell^{grow}$ of §15.",
      "notes": "Not explicitly redefined within this file; no defining_relation is stated here so none is given. Also appears in §31's ρ_sel = [m_sel]_+/(A_ℓ+[m_sel]_+)."
    },
    {
      "id": "ns.c6.c6c.eta_grow",
      "latex": "\\eta_\\ell^{grow} = \\frac{[A_\\ell(t_1)-A_\\ell(t_0)]_+}{\\mathfrak C_\\ell} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "峰值成長效率",
      "label_en": "Peak-growth efficiency",
      "definition_zh": "§15 定義，量測導數振幅淨成長相對全域容量的比例。C6-C.3（§16）證明其被 Duhamel coherence 上界控制：$\\eta_\\ell^{grow}\\le\\Gamma_\\ell$；故若某世代以效率 $\\eta_\\ell^{grow}\\ge\\eta_0>0$ 再生正峰值，自動要求 $\\Gamma_\\ell\\ge\\eta_0$——真正的成長排除任意弱的 Duhamel coherence。",
      "definition_en": "Defined in §15, measuring net derivative-amplitude growth relative to global capacity. C6-C.3 (§16) proves $\\eta_\\ell^{grow}\\le\\Gamma_\\ell$; hence a generation truly regenerating a positive peak with efficiency $\\eta_\\ell^{grow}\\ge\\eta_0>0$ automatically forces $\\Gamma_\\ell\\ge\\eta_0$ — actual growth rules out arbitrarily weak Duhamel coherence.",
      "defining_relation": "\\eta_\\ell^{grow} = \\frac{[A_\\ell(t_1)-A_\\ell(t_0)]_+}{\\mathfrak C_\\ell}",
      "notes": "Threshold η_0 (§16) is the standing nondegenerate-growth hypothesis; combined with λ_Z in §23 gives χ_E,γ_E ≥ λ_Z η_0. Written η^grow (subscript suppressed) in §0 and §63/64."
    },
    {
      "id": "ns.c6.c6c.set_e",
      "latex": "E = \\left\\{x:\\sigma Z_{\\ell,i}(x)\\ge\\lambda_Z\\|Z_\\ell\\|_\\infty\\right\\}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "響應符號厚集",
      "label_en": "Response sign-thick target set",
      "definition_zh": "§17 定義，為響應 $Z_\\ell$ 在閾值 $\\lambda_Z$ 下達到「符號厚」的空間集合，要求 $|E|>0$。是 §18-§25 與 C6-C.4（§21）thick-target source coherence 定理的目標集合；鏈尺度壞核情形供給其下界體積密度（§24：$|E|\\ge c_3\\delta^3r^3$）。",
      "definition_en": "Defined in §17 as the spatial set where $Z_\\ell$ is \"sign-thick\" at threshold $\\lambda_Z$, required to have $|E|>0$. The target region for §18-§25 and the Thick-Target Source Coherence Theorem (C6-C.4, §21); the chain-scale bad-core case later supplies a lower volume-density bound (§24: $|E|\\ge c_3\\delta^3r^3$).",
      "defining_relation": "E = \\{x:\\sigma Z_{\\ell,i}(x)\\ge\\lambda_Z\\|Z_\\ell\\|_\\infty\\}"
    },
    {
      "id": "ns.c6.c6c.lambda_z",
      "latex": "\\lambda_Z",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "符號厚度閾值參數",
      "label_en": "Sign-thickness threshold parameter",
      "definition_zh": "§17 定義符號厚集 $E$ 時引入的閾值常數，貫穿全篇作為 thick-target coherence 不等式 $\\chi_E\\gamma_E\\ge\\lambda_Z\\Gamma^{Duh}$（C6-C.4，§21）與 dominance reserve 臨界比 $\\epsilon_{\\rm crit}=(\\lambda_Z-\\lambda)/(1+\\lambda)$（§29）的核心閾值。",
      "definition_en": "The threshold constant introduced in §17's definition of $E$; recurs as the key threshold in the thick-target coherence inequality $\\chi_E\\gamma_E\\ge\\lambda_Z\\Gamma^{Duh}$ (C6-C.4, §21) and in the dominance critical ratio $\\epsilon_{\\rm crit}=(\\lambda_Z-\\lambda)/(1+\\lambda)$ (§29).",
      "notes": "§29 assumes λ_Z > λ for a separate constant λ presupposed from C6-B's dominance framework, not independently defined in this file."
    },
    {
      "id": "ns.c6.c6c.capacity_e",
      "latex": "\\mathfrak C_E = \\int_{t_0}^{t_1}\\int_E |q_{\\ell,i}(s,x)|dx\\,ds",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "目標集上的源容量",
      "label_en": "Source capacity over the target set E",
      "definition_zh": "§18 定義，為 Duhamel 被積式絕對值在整個符號厚集 $E$ 與時間窗上的積分，滿足 $\\mathfrak C_E\\le|E|\\mathfrak C_\\ell$，是 $\\chi_E$ 的分子，也是 §24-25 source-slab toll 估計的核心量。",
      "definition_en": "Defined in §18 as the integral of the integrand's absolute value over the whole sign-thick set $E$ and time window, satisfying $\\mathfrak C_E\\le|E|\\mathfrak C_\\ell$; the numerator of $\\chi_E$ and the central quantity in the §24-25 source-slab toll estimates.",
      "defining_relation": "\\mathfrak C_E = \\int_{t_0}^{t_1}\\int_E |q_{\\ell,i}(s,x)|dx\\,ds"
    },
    {
      "id": "ns.c6.c6c.chi_e",
      "latex": "\\chi_E = \\frac{\\mathfrak C_E}{|E|\\mathfrak C_\\ell} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "空間目標集中度（集合版）",
      "label_en": "Spatial target concentration (over E)",
      "definition_zh": "§19 定義，為全域 forcing 容量在整個目標集 $E$ 上可見的平均比例（$\\chi_\\ast^{target}$ 的集合版）。C6-C.4（§21）證明 $\\chi_E\\gamma_E\\ge\\lambda_Z\\Gamma_\\ell$，並得獨立下界 $\\chi_E\\ge\\lambda_Z\\Gamma_\\ell$（§22）。",
      "definition_en": "Defined in §19 as the average fraction of global forcing capacity visible across the entire target set $E$ (set-valued analogue of $\\chi_\\ast^{target}$). C6-C.4 (§21) proves $\\chi_E\\gamma_E\\ge\\lambda_Z\\Gamma_\\ell$, giving the separate bound $\\chi_E\\ge\\lambda_Z\\Gamma_\\ell$ (§22).",
      "defining_relation": "\\chi_E = \\frac{\\mathfrak C_E}{|E|\\mathfrak C_\\ell}"
    },
    {
      "id": "ns.c6.c6c.gamma_e",
      "latex": "\\gamma_E = \\frac{\\int_{t_0}^{t_1}\\int_E \\sigma q_{\\ell,i}(s,x)dx\\,ds}{\\mathfrak C_E} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "時間符號一致性（集合版）",
      "label_en": "Temporal sign coherence (over E)",
      "definition_zh": "§20 定義（經 Fubini 定理），為目標集 $E$ 上被積式維持同號的比例（$\\gamma_\\ast^{time}$ 的集合版），分子因 $E$ 為選定響應符號高集而恆正。與 $\\chi_E$ 共同滿足 $\\chi_E\\gamma_E\\ge\\lambda_Z\\Gamma_\\ell$，並得獨立下界 $\\gamma_E\\ge\\lambda_Z\\Gamma_\\ell$（§22）。",
      "definition_en": "Defined in §20 (via Fubini) as the same-sign-persistence fraction of the integrand over $E$ (set-valued analogue of $\\gamma_\\ast^{time}$); numerator positive because $E$ is the selected response-sign-high set. Together with $\\chi_E$ satisfies $\\chi_E\\gamma_E\\ge\\lambda_Z\\Gamma_\\ell$, yielding $\\gamma_E\\ge\\lambda_Z\\Gamma_\\ell$ (§22).",
      "defining_relation": "\\gamma_E = \\frac{\\int_{t_0}^{t_1}\\int_E \\sigma q_{\\ell,i}(s,x)dx\\,ds}{\\mathfrak C_E}"
    },
    {
      "id": "ns.c6.c6c.source_slab_toll",
      "latex": "\\frac{\\mathfrak C_E}{r^3\\mathfrak C_\\ell} \\ge c_3\\delta^3\\lambda_Z\\Gamma_\\ell",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "一致性源片代價",
      "label_en": "Coherent Source-Slab Toll",
      "definition_zh": "§24 明確 boxed 命名的量：當符號厚集 $E\\subset B_r(x_0)$ 且鏈尺度體積密度 $|E|\\ge c_3\\delta^3r^3$（承接 C5 volume-to-line contrapositive）成立時，正規化源容量比 $\\mathfrak C_E/(r^3\\mathfrak C_\\ell)$ 至少為 $c_3\\delta^3\\lambda_Z\\Gamma_\\ell$（若 $\\Gamma_\\ell\\ge\\gamma_0$ 則為 $c_3\\delta^3\\lambda_Z\\gamma_0$）。代表非退化再入需要固定正規化源片容量，是 §43 uniform coherent branch「固定正規化源片債務」的原型。",
      "definition_en": "The quantity explicitly boxed and named in §24: when $E\\subset B_r(x_0)$ has chain-scale volume density $|E|\\ge c_3\\delta^3r^3$ (C5 volume-to-line contrapositive), the normalized source-capacity ratio $\\mathfrak C_E/(r^3\\mathfrak C_\\ell)$ is at least $c_3\\delta^3\\lambda_Z\\Gamma_\\ell$ (or $c_3\\delta^3\\lambda_Z\\gamma_0$ under $\\Gamma_\\ell\\ge\\gamma_0$). Formalizes that nondegenerate re-entry requires a fixed normalized source-slab capacity, prefiguring the fixed normalized debt of §43's uniform coherent branch.",
      "defining_relation": "\\frac{\\mathfrak C_E}{r^3\\mathfrak C_\\ell} \\ge c_3\\delta^3\\lambda_Z\\Gamma_\\ell",
      "notes": "c_3, δ, r, x_0 are inherited chain-scale/structural constants from the C5 series, not freshly defined here. §25 additionally states an unboxed \"absolute\" (non-normalized) version, 𝔠_E ≳ λ_Z δ^3 ||Z_ℓ||∞ r^3, described in prose as the absolute source-slab toll but not itself given a separate boxed name."
    },
    {
      "id": "ns.c6.c6c.epsilon_dom",
      "latex": "\\epsilon = \\frac{\\|Y_\\ell\\|_\\infty}{\\|Z_\\ell\\|_\\infty}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "繼承場對響應比",
      "label_en": "Inherited-field-to-response ratio",
      "definition_zh": "§29 開頭以「C6-B used」引入，承接自 C6-B 而非本輪新定義，但為本輪 $\\epsilon_{\\rm crit}$ 與 $\\rho_{\\rm dom}$ 定義所必需：量測熱繼承項 $Y_\\ell$ 相對 Duhamel 響應 $Z_\\ell$ 的相對大小，用於判斷一次性符號繼承是否成立。",
      "definition_en": "Stated at the start of §29 as \"used by C6-B\" — inherited from C6-B rather than freshly defined here, but necessary for this round's $\\epsilon_{\\rm crit}$ and $\\rho_{\\rm dom}$: measures the size of the heat-inherited term $Y_\\ell$ relative to the Duhamel response $Z_\\ell$, used to judge one-time sign inheritance.",
      "defining_relation": "\\epsilon = \\frac{\\|Y_\\ell\\|_\\infty}{\\|Z_\\ell\\|_\\infty}"
    },
    {
      "id": "ns.c6.c6c.epsilon_crit",
      "latex": "\\epsilon_{\\rm crit} := \\frac{\\lambda_Z-\\lambda}{1+\\lambda}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "臨界主導比閾值",
      "label_en": "Critical dominance ratio threshold",
      "definition_zh": "§29 由不等式 $\\lambda_Z-\\epsilon>\\lambda(1+\\epsilon)$ 解出並 boxed 定義（假設 $\\lambda_Z>\\lambda$）：一次性 response-to-actual-field 符號繼承成立的 $\\epsilon$ 臨界閾值，是正規化 dominance reserve $\\rho_{\\rm dom}$ 的分母基準。",
      "definition_en": "Defined (boxed) in §29 by solving $\\lambda_Z-\\epsilon>\\lambda(1+\\epsilon)$ (assuming $\\lambda_Z>\\lambda$): the critical threshold on $\\epsilon$ below which one-time response-to-actual-field sign inheritance holds; the normalizing benchmark in $\\rho_{\\rm dom}$'s definition.",
      "defining_relation": "\\epsilon_{\\rm crit} := \\frac{\\lambda_Z-\\lambda}{1+\\lambda}"
    },
    {
      "id": "ns.c6.c6c.rho_dom",
      "latex": "\\rho_{\\rm dom} = \\left[1-\\frac{\\epsilon}{\\epsilon_{\\rm crit}}\\right]_+ \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "主導性儲備",
      "label_en": "(Normalized) dominance reserve",
      "definition_zh": "§29 定義，為 $\\epsilon$ 相對臨界閾值 $\\epsilon_{\\rm crit}$ 的正規化餘裕，是 re-entry reserve vector $\\mathbf R^{re}$ 的一個座標。若 $\\rho_{\\rm dom}\\to0$（§30），非線性響應不再足夠強地主導繼承熱場以證實真正符號繼承，此即 §30 boxed 命名的「Inherited-Field Takeover / Dominance Saturation」（邊界代碼 DOM，§41 C-B2）：邊界不應再被解讀為真正 $F_{\\rm NL}\\to H$，目標 $H$ 可能反而繼承自前一狀態。",
      "definition_en": "Defined in §29 as the normalized margin of $\\epsilon$ below $\\epsilon_{\\rm crit}$; one coordinate of $\\mathbf R^{re}$. If $\\rho_{\\rm dom}\\to0$ (§30), the nonlinear response no longer dominates the inherited heat field enough to certify genuine sign inheritance — named \"Inherited-Field Takeover / Dominance Saturation\" (boxed, §30; boundary code DOM, §41 C-B2): the edge should no longer be read as genuine $F_{\\rm NL}\\to H$, since $H$ may instead be inherited from the previous state.",
      "defining_relation": "\\rho_{\\rm dom} = \\left[1-\\frac{\\epsilon}{\\epsilon_{\\rm crit}}\\right]_+"
    },
    {
      "id": "ns.c6.c6c.m_sel",
      "latex": "m_{\\rm sel} = \\sigma D^\\ell u_i(x_0) - \\max_{(\\zeta',j)\\ne(\\zeta,i)} |D^{\\zeta'}u_j(x_0)|",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "分量選擇邊際",
      "label_en": "Component-selection margin",
      "definition_zh": "§31 於候選壞點 $x_0$ 處定義，為選定的帶號導數分量與所有其他導數分量/符號中最大者之間的邊際差，是正規化選擇儲備 $\\rho_{\\rm sel}$ 經 $[\\cdot]_+$ 截斷後的分子。",
      "definition_en": "Defined in §31 at the candidate bad point $x_0$ as the margin between the selected signed derivative component and the maximum over all other derivative components/signs; the (positive-part) numerator of the normalized selection reserve $\\rho_{\\rm sel}$.",
      "defining_relation": "m_{\\rm sel} = \\sigma D^\\ell u_i(x_0) - \\max_{(\\zeta',j)\\ne(\\zeta,i)} |D^{\\zeta'}u_j(x_0)|"
    },
    {
      "id": "ns.c6.c6c.rho_sel",
      "latex": "\\rho_{\\rm sel} = \\frac{[m_{\\rm sel}]_+}{A_\\ell+[m_{\\rm sel}]_+} \\in[0,1)",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "分量選擇儲備",
      "label_en": "Component-selection reserve",
      "definition_zh": "§31 定義，將選擇邊際 $m_{\\rm sel}$ 正規化到 $[0,1)$，是 $\\mathbf R^{re}$ 的一個座標；嚴格選擇一致性要求 $\\rho_{\\rm sel}>0$。若 $\\rho_{\\rm sel}\\to0$（§32），生成的壞分量趨近與另一導數分量/符號打平，C6-B 單分量再入證書隨之失去穩定性，此即 §32 boxed 命名的「Selection Degeneration」（邊界代碼 SEL，§41 C-B3）；不蘊含正則性，但特定型別的 $F_{\\rm NL}\\to H$ 邊失去連續性。",
      "definition_en": "Defined in §31 by normalizing $m_{\\rm sel}$ to $[0,1)$; one coordinate of $\\mathbf R^{re}$, with strict selection coherence requiring $\\rho_{\\rm sel}>0$. If $\\rho_{\\rm sel}\\to0$ (§32), the generated bad component approaches a tie with another derivative component/sign and the C6-B one-component re-entry certificate loses stability — \"Selection Degeneration\" (boxed, §32; boundary code SEL, §41 C-B3). Does not by itself imply regularity, but the specific typed $F_{\\rm NL}\\to H$ edge loses continuity.",
      "defining_relation": "\\rho_{\\rm sel} = \\frac{[m_{\\rm sel}]_+}{A_\\ell+[m_{\\rm sel}]_+}"
    },
    {
      "id": "ns.c6.c6c.beta_z",
      "latex": "\\beta_Z",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "響應符號高佔比",
      "label_en": "Response sign-high occupancy",
      "definition_zh": "§33 以「Let: $\\beta_Z$ be...」定義，為目標鏈尺度壞核上響應符號高的佔比，是 harmonic sign reserve $\\rho_{\\rm sign}$ 的分子基礎，並在 §43 uniform coherent branch 中被下界估計為 $\\beta_Z\\ge\\delta+(1-\\delta)b_0$。",
      "definition_en": "Defined in §33 (\"Let: $\\beta_Z$ be...\") as the response sign-high occupancy fraction on the target chain-scale bad core; the basis for the harmonic sign reserve $\\rho_{\\rm sign}$, bounded below by $\\beta_Z\\ge\\delta+(1-\\delta)b_0$ on the uniform coherent branch (§43).",
      "notes": "δ is an inherited chain-scale/sign-thickness threshold from the C5 series (e.g. C5-L), not freshly defined in this file."
    },
    {
      "id": "ns.c6.c6c.rho_sign",
      "latex": "\\rho_{\\rm sign} = \\left[\\frac{\\beta_Z-\\delta}{1-\\delta}\\right]_+ \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "諧波符號儲備",
      "label_en": "Harmonic sign reserve",
      "definition_zh": "§33 boxed 定義，將 $\\beta_Z$ 相對閾值 $\\delta$ 正規化，是 $\\mathbf R^{re}$ 的一個座標；$\\rho_{\\rm sign}>0$ 表響應具嚴格符號厚度餘裕。若 $\\rho_{\\rm sign}\\to0$（§34），再入趨近諧波空間閾值，即「harmonic critical saturation」（邊界代碼 SIGN，§41 C-B4）；C5-L 顯示此時 $\\beta^{win}\\downarrow\\delta$ 仍保有正下降係數 $(1+\\lambda)\\delta-1>0$，故諧波符號飽和不會抹除下游導數債務（§34 boxed 命名）。",
      "definition_en": "Defined (boxed) in §33 by normalizing $\\beta_Z$ against $\\delta$; one coordinate of $\\mathbf R^{re}$, with $\\rho_{\\rm sign}>0$ indicating strict sign-thickness margin. If $\\rho_{\\rm sign}\\to0$ (§34), the re-entry approaches the harmonic spatial threshold — \"harmonic critical saturation\" (boundary code SIGN, §41 C-B4); C5-L shows the descent coefficient $(1+\\lambda)\\delta-1>0$ stays positive even as $\\beta^{win}\\downarrow\\delta$, so harmonic sign saturation does not erase the downstream derivative debt (named, boxed, §34).",
      "defining_relation": "\\rho_{\\rm sign} = \\left[\\frac{\\beta_Z-\\delta}{1-\\delta}\\right]_+"
    },
    {
      "id": "ns.c6.c6c.v_time",
      "latex": "\\mathfrak V_{\\rm time} = \\sup_{s\\in I_\\ell} \\|D^\\ell u(s)-D^\\ell u(t_\\ast)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "時間變動量",
      "label_en": "Temporal variation",
      "definition_zh": "§35 boxed 定義，量測 $\\ell$ 階導數在窗口 $I_\\ell$ 內相對某參考時刻 $t_\\ast$ 的最大偏移；C6-B persistence 要求 $(1+\\lambda)\\mathfrak V_{\\rm time}<m_{\\rm thr}$（$m_{\\rm thr}$ 為一次性實際符號邊際，承接 C6-B）。是 persistence reserve $\\rho_{\\rm time}$ 的核心輸入，且由 §36 知其受 $\\int(C\\nu A_{\\ell+2}+\\mathcal N_\\ell^{proj})dt$ 控制，故 persistence collapse 會回歸黏性/非線性時間 forcing。",
      "definition_en": "Defined (boxed) in §35 as the supremum, over window $I_\\ell$, of the deviation of the $\\ell$-th derivative from its value at reference time $t_\\ast$; C6-B persistence requires $(1+\\lambda)\\mathfrak V_{\\rm time}<m_{\\rm thr}$ ($m_{\\rm thr}$ the one-time actual sign margin, from C6-B). Key input to the persistence reserve $\\rho_{\\rm time}$; §36 notes it is controlled by $\\int(C\\nu A_{\\ell+2}+\\mathcal N_\\ell^{proj})dt$, so persistence collapse routes back to viscous/nonlinear temporal forcing rather than being pure geometric noise.",
      "defining_relation": "\\mathfrak V_{\\rm time} = \\sup_{s\\in I_\\ell} \\|D^\\ell u(s)-D^\\ell u(t_\\ast)\\|_\\infty",
      "notes": "m_thr > 0 (§35, \"one-time actual sign margin\") is an inherited C6-B threshold used alongside 𝔙_time; not independently boxed-defined in this file."
    },
    {
      "id": "ns.c6.c6c.rho_time",
      "latex": "\\rho_{\\rm time} = \\left[1-\\frac{(1+\\lambda)\\mathfrak V_{\\rm time}}{m_{\\rm thr}}\\right]_+ \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "持續性儲備",
      "label_en": "Persistence reserve",
      "definition_zh": "§35 boxed 定義，將時間變動量 $\\mathfrak V_{\\rm time}$ 相對持續性條件正規化，是 $\\mathbf R^{re}$ 的一個座標。若 $\\rho_{\\rm time}\\to0$（§36），壞幾何失去整窗持續性儲備，即「persistence collapse」（邊界代碼 TIME，§41 C-B5），但因 $\\mathfrak V_{\\rm time}$ 受黏性/投影非線性項控制，collapse 會路由回「viscous/nonlinear temporal forcing」（§36 boxed 命名），而非純幾何雜訊。",
      "definition_en": "Defined (boxed) in §35 by normalizing $\\mathfrak V_{\\rm time}$ against the persistence condition; one coordinate of $\\mathbf R^{re}$. If $\\rho_{\\rm time}\\to0$ (§36), the bad geometry loses its whole-window persistence reserve — \"persistence collapse\" (boundary code TIME, §41 C-B5) — but since $\\mathfrak V_{\\rm time}$ is controlled by viscous/projected-nonlinear terms, it routes back toward \"viscous/nonlinear temporal forcing\" (boxed, §36) rather than being pure geometric noise.",
      "defining_relation": "\\rho_{\\rm time} = \\left[1-\\frac{(1+\\lambda)\\mathfrak V_{\\rm time}}{m_{\\rm thr}}\\right]_+"
    },
    {
      "id": "ns.c6.c6c.rho_setup",
      "latex": "\\rho_{\\rm setup} \\in\\{0,1\\}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "設定合法性儲備",
      "label_en": "Setup (legality) reserve",
      "definition_zh": "§37 boxed 定義為二元指示變數，編碼目標階數/時間對是否合法滿足 Grujić–Xu 定理入口所需設定，是 $\\mathbf R^{re}$ 的一個座標。若 $\\rho_{\\rm setup}=0$，型別化再入退出至類別 $\\mathsf A$ 而非 $H$（邊界代碼 SETUP，§41 C-B6，路由至合法性類別）。",
      "definition_en": "Defined (boxed) in §37 as a binary indicator encoding whether the target order/time pair legally satisfies the required Grujić–Xu theorem-entry setup; one coordinate of $\\mathbf R^{re}$. If $\\rho_{\\rm setup}=0$, the typed re-entry exits to class $\\mathsf A$ rather than $H$ (boundary code SETUP, §41 C-B6, routing to the legality class).",
      "defining_relation": "\\rho_{\\rm setup} \\in\\{0,1\\}"
    },
    {
      "id": "ns.c6.c6c.r_re_vector",
      "latex": "\\mathbf R^{re} = \\left(\\Gamma^{Duh},\\chi_E,\\gamma_E,\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup}\\right) \\in[0,1]^7\\times\\{0,1\\}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "再入儲備向量",
      "label_en": "Re-entry reserve vector",
      "definition_zh": "§38 boxed 定義，將本輪所有獨立再入儲備座標打包成一個向量：Duhamel coherence、空間/時間集合一致性、繼承場主導、分量選擇、諧波符號、時間持續性、設定合法性。因 $\\chi_E,\\gamma_E\\ge\\lambda_Z\\Gamma^{Duh}$（符號厚響應上），部分座標受約束而非完全獨立。是 §39 bottleneck $b^{re}$ 與 C6-C.6 有限再入瓶頸定理（§40）的直接輸入。",
      "definition_en": "Defined (boxed) in §38, packaging every independent re-entry reserve coordinate into one vector: Duhamel coherence, spatial/temporal set coherence, inherited-field dominance, component selection, harmonic sign, temporal persistence, and setup legality. Because $\\chi_E,\\gamma_E\\ge\\lambda_Z\\Gamma^{Duh}$ holds on a sign-thick response, some coordinates are constrained rather than fully independent. Directly feeds the bottleneck $b^{re}$ (§39) and the Finite Re-entry Bottleneck Theorem C6-C.6 (§40).",
      "defining_relation": "\\mathbf R^{re} = \\left(\\Gamma^{Duh},\\chi_E,\\gamma_E,\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup}\\right)"
    },
    {
      "id": "ns.c6.c6c.b_re_bottleneck",
      "latex": "b^{re} = \\min\\left\\{\\Gamma^{Duh},\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup}\\right\\}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "再入瓶頸",
      "label_en": "Re-entry bottleneck",
      "definition_zh": "§39 boxed 定義，取六個主要 reserve 座標之最小值（不含由其餘座標約束推得的 $\\chi_E,\\gamma_E$）。$b^{re}>0$ 表所有主要型別化再入閘門具嚴格餘裕。C6-C.6（§40，Finite Re-entry Bottleneck Theorem）證明：對無限多候選再入世代，取子序列後恰有 C-UNIFORM（存在 $b_0>0$ 使 $b_n^{re}\\ge b_0$）或 C-BOUNDARY（$b_n^{re}\\to0$，因座標有限，進一步子序列後至少一 reserve 座標趨零）二者之一。C6-C.7（§53）重述為 Type U / Type S。§50 另定義「cycle-critically saturated」：$b_n^{re}\\to0$ 但每個有限世代仍成功再入 $H$。",
      "definition_en": "Defined (boxed) in §39 as the minimum of the six primary reserve coordinates (excluding χ_E, γ_E which are already constrained). $b^{re}>0$ means all major typed re-entry gates have strict reserve. The Finite Re-entry Bottleneck Theorem (C6-C.6, §40) proves that for infinitely many candidate generations, after subsequence exactly one of C-UNIFORM (some $b_0>0$ with $b_n^{re}\\ge b_0$) or C-BOUNDARY ($b_n^{re}\\to0$, with a further subsequence isolating one vanishing coordinate) holds. C6-C.7 (§53) restates this as Type U / Type S. §50 further defines a sequence \"cycle-critically saturated\" if $b_n^{re}\\to0$ while every finite generation still re-enters $H$.",
      "defining_relation": "b^{re} = \\min\\left\\{\\Gamma^{Duh},\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup}\\right\\}"
    },
    {
      "id": "ns.c6.c6c.uniform_coherence_branch",
      "latex": "b_n^{re}\\ge b_0>0",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "均勻時空非線性一致性分支",
      "label_en": "Uniform Spatiotemporal Nonlinear Coherence Branch",
      "definition_zh": "§42 boxed 命名（並於 §0 結果摘要中預告）的分支：當 $b_n^{re}\\ge b_0>0$ 沿無限多世代成立時，每一世代同時具備非退化 Duhamel 響應、空間目標集中度、時間源符號一致性、非線性對繼承熱場的主導、穩定選定分量、嚴格符號厚度餘裕、嚴格持續性餘裕、定理合法性——是唯一仍存活的真正一致 $F_{\\rm NL}\\to H$ 循環候選。§43 證明此分支每一再入世代皆攜帶固定正規化源片債務，但 §44 指出目前無已知全域有限預算能對所有世代求和以排除此分支，故 C6-C 未能排除它（§54）。",
      "definition_en": "The branch named (boxed) in §42 and previewed in the §0 result summary: when $b_n^{re}\\ge b_0>0$ holds along infinitely many generations, every generation has nondegenerate Duhamel response, spatial target concentration, temporal source sign coherence, nonlinear dominance over inherited heat, a stable selected component, strict sign-thickness margin, strict persistence margin, and theorem legality — the only remaining genuinely coherent $F_{\\rm NL}\\to H$ cycle candidate. §43 shows every generation on this branch carries a fixed normalized source-slab debt, but §44 notes no known global finite budget currently sums this across all generations to exclude the branch — so C6-C does not eliminate it (§54).",
      "defining_relation": "b_n^{re}\\ge b_0>0",
      "notes": "Equivalent to \"Type U\" in C6-C.7 (§53) and to the negation of \"cycle-critically saturated\" (§50)."
    },
    {
      "id": "ns.c6.c6c.re_entry_map",
      "latex": "\\mathscr R_n: \\Theta_{H,n} \\mapsto \\Theta_{F_{\\rm NL},n} \\mapsto \\mathbf R_n^{re} \\mapsto \\Theta_{H,n+1}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "再入映射",
      "label_en": "Re-entry map",
      "definition_zh": "§52 boxed 定義，將一次型別化非線性再入世代寫成映射鏈：從第 $n$ 代 $H$-型態 $\\Theta_{H,n}$，到非線性 forcing 型態 $\\Theta_{F_{\\rm NL},n}$，到再入儲備向量 $\\mathbf R_n^{re}$，再到下一代 $H$-型態 $\\Theta_{H,n+1}$。遞迴循環要求 $\\Theta_{H,n+1}$ 保持在能支持下一代 forcing 的子型別中，此子型別遞迴性質仍是開放問題。",
      "definition_en": "Defined (boxed) in §52, writing one typed nonlinear re-entry generation as a chain of maps: from generation $n$'s $H$-type state $\\Theta_{H,n}$, to the nonlinear forcing state $\\Theta_{F_{\\rm NL},n}$, to the reserve vector $\\mathbf R_n^{re}$, to the next generation's $H$-type state $\\Theta_{H,n+1}$. A recurrent cycle requires $\\Theta_{H,n+1}$ to remain in the forcing-producing subtype needed for the next generation; this subtype recurrence remains open.",
      "defining_relation": "\\mathscr R_n: \\Theta_{H,n} \\mapsto \\Theta_{F_{\\rm NL},n} \\mapsto \\mathbf R_n^{re} \\mapsto \\Theta_{H,n+1}",
      "notes": "Θ_{H,n} and Θ_{F_NL,n} are typed ETN state labels presupposed from the broader C6 series (C6-A/B), not independently defined in this file."
    },
    {
      "id": "ns.c6.c6c.k_fnl_coh",
      "latex": "\\mathcal K_{F_{\\rm NL}}^{coh} = \\{\\Theta_F:\\mathbf R^{re}\\text{ satisfies re-entry gates}\\}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "一致性定義域",
      "label_en": "Coherent domain (of the typed F_NL→H relation)",
      "definition_zh": "§56 boxed 定義，取代粗糙邊 $F_{\\rm NL}\\to H$ 的定義域：僅包含使再入儲備向量 $\\mathbf R^{re}$ 滿足全部再入閘門的 forcing 型態 $\\Theta_F$ 集合。此定義域的邊界即 §41 定義的有限字母表。",
      "definition_en": "Defined (boxed) in §56, replacing the coarse edge $F_{\\rm NL}\\to H$ with a typed relation whose domain is exactly the set of forcing states $\\Theta_F$ for which $\\mathbf R^{re}$ satisfies all re-entry gates. The boundary of this domain is the finite alphabet defined in §41.",
      "defining_relation": "\\mathcal K_{F_{\\rm NL}}^{coh} = \\{\\Theta_F:\\mathbf R^{re}\\text{ satisfies re-entry gates}\\}"
    },
    {
      "id": "ns.c6.c6c.theta_re_state",
      "latex": "\\Theta^{C6C}_{re} = \\left\\langle \\Gamma^{Duh},\\nu^{coh},\\chi_E,\\gamma_E,\\eta^{grow},\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup} \\right\\rangle",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "C6-C 再入狀態（ETN 更新）",
      "label_en": "C6-C re-entry state (ETN update)",
      "definition_zh": "§62「True ETN update」boxed 定義，將本輪核心量彙整成十元組狀態：Duhamel coherence、coherence 測度、集合版空間/時間一致性、成長效率、以及五個正規化 reserve 座標。是本輪對整體 C6 Extended Type Network（ETN）狀態表示法的正式更新。",
      "definition_en": "Defined (boxed) in §62 (\"True ETN update\"), assembling this round's core quantities into a ten-component state tuple: Duhamel coherence, the coherence measure, set-valued spatial/temporal coherence, growth efficiency, and five normalized reserve coordinates. This round's formal update to the overall C6 Extended Type Network (ETN) state representation.",
      "defining_relation": "\\Theta^{C6C}_{re} = \\left\\langle \\Gamma^{Duh},\\nu^{coh},\\chi_E,\\gamma_E,\\eta^{grow},\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup} \\right\\rangle"
    },
    {
      "id": "ns.c6.c6c.boundary_alphabet",
      "latex": "\\partial\\mathcal K_{re} = \\{\\text{DIFF},\\text{CANCEL},\\text{CAPACITY},\\text{DOM},\\text{SEL},\\text{SIGN},\\text{TIME},\\text{SETUP}\\}",
      "series": "NS",
      "first_appearance": "C6-C",
      "label_zh": "循環邊界字母表",
      "label_en": "Cycle boundary state (alphabet)",
      "definition_zh": "§62 boxed 定義，為循環一致性定義域 $\\mathcal K_{F_{\\rm NL}}^{coh}$ 邊界的正式八元素字母表，對應 §41 所列六種邊界情形（C-B1 精煉為 DIFF/CANCEL/CAPACITY 三支，另加 C-B2–C-B6 對應 DOM/SEL/SIGN/TIME/SETUP）。任一再入世代若 $b_n^{re}\\to0$，取進一步子序列後恰進入此字母表中的一種。",
      "definition_en": "Defined (boxed) in §62 as the formal eight-element alphabet for the boundary of the coherent domain $\\mathcal K_{F_{\\rm NL}}^{coh}$, corresponding to the six boundary cases of §41 (C-B1 refines into DIFF/CANCEL/CAPACITY; C-B2–C-B6 give DOM/SEL/SIGN/TIME/SETUP). If $b_n^{re}\\to0$ along a re-entry sequence, a further subsequence enters exactly one letter of this alphabet.",
      "defining_relation": "\\partial\\mathcal K_{re} = \\{\\text{DIFF},\\text{CANCEL},\\text{CAPACITY},\\text{DOM},\\text{SEL},\\text{SIGN},\\text{TIME},\\text{SETUP}\\}",
      "notes": "DIFF/CANCEL correspond to C-DIFF/C-CANCEL (§28, refining C-B1 \"Γ^Duh→0\"); CAPACITY corresponds to \"Forcing-Capacity Inflation\" (§45); DOM/SEL/SIGN/TIME/SETUP correspond to ρ_dom→0, ρ_sel→0, ρ_sign→0, ρ_time→0, ρ_setup=0 (C-B2–C-B6, §41)."
    },
    {
      "id": "ns.c6.c6d.tau_e",
      "latex": "\\tau_e \\in \\{S, D, E\\}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "邊時間語義標籤",
      "label_en": "edge time-semantics tag",
      "definition_zh": "C6-D 為每條圖邊新增的第二個獨立標籤,區分靜態同事件關係(S)、動態時間演化(D)、與終止於正則性定理的外部終止邊(E)。",
      "definition_en": "A second independent tag C6-D assigns to every graph edge, distinguishing a static same-event relation (S), a dynamic transition to a later generation (D), and an external kill ending in a regularity-theorem sink (E)."
    },
    {
      "id": "ns.c6.c6d.c_gp",
      "latex": "\\mathcal C_{GP} \\subset \\mathcal K_G \\times \\mathcal K_P",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "同事件幾何-壓力相容關係",
      "label_en": "same-event geometry-pressure compatibility relation",
      "definition_zh": "定義於緊化幾何狀態空間 K_G 與壓力狀態空間 K_P 乘積上的子集,收集在同一事件 (t,x,R) 成立的幾何-壓力組合,精煉為 G→P 與 P→G 兩靜態關係的交集。",
      "definition_en": "A subset of the product of the compactified geometry state space K_G and pressure state space K_P, collecting geometry-pressure pairs compatible at one shared event (t,x,R); refined as the intersection of the static relations R_{G→P} and R_{P→G}.",
      "defining_relation": "\\mathcal C_{GP} = R_{G\\to P}\\cap R_{P\\to G} \\subset \\mathcal K_G\\times\\mathcal K_P",
      "notes": "第 31 節在完整聯合狀態 Θ_GP 層級進一步定義精煉版 K_GP^comp。"
    },
    {
      "id": "ns.c6.c6d.phi_gp",
      "latex": "\\Phi_{GP}: \\mathcal C_{GP,n} \\dashrightarrow \\mathcal C_{GP,n+1}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "聯合狀態動態回歸映射",
      "label_en": "joint-state dynamic return map",
      "definition_zh": "真正週期性動力循環所需的額外映射,把第 n 代的同事件相容纖維送往第 n+1 代;C5-D/F 從未證明其存在,是本輪拒絕粗略 G↔P 循環的核心理由。",
      "definition_en": "The additional map a genuine recurrent dynamical cycle requires, sending generation n's compatibility fiber to generation n+1's; C5-D/F never established its existence, the core reason C6-D rejects the coarse G↔P cycle.",
      "notes": "第 32 節在聯合狀態層級將其重新表述為 Θ_{GP,n} 到 Θ_{GP,n+1} 的映射。"
    },
    {
      "id": "ns.c6.c6d.q_tensor",
      "latex": "Q = S^2 + \\frac14\\omega\\otimes\\omega - \\frac14|\\omega|^2I",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "局部二次張量",
      "label_en": "local quadratic tensor",
      "definition_zh": "由應變平方與渦度二次項組成的對稱張量,是應變演化方程中驅動幾何的二次強迫項,也是 A_χ^Q、B_χ^Q 的基礎。",
      "definition_en": "A symmetric tensor built from the strain-square and vorticity-quadratic terms; it is the quadratic forcing term driving geometry in the strain-evolution equation, and the basis for A_χ^Q and B_χ^Q."
    },
    {
      "id": "ns.c6.c6d.a_chi_q",
      "latex": "A_\\chi^Q = \\int \\chi|Q|\\,dx",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "二次絕對強度",
      "label_en": "quadratic absolute intensity",
      "definition_zh": "核心截止函數 χ 下對 |Q| 的加權積分,量化局部二次強迫的總體大小,是本輪各不等式的標準右側尺度。",
      "definition_en": "The cutoff-weighted integral of |Q| over the selected core, quantifying the overall size of the local quadratic forcing; it is the standard right-hand-side scale in the round's inequalities."
    },
    {
      "id": "ns.c6.c6d.b_chi_q",
      "latex": "B_\\chi^Q = \\int \\chi Q\\,dx",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "二次平均",
      "label_en": "quadratic mean",
      "definition_zh": "核心內 Q 的加權平均(未取絕對值),用於定義伴隨平均應變演化 M_χ'。",
      "definition_en": "The cutoff-weighted mean of Q over the core (without absolute value), used to define the adjoint mean-strain evolution M_χ'."
    },
    {
      "id": "ns.c6.c6d.m_chi_prime",
      "latex": "M_\\chi' = -B_\\chi^Q - P_\\chi",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "伴隨平均應變演化",
      "label_en": "adjoint mean-strain evolution",
      "definition_zh": "核心平均應變狀態的演化率,等於負二次平均與負壓力 Hessian 平均之和;第 7 節平均穩定性閘門假設其大小被 ε A_χ^Q 控制。",
      "definition_en": "The evolution rate of the core-averaged strain state, equal to minus the quadratic mean minus the pressure-Hessian mean; the Section 7 mean-stability gate assumes its size is controlled by ε A_χ^Q.",
      "defining_relation": "|M_\\chi'| \\le \\epsilon A_\\chi^Q, \\quad 0\\le\\epsilon<\\gamma_K"
    },
    {
      "id": "ns.c6.c6d.p_chi",
      "latex": "P_\\chi = \\int \\chi\\nabla^2p\\,dx",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "總局部化壓力 Hessian 平均",
      "label_en": "total localized pressure-Hessian mean",
      "definition_zh": "核心內壓力 Hessian 的加權積分,是強中間幾何直接產生的壓力量;第 8 節強調它僅是「總壓力」,不預設遠場主導、共同調和矩陣或特定符號等更強性質。",
      "definition_en": "The cutoff-weighted integral of the pressure Hessian over the core — the pressure quantity strong-middle geometry directly produces; Section 8 stresses it is only the \"total\" pressure, not implying far-field dominance, a common harmonic matrix, or a specific signature.",
      "notes": "在第 8、47、58 節及 no-go 清單中亦記作 P_{\\rm total}。"
    },
    {
      "id": "ns.c6.c6d.r_p",
      "latex": "R_P = -\\widehat H_K:P_\\chi",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "總定向壓力響應",
      "label_en": "total oriented pressure response",
      "definition_zh": "壓力 Hessian 平均沿標準化強中間軸 Ĥ_K 方向的負向投影,量化壓力對抗二次平均強迫的程度。",
      "definition_en": "The negative projection of the pressure-Hessian mean onto the normalized strong-middle direction Ĥ_K, quantifying how strongly pressure opposes the quadratic mean forcing.",
      "defining_relation": "R_P \\ge r_P A_\\chi^Q, \\quad r_P:=\\gamma_K-\\epsilon>0"
    },
    {
      "id": "ns.c6.c6d.pressure_provenance_split",
      "latex": "P_\\chi = P_\\chi^{loc} + P_\\chi^{far}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "壓力來源分裂引理(C6-D.2)",
      "label_en": "Pressure-Provenance Split Lemma (C6-D.2)",
      "definition_zh": "依 Bradshaw–Tsai 局部壓力展開,將總定向響應拆成局部項 a_loc 與遠場項 a_far,證明兩者之和等於 R_P,故循環必須選擇一個壓力來源分支(局部或遠場)。",
      "definition_en": "Using the Bradshaw–Tsai local pressure expansion, the total oriented response splits into a local part a_loc and a far part a_far summing to R_P, forcing any candidate cycle to choose a pressure-provenance branch (local or far).",
      "defining_relation": "a_{\\rm loc}+a_{\\rm far}=R_P\\ge r_PA_\\chi^Q \\ \\Rightarrow\\ a_{\\rm loc}\\ge\\tfrac12r_PA_\\chi^Q \\text{ or } a_{\\rm far}\\ge\\tfrac12r_PA_\\chi^Q",
      "notes": "a_loc := -Ĥ_K:P_χ^loc,a_far := -Ĥ_K:P_χ^far。"
    },
    {
      "id": "ns.c6.c6d.f_chi",
      "latex": "F_\\chi = \\frac{1}{m_\\chi}P_\\chi^{far}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "加權遠場壓力矩陣",
      "label_en": "weighted far-pressure matrix",
      "definition_zh": "遠場壓力 Hessian 平均除以核心測度 m_χ=∫χdx 所得的矩陣;因 p_far 在核心內調和,F_χ 屬無跡對稱矩陣空間 Sym_0(3),是後續符號分類與軸向鎖定分析的核心對象。",
      "definition_en": "The far-pressure Hessian mean divided by the core measure m_χ = ∫χdx; because p_far is harmonic inside the core, F_χ lies in the trace-free symmetric space Sym_0(3), and is the central object for the signature and axis-locking analysis that follow.",
      "defining_relation": "-e_1^TF_\\chi e_1 \\ge \\frac{|H_K|}{2m_\\chi}r_PA_\\chi^Q"
    },
    {
      "id": "ns.c6.c6d.f_hat",
      "latex": "\\widehat F = F_\\chi/|F_\\chi|_F \\in S^4\\cap\\operatorname{Sym}_0(3)",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "標準化遠場壓力矩陣",
      "label_en": "normalized far-pressure matrix",
      "definition_zh": "F_χ 除以其 Frobenius 範數所得的單位無跡對稱矩陣,用來定義軸向響應邊界 μ_axis,並作為第 27 節遠場矩陣繼承性比較的基準。",
      "definition_en": "F_χ divided by its Frobenius norm, a unit trace-free symmetric matrix; it defines the axis response margin μ_axis and serves as the reference for the far-matrix heredity comparison in Section 27."
    },
    {
      "id": "ns.c6.c6d.mu_axis",
      "latex": "\\mu_{\\rm axis} = -e_1^T\\widehat F e_1 \\in[-1,1]",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "軸向響應邊界",
      "label_en": "axis response margin",
      "definition_zh": "標準化遠場矩陣沿當前壓縮軸 e_1 的負向二次型值,衡量遠場壓力鎖定該軸的強度;遠場分支成立時 μ_axis>0,其趨近 0 是循環逃逸單負號軸鎖定不相容的邊界之一(GP-B6)。",
      "definition_en": "The negative quadratic form of the normalized far matrix along the current compressive axis e_1, measuring how strongly far pressure locks that axis; the far branch gives μ_axis > 0, and its approach to 0 is one way (boundary GP-B6) a recurrence candidate escapes the one-negative axis-lock incompatibility."
    },
    {
      "id": "ns.c6.c6d.signature_classification",
      "latex": "(-,+,+) \\text{ [one-negative]}, \\qquad (-,-,+) \\text{ [two-negative]}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "遠場壓力矩陣符號分類",
      "label_en": "far-pressure matrix signature classification",
      "definition_zh": "非零無跡對稱矩陣 F 的非退化符號分兩類:單負號 (-,+,+) 可鎖定單一投影錐;雙負號 (-,-,+) 只給出負向平面/帶狀幾何,不鎖定單軸;det F=0 是兩者間的符號邊界。",
      "definition_en": "The nondegenerate signatures of a nonzero trace-free symmetric F split into two classes: one-negative (-,+,+) can lock a single projective cap, while two-negative (-,-,+) gives only a negative belt/plane geometry without locking one axis; det F=0 is the boundary between them.",
      "defining_relation": "\\det F<0 \\Leftrightarrow (-,+,+); \\qquad \\det F>0 \\Leftrightarrow (-,-,+); \\qquad \\det F=0 \\text{ boundary}",
      "notes": "亦記作 sig F;det F=0 對應 C5-G 的 Pressure Signature-Gap Defect。"
    },
    {
      "id": "ns.c6.c6d.theta_gp",
      "latex": "\\Theta_{GP} = \\left\\langle K,\\vartheta,[e_1],A_\\chi^Q,\\epsilon_{\\rm mean},P_\\chi^{loc},P_\\chi^{far},\\phi_{\\rm far},\\Gamma_P^{prov},\\widehat F,\\operatorname{sig}F,\\mu_{\\rm axis}\\right\\rangle",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "幾何-壓力聯合狀態",
      "label_en": "joint geometry-pressure state",
      "definition_zh": "取代舊有兩節點靜態循環的正確 C6 狀態,將強中間幾何、二次強迫強度、平均旋轉、局部/遠場壓力、來源分數、來源相干度、標準化遠場矩陣、符號與軸向邊界等座標打包成單一聯合事件描述。",
      "definition_en": "The correct C6 state replacing the old two-node static loop; it packages the strong-middle geometry, quadratic-forcing intensity, mean rotation, local/far pressure, far capture fraction, provenance coherence, normalized far matrix, signature, and axis margin into one joint-event description.",
      "notes": "第 56 節以 Θ_GP^{C6D} 給出同一物件的文字化版本。"
    },
    {
      "id": "ns.c6.c6d.gamma_p_prov",
      "latex": "\\Gamma_P^{prov} = \\frac{R_P}{C_P} \\in(0,1]",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "壓力來源相干度",
      "label_en": "pressure provenance coherence",
      "definition_zh": "總定向響應 R_P 與絕對定向壓力容量 C_P 之比,揭露局部與遠場貢獻是否大幅互相抵消;比值趨近 0 代表來源抵消邊界(GP-B4)。",
      "definition_en": "The ratio of the total oriented response R_P to the absolute oriented pressure capacity C_P, exposing whether local and far contributions substantially cancel each other; the ratio approaching 0 marks the provenance-cancellation boundary (GP-B4).",
      "defining_relation": "C_P = |\\widehat H:P_\\chi^{loc}| + |\\widehat H:P_\\chi^{far}|"
    },
    {
      "id": "ns.c6.c6d.phi_far",
      "latex": "\\phi_{\\rm far} = \\frac{a_{\\rm far}^{+}}{a_{\\rm loc}^{+}+a_{\\rm far}^{+}} \\in[0,1]",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "遠場捕獲分數",
      "label_en": "far capture fraction",
      "definition_zh": "遠場正定向壓力貢獻佔局部與遠場正貢獻總和的比例;φ_far≈1 代表遠場主導,φ_far≈0 代表局部壓力接管(GP-B3),遠場回歸測試要求此值非退化。",
      "definition_en": "The share of positive far-pressure oriented contribution within the sum of positive local and far contributions; φ_far ≈ 1 means far-dominated compensation, φ_far ≈ 0 means local-pressure takeover (GP-B3), and a far-pressure return test requires this to stay nondegenerate.",
      "defining_relation": "a_{\\rm loc}^{+}=[-\\widehat H:P_\\chi^{loc}]_+, \\quad a_{\\rm far}^{+}=[-\\widehat H:P_\\chi^{far}]_+"
    },
    {
      "id": "ns.c6.c6d.rho_g_her",
      "latex": "\\rho_G^{her} = \\left[1-\\frac{d_e(n,n+1)}{d_0}\\right]_+ \\cdot \\frac{\\min(\\vartheta_n,\\vartheta_{n+1})}{\\delta_0+\\min(\\vartheta_n,\\vartheta_{n+1})}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "幾何繼承性儲備",
      "label_en": "normalized geometry persistence reserve",
      "definition_zh": "結合投影軸距離 d_e(n,n+1) 與相鄰兩代最小中間間隙,量化壓縮軸方向與強中間幾何在世代間保持的程度;僅為 metadata,並非已證定理量。",
      "definition_en": "Combines the projective axis distance d_e(n,n+1) with the minimum middle gap of two adjacent generations to quantify how well the compressive axis and strong-middle geometry persist across generations; it is metadata, not a proven theorem quantity.",
      "defining_relation": "d_e(n,n+1) = \\|e_n\\otimes e_n - e_{n+1}\\otimes e_{n+1}\\|_F"
    },
    {
      "id": "ns.c6.c6d.rho_f_her",
      "latex": "\\rho_F^{her} = \\left[1-\\frac{d_F(n,n+1)}{d_{F,0}}\\right]_+",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "遠場壓力繼承性儲備",
      "label_en": "far-pressure heredity reserve",
      "definition_zh": "由標準化遠場矩陣的世代間距離 d_F(n,n+1) 定義的正規化儲備量,量化遠場壓力矩陣結構跨世代保持的程度。",
      "definition_en": "Defined from the generation-to-generation distance d_F(n,n+1) between normalized far matrices, quantifying how well the far-pressure matrix structure persists across generations.",
      "defining_relation": "d_F(n,n+1) = \\|\\widehat F_n - \\widehat F_{n+1}\\|_F"
    },
    {
      "id": "ns.c6.c6d.r_gp_vector",
      "latex": "\\mathbf R^{GP} = \\left(\\rho_{\\rm geom},\\rho_{\\rm mean},\\rho_{\\rm far},\\rho_{\\rm prov},\\rho_{\\rm sig},\\rho_{\\rm axis},\\rho_{F}^{her},\\rho_G^{her}\\right)",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "幾何-壓力回歸儲備向量",
      "label_en": "geometry-pressure recurrence reserve vector",
      "definition_zh": "彙集八個儲備座標(幾何、平均旋轉耗盡、遠場捕獲、來源相干、符號邊界距離、軸向邊界、遠場矩陣繼承、幾何繼承)的向量;C6-D.7 有限儲備瓶頸定理證明沿任何候選回歸序列的子序列,此向量若非全體一致有下界,就必有至少一分量趨近 0。",
      "definition_en": "A vector collecting eight reserve coordinates (geometry, mean-rotation depletion, far capture, provenance coherence, distance from the signature boundary, axis margin, far-matrix heredity, geometry heredity); the C6-D.7 Finite Recurrence Bottleneck Theorem proves a subsequence either keeps it uniformly bounded below or has at least one coordinate tend to 0.",
      "notes": "對應第 56 節的循環邊界字母表 ∂K_GP = {GAP, MEAN, LOCAL, PROV, SIG, AXIS, F-HER, G-HER, REG}。"
    },
    {
      "id": "ns.c6.c6d.gp_1neg_strong",
      "latex": "GP_{1-}^{strong}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "強單負號聯合分支",
      "label_en": "strong one-negative joint branch",
      "definition_zh": "遠場定向補償、符號為 (-,+,+)、軸向邊界與中間間隙皆非退化、且逐點強中間幾何成立的聯合子型;同事件中壓縮軸被鎖定且 Q 零重心被排除,是剛性聯合狀態,但僅為候選回歸狀態而非已證循環。",
      "definition_en": "The joint subtype with far-oriented compensation, signature (-,+,+), nondegenerate axis margin and middle gap, and pointwise strong-middle geometry; at the same event the compressive axis is locked and Q zero-barycenter is excluded, a rigid joint state but only a candidate recurrent state, not a certified cycle.",
      "notes": "對比分支 GP_{2-}(符號 (-,-,+))避開單軸鎖定不相容,但同樣未證回歸定理。"
    },
    {
      "id": "ns.c6.c6d.gp_2neg",
      "latex": "GP_{2-}",
      "series": "NS",
      "first_appearance": "C6-D",
      "label_zh": "雙負號聯合分支",
      "label_en": "two-negative joint branch",
      "definition_zh": "遠場符號為 (-,-,+) 的聯合分支,允許更寬的壓縮軸支撐,因而避開單軸鎖定不相容;但避開不相容不等於產生回歸幾何,GP_{2-,n}→GP_{2-,n+1} 仍未被證明。",
      "definition_en": "The joint branch with far signature (-,-,+), permitting wider compressive-axis support and thus avoiding the one-negative axis-lock obstruction; but avoiding an incompatibility is not the same as generating the return geometry, and GP_{2-,n} → GP_{2-,n+1} remains unproved."
    },
    {
      "id": "ns.c6.c6e.lambda2_plus",
      "latex": "\\lambda_2^+",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "中間特徵值（正部）",
      "label_en": "Middle eigenvalue (positive part)",
      "definition_zh": "應變張量 S(x,t) 的中間特徵值（正部），源自 Miller 的中間特徵值正則性判準，是尺度臨界的應變量。C6-E 第1.1節重新引入此量作為中間時間負荷 m(t) 被積函數的核心成分，用以說明 m(t) 的積分核本身是一個真正的正值空間應變密度，而非抽象純量。",
      "definition_en": "The (positive part of the) middle eigenvalue of the strain tensor S(x,t), drawn from Miller's middle-eigenvalue regularity criterion as a scale-critical strain quantity. C6-E section 1.1 reintroduces it as the core ingredient of the integrand defining the middle temporal load m(t), establishing that m(t)'s integrand is a genuine positive spatial strain density rather than an abstract scalar.",
      "notes": "Inherited from Miller's criterion (References #1) and used throughout C4/C5/C6; not re-derived in this file."
    },
    {
      "id": "ns.c6.c6e.m_t",
      "latex": "m(t)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "中間時間負荷",
      "label_en": "Middle temporal load",
      "definition_zh": "定義為 $m(t)=\\int_{\\mathbb R^3}\\lambda_2^+(x,t)|S(x,t)|^2dx$，是 C4/C5 貫穿使用的中間時間負荷純量。C6-E 的核心論點在於指出此純量的被積函數 $a_M(t,x)$ 是一個正值空間源密度，因此 m(t) 其實是空間源測度的時間邊際（marginal），而非獨立完整的物理狀態。",
      "definition_en": "Defined as $m(t)=\\int_{\\mathbb R^3}\\lambda_2^+(x,t)|S(x,t)|^2dx$, the scalar middle temporal load used throughout C4/C5. C6-E's central move is to show its integrand $a_M(t,x)$ is a genuine positive spatial source density, so m(t) is in fact the temporal marginal of a spacetime source measure rather than a self-contained physical state.",
      "defining_relation": "m(t) = \\int_{\\mathbb R^3}\\lambda_2^+(x,t)|S(x,t)|^2dx"
    },
    {
      "id": "ns.c6.c6e.q_sv",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "應變-渦度算子",
      "label_en": "Strain-vorticity operator",
      "definition_zh": "定義為 $\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)$，是投影後應變演化方程中的算子項，滿足 $H^1$ 應變成長恆等式 $\\frac12\\frac d{dt}\\|S\\|_{\\dot H^1}^2+\\nu\\|\\Delta S\\|_2^2=-\\langle\\mathcal Q_{SV},-\\Delta S\\rangle$，且與渦度外積正交：$\\langle-\\Delta S,\\omega\\otimes\\omega\\rangle=0$。C6-E 第1.2節用它說明算子時間成長負荷同樣來自一個可空間積分的帶號密度。",
      "definition_en": "Defined as $\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)$, the operator term in the projected strain-evolution equation, entering the exact $H^1$ strain-growth identity $\\frac12\\frac d{dt}\\|S\\|_{\\dot H^1}^2+\\nu\\|\\Delta S\\|_2^2=-\\langle\\mathcal Q_{SV},-\\Delta S\\rangle$ and satisfying the orthogonality $\\langle-\\Delta S,\\omega\\otimes\\omega\\rangle=0$. C6-E section 1.2 uses it to show the operator temporal growth load likewise arises from a spatially integrable signed density.",
      "defining_relation": "\\mathcal Q_{SV} = P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)",
      "notes": "P_{st} denotes the projection operator from Miller's strain-vorticity formalism (References #2); not re-derived in this file. Same symbol as ns.c6.QSV_criterion (filed under C6-Q); this C6-E entry captures its first appearance in the TS/GP/HF sub-thread, before C6-Q's later distinction from g_O^loc."
    },
    {
      "id": "ns.c6.c6e.a_m",
      "latex": "a_M(t,x)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "中間空間源密度",
      "label_en": "Middle spatial source density",
      "definition_zh": "定義為 $a_M(t,x)=\\lambda_2^+(S(t,x))|S(t,x)|^2\\ge0$，即 m(t) 的被積函數本身，是本輪第2節引入的正值空間密度，滿足 $m(t)=\\int_{\\mathbb R^3}a_M(t,x)dx$。它是後續建構標準中間時空提升 $\\Pi_J^M$（C6-E.1）的原始素材。",
      "definition_en": "Defined as $a_M(t,x)=\\lambda_2^+(S(t,x))|S(t,x)|^2\\ge0$, the integrand of m(t) itself, introduced in section 2 as a nonnegative spatial density satisfying $m(t)=\\int_{\\mathbb R^3}a_M(t,x)dx$. It is the raw material used to build the canonical middle spacetime lift $\\Pi_J^M$ in Theorem C6-E.1.",
      "defining_relation": "a_M(t,x) = \\lambda_2^+(S(t,x))|S(t,x)|^2 \\ge 0"
    },
    {
      "id": "ns.c6.c6e.j_window",
      "latex": "J=(t_-,t_+)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "活躍記錄窗口",
      "label_en": "Active record window",
      "definition_zh": "固定的時間區間 $J=(t_-,t_+)$，是本輪幾乎所有量（$M_J$、$P_J$、$\\Pi_J^M$、$\\Pi_J^O$、$\\Omega_T(J)$、$\\Omega_{ST}(J)$ 等）共同的下標與定義域，代表被審核的一段活躍時間窗口。",
      "definition_en": "The fixed time interval $J=(t_-,t_+)$ that serves as the common subscript/domain for nearly every quantity in this round ($M_J$, $P_J$, $\\Pi_J^M$, $\\Pi_J^O$, $\\Omega_T(J)$, $\\Omega_{ST}(J)$, etc.), denoting the active record window under audit.",
      "defining_relation": "J = (t_-, t_+)"
    },
    {
      "id": "ns.c6.c6e.m_j",
      "latex": "M_J",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "窗口中間總質量",
      "label_en": "Total middle mass over window",
      "definition_zh": "定義為 $M_J=\\int_J m(t)dt>0$，即中間負荷 m(t) 在記錄窗口 J 上的總質量，作為把 $a_M$、m(t) 正規化為機率測度（$\\Pi_J^M$、$\\mu_J^M$）時所用的分母。",
      "definition_en": "Defined as $M_J=\\int_J m(t)dt>0$, the total mass of the middle load m(t) accumulated over the record window J, serving as the normalizing denominator that turns $a_M$ and m(t) into the probability measures $\\Pi_J^M$ and $\\mu_J^M$.",
      "defining_relation": "M_J = \\int_J m(t)dt > 0"
    },
    {
      "id": "ns.c6.c6e.pi_j_m",
      "latex": "\\Pi_J^M",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "標準中間時空提升（C6-E.1）",
      "label_en": "Canonical Middle Spacetime Lift (C6-E.1)",
      "definition_zh": "定義為 $d\\Pi_J^M(t,x)=\\frac{a_M(t,x)}{M_J}dx\\,dt\\in\\mathcal P(J\\times\\mathbb R^3)$，是定理 C6-E.1 的核心對象：中間時間負荷的正規化空間源密度被提升為一個時空機率測度，其時間邊際恰為 C5 使用的中間時間負荷機率 $\\mu_J^M=(\\pi_t)_\\#\\Pi_J^M$。本節結論是「中間時間狀態其實就是空間源測度的邊際」。",
      "definition_en": "Defined as $d\\Pi_J^M(t,x)=\\frac{a_M(t,x)}{M_J}dx\\,dt\\in\\mathcal P(J\\times\\mathbb R^3)$, the central object of Theorem C6-E.1: the normalized middle spatial source density lifted into a spacetime probability measure, whose temporal marginal $\\mu_J^M=(\\pi_t)_\\#\\Pi_J^M$ recovers exactly the C5 middle-load probability. The section's conclusion is that \"the middle temporal state is literally a spatial source measure marginal.\"",
      "defining_relation": "d\\Pi_J^M(t,x) = \\frac{a_M(t,x)}{M_J}dx\\,dt",
      "notes": "π_t(t,x)=t denotes the time-projection map used in the pushforward (\\pi_t)_\\#."
    },
    {
      "id": "ns.c6.c6e.mu_j_m",
      "latex": "\\mu_J^M",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "中間時間邊際機率",
      "label_en": "Middle temporal marginal probability",
      "definition_zh": "定義為 $d\\mu_J^M(t)=\\frac{m(t)}{M_J}dt$，即 C5 所用的中間時間負荷機率，並在本輪被證明恰為 $\\Pi_J^M$ 沿時間軸的推前邊際：$\\mu_J^M=(\\pi_t)_\\#\\Pi_J^M$。",
      "definition_en": "Defined as $d\\mu_J^M(t)=\\frac{m(t)}{M_J}dt$, the C5 middle-load temporal probability, shown here to be exactly the pushforward marginal of $\\Pi_J^M$ along the time axis: $\\mu_J^M=(\\pi_t)_\\#\\Pi_J^M$.",
      "defining_relation": "d\\mu_J^M(t) = \\frac{m(t)}{M_J}dt"
    },
    {
      "id": "ns.c6.c6e.e1_t",
      "latex": "E_1(t)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "H¹ 應變能量",
      "label_en": "H^1 strain energy",
      "definition_zh": "定義為 $E_1(t)=\\frac12\\|S(t)\\|_{\\dot H^1}^2$，是應變場的 $\\dot H^1$ 能量，其時間導數 $h(t)=E_1'(t)$ 即為本輪第4節起分析的算子時間成長負荷。",
      "definition_en": "Defined as $E_1(t)=\\frac12\\|S(t)\\|_{\\dot H^1}^2$, the $\\dot H^1$ energy of the strain field, whose time derivative $h(t)=E_1'(t)$ is the operator temporal growth load analyzed from section 4 onward.",
      "defining_relation": "E_1(t) = \\frac12\\|S(t)\\|_{\\dot H^1}^2"
    },
    {
      "id": "ns.c6.c6e.h_t",
      "latex": "h(t)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "算子時間成長負荷",
      "label_en": "Operator temporal growth load",
      "definition_zh": "定義為 $h(t)=E_1'(t)$，利用精確算子恆等式可寫成 $h(t)=-\\langle\\mathcal Q_{SV},-\\Delta S\\rangle-\\nu\\|\\Delta S\\|_2^2$，並等於帶號局部密度 $g_O(t,x)$ 的空間積分。是 C6-E.2 標準正值算子時空提升的出發點。",
      "definition_en": "Defined as $h(t)=E_1'(t)$, rewritten via the exact operator identity as $h(t)=-\\langle\\mathcal Q_{SV},-\\Delta S\\rangle-\\nu\\|\\Delta S\\|_2^2$, and equal to the spatial integral of the signed local density $g_O(t,x)$. It is the starting point for the Theorem C6-E.2 canonical positive-operator spacetime lift.",
      "defining_relation": "h(t) = E_1'(t) = \\int_{\\mathbb R^3}g_O(t,x)dx",
      "notes": "The intermediate identity h(t) = -\\langle\\mathcal Q_{SV},-\\Delta S\\rangle - \\nu\\|\\Delta S\\|_2^2 is stated in section 4 immediately before g_O is introduced."
    },
    {
      "id": "ns.c6.c6e.g_o",
      "latex": "g_O(t,x)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "算子帶號局部密度",
      "label_en": "Operator signed local density",
      "definition_zh": "定義為 $g_O(t,x)=-\\mathcal Q_{SV}(t,x):(-\\Delta S(t,x))-\\nu|\\Delta S(t,x)|^2$，是 h(t) 的空間被積函數，滿足 $h(t)=\\int_{\\mathbb R^3}g_O(t,x)dx$。其正部 $[g_O]_+$ 用以定義正值局部算子容量 $c_O(t)$。",
      "definition_en": "Defined as $g_O(t,x)=-\\mathcal Q_{SV}(t,x):(-\\Delta S(t,x))-\\nu|\\Delta S(t,x)|^2$, the spatial integrand of h(t), satisfying $h(t)=\\int_{\\mathbb R^3}g_O(t,x)dx$. Its positive part $[g_O]_+$ is used to define the positive local operator capacity $c_O(t)$.",
      "defining_relation": "g_O(t,x) = -\\mathcal Q_{SV}(t,x):(-\\Delta S(t,x)) - \\nu|\\Delta S(t,x)|^2",
      "notes": "This is the quantity C6-Q later renames/reframes as g_O^{proj} (ns.c6.gOproj) once C6-Q introduces the projection-free alternative g_O^{loc} (ns.c6.gOloc) and needs to distinguish the two carrier models of the same temporal marginal."
    },
    {
      "id": "ns.c6.c6e.c_o_t",
      "latex": "c_O(t)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "正值局部算子容量",
      "label_en": "Positive local operator capacity",
      "definition_zh": "定義為 $c_O(t)=\\int_{\\mathbb R^3}[g_O(t,x)]_+dx$，滿足 $[h(t)]_+\\le c_O(t)$，是每個時刻 t 上局部正值 $H^1$ 成長容量的總量，用以構造條件空間分布 $p_O(x|t)$，並經由 $C_J^O$ 進入 $\\Gamma_J^O$ 的定義，作為「可用容量」的度量。",
      "definition_en": "Defined as $c_O(t)=\\int_{\\mathbb R^3}[g_O(t,x)]_+dx$, satisfying $[h(t)]_+\\le c_O(t)$; it is the total local positive $H^1$-growth capacity at time t, used to build the conditional spatial distribution $p_O(x|t)$ and, via $C_J^O$, to measure \"available capacity\" in the definition of $\\Gamma_J^O$.",
      "defining_relation": "c_O(t) = \\int_{\\mathbb R^3}[g_O(t,x)]_+dx"
    },
    {
      "id": "ns.c6.c6e.p_o_cond",
      "latex": "p_O(x|t)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "條件正值空間分布",
      "label_en": "Conditional positive spatial distribution",
      "definition_zh": "對 $[h(t)]_+>0$ 的時刻定義為 $p_O(x|t)=\\frac{[g_O(t,x)]_+}{c_O(t)}$，是一個空間機率密度；當 $[h(t)]_+=0$ 時可任意選取一個固定的「墓地」機率分布，該時刻在後續加權中得零時間權重。此密度是建構 C6-E.2 標準正值算子時空提升 $\\Pi_J^O$ 的空間條件層。",
      "definition_en": "For times with $[h(t)]_+>0$, defined as $p_O(x|t)=\\frac{[g_O(t,x)]_+}{c_O(t)}$, a spatial probability density; at times where $[h(t)]_+=0$ one may pick any fixed \"cemetery\" probability distribution, which receives zero temporal weight downstream. This is the spatial conditional layer used to build the Theorem C6-E.2 canonical positive-operator spacetime lift $\\Pi_J^O$.",
      "defining_relation": "p_O(x|t) = \\frac{[g_O(t,x)]_+}{c_O(t)}"
    },
    {
      "id": "ns.c6.c6e.p_j",
      "latex": "P_J",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "窗口正值算子總質量",
      "label_en": "Positive operator temporal mass over window",
      "definition_zh": "定義為 $P_J=\\int_J[h(t)]_+dt>0$，是正值算子時間成長在窗口 J 上的總質量，作為正規化 $\\mu_J^O$ 及定義 $\\Gamma_J^O$ 的分母。",
      "definition_en": "Defined as $P_J=\\int_J[h(t)]_+dt>0$, the total mass of positive operator temporal growth accumulated over window J, serving as the normalizing denominator for $\\mu_J^O$ and for defining $\\Gamma_J^O$.",
      "defining_relation": "P_J = \\int_J [h(t)]_+dt > 0"
    },
    {
      "id": "ns.c6.c6e.mu_j_o",
      "latex": "\\mu_J^O",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "算子時間邊際機率",
      "label_en": "Operator temporal marginal probability",
      "definition_zh": "定義為 $d\\mu_J^O(t)=\\frac{[h(t)]_+}{P_J}dt$，是正值算子成長負荷在窗口 J 上正規化後的時間機率測度，為 $\\Pi_J^O$ 的時間邊際。",
      "definition_en": "Defined as $d\\mu_J^O(t)=\\frac{[h(t)]_+}{P_J}dt$, the normalized temporal probability measure of the positive operator growth load over J, serving as the temporal marginal of $\\Pi_J^O$.",
      "defining_relation": "d\\mu_J^O(t) = \\frac{[h(t)]_+}{P_J}dt"
    },
    {
      "id": "ns.c6.c6e.pi_j_o",
      "latex": "\\Pi_J^O",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "標準正值算子時空提升（C6-E.2）",
      "label_en": "Canonical Positive-Operator Spacetime Lift (C6-E.2)",
      "definition_zh": "定義為 $d\\Pi_J^O(t,x)=d\\mu_J^O(t)\\,p_O(x|t)dx\\in\\mathcal P(J\\times\\mathbb R^3)$，滿足 $(\\pi_t)_\\#\\Pi_J^O=\\mu_J^O$，是定理 C6-E.2 的核心對象：正值算子時間成長狀態利用局部正值 $H^1$ 成長容量所得的標準空間提升。",
      "definition_en": "Defined as $d\\Pi_J^O(t,x)=d\\mu_J^O(t)\\,p_O(x|t)dx\\in\\mathcal P(J\\times\\mathbb R^3)$, satisfying $(\\pi_t)_\\#\\Pi_J^O=\\mu_J^O$; the central object of Theorem C6-E.2 — the positive temporal operator-growth state's canonical spatial lift, built from the local positive $H^1$ growth capacity.",
      "defining_relation": "d\\Pi_J^O(t,x) = d\\mu_J^O(t)\\,p_O(x|t)dx"
    },
    {
      "id": "ns.c6.c6e.c_j_o",
      "latex": "C_J^O",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "限制正值時段之局部容量",
      "label_en": "Restricted positive-time local capacity",
      "definition_zh": "定義為 $C_J^O=\\int_{J\\cap\\{h>0\\}}c_O(t)dt$，滿足 $P_J\\le C_J^O$，是把局部正值容量 $c_O(t)$ 限制在全域算子成長為正的時刻上所得的總量，用於定義算子取消效率 $\\Gamma_J^O$，並在第40節的容量膨脹討論中以 $C_J^O/P_J\\to\\infty$ 再度出現。",
      "definition_en": "Defined as $C_J^O=\\int_{J\\cap\\{h>0\\}}c_O(t)dt$, satisfying $P_J\\le C_J^O$; the total local positive capacity $c_O(t)$ restricted to times of globally positive operator growth, used to define the operator cancellation efficiency $\\Gamma_J^O$, and reappearing in section 40's capacity-inflation discussion as $C_J^O/P_J\\to\\infty$.",
      "defining_relation": "C_J^O = \\int_{J\\cap\\{h>0\\}} c_O(t)dt"
    },
    {
      "id": "ns.c6.c6e.gamma_j_o",
      "latex": "\\Gamma_J^O",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "算子取消效率（C6-E.3）",
      "label_en": "Operator cancellation efficiency (C6-E.3)",
      "definition_zh": "定義為 $\\Gamma_J^O=\\frac{P_J}{C_J^O}\\in(0,1]$（當 $C_J^O>0$）。定理 C6-E.3（Operator Capacity-Inflation Identity）給出精確恆等式 $\\frac{C_J^O}{P_J}=\\frac1{\\Gamma_J^O}$：$\\Gamma_J^O\\approx1$ 表示正值局部成長容量幾乎未被抵消即成為全域正值成長；$\\Gamma_J^O\\ll1$ 表示大量局部正值容量被同時刻的負值空間貢獻抵消，稱為「Operator Spatial-Cancellation / Capacity-Inflation Defect」。在儲備向量 $\\mathbf R^{TS}$ 中以 $\\Gamma_O^O:=\\Gamma_J^O$ 記號重複使用。",
      "definition_en": "Defined as $\\Gamma_J^O=\\frac{P_J}{C_J^O}\\in(0,1]$ (when $C_J^O>0$). Theorem C6-E.3 (Operator Capacity-Inflation Identity) gives the exact identity $\\frac{C_J^O}{P_J}=\\frac1{\\Gamma_J^O}$: $\\Gamma_J^O\\approx1$ means positive local growth capacity survives to global positive growth with little cancellation; $\\Gamma_J^O\\ll1$ means a large amount of local positive capacity is cancelled by simultaneous negative spatial contributions — the \"Operator Spatial-Cancellation / Capacity-Inflation Defect.\" It is reused in the reserve vector $\\mathbf R^{TS}$ under the notation $\\Gamma_O^O:=\\Gamma_J^O$.",
      "defining_relation": "\\Gamma_J^O = \\frac{P_J}{C_J^O} \\in (0,1]",
      "notes": "Also written Γ_O^O in section 35's reserve-vector notation and in the TS-B3 boundary condition (section 38); same quantity."
    },
    {
      "id": "ns.c6.c6e.omega_t",
      "latex": "\\Omega_T(J)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "時間重疊係數",
      "label_en": "Temporal overlap coefficient",
      "definition_zh": "定義為 $\\Omega_T(J)=1-d_{TV}(\\mu_J^M,\\mu_J^O)$，對密度可寫成 $\\Omega_T(J)=\\int_J\\min\\left\\{\\frac{m(t)}{M_J},\\frac{[h(t)]_+}{P_J}\\right\\}dt\\in[0,1]$，衡量中間與算子兩個正規化時間邊際之間的機率重疊，量化「temporal coactivation」，並在 TS-B1 邊界（$\\Omega_T\\to0$，時間相位分離）中扮演角色。",
      "definition_en": "Defined as $\\Omega_T(J)=1-d_{TV}(\\mu_J^M,\\mu_J^O)$, or in density form $\\Omega_T(J)=\\int_J\\min\\left\\{\\frac{m(t)}{M_J},\\frac{[h(t)]_+}{P_J}\\right\\}dt\\in[0,1]$; it measures the probability overlap between the normalized middle and operator temporal marginals, quantifying \"temporal coactivation,\" and appears as the TS-B1 boundary ($\\Omega_T\\to0$, temporal phase segregation).",
      "defining_relation": "\\Omega_T(J) = 1 - d_{TV}(\\mu_J^M,\\mu_J^O) = \\int_J \\min\\left\\{\\frac{m(t)}{M_J},\\frac{[h(t)]_+}{P_J}\\right\\}dt"
    },
    {
      "id": "ns.c6.c6e.omega_st",
      "latex": "\\Omega_{ST}(J)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "時空共享源重疊係數",
      "label_en": "Spacetime shared-source overlap coefficient",
      "definition_zh": "定義為 $\\Omega_{ST}(J)=1-d_{TV}(\\Pi_J^M,\\Pi_J^O)=\\int\\min\\{d\\Pi_J^M,d\\Pi_J^O\\}$，是兩個時空提升測度之間的機率重疊，衡量「真正共享空間源」的程度。定理 C6-E.4（Temporal Projection Contraction Theorem）證明總變差在可測推前映射下收縮，故恆有 $\\Omega_{ST}(J)\\le\\Omega_T(J)$：共享時空源是比時間共活躍更強的條件。",
      "definition_en": "Defined as $\\Omega_{ST}(J)=1-d_{TV}(\\Pi_J^M,\\Pi_J^O)=\\int\\min\\{d\\Pi_J^M,d\\Pi_J^O\\}$, the probability overlap between the two spacetime lift measures, quantifying genuine shared spatial source. Theorem C6-E.4 (Temporal Projection Contraction Theorem) shows total variation contracts under measurable push-forward, giving $\\Omega_{ST}(J)\\le\\Omega_T(J)$: shared spacetime source is a strictly stronger condition than temporal coactivation.",
      "defining_relation": "\\Omega_{ST}(J) = 1 - d_{TV}(\\Pi_J^M,\\Pi_J^O) = \\int \\min\\{d\\Pi_J^M, d\\Pi_J^O\\}"
    },
    {
      "id": "ns.c6.c6e.sigma_space",
      "latex": "\\Sigma_{\\rm space}(J)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "空間源分離缺陷",
      "label_en": "Spatial source-segregation defect",
      "definition_zh": "定義為 $\\Sigma_{\\rm space}(J)=\\Omega_T(J)-\\Omega_{ST}(J)\\ge0$，是時間重疊超出時空共享重疊的部分，量化「同時活躍但空間上分離」的程度，是本輪提出的新型 typed coupling coordinate。",
      "definition_en": "Defined as $\\Sigma_{\\rm space}(J)=\\Omega_T(J)-\\Omega_{ST}(J)\\ge0$, the amount by which temporal overlap exceeds spacetime shared overlap, quantifying \"coactive in time yet spatially segregated\" activity; introduced as a new typed coupling coordinate.",
      "defining_relation": "\\Sigma_{\\rm space}(J) = \\Omega_T(J) - \\Omega_{ST}(J) \\ge 0"
    },
    {
      "id": "ns.c6.c6e.rho_share",
      "latex": "\\rho_{\\rm share}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享比率",
      "label_en": "Share ratio",
      "definition_zh": "當 $\\Omega_T>0$ 時定義為 $\\rho_{\\rm share}=\\frac{\\Omega_{ST}}{\\Omega_T}\\in[0,1]$：$\\rho_{\\rm share}\\approx1$ 表示時間重疊大致存活於空間提升之後；$\\rho_{\\rm share}\\ll1$ 表示同時活動其實居於空間上分離的源。是儲備向量 $\\mathbf R^{TS}$ 的分量之一，亦是 TS-B2 邊界（$\\rho_{\\rm share}\\to0$）所監控的量。",
      "definition_en": "Defined for $\\Omega_T>0$ as $\\rho_{\\rm share}=\\frac{\\Omega_{ST}}{\\Omega_T}\\in[0,1]$: $\\rho_{\\rm share}\\approx1$ means temporal overlap largely survives spatial lifting; $\\rho_{\\rm share}\\ll1$ means same-time activity lives on spatially segregated sources. It is a component of the reserve vector $\\mathbf R^{TS}$ and the quantity monitored by the TS-B2 boundary ($\\rho_{\\rm share}\\to0$).",
      "defining_relation": "\\rho_{\\rm share} = \\frac{\\Omega_{ST}}{\\Omega_T} \\in [0,1]"
    },
    {
      "id": "ns.c6.c6e.pi_j_cap",
      "latex": "\\Pi_J^\\cap",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享中間-算子時空源測度",
      "label_en": "Shared Middle–Operator Spacetime Source Measure",
      "definition_zh": "在 $\\Omega_{ST}>0$ 的假設下，設 $f_M,f_O$ 為 $\\Pi_J^M,\\Pi_J^O$ 對共同支配測度 λ 的密度，定義 $d\\Pi_J^\\cap=\\frac{\\min(f_M,f_O)}{\\Omega_{ST}}d\\lambda\\in\\mathcal P(J\\times\\mathbb R^3)$。此測度是第18節起分析的核心對象：其支撐上幾乎每一點同時具有正的中間密度與正的局部算子成長容量，是比時間重疊更強的條件，也是後續中間間隙分析與方向錐提取（C6-E.6）的基礎。",
      "definition_en": "Under the assumption $\\Omega_{ST}>0$, with $f_M,f_O$ the densities of $\\Pi_J^M,\\Pi_J^O$ against a common dominating measure λ, defined as $d\\Pi_J^\\cap=\\frac{\\min(f_M,f_O)}{\\Omega_{ST}}d\\lambda\\in\\mathcal P(J\\times\\mathbb R^3)$. This is the central object from section 18 onward: at almost every point in its support, both the middle density and the local positive operator-growth capacity are positive — strictly stronger than temporal overlap — and it underlies the later middle-gap analysis and the directional-cone extraction (C6-E.6).",
      "defining_relation": "d\\Pi_J^\\cap = \\frac{\\min(f_M,f_O)}{\\Omega_{ST}}d\\lambda"
    },
    {
      "id": "ns.c6.c6e.vartheta_s",
      "latex": "\\vartheta(S)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享中間間隙變量",
      "label_en": "Shared middle-gap variable",
      "definition_zh": "對 $S\\ne0$ 定義為 $\\vartheta(S)=\\frac{\\lambda_2^+(S)\\lambda_3(S)}{|S|^2}$。在 $\\Pi^\\cap$ 的支撐上恆有 $\\lambda_2^+>0$，故除了在中間間隙邊界的極限點外恆有 $\\vartheta>0$。此量用以追蹤共享源上強中間結構是否退化，是後續 $\\rho_{\\rm gap}$、方向錐提取（C6-E.6）與 TS-B4 中間間隙崩潰邊界的核心變量。",
      "definition_en": "For $S\\ne0$, defined as $\\vartheta(S)=\\frac{\\lambda_2^+(S)\\lambda_3(S)}{|S|^2}$. On the support of $\\Pi^\\cap$, $\\lambda_2^+>0$ always holds, so $\\vartheta>0$ except at a limiting middle-gap boundary. It tracks whether the shared source retains a strong-middle structure, and underlies $\\rho_{\\rm gap}$, the directional-cone extraction (C6-E.6), and the TS-B4 middle-gap-collapse boundary.",
      "defining_relation": "\\vartheta(S) = \\frac{\\lambda_2^+(S)\\lambda_3(S)}{|S|^2}",
      "notes": "λ_3(S) denotes the strain tensor's third eigenvalue per the Miller/Grujić–Xu eigenvalue conventions used across the C-series; not independently redefined in this file."
    },
    {
      "id": "ns.c6.c6e.rho_gap",
      "latex": "\\rho_{\\rm gap}(\\delta)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享間隙儲備",
      "label_en": "Shared gap reserve",
      "definition_zh": "對固定 $\\delta>0$ 定義為 $\\rho_{\\rm gap}(\\delta)=\\Pi_J^\\cap\\{\\vartheta(S)\\ge\\delta\\}$：$\\rho_{\\rm gap}\\ll1$ 表示共享源集中於中間間隙邊界附近；$\\rho_{\\rm gap}\\ge g_0>0$ 表示共享源有不退化比例具強中間形狀。若沿遞迴序列對每個固定 $\\delta$ 皆有 $\\rho_{{\\rm gap},n}(\\delta)\\to0$，則 $\\vartheta\\to0$（共享源機率意義下），恰為 C5-E 的 Middle-Gap Defect，路由至類別 G。",
      "definition_en": "For fixed $\\delta>0$, defined as $\\rho_{\\rm gap}(\\delta)=\\Pi_J^\\cap\\{\\vartheta(S)\\ge\\delta\\}$: $\\rho_{\\rm gap}\\ll1$ means shared source concentrates near the middle-gap boundary; $\\rho_{\\rm gap}\\ge g_0>0$ means a nondegenerate fraction of shared source has strong-middle shape. If $\\rho_{{\\rm gap},n}(\\delta)\\to0$ for every fixed $\\delta$ along a recurrent sequence, then $\\vartheta\\to0$ in shared-source probability — exactly the C5-E Middle-Gap Defect, routing to class G.",
      "defining_relation": "\\rho_{\\rm gap}(\\delta) = \\Pi_J^\\cap\\{\\vartheta(S)\\ge\\delta\\}",
      "notes": "This is the TS-B4 boundary quantity (section 38): ρ_gap → 0 routes to class G. Section 35 glosses it as \"nondegenerate shared middle-gap mass\" when listed as an R^TS component."
    },
    {
      "id": "ns.c6.c6e.v_tx",
      "latex": "V(t,x)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "正規化應變方向",
      "label_en": "Normalized strain direction",
      "definition_zh": "在 $\\Pi^\\cap$ 的支撐上定義為 $V(t,x)=\\frac{S(t,x)}{|S(t,x)|}\\in S^4\\subset\\operatorname{Sym}_0(3)$，即應變張量的單位方向，是第22-25節（C6-E.6 方向錐提取引理）緊緻性論證中的核心物件：固定 $\\delta$ 之下集合 $\\mathcal S_\\delta=\\{V\\in S^4:\\vartheta(V)\\ge\\delta\\}$ 為緊集，使有限覆蓋論證得以進行。",
      "definition_en": "On the support of $\\Pi^\\cap$, defined as $V(t,x)=\\frac{S(t,x)}{|S(t,x)|}\\in S^4\\subset\\operatorname{Sym}_0(3)$, the unit direction of the strain tensor; the central object in the compactness argument of sections 22-25 (the C6-E.6 directional-cone extraction lemma), where $\\mathcal S_\\delta=\\{V\\in S^4:\\vartheta(V)\\ge\\delta\\}$ is compact, enabling a finite-cover argument.",
      "defining_relation": "V(t,x) = \\frac{S(t,x)}{|S(t,x)|} \\in S^4 \\subset \\operatorname{Sym}_0(3)"
    },
    {
      "id": "ns.c6.c6e.s_delta",
      "latex": "\\mathcal S_\\delta",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "δ-強中間方向緊集",
      "label_en": "δ-strong-middle directional compact set",
      "definition_zh": "定義為 $\\mathcal S_\\delta=\\{V\\in S^4:\\vartheta(V)\\ge\\delta\\}\\subset S^4$，是正規化應變方向球面上「中間間隙係數不小於 δ」的子集，被證明為緊集，從而可用有限個 $\\varepsilon$-球 $B_\\varepsilon(K_j)$（$j=1,\\dots,N_{\\delta,\\varepsilon}<\\infty$）覆蓋，此有限方向覆蓋是 C6-E.6 Shared Directional-Cone Extraction Lemma 的關鍵步驟。",
      "definition_en": "Defined as $\\mathcal S_\\delta=\\{V\\in S^4:\\vartheta(V)\\ge\\delta\\}\\subset S^4$, the subset of the normalized-strain-direction sphere where the middle-gap coefficient is at least δ; shown to be compact, hence coverable by finitely many $\\varepsilon$-balls $B_\\varepsilon(K_j)$ ($j=1,\\dots,N_{\\delta,\\varepsilon}<\\infty$) — the key finite-cover step in the C6-E.6 Shared Directional-Cone Extraction Lemma.",
      "defining_relation": "\\mathcal S_\\delta = \\{V\\in S^4 : \\vartheta(V)\\ge\\delta\\}"
    },
    {
      "id": "ns.c6.c6e.mathfrak_q_j",
      "latex": "\\mathfrak Q_J(L)",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享核心集中度",
      "label_en": "Shared-core concentration",
      "definition_zh": "對給定合法參考尺度 $r_J>0$ 及 $L\\ge1$，定義為 $\\mathfrak Q_J(L)=\\sup_{x_0\\in\\mathbb R^3}\\Pi_J^\\cap\\left(\\{\\vartheta\\ge\\delta,|V-K|\\le\\varepsilon\\}\\cap[J\\times B_{Lr_J}(x_0)]\\right)$，衡量尺度 $Lr_J$ 的空間球能否承擔不退化比例的共享方向源質量。分兩種局部化狀態：E-CORE（$\\mathfrak Q_J(L)\\ge q_0$，某核心尺度球承擔不退化份額）與 E-DIFF（對每個固定 L 皆 $\\mathfrak Q_{J_n}(L)\\to0$，稱為 Shared-Source Spatial Diffusion / Multiplicity，即 TS-B5 邊界）。",
      "definition_en": "For a given legal reference scale $r_J>0$ and $L\\ge1$, defined as $\\mathfrak Q_J(L)=\\sup_{x_0\\in\\mathbb R^3}\\Pi_J^\\cap\\left(\\{\\vartheta\\ge\\delta,|V-K|\\le\\varepsilon\\}\\cap[J\\times B_{Lr_J}(x_0)]\\right)$, measuring whether some ball of scale $Lr_J$ can carry a nondegenerate fraction of shared directional source mass. It splits into two localization regimes: E-CORE ($\\mathfrak Q_J(L)\\ge q_0$, a core-scale ball carries a nondegenerate share) and E-DIFF (for every fixed L, $\\mathfrak Q_{J_n}(L)\\to0$, termed Shared-Source Spatial Diffusion/Multiplicity — the TS-B5 boundary).",
      "defining_relation": "\\mathfrak Q_J(L) = \\sup_{x_0\\in\\mathbb R^3} \\Pi_J^\\cap\\left(\\{\\vartheta\\ge\\delta,\\ |V-K|\\le\\varepsilon\\}\\cap[J\\times B_{Lr_J}(x_0)]\\right)",
      "notes": "r_J is a legal reference spatial scale supplied externally (UV ancestry, a theorem window, a selected pressure core, or another certified scale); without it, localization statements have no blow-up-scale meaning and are routed to the legality class A (section 28)."
    },
    {
      "id": "ns.c6.c6e.rho_her_ts",
      "latex": "\\rho_{\\rm her}^{TS}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享源遺傳儲備",
      "label_en": "Shared-source heredity reserve",
      "definition_zh": "在為相鄰事件 $J_n,J_{n+1}$ 選定合法正規化/重新定心後，令 $d_{\\rm src}(\\Pi_n^\\cap,\\Pi_{n+1}^\\cap)$ 為緊化共享源狀態空間上弱收斂的某固定度量化，定義 $\\rho_{\\rm her}^{TS}=\\left[1-\\frac{d_{\\rm src}}{d_0}\\right]_+$。此為型別化的遞迴元資料，非新的普適 PDE 估計，用以監控共享核心能否跨世代相容遞迴（TS-B6 邊界：$\\rho_{\\rm her}^{TS}\\to0$ 表示遺傳崩潰）。",
      "definition_en": "After choosing a legal normalization/recentering for adjacent events $J_n,J_{n+1}$, with $d_{\\rm src}(\\Pi_n^\\cap,\\Pi_{n+1}^\\cap)$ any fixed metrization of weak convergence on the compactified shared-source state space, defined as $\\rho_{\\rm her}^{TS}=\\left[1-\\frac{d_{\\rm src}}{d_0}\\right]_+$. This is typed recurrence metadata rather than a new universal PDE estimate, monitoring whether the shared core recurs compatibly across generations (TS-B6 boundary: $\\rho_{\\rm her}^{TS}\\to0$ signals heredity collapse).",
      "defining_relation": "\\rho_{\\rm her}^{TS} = \\left[1 - \\frac{d_{\\rm src}}{d_0}\\right]_+"
    },
    {
      "id": "ns.c6.c6e.t_state",
      "latex": "T",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "孤立時間陷阱狀態 T",
      "label_en": "Isolated temporal trap state T",
      "definition_zh": "C6-A 提出的三個粗粒候選陷阱之一（另二為 $G\\leftrightarrow P$、$H\\leftrightarrow F$），原以純時間相位/負荷資料（$\\mu^M,\\mu^O$ 及 Young/concentration 元資料）表徵自環 $T\\looparrowright T$。定理 C6-E.7（Pure-Temporal State Completeness No-Go）證明：兩組時空源對 $(\\Pi_1^M,\\Pi_1^O)$ 與 $(\\Pi_2^M,\\Pi_2^O)$ 可有相同時間邊際 $\\mu^M,\\mu^O$ 卻有完全不同的空間重疊、間隙幾何、方向集中、核心局部化與遺傳性質，故 T 不足以決定物理源耦合狀態。本輪最終判定：T 作為完整物理自環狀態被拒絕（REJECTED），應由聯合節點 TS 取代。",
      "definition_en": "One of the three coarse candidate traps proposed in C6-A (alongside $G\\leftrightarrow P$ and $H\\leftrightarrow F$), originally represented purely by temporal phase/load data ($\\mu^M,\\mu^O$ plus Young/concentration metadata) as a self-loop $T\\looparrowright T$. Theorem C6-E.7 (Pure-Temporal State Completeness No-Go) shows two spacetime source pairs $(\\Pi_1^M,\\Pi_1^O)$ and $(\\Pi_2^M,\\Pi_2^O)$ can share identical temporal marginals $\\mu^M,\\mu^O$ while differing completely in spatial overlap, middle-gap geometry, directional concentration, core localization, and heredity — so T cannot determine the physical source-coupling state. This round's final verdict: T is REJECTED as a complete physical self-cycle state and must be replaced by the joint node TS.",
      "notes": "T itself originates in C6-A (NS_C6A_CertifiedDefectGraph...); C6-E is the round that rejects it as complete, and is where the no-go theorem C6-E.7 has T as its central object."
    },
    {
      "id": "ns.c6.c6e.ts_state",
      "latex": "TS",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "聯合時空狀態 TS",
      "label_en": "Joint temporal-spatial state TS",
      "definition_zh": "定義為 $TS=(\\text{temporal phase},\\text{spacetime source coupling})$，是取代孤立 T 節點的正確候選物件。真正的遞迴陷阱需要 $TS_n$ 憑藉時空源遺傳遞迴至 $TS_{n+1}$；舊有的 $T\\looparrowright T$ 僅是此遞迴問題在時間軸上的投影。第46節將圖中舊候選 T 替換為聯合節點 TS，區分靜態投影 $TS\\to T$ 與有條件的動態遞迴；第47節進一步精煉為 $TS_{\\rm hereditary}$，作為 C6-A 三個粗粒陷阱之一（其餘為 $HF_{\\rm coherent}$、$GP_{\\rm hereditary}$）的更新候選，狀態仍為 open、未被證實。",
      "definition_en": "Defined as $TS=(\\text{temporal phase},\\text{spacetime source coupling})$, the correct candidate replacing the isolated T node. A genuine recurrent trap needs $TS_n$ to recur to $TS_{n+1}$ via spatiotemporal source heredity; the old $T\\looparrowright T$ is only this recurrence problem's projection onto the time axis. Section 46 replaces the old candidate T with the joint node TS in the defect graph, distinguishing the static projection $TS\\to T$ from the conditional dynamic recurrence; section 47 further refines it to $TS_{\\rm hereditary}$, the updated form of one of C6-A's three coarse traps (alongside $HF_{\\rm coherent}$ and $GP_{\\rm hereditary}$) — status open, not certified.",
      "defining_relation": "TS = (\\text{temporal phase},\\ \\text{spacetime source coupling})",
      "notes": "First full definition of TS as a joint node, replacing the isolated T candidate. ns.framework.TS is the framework-level summary; ns.c6.c6g.ts_node is C6-G's further refinement (V_int, TS-circ, cross-domain TS-circ-X)."
    },
    {
      "id": "ns.c6.c6e.r_ts_vector",
      "latex": "\\mathbf R^{TS}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "共享源耦合儲備向量",
      "label_en": "Shared-source coupling reserve vector",
      "definition_zh": "對一個時間/共享源事件定義為 $\\mathbf R^{TS}=(\\Omega_T,\\rho_{\\rm share},\\Gamma_O^O,\\rho_{\\rm gap},\\rho_{\\rm core},\\rho_{\\rm her}^{TS},\\rho_{\\rm scale})$，七個分量分別對應本輪各個儲備量。若沿子序列所有分量皆 $\\ge r_0>0$，稱為「Uniformly Shared-Source Coherent」（第36節）；否則由有限維緊緻性，必有至少一個固定分量趨於零，對應第38節的七類邊界 TS-B1至TS-B7，此為定理 C6-E.8（Finite Temporal–Spatial Coupling Bottleneck Theorem）二分法的基礎。",
      "definition_en": "For a temporal/shared-source event, defined as $\\mathbf R^{TS}=(\\Omega_T,\\rho_{\\rm share},\\Gamma_O^O,\\rho_{\\rm gap},\\rho_{\\rm core},\\rho_{\\rm her}^{TS},\\rho_{\\rm scale})$, seven components corresponding to this round's reserve quantities. If all components stay $\\ge r_0>0$ along a subsequence, the sequence is \"Uniformly Shared-Source Coherent\" (section 36); otherwise finite-dimensional compactness forces at least one fixed component to tend to zero, corresponding to the seven boundary types TS-B1–TS-B7 of section 38 — the basis of the dichotomy in Theorem C6-E.8 (Finite Temporal–Spatial Coupling Bottleneck Theorem).",
      "defining_relation": "\\mathbf R^{TS} = \\left(\\Omega_T,\\ \\rho_{\\rm share},\\ \\Gamma_O^O,\\ \\rho_{\\rm gap},\\ \\rho_{\\rm core},\\ \\rho_{\\rm her}^{TS},\\ \\rho_{\\rm scale}\\right)"
    },
    {
      "id": "ns.c6.c6e.rho_core",
      "latex": "\\rho_{\\rm core}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "參考尺度核心局部化儲備",
      "label_en": "Reference-scale core-localization reserve",
      "definition_zh": "儲備向量 $\\mathbf R^{TS}$ 的分量之一，代表「reference-scale shared-core localization」，即共享方向源質量能否被某個固定倍率 L 的參考尺度球（透過 $\\mathfrak Q_J(L)$ 的 E-CORE 狀態）承擔的程度。文中未給出獨立公式，其退化情形即 TS-B5 邊界（$\\rho_{\\rm core}\\to0$，對應 E-DIFF/Shared-Source Spatial Diffusion）。",
      "definition_en": "A component of the reserve vector $\\mathbf R^{TS}$, glossed as \"reference-scale shared-core localization\" — the extent to which shared directional source mass can be carried by a reference-scale ball of some fixed multiple L (via the E-CORE regime of $\\mathfrak Q_J(L)$). No independent formula is given in the text; its degeneration is the TS-B5 boundary ($\\rho_{\\rm core}\\to0$, corresponding to the E-DIFF/Shared-Source Spatial Diffusion regime).",
      "notes": "Defined only by gloss/list entry (section 35), not by an explicit standalone formula; its meaning is carried by the E-CORE/E-DIFF dichotomy of \\mathfrak Q_J(L) (sections 26-27)."
    },
    {
      "id": "ns.c6.c6e.rho_scale",
      "latex": "\\rho_{\\rm scale}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "合法參考尺度/設定儲備",
      "label_en": "Legal reference-scale/setup reserve",
      "definition_zh": "儲備向量 $\\mathbf R^{TS}$ 的分量之一，代表「legal reference-scale/setup reserve」：若缺乏合法比較尺度 $r_J$，「機率測度落在某有限球內」這類敘述便無爆破尺度意義，因此被路由至合法性/設定類別 A（第28節）。其退化即 TS-B7 邊界（$\\rho_{\\rm scale}\\to0$，路由至 A）。",
      "definition_en": "A component of the reserve vector $\\mathbf R^{TS}$, glossed as the \"legal reference-scale/setup reserve\": without a legal comparison scale $r_J$, a statement like \"the probability measure lies in some finite ball\" carries no blow-up-scale meaning, so the absence of one is routed to the legality/setup class A (section 28). Its degeneration is the TS-B7 boundary ($\\rho_{\\rm scale}\\to0$, routing to A).",
      "notes": "Defined only by gloss/list entry (section 35), not by an explicit standalone formula."
    },
    {
      "id": "ns.c6.c6e.theta_ts_c6e",
      "latex": "\\Theta_{TS}^{C6E}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "C6-E 聯合源狀態（ETN更新）",
      "label_en": "C6-E joint source state (ETN update)",
      "definition_zh": "定義為 $\\Theta_{TS}^{C6E}=\\left\\langle\\mu^M,\\mu^O,\\Pi^M,\\Pi^O,\\Omega_T,\\Omega_{ST},\\Gamma_O^O,\\Pi^\\cap,\\vartheta,V,\\mathfrak Q,\\rho_{\\rm her}^{TS}\\right\\rangle$，第54節「True ETN update」中彙整本輪所有核心對象而成的單一狀態元組，作為 C6-E 對 Extended Typed Notation（ETN）系統的更新。",
      "definition_en": "Defined as $\\Theta_{TS}^{C6E}=\\left\\langle\\mu^M,\\mu^O,\\Pi^M,\\Pi^O,\\Omega_T,\\Omega_{ST},\\Gamma_O^O,\\Pi^\\cap,\\vartheta,V,\\mathfrak Q,\\rho_{\\rm her}^{TS}\\right\\rangle$, section 54's \"True ETN update\" — a single state tuple aggregating all of this round's core objects, C6-E's update to the Extended Typed Notation (ETN) system.",
      "defining_relation": "\\Theta_{TS}^{C6E} = \\left\\langle \\mu^M,\\mu^O,\\Pi^M,\\Pi^O,\\Omega_T,\\Omega_{ST},\\Gamma_O^O,\\Pi^\\cap,\\vartheta,V,\\mathfrak Q,\\rho_{\\rm her}^{TS} \\right\\rangle",
      "notes": "\"ETN\" is not spelled out beyond the section heading in this file; treated here as this round's running formal-state ledger, consistent with its use across the C6 sub-series."
    },
    {
      "id": "ns.c6.c6e.partial_k_ts",
      "latex": "\\partial\\mathcal K_{TS}",
      "series": "NS",
      "first_appearance": "C6-E",
      "label_zh": "耦合邊界字母表",
      "label_en": "Coupling-boundary alphabet",
      "definition_zh": "定義為 $\\partial\\mathcal K_{TS}=\\{\\text{TEMP},\\text{SPSEG},\\text{OPCAP},\\text{GAP},\\text{DIFF},\\text{HER},\\text{SETUP}\\}$，是第38節七類 TS-B1至TS-B7 邊界（依序對應 $\\Omega_T\\to0$、$\\rho_{\\rm share}\\to0$、$\\Gamma_O^O\\to0$、$\\rho_{\\rm gap}\\to0$、$\\rho_{\\rm core}\\to0$、$\\rho_{\\rm her}^{TS}\\to0$、$\\rho_{\\rm scale}\\to0$）的符號化字母表，與定理 C6-E.8 的 TS-BOUNDARY 分支對應。",
      "definition_en": "Defined as $\\partial\\mathcal K_{TS}=\\{\\text{TEMP},\\text{SPSEG},\\text{OPCAP},\\text{GAP},\\text{DIFF},\\text{HER},\\text{SETUP}\\}$, the symbolic alphabet for the seven TS-B1–TS-B7 boundary cases of section 38 (corresponding respectively to $\\Omega_T\\to0$, $\\rho_{\\rm share}\\to0$, $\\Gamma_O^O\\to0$, $\\rho_{\\rm gap}\\to0$, $\\rho_{\\rm core}\\to0$, $\\rho_{\\rm her}^{TS}\\to0$, $\\rho_{\\rm scale}\\to0$), matching the TS-BOUNDARY branch of Theorem C6-E.8 (Finite Temporal–Spatial Coupling Bottleneck Theorem).",
      "defining_relation": "\\partial\\mathcal K_{TS} = \\{\\text{TEMP}, \\text{SPSEG}, \\text{OPCAP}, \\text{GAP}, \\text{DIFF}, \\text{HER}, \\text{SETUP}\\}"
    },
    {
      "id": "ns.c6.c6f.q_sv",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "Miller 應變-渦度運算子",
      "label_en": "Miller Strain-Vorticity Operator",
      "definition_zh": "Section 1.2 引入的 Miller 應變 H¹ 恆等式強制項，為 (u·∇)S+S²+¾ω⊗ω 的 Leray 投影 P_st(·)；本輪將其重新操作化為運算子側正密度 g_O 的核心成分，驅動整個 operator-toll 分支。",
      "definition_en": "The forcing term in Miller's strain-H¹ energy identity (Section 1.2), the Leray projection P_st((u·∇)S+S²+¾ω⊗ω); this round re-grounds it from the primary source and makes it the core ingredient of the operator-side positive density g_O that drives the entire operator-toll branch.",
      "defining_relation": "\\mathcal Q_{SV} = P_{st}\\left((u\\cdot\\nabla)S + S^2 + \\frac34\\omega\\otimes\\omega\\right)",
      "notes": "運算子本身取自 Miller (arXiv:2407.02691) 原始構造；C6-F 的 Section 1.2「Fresh primary-source audit」重新確立其角色，並自 Section 3 起實際用於定義 g_O、O_*、F-OP 等本輪新物件。 Same symbol as ns.c6.QSV_criterion (filed under C6-Q) and ns.c6.c6e.q_sv; this C6-F entry documents its re-grounding from the primary source and operational use in g_O."
    },
    {
      "id": "ns.c6.c6f.g_o",
      "latex": "g_O",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "正運算子局部密度",
      "label_en": "Positive Operator Local Density",
      "definition_zh": "Section 3 定義的逐點密度 g_O=-Q_SV:(-ΔS)-ν|ΔS|²，其空間積分即應變能量泛函的時間導數 h(t)=E_1'(t)；正部 [g_O]_+ 是運算子側機率提升 f_O 與正質量 P_J 的來源。",
      "definition_en": "The pointwise density g_O=-Q_SV:(-ΔS)-ν|ΔS|² defined in Section 3, whose spatial integral is h(t)=E_1'(t), the time-derivative of the strain energy functional; its positive part [g_O]_+ generates the operator-side probability lift f_O and positive mass P_J.",
      "defining_relation": "g_O = -\\mathcal Q_{SV}:(-\\Delta S) - \\nu|\\Delta S|^2",
      "notes": "Same quantity as ns.c6.c6e.g_o; later renamed/reframed as g_O^{proj} (ns.c6.gOproj) in C6-Q once the projection-free g_O^{loc} (ns.c6.gOloc) is introduced."
    },
    {
      "id": "ns.c6.c6f.p_j",
      "latex": "P_J",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "正運算子淨質量",
      "label_en": "Positive Net Operator Mass",
      "definition_zh": "Section 3 定義為 [h(t)]_+ 在時間窗 J 上的積分（假設 >0），是 C6-E 中段時間質量 M_J 在運算子側的新對應物，出現在本輪幾乎所有 operator-side 物理稅收下界中（C6-F.2、F.5、F.6 等）。",
      "definition_en": "Defined in Section 3 as the J-integral of [h(t)]_+ (assumed positive); the operator-side counterpart to the middle temporal mass M_J from C6-E, appearing in nearly every operator-side physical-toll lower bound of this round (C6-F.2, F.5, F.6, etc.).",
      "defining_relation": "P_J = \\int_J [h(t)]_+\\,dt"
    },
    {
      "id": "ns.c6.c6f.omega_st",
      "latex": "\\Omega_{ST}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "共享時空源重疊質量",
      "label_en": "Shared Spacetime-Source Overlap Mass",
      "definition_zh": "Section 5 定義，中段機率密度 f_M 與運算子機率密度 f_O 逐點最小值的時空積分，衡量兩者真正共享而非僅時間重疊 Ω_T 的機率質量；假設 Ω_ST>0 是本輪一切定理的共同前提。",
      "definition_en": "Defined in Section 5 as the spacetime integral of the pointwise minimum of the middle density f_M and operator density f_O, measuring genuinely shared probability mass rather than mere temporal overlap Ω_T; Ω_ST>0 is the standing hypothesis for every theorem in this round.",
      "defining_relation": "\\Omega_{ST} = \\int \\min(f_M,f_O)\\,dx\\,dt"
    },
    {
      "id": "ns.c6.c6f.w_cap",
      "latex": "w_\\cap",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "正規化共享源密度",
      "label_en": "Normalized Shared-Source Density",
      "definition_zh": "Section 5 定義為 min(f_M,f_O)/Ω_ST，是共享測度 Π_J^∩ 的密度；滿足 f_M≥Ω_ST w_∩ 且 f_O≥Ω_ST w_∩（C6-F.1 逐點支配定理），是本輪核心抽取（D_J、t_*、E_*）所依據的權重。",
      "definition_en": "Defined in Section 5 as min(f_M,f_O)/Ω_ST, the density of the shared measure Π_J^∩; satisfies f_M≥Ω_ST w_∩ and f_O≥Ω_ST w_∩ pointwise (the C6-F.1 domination theorem), and is the weight underlying every core-extraction step of this round (D_J, t_*, E_*).",
      "defining_relation": "w_\\cap(t,x) = \\dfrac{\\min(f_M,f_O)}{\\Omega_{ST}}"
    },
    {
      "id": "ns.c6.c6f.pi_cap",
      "latex": "\\Pi_J^\\cap",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "共享源機率測度",
      "label_en": "Shared-Source Probability Measure",
      "definition_zh": "密度為 w_∩ 的機率測度（Section 5），是 C6-E 命名之「共享中段-運算子時空源測度」在本輪由 f_M、f_O、Ω_ST 顯式構造出的版本；其在核心圓柱 D_J 上的測度 Π_J^∩(D_J)≥q_0 貫穿本輪全部物理稅收定理。",
      "definition_en": "The probability measure with density w_∩ (Section 5); the explicit f_M/f_O/Ω_ST construction of the shared middle-operator spacetime source measure named in C6-E. Its value Π_J^∩(D_J)≥q_0 on the core-cylinder D_J underlies every physical-toll theorem in this round.",
      "notes": "名稱與概念源自 C6-E（Section 0 稱其為 C6-E 最重要的新 object），本輪 Section 3-5 首次給出經由 f_M、f_O、Ω_ST 的完整顯式構造。"
    },
    {
      "id": "ns.c6.c6f.rho_m",
      "latex": "\\rho_M",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "絕對中段負載儲備",
      "label_en": "Absolute Middle-Load Reserve",
      "definition_zh": "Section 8 定義為 min{1, M_J/M_J^ref}，防止機率正規化掩蓋絕對物理負載退化（即 M_J→0 而 Π^∩、Ω_ST 不變）的守門量；是交叉領域儲備向量 R^X 的第一分量。",
      "definition_en": "Defined in Section 8 as min{1, M_J/M_J^ref}; a guard quantity preventing probability normalization from masking a collapse of absolute physical load (M_J→0 while Π^∩ and Ω_ST remain fixed). The first coordinate of the cross-domain reserve vector R^X.",
      "defining_relation": "\\rho_M = \\min\\left\\{1, \\dfrac{M_J}{M_J^{ref}}\\right\\}"
    },
    {
      "id": "ns.c6.c6f.rho_p",
      "latex": "\\rho_P",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "絕對運算子負載儲備",
      "label_en": "Absolute Operator-Load Reserve",
      "definition_zh": "Section 8 定義為 min{1, P_J/P_J^ref}，為 ρ_M 在運算子側的對應物；ρ_M→0 或 ρ_P→0 構成 Section 9 的「絕對負載臨界飽和」，即邊界字母表 X-B1（Section 44）的觸發條件。",
      "definition_en": "Defined in Section 8 as min{1, P_J/P_J^ref}, the operator-side counterpart to ρ_M; ρ_M→0 or ρ_P→0 constitutes the Absolute-Load Critical Saturation of Section 9, the trigger condition for boundary letter X-B1 (Section 44).",
      "defining_relation": "\\rho_P = \\min\\left\\{1, \\dfrac{P_J}{P_J^{ref}}\\right\\}"
    },
    {
      "id": "ns.c6.c6f.e_kde",
      "latex": "E_{K,\\delta,\\varepsilon}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "共享方向核心區域",
      "label_en": "Shared Directional Core Region",
      "definition_zh": "Section 10 定義，由中段間隙門檻 δ、單位無跡對稱錐心 K、錐寬 ε 界定的時空子集：{(t,x): θ(S)≥δ, |S/|S|-K|≤ε}；是後續核心圓柱 D_J 與同時刻核心 E_* 的共同幾何原型。",
      "definition_en": "Defined in Section 10 as the spacetime set {(t,x): θ(S)≥δ, |S/|S|-K|≤ε}, cut out by the middle-gap threshold δ, a unit-norm trace-free symmetric cone center K, and cone width ε; the geometric prototype for the core-cylinder D_J and the same-time core E_*."
    },
    {
      "id": "ns.c6.c6f.d_j",
      "latex": "D_J",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "共享核心圓柱",
      "label_en": "Shared Core-Cylinder",
      "definition_zh": "Section 11 定義為時空圓柱 J×B_{Lr_J}(x_J) 與方向核心區域 E_{K,δ,ε} 的交集；本輪假設共享測度 Π_J^∩(D_J)≥q_0>0，作為 C6-E 核心局部化共享源前提在本輪的具體載體。",
      "definition_en": "Defined in Section 11 as the intersection of the spacetime cylinder J×B_{Lr_J}(x_J) with the directional core region E_{K,δ,ε}; assumed to carry shared measure Π_J^∩(D_J)≥q_0>0, serving as this round's concrete vehicle for the C6-E core-localized shared-source antecedent."
    },
    {
      "id": "ns.c6.c6f.q_0",
      "latex": "q_0",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "核心局部化質量下界",
      "label_en": "Core-Localization Mass Lower Bound",
      "definition_zh": "Section 11 假設的正常數，滿足 Π_J^∩(D_J)≥q_0>0；出現於本輪幾乎每個主要不等式（C6-F.2 至 C6-F.7）及交叉領域儲備向量 R^X 中，量化共享源核心的最小機率權重。",
      "definition_en": "The positive constant hypothesized in Section 11 satisfying Π_J^∩(D_J)≥q_0>0; appears in nearly every major inequality of this round (C6-F.2 through C6-F.7) and as a coordinate of the reserve vector R^X, quantifying the minimal probability weight of the shared-source core."
    },
    {
      "id": "ns.c6.c6f.t_star",
      "latex": "t_\\ast",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "同一抽取時刻",
      "label_en": "Same-Time Extraction Instant",
      "definition_zh": "Section 13-14 由 Fubini/平均原理證得存在的時刻，滿足 w_D(t_*)≥q_0/|J|；是本輪「同一時間同時取得中段與運算子物理核心」（C6-F.3）的關鍵存在性物件，解決了時間同步問題。",
      "definition_en": "The instant whose existence follows from the Fubini/averaging argument in Sections 13-14, satisfying w_D(t_*)≥q_0/|J|; the key existential object behind this round's same-time middle-and-operator core extraction (C6-F.3), genuinely resolving the temporal-synchronization problem.",
      "defining_relation": "w_D(t_\\ast) \\ge \\dfrac{q_0}{|J|}"
    },
    {
      "id": "ns.c6.c6f.e_star",
      "latex": "E_\\ast",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "同時刻共享核心區域",
      "label_en": "Same-Time Shared Core Region",
      "definition_zh": "Section 14 定義，在抽取時刻 t_* 上取的空間集合 B_{Lr_J}(x_J)∩{θ≥δ, |S/|S|-K|≤ε}；是 C6-F.3 至 C6-F.6 一系列「同一時間、同一區域」中段與運算子物理稅收下界的共同積分域。",
      "definition_en": "Defined in Section 14 as the spatial set B_{Lr_J}(x_J)∩{θ≥δ, |S/|S|-K|≤ε} evaluated at the extracted instant t_*; the common domain of integration for the same-time, same-region middle and operator physical-toll lower bounds of C6-F.3 through C6-F.6."
    },
    {
      "id": "ns.c6.c6f.o_star",
      "latex": "\\mathfrak O_\\ast",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "黏性尺度化運算子範數",
      "label_en": "Viscosity-Scaled Operator Norm",
      "definition_zh": "Section 19 定義為 ν^{-1/2}||Q_SV||_{L²(E_*)}，即運算子強制在同時刻核心上的黏性尺度化 L² 範數；與 𝔇_* 之積下界為 P_J Ω_ST q_0/|J|，構成 C6-F.6 二分定理的 F-OP 分支核心量。",
      "definition_en": "Defined in Section 19 as ν^{-1/2}||Q_SV||_{L²(E_*)}, the viscosity-rescaled L² norm of the operator forcing on the same-time core; its product with 𝔇_* is bounded below by P_J Ω_ST q_0/|J|, forming the core quantity of the F-OP branch of the C6-F.6 dichotomy.",
      "defining_relation": "\\mathfrak O_\\ast = \\nu^{-1/2}\\|\\mathcal Q_{SV}\\|_{L^2(E_\\ast)}"
    },
    {
      "id": "ns.c6.c6f.d_star",
      "latex": "\\mathfrak D_\\ast",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "黏性尺度化導數範數",
      "label_en": "Viscosity-Scaled Derivative Norm",
      "definition_zh": "Section 19 定義為 ν^{1/2}||ΔS||_{L²(E_*)}，即同時刻核心上三階應變導數的黏性尺度化 L² 範數；為 C6-F.6 二分定理 F-DER 分支的核心量，且在 L² 意義下等價於速度場三階導數 D³u（Section 21）。",
      "definition_en": "Defined in Section 19 as ν^{1/2}||ΔS||_{L²(E_*)}, the viscosity-rescaled L² norm of the third-order strain derivative on the same-time core; the core quantity of the F-DER branch of the C6-F.6 dichotomy, and L²-equivalent to the third velocity derivative D³u (Section 21).",
      "defining_relation": "\\mathfrak D_\\ast = \\nu^{1/2}\\|\\Delta S\\|_{L^2(E_\\ast)}"
    },
    {
      "id": "ns.c6.c6f.f_op",
      "latex": "F\\text{-OP}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "運算子強制分支",
      "label_en": "Operator-Forcing Branch",
      "definition_zh": "C6-F.6（Section 20）二分定理的第一分支：ν^{-1/2}||Q_SV||_{2,E_*} ≳ (P_J Ω_ST q_0/|J|)^{1/2}；此分支成立時，共享運算子核心自然導向非線性強制類 F（Section 31），須經 C6-C 再入儲備才能成為 HF_coherent 候選。",
      "definition_en": "The first branch of the C6-F.6 dichotomy (Section 20): ν^{-1/2}||Q_SV||_{2,E_*} ≳ (P_J Ω_ST q_0/|J|)^{1/2}. When it holds, the shared operator core is naturally routed to the nonlinear-forcing class F (Section 31), requiring the C6-C re-entry reserves to become an HF_coherent candidate."
    },
    {
      "id": "ns.c6.c6f.f_der",
      "latex": "F\\text{-DER}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "高階導數分支",
      "label_en": "High-Derivative Branch",
      "definition_zh": "C6-F.6（Section 20）二分定理的第二分支：ν^{1/2}||ΔS||_{2,E_*} ≳ (P_J Ω_ST q_0/|J|)^{1/2}；是通往 Grujić–Xu H 狀態的 pre-H 座標，須另加導數實現閘門 ρ_der 才可能真正進入 H（Section 22、33）。",
      "definition_en": "The second branch of the C6-F.6 dichotomy (Section 20): ν^{1/2}||ΔS||_{2,E_*} ≳ (P_J Ω_ST q_0/|J|)^{1/2}; a pre-H coordinate toward the Grujić-Xu H state, which requires the additional derivative-realization gate ρ_der to genuinely reach H (Sections 22, 33)."
    },
    {
      "id": "ns.c6.c6f.epsilon_q",
      "latex": "\\epsilon_Q",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "Q 加權錐洩漏量",
      "label_en": "Q-Weighted Cone Leakage",
      "definition_zh": "Section 25 定義，強中段良好集 G_K 補集上 |Q| 加權質量佔總加權質量 A_χ*^Q 的比例；量化「共享源方向錐」與 C5-D 所需「Q/場加權強中段錐」間的落差，決定 TS→GP 能否經 C5-D 定理實現。",
      "definition_en": "Defined in Section 25 as the fraction of Q-weighted mass A_χ*^Q lying outside the strong-middle good set G_K; quantifies the gap between the source-weighted directional cone and the Q/field-weighted strong-middle cone required by the C5-D theorem, governing whether TS→GP can be realized.",
      "defining_relation": "\\epsilon_Q = \\dfrac{\\int_{G_K^c}\\chi_\\ast|Q|\\,dx}{A_{\\chi_\\ast}^{Q}}"
    },
    {
      "id": "ns.c6.c6f.rho_qcap",
      "latex": "\\rho_{Qcap}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "源至場捕獲儲備",
      "label_en": "Source-to-Field Capture Reserve",
      "definition_zh": "Section 26 定義為 1-ε_Q；當 ρ_Qcap≈1（即洩漏低於 C5-D 定量閾值 γ_K/[2(1+γ_K)]）時，共享源方向相干性可提升為完整局部二次場的一致相干性（Section 27，C6-F.7 假設4）。",
      "definition_en": "Defined in Section 26 as 1-ε_Q; when ρ_Qcap≈1 (leakage below the C5-D quantitative threshold γ_K/[2(1+γ_K)]), the shared-source directional coherence upgrades to nondegenerate coherence of the full local quadratic field (Section 27; hypothesis 4 of C6-F.7).",
      "defining_relation": "\\rho_{Qcap} = 1-\\epsilon_Q"
    },
    {
      "id": "ns.c6.c6f.rho_mean",
      "latex": "\\rho_{\\rm mean}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "平均旋轉儲備",
      "label_en": "Mean-Rotation Reserve",
      "definition_zh": "Section 28 定義為 [1-|M_χ*'|/((γ_K/2)A_χ*^Q)]_+；ρ_mean>0 表示平均旋轉不足以吸收相干二次強制，此時觸發 C5-D 定向壓力再入定理，是 C6-F.7 條件式 TS→GP 定理的假設5。",
      "definition_en": "Defined in Section 28 as [1-|M_χ*'|/((γ_K/2)A_χ*^Q)]_+; positivity means mean rotation cannot absorb the coherent quadratic forcing, triggering the C5-D oriented-pressure re-entry theorem — hypothesis 5 of the conditional C6-F.7 TS-core→GP theorem."
    },
    {
      "id": "ns.c6.c6f.rho_der",
      "latex": "\\rho_{\\rm der}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "導數實現旗標",
      "label_en": "Derivative-Realization Flag",
      "definition_zh": "Section 32 定義的二元旗標 ρ_der∈{0,1}，指示由 F-DER 活動生成的選定導數階數/時刻是否合法進入 Grujić–Xu 定理/逃逸時間框架；僅當 ρ_der=1 才觸發 Section 33 的 H∨REG 二分結果。",
      "definition_en": "The binary flag ρ_der∈{0,1} defined in Section 32, indicating whether a selected derivative order/time generated from F-DER activity legally enters the Grujić-Xu theorem/escape-time framework; only ρ_der=1 triggers the H∨REG dichotomy of Section 33."
    },
    {
      "id": "ns.c6.c6f.r_x",
      "latex": "\\mathbf R^{X}",
      "series": "NS",
      "first_appearance": "C6-F",
      "label_zh": "交叉領域儲備向量",
      "label_en": "Cross-Domain Reserve Vector",
      "definition_zh": "Section 42 定義的十三分量向量 (ρ_M,ρ_P,Ω_ST,q_0,ρ_gap,ρ_cone,ρ_thick,ρ_Qcap,ρ_mean,ρ_prov,ρ_der,ρ_her^TS,ρ_scale)，彙整本輪及先前各輪全部橋接儲備；C6-F.9 據此二分為 X-UNIFORM（全體一致 ≥r_0）或 X-BOUNDARY（某分量→0）。",
      "definition_en": "The 13-component vector (ρ_M,ρ_P,Ω_ST,q_0,ρ_gap,ρ_cone,ρ_thick,ρ_Qcap,ρ_mean,ρ_prov,ρ_der,ρ_her^TS,ρ_scale) defined in Section 42, aggregating every bridge reserve from this and prior rounds; the C6-F.9 theorem dichotomizes on it into X-UNIFORM (all coordinates uniformly ≥r_0) or X-BOUNDARY (some coordinate →0).",
      "defining_relation": "\\mathbf R^{X} = \\left(\\rho_M,\\rho_P,\\Omega_{ST},q_0,\\rho_{\\rm gap},\\rho_{\\rm cone},\\rho_{\\rm thick},\\rho_{Qcap},\\rho_{\\rm mean},\\rho_{\\rm prov},\\rho_{\\rm der},\\rho_{\\rm her}^{TS},\\rho_{\\rm scale}\\right)"
    },
    {
      "id": "ns.c6.c6g.sigma_e",
      "latex": "\\sigma_e",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊的證明狀態標籤",
      "label_en": "Edge proof-status tag",
      "definition_zh": "邊的證明狀態標籤，取值 $\\{I,C,N,E\\}$（蘊含、條件蘊含、非互斥、外部終止），與 $\\tau_e$ 共同判定邊能否用於動態SCC。",
      "definition_en": "The proof-status tag on each edge, valued in $\\{I,C,N,E\\}$ (implication, conditional implication, non-exclusion, external kill); with $\\tau_e$ it determines which edges can compose a dynamic SCC.",
      "defining_relation": "\\sigma_e\\in\\{I,C,N,E\\}"
    },
    {
      "id": "ns.c6.c6g.tau_e",
      "latex": "\\tau_e",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊的時間語意標籤",
      "label_en": "Edge time-semantics tag",
      "definition_zh": "邊的時間語意標籤，取值 $\\{S,D,E\\}$（同事件/靜態、真實世代轉移、外部終止）；唯有 $D$ 邊能構成動態遞迴SCC，靜態邊須先quotient壓縮。",
      "definition_en": "The time-semantics tag on each edge, valued in $\\{S,D,E\\}$ (same-event/static, genuine generation transition, external kill); only $D$-edges can build a dynamic recurrent SCC, so static edges are quotient-collapsed first.",
      "defining_relation": "\\tau_e\\in\\{S,D,E\\}"
    },
    {
      "id": "ns.c6.c6g.gp_node",
      "latex": "GP",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "幾何-壓力聯合節點",
      "label_en": "Geometry-pressure joint node",
      "definition_zh": "由 $G\\overset S\\leftrightarrow P$ 之靜態相容關係（$(G,P)\\in\\mathcal C_{GP}$，即 $G\\sim_SP$）quotient而成的聯合節點，取代舊圖中 $G,P$ 分離節點的處理；其非退化版 $GP^\\circ$ 之自遞迴仍為未證候選。",
      "definition_en": "The joint node obtained by quotienting the static compatibility relation $G\\overset S\\leftrightarrow P$ (i.e. $(G,P)\\in\\mathcal C_{GP}$, so $G\\sim_SP$), replacing separate dynamic $G,P$ nodes; its nondegenerate form $GP^\\circ$'s self-recurrence remains an uncertified candidate.",
      "defining_relation": "G\\sim_S P \\Rightarrow GP",
      "notes": "Detailed C6-G definition of the joint node GP; ns.framework.GP is the framework-level summary entry for the same defect class."
    },
    {
      "id": "ns.c6.c6g.hf_node",
      "latex": "HF",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "H/F 非線性相干再入節點",
      "label_en": "H/F nonlinear coherent re-entry node",
      "definition_zh": "因 $H\\not\\Longleftrightarrow F$ 非普遍類別循環，僅相干非線性再入子型別可能遞迴，故定義 $HF:=HF_{\\rm nonlinear\\ coherent}$；其非退化版 $HF^\\circ$ 自遞迴同為未證候選。",
      "definition_en": "Since $H\\not\\Longleftrightarrow F$ is not a universal class cycle, only the coherent nonlinear re-entry subtype can recur, giving $HF:=HF_{\\rm nonlinear\\ coherent}$; its nondegenerate form $HF^\\circ$'s self-recurrence is likewise an uncertified candidate.",
      "defining_relation": "HF := HF_{\\rm nonlinear\\ coherent}",
      "notes": "Detailed C6-G definition of the joint node HF; ns.framework.HF is the framework-level summary entry for the same defect class."
    },
    {
      "id": "ns.c6.c6g.ts_node",
      "latex": "TS",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "時空共源聯合節點",
      "label_en": "Temporal-spatial shared-source joint node",
      "definition_zh": "$T$ 被證明只是時空源測度 $(\\Pi^M,\\Pi^O)$ 的時間邊際，故提升為完整狀態 $TS$，是跨領域交界狀態；C6-G.3證明 $TS$ 不能作為孤立最小內部倖存者候選。",
      "definition_en": "$T$ is shown to be only the temporal marginal of the spacetime source measures $(\\Pi^M,\\Pi^O)$, so the full state is lifted to $TS$, a cross-domain junction state; C6-G.3 proves $TS$ cannot be an isolated minimal interior survivor candidate.",
      "defining_relation": "T = \\text{temporal marginal of }(\\Pi^M,\\Pi^O) \\Rightarrow TS",
      "notes": "Further refinement of the TS joint node first defined in ns.c6.c6e.ts_state; ns.framework.TS is the framework-level summary entry."
    },
    {
      "id": "ns.c6.c6g.pi_m_pi_o",
      "latex": "(\\Pi^M,\\Pi^O)",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "時空源測度對",
      "label_en": "Spacetime source measure pair",
      "definition_zh": "$T$ 是此測度對的時間邊際投影；此對測度是 $TS$ 作為時空共源狀態的底層物件，上標未於文中明文展開。",
      "definition_en": "$T$ is the temporal-marginal projection of this measure pair, the underlying object behind treating $TS$ as a genuine spacetime shared-source state; the superscripts are not spelled out in the text.",
      "defining_relation": "T \\to (\\Pi^M,\\Pi^O)"
    },
    {
      "id": "ns.c6.c6g.v_int",
      "latex": "V_{\\rm int}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "精細內部節點集",
      "label_en": "Refined interior node set",
      "definition_zh": "$V_{\\rm int}=\\{TS^\\circ,GP^\\circ,HF^\\circ\\}$，取代C5-M的 $\\{A,T,G,P,H,F\\}$，是C6-G.1「內部SCC解體定理」的討論對象。",
      "definition_en": "$V_{\\rm int}=\\{TS^\\circ,GP^\\circ,HF^\\circ\\}$, replacing the six coarse C5-M nodes; this is the set analyzed by the C6-G.1 Interior SCC Dissolution Theorem.",
      "defining_relation": "V_{\\rm int}=\\{TS^\\circ,GP^\\circ,HF^\\circ\\}"
    },
    {
      "id": "ns.c6.c6g.circ_superscript",
      "latex": "\\circ",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "內部（非退化）上標記號",
      "label_en": "Interior (nondegenerate) superscript marker",
      "definition_zh": "附於節點右上方，表示定義該節點內部區域的所有儲備座標皆嚴格為正/非退化，是全篇區分一般節點與內部節點的核心約定。",
      "definition_en": "A superscript on a node symbol meaning every reserve coordinate defining its interior regime is strictly positive/nondegenerate; the core convention distinguishing a bare node from its interior version throughout.",
      "defining_relation": "X^\\circ \\iff \\text{all interior-regime reserves of } X \\text{ strictly positive}"
    },
    {
      "id": "ns.c6.c6g.r_ts",
      "latex": "\\mathbf R_{TS}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "TS 內部儲備向量",
      "label_en": "TS interior reserve vector",
      "definition_zh": "代表性TS內部狀態的九項儲備座標向量，刻劃 $TS^\\circ$ 內部區域的具體座標。",
      "definition_en": "The nine-coordinate reserve vector of a representative TS interior state, specifying the coordinates of the $TS^\\circ$ interior regime.",
      "defining_relation": "\\mathbf R_{TS}=\\left(\\rho_M,\\rho_P,\\Omega_{ST},q_0,\\rho_{\\rm gap},\\rho_{\\rm cone},\\rho_{\\rm thick},\\rho_{\\rm her}^{TS},\\rho_{\\rm scale}\\right)"
    },
    {
      "id": "ns.c6.c6g.ts_circ_x",
      "latex": "TS^\\circ_X",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "完整跨領域（X-）內部 TS 狀態",
      "label_en": "Full cross-domain (X-)interior TS state",
      "definition_zh": "$\\mathbf R_{TS}$ 外加四項跨領域路由儲備後之完整內部狀態，表示所有C6-F橋接所需儲備皆一致為正；是第10節已證跨領域邊與C6-G.3 M1/M2的前提物件。",
      "definition_en": "The full interior state adding four cross-domain routing reserves to $\\mathbf R_{TS}$, meaning every reserve required by the C6-F bridges is uniformly positive; the hypothesis behind §10's certified cross-domain edges and C6-G.3's M1/M2 branches.",
      "defining_relation": "TS^\\circ_X \\iff \\mathbf R_{TS}\\text{ and }(\\rho_{Qcap},\\rho_{\\rm mean},\\rho_{\\rm prov},\\rho_{\\rm der})\\text{ uniformly positive}"
    },
    {
      "id": "ns.c6.c6g.r_gp",
      "latex": "\\mathbf R_{GP}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "GP 內部儲備向量",
      "label_en": "GP interior reserve vector",
      "definition_zh": "代表性GP狀態的八項儲備向量；$GP^\\circ$ 表示遠離中段間隙、平均旋轉接管、局部/遠場壓力接管、簽名與軸邊際崩潰等邊界。",
      "definition_en": "The eight-coordinate reserve vector for a representative GP state; $GP^\\circ$ means staying away from middle-gap, mean-rotation-takeover, pressure-takeover, signature- and axis-margin-collapse boundaries.",
      "defining_relation": "\\mathbf R_{GP}=\\left(\\rho_{\\rm geom},\\rho_{\\rm mean},\\rho_{\\rm far},\\rho_{\\rm prov},\\rho_{\\rm sig},\\rho_{\\rm axis},\\rho_F^{her},\\rho_G^{her}\\right)"
    },
    {
      "id": "ns.c6.c6g.r_hf",
      "latex": "\\mathbf R_{HF}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "HF 內部儲備向量",
      "label_en": "HF interior reserve vector",
      "definition_zh": "代表性HF狀態的六項儲備向量；$HF^\\circ$ 表示此相干非線性再入狀態遠離所有C6-C再入邊界。",
      "definition_en": "The six-coordinate reserve vector for a representative HF state; $HF^\\circ$ means this coherent nonlinear re-entry state stays away from all C6-C re-entry boundaries.",
      "defining_relation": "\\mathbf R_{HF}=\\left(\\Gamma^{Duh},\\rho_{\\rm dom},\\rho_{\\rm sel},\\rho_{\\rm sign},\\rho_{\\rm time},\\rho_{\\rm setup}\\right)"
    },
    {
      "id": "ns.c6.c6g.f_op_der",
      "latex": "F_{\\rm OP},\\ F_{\\rm DER}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "算子/導數高階介面狀態",
      "label_en": "Operator/derivative high-order interface states",
      "definition_zh": "TS算子與高階導數代價分岔出的兩中介型別；合法導數實現則 $F_{\\rm DER}\\to H\\vee\\mathrm{REG}$，滿足再入相干性則 $F_{\\rm OP}\\to HF^\\circ$。",
      "definition_en": "Two intermediate types the TS operator/high-derivative toll splits into; legal derivative realization gives $F_{\\rm DER}\\to H\\vee\\mathrm{REG}$, and re-entry coherence gives $F_{\\rm OP}\\to HF^\\circ$.",
      "defining_relation": "TS^\\circ_X \\to F_{\\rm OP}\\vee F_{\\rm DER};\\ F_{\\rm DER}\\to H\\vee\\mathrm{REG};\\ F_{\\rm OP}\\to HF^\\circ"
    },
    {
      "id": "ns.c6.c6g.delta_cert",
      "latex": "\\delta_{\\rm cert}(C)",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "循環的證明缺口數",
      "label_en": "Certification deficit of a cycle",
      "definition_zh": "候選循環 $C$ 中缺乏已證動態轉移之邊數；已證循環須 $\\delta_{\\rm cert}=0$，C6-G.2證明現行每條候選循環皆 $\\ge1$。",
      "definition_en": "The count of edges in candidate cycle $C$ lacking a certified dynamic transition; a certified cycle needs $\\delta_{\\rm cert}=0$, and C6-G.2 proves every current candidate cycle has $\\delta_{\\rm cert}(C)\\ge1$.",
      "defining_relation": "\\delta_{\\rm cert}(C)=\\#\\{e_j: e_j \\text{ lacks a certified composable dynamic transition}\\}"
    },
    {
      "id": "ns.c6.c6g.partial_k_c6",
      "latex": "\\partial\\mathcal K_{C6}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "C6 全域臨界邊界聯集",
      "label_en": "C6 global critical boundary union",
      "definition_zh": "三內部節點各自臨界邊界之聯集，隨後被quotient成十類全域邊界字母表 $\\mathfrak B$。",
      "definition_en": "The union of the three interior nodes' own critical boundaries, subsequently quotiented into the ten-member global boundary alphabet $\\mathfrak B$.",
      "defining_relation": "\\partial\\mathcal K_{C6}=\\partial\\mathcal K_{TS}\\cup\\partial\\mathcal K_{GP}\\cup\\partial\\mathcal K_{HF}"
    },
    {
      "id": "ns.c6.c6g.b_load",
      "latex": "\\mathsf B_{LOAD}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：物理負荷臨界飽和",
      "label_en": "Boundary superclass: physical-load critical saturation",
      "definition_zh": "涵蓋TS絕對負荷崩潰、HF響應優勢崩潰等正規化振幅趨零情形；正規化形狀可能仍相干，但絕對PDE代價已消失。",
      "definition_en": "Covers TS absolute load collapse, HF response-dominance collapse, and other normalized amplitudes tending to zero; the normalized shape may stay coherent even as absolute PDE toll vanishes.",
      "defining_relation": "\\mathsf B_{LOAD}=\\text{physical-load critical saturation}"
    },
    {
      "id": "ns.c6.c6g.b_coh",
      "latex": "\\mathsf B_{COH}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：相干性崩潰",
      "label_en": "Boundary superclass: coherence collapse",
      "definition_zh": "涵蓋HF Duhamel相干性崩潰、時間符號抵消、TS算子容量抵消、GP壓力抵消；響應非退化時依§30.1有 $B_{COH}\\to B_{CAP^\\infty}$。",
      "definition_en": "Covers HF Duhamel coherence collapse, temporal sign cancellation, TS operator-capacity cancellation, and GP pressure cancellation; per §30.1, nondegenerate response gives $B_{COH}\\to B_{CAP^\\infty}$.",
      "defining_relation": "B_{COH}\\to B_{CAP^\\infty}\\ \\text{(if response stays nondegenerate)}"
    },
    {
      "id": "ns.c6.c6g.b_seg",
      "latex": "\\mathsf B_{SEG}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：分離/隔離",
      "label_en": "Boundary superclass: segregation",
      "definition_zh": "涵蓋TS時間/空間源分離、HF目標擴散、核心尺度多重性；質量持續存在但無法停留於單一可組合載體。",
      "definition_en": "Covers TS temporal/spatial source segregation, HF target diffusion, and core-scale multiplicity; mass persists but cannot stay on one composable carrier.",
      "defining_relation": "B_{SEG} = \\text{family where mass persists but disperses off a single composable carrier}"
    },
    {
      "id": "ns.c6.c6g.b_geom",
      "latex": "\\mathsf B_{GEOM}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：幾何臨界性",
      "label_en": "Boundary superclass: geometric criticality",
      "definition_zh": "涵蓋中段間隙崩潰、方向錐退化、HF諧波符號飽和、GP軸/簽名邊際崩潰；多屬非零代價邊界而非自由出口（§30.3, §30.5）。",
      "definition_en": "Covers middle-gap collapse, directional cone degeneration, HF harmonic sign-threshold saturation, and GP axis/signature-margin collapse; mostly nonzero-debt boundaries, not free exits (§30.3, §30.5).",
      "defining_relation": "B_{GEOM}^{gap}\\to G\\text{-geometry defect (known route, §30.3)}"
    },
    {
      "id": "ns.c6.c6g.b_field",
      "latex": "\\mathsf B_{FIELD}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：源-場去耦合",
      "label_en": "Boundary superclass: source-field decoupling",
      "definition_zh": "涵蓋TS源至場捕獲失敗、導數實現失敗、分量選擇退化；度量未能將源/響應物件轉為下一介面所需完整PDE場物件的失敗。",
      "definition_en": "Covers TS source-to-field capture failure, derivative-realization failure, and component-selection degeneration; measures failure to realize a source/response object as the required full PDE field object.",
      "defining_relation": "B_{FIELD} = \\text{failure to realize a source/response object as the required PDE field object}"
    },
    {
      "id": "ns.c6.c6g.b_mean",
      "latex": "\\mathsf B_{MEAN}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：平均旋轉接管",
      "label_en": "Boundary superclass: mean-rotation takeover",
      "definition_zh": "GP路由中相干二次強迫可能被吸收進 $M_\\chi'$ 而非壓力項，構成與壓力路徑競爭的補償通道。",
      "definition_en": "In GP routing, coherent quadratic forcing may be absorbed into $M_\\chi'$ rather than pressure, a genuine compensation channel competing with the pressure route.",
      "defining_relation": "\\text{quadratic forcing}\\to M_\\chi'\\ \\text{(instead of pressure)}"
    },
    {
      "id": "ns.c6.c6g.b_prov",
      "latex": "\\mathsf B_{PROV}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：來源正當性崩潰",
      "label_en": "Boundary superclass: provenance collapse",
      "definition_zh": "涵蓋局部/遠場壓力接管、壓力簽名邊界 $\\det F\\to0$、源正當性碎裂、壓力遺傳流失；記錄壓力/源物件是否具正確來源脈絡。",
      "definition_en": "Covers local/far pressure takeover, the pressure signature boundary $\\det F\\to0$, source fragmentation, and heredity loss; records whether the pressure/source object has correct provenance.",
      "defining_relation": "\\det F\\to0\\ \\text{(pressure signature boundary instance)}"
    },
    {
      "id": "ns.c6.c6g.b_her",
      "latex": "\\mathsf B_{HER}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：遺傳性崩潰",
      "label_en": "Boundary superclass: heredity collapse",
      "definition_zh": "涵蓋HF窗口持續性崩潰、GP幾何/遠場壓力遺傳崩潰、TS共享源遺傳崩潰；是主要的循環組成邊界——狀態個體強但無法產出下一代。",
      "definition_en": "Covers HF window-persistence collapse and GP/TS heredity collapse; the principal cycle-composition boundary — a state may be individually strong yet fail to produce the next generation.",
      "defining_relation": "B_{HER}^{HF}\\to F/\\text{temporal forcing (known route, §30.2)}"
    },
    {
      "id": "ns.c6.c6g.b_setup",
      "latex": "\\mathsf B_{SETUP}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界超類：定理設置失敗",
      "label_en": "Boundary superclass: setup failure",
      "definition_zh": "涵蓋Grujić–Xu定理設置失敗、參考尺度/來源合法性失敗；路由至合法性出口節點 $A$ 而非物理遞迴節點。",
      "definition_en": "Covers Grujić–Xu theorem setup failure and reference-scale/provenance legality failure; routes to the legality-exit node $A$ rather than a physical recurrent node.",
      "defining_relation": "B_{SETUP}\\to A"
    },
    {
      "id": "ns.c6.c6g.b_cap_infty",
      "latex": "\\mathsf B_{CAP^\\infty}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "無窮遠邊界（容量發散）",
      "label_en": "Boundary-at-infinity (divergent capacity)",
      "definition_zh": "由發散容量/響應比 $\\Gamma^{-1}\\to\\infty$ 定義，經緊化 $\\widehat C=C/(1+C)\\in[0,1]$ 轉為有限條件 $\\widehat C\\to1$；是無窮遠處邊界而非趨零儲備面。",
      "definition_en": "Defined from a divergent capacity/response ratio $\\Gamma^{-1}\\to\\infty$; via compactification $\\widehat C=C/(1+C)\\in[0,1]$ this becomes the finite condition $\\widehat C\\to1$ — a boundary at infinity, not a vanishing-reserve face.",
      "defining_relation": "\\Gamma^{-1}\\to\\infty;\\ \\widehat C=\\frac{C}{1+C}\\in[0,1];\\ \\mathsf B_{CAP^\\infty} := (\\widehat C\\to1)"
    },
    {
      "id": "ns.c6.c6g.boundary_alphabet",
      "latex": "\\mathfrak B",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "全域邊界字母表",
      "label_en": "Global boundary alphabet",
      "definition_zh": "C6-C/D/E/F儲備退化面壓縮成的十類全域邊界超類集合；$\\mathrm{REG}$刻意排除在外，是C6-G.3 M3分支的核心物件。",
      "definition_en": "The ten-member global boundary superclass set compressing all C6-C/D/E/F reserve-degeneration faces; $\\mathrm{REG}$ is deliberately excluded, and this set is central to C6-G.3's M3 branch.",
      "defining_relation": "\\mathfrak B=\\{B_{LOAD},B_{COH},B_{SEG},B_{GEOM},B_{FIELD},B_{MEAN},B_{PROV},B_{HER},B_{SETUP},B_{CAP^\\infty}\\}"
    },
    {
      "id": "ns.c6.c6g.boundary_saturated_survivor",
      "latex": "\\textbf{Boundary-Saturated Survivor}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界飽和倖存者",
      "label_en": "Boundary-saturated survivor",
      "definition_zh": "無窮倖存者序列沿子序列逼近某固定 $B_*\\in\\mathfrak B$；是C6-G.3的M3分支，也是本輪標題核心概念，十個超類代表不同失效機制、不可視為單一終態。",
      "definition_en": "An infinite survivor sequence approaching, along a subsequence, some fixed $B_*\\in\\mathfrak B$; branch M3 of C6-G.3 and the concept named in this round's title — the ten superclasses are distinct failure mechanisms, not one anonymous terminal state.",
      "defining_relation": "d(\\Theta_{n_j},B_*)\\to0\\ \\text{for some fixed } B_*\\in\\mathfrak B"
    },
    {
      "id": "ns.c6.c6g.d_b",
      "latex": "D_B",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "邊界事件債務向量",
      "label_en": "Boundary event debt vector",
      "definition_zh": "對邊界事件 $B$ 定義的八項示意債務向量，不同面啟動不同分量，是判定哪些面攜帶強制性有限債務的前置物件。",
      "definition_en": "The eight-coordinate schematic debt vector for a boundary event $B$; different faces activate different coordinates, and it underlies determining which faces carry a coercive, finite debt.",
      "defining_relation": "D_B=\\left(D_{\\rm energy},D_{\\rm diss},D_{\\rm pressure},D_{\\rm middle},D_{\\rm derivative},D_{\\rm forcing},D_{\\rm capacity},D_{\\rm time}\\right)"
    },
    {
      "id": "ns.c6.c6g.globally_coercive",
      "latex": "\\text{globally coercive}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "（邊界面）全域強制性",
      "label_en": "Globally coercive (boundary face property)",
      "definition_zh": "邊界面 $B$ 若每一靠近邊界之世代支付 $d_B(n)\\ge\\epsilon_B>0$ 且 $\\sum_n d_B(n)<\\infty$，則稱其全域強制，不能無窮遞迴——是C6-A有限預算引理的邊界類比。",
      "definition_en": "A boundary face $B$ where every near-boundary generation pays $d_B(n)\\ge\\epsilon_B>0$ with $\\sum_n d_B(n)<\\infty$ is globally coercive and cannot recur infinitely — the boundary-level analogue of the C6-A finite-budget lemma.",
      "defining_relation": "d_B(n)\\ge\\epsilon_B>0\\ \\text{with}\\ \\sum_n d_B(n)<\\infty \\Rightarrow B\\text{ cannot recur infinitely often}"
    },
    {
      "id": "ns.c6.c6g.minimal_survivor_reduction",
      "latex": "GP_{\\rm uniform\\ hereditary}\\vee HF_{\\rm uniform\\ coherent}\\vee\\text{Boundary-Saturated Survivor}\\vee A",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "最小倖存者化約（四分支析取）",
      "label_en": "Minimal survivor reduction (four-way disjunction)",
      "definition_zh": "C6-G.3定理結論：避開$\\mathrm{REG}$之無窮倖存者取子序列後必為M1一致遺傳GP、M2一致相干HF、M3邊界飽和、或M4合法性出口$A$之一；關鍵推論是$TS$被排除於候選名單外。",
      "definition_en": "The C6-G.3 conclusion: any infinite survivor avoiding $\\mathrm{REG}$ must, after a subsequence, be one of M1 uniform hereditary GP, M2 uniform coherent HF, M3 boundary-saturated, or M4 legality exit into $A$; crucially, $TS$ is thereby removed from the candidate list.",
      "defining_relation": "GP^\\circ_n\\ (M1) \\vee HF^\\circ_n\\ (M2) \\vee [d(\\Theta_{n_j},B_*)\\to0]\\ (M3) \\vee [\\text{enters } A]\\ (M4)"
    },
    {
      "id": "ns.c6.c6g.g_c6_typed",
      "latex": "\\mathcal G_{C6}^{typed}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "C6 現行已型別化動態圖",
      "label_en": "C6 current typed dynamic graph",
      "definition_zh": "C6-G.4「零已證循環稽核」之對象：靜態quotient與儲備型別化後之現行圖，不存在完全由已證邊組成的非平凡遞迴循環；此為現行研究圖定理，非N-S流本身之證明。",
      "definition_en": "The object of the C6-G.4 Zero-Certified-Cycle Audit: the current typed graph, after static quotient and reserve typing, contains no nontrivial recurrent cycle of certified edges — a current research-graph theorem, not a proof about the N-S flow itself.",
      "defining_relation": "\\mathcal G_{C6}^{typed}: \\delta_{\\rm cert}(C)\\ge1\\ \\forall C"
    },
    {
      "id": "ns.c6.c6g.k_bar_c6",
      "latex": "\\overline{\\mathcal K}_{C6}",
      "series": "NS",
      "first_appearance": "C6-G",
      "label_zh": "C6 全域緊化狀態空間",
      "label_en": "C6 global compactified state space",
      "definition_zh": "內部狀態空間 $\\mathcal K_{\\rm int}=\\mathcal K_{TS}\\sqcup\\mathcal K_{GP}\\sqcup\\mathcal K_{HF}$ 與邊界聯集及出口節點 $\\{A,\\mathrm{REG}\\}$ 之聯集，是本輪對整個typed狀態系統的正式總結。",
      "definition_en": "The union of the interior state space $\\mathcal K_{\\rm int}=\\mathcal K_{TS}\\sqcup\\mathcal K_{GP}\\sqcup\\mathcal K_{HF}$, the boundary union, and exit nodes $\\{A,\\mathrm{REG}\\}$ — this round's formal summary of the entire typed state system.",
      "defining_relation": "\\overline{\\mathcal K}_{C6}=\\mathcal K_{\\rm int}\\cup\\partial\\mathcal K_{C6}\\cup\\{A,\\mathrm{REG}\\}"
    },
    {
      "id": "ns.c6.c6h.mathfrak_b_original",
      "latex": "\\mathfrak B",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "粗粒化邊界面字母表（原始十類）",
      "label_en": "Coarse-grained boundary-face alphabet (original ten)",
      "definition_zh": "C6-G 將所有 critical faces 粗粒化後得到的十元邊界面集合，涵蓋 $B_{LOAD}, B_{COH}, B_{SEG}, B_{GEOM}, B_{FIELD}, B_{MEAN}, B_{PROV}, B_{HER}, B_{SETUP}, B_{CAP^\\infty}$。C6-H 全篇的核心任務即是對此集合進行語意分類與逐步約簡，最終縮至六元終端物理面字母表 $\\mathfrak B_{phys}^{(2)}$。",
      "definition_en": "The ten-element set of boundary faces obtained by coarse-graining all critical faces in C6-G, comprising $B_{LOAD}, B_{COH}, B_{SEG}, B_{GEOM}, B_{FIELD}, B_{MEAN}, B_{PROV}, B_{HER}, B_{SETUP}, B_{CAP^\\infty}$. The entire C6-H round is organized around semantically classifying and progressively reducing this alphabet down to the six-element terminal physical alphabet $\\mathfrak B_{phys}^{(2)}$.",
      "defining_relation": "\\mathfrak B = \\{ B_{LOAD}, B_{COH}, B_{SEG}, B_{GEOM}, B_{FIELD}, B_{MEAN}, B_{PROV}, B_{HER}, B_{SETUP}, B_{CAP^\\infty} \\}",
      "notes": "Originates in the prior round C6-G, not newly defined by C6-H; restated here as the starting point for this round's reduction."
    },
    {
      "id": "ns.c6.c6h.b_field",
      "latex": "B_{FIELD}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "場捕獲邊失效面",
      "label_en": "Field-capture edge-failure face",
      "definition_zh": "Type E（edge-failure boundary）的一個例子，代表 source-to-field capture 失敗、response dominance 失敗、derivative realization 失敗，或所選分量喪失嚴格 margin。依 C6-H.1 的結論，由於此類敘述僅說明「所嘗試的邊未能複合」而不指定唯一新的 PDE 物理狀態，$B_{FIELD}$ 被正式自物理邊界 SCC node list 中移除，僅保留作為 transition-failure metadata。",
      "definition_en": "An instance of a Type E (edge-failure) boundary, meaning source-to-field capture fails, response dominance fails, derivative realization fails, or the selected component loses strict margin. By the conclusion of C6-H.1, since such statements only say \"the attempted edge did not compose\" without specifying a unique new PDE physical state, $B_{FIELD}$ is formally removed from the physical boundary SCC node list and retained only as transition-failure metadata.",
      "notes": "Classification table in §36 records its status as \"REMOVED AS NODE.\""
    },
    {
      "id": "ns.c6.c6h.b_her",
      "latex": "B_{HER}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "遺傳性邊失效面",
      "label_en": "Heredity edge-failure face",
      "definition_zh": "另一個 Type E（edge-failure boundary）例子，代表 geometry heredity 失敗、pressure heredity 失敗、window persistence 失敗，或 shared-source heredity 失敗。與 $B_{FIELD}$ 同理，依 C6-H.1 之 Edge-Boundary Node No-Go，$B_{HER}$ 亦自物理邊界 SCC 節點清單中移除，僅作為 transition-failure metadata 保留。",
      "definition_en": "Another instance of a Type E (edge-failure) boundary: geometry heredity fails, pressure heredity fails, window persistence fails, or shared-source heredity fails. Like $B_{FIELD}$, by the Edge-Boundary Node No-Go of C6-H.1, $B_{HER}$ is likewise removed from the physical boundary SCC node list and retained only as transition-failure metadata.",
      "notes": "Classification table in §36 records its status as \"REMOVED AS NODE.\""
    },
    {
      "id": "ns.c6.c6h.b_load",
      "latex": "B_{LOAD}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "負載崩塌／正規化面",
      "label_en": "Load-collapse / normalization face",
      "definition_zh": "屬於 Type N（Normalization boundary）的邊界面，意指某條 cycle/generation 邊上實現的 physical toll 相對於所選正規化尺度趨於零。它既非純粹的物理形狀面，也非 edge failure，而是依賴合法正規化尺度的極限現象；在 C6-H.5（Coherence Dichotomy）與 C6-H.6（Middle-Gap Dichotomy）中皆作為兩個分支之一出現，並保留在最終六元物理邊界字母表 $\\mathfrak B_{phys}^{(2)}$ 中，狀態為 OPEN（未被 uniform finite-energy budget 消除）。",
      "definition_en": "A Type N (normalization) boundary face: a physical toll realized along a chosen cycle/generation edge tends to zero relative to the legal normalization scale. It is neither a pure physical-shape face nor an edge failure, but a genuine limiting regime tied to the normalization scale; it appears as one branch in both the Coherence Dichotomy (C6-H.5) and the Middle-Gap Dichotomy (C6-H.6), and survives into the final six-face alphabet $\\mathfrak B_{phys}^{(2)}$ with status OPEN (not coercively eliminated by the finite-energy budget).",
      "notes": "Section 34 stresses $B_{LOAD}$ is \"not an external regularity sink in general\" — a survivor may reroute, change edge, or move to another face after load-collapse."
    },
    {
      "id": "ns.c6.c6h.b_cap_infty",
      "latex": "B_{CAP^\\infty}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "無窮容量邊界面",
      "label_en": "Boundary-at-infinity / capacity-at-infinity face",
      "definition_zh": "Type $\\infty$（Boundary at infinity）邊界面，意指一個正規化容量／回應比或物理臨界容量發散，僅能在緊化（compactification）後表示，例如 $\\widehat C=C/(1+C)\\to1$。它是 C6-H.5 與 C6-H.6 兩個二分法定理中「容量發散」分支的共同終點，也是最終六元物理字母表 $\\mathfrak B_{phys}^{(2)}$ 的成員；§33 強調其發散並不自動構成矛盾，狀態維持 OPEN。",
      "definition_en": "The Type $\\infty$ (boundary-at-infinity) face: a normalized capacity/response ratio or physical critical capacity diverges, representable only after compactification, e.g. $\\widehat C=C/(1+C)\\to1$. It is the common terminus of the \"capacity diverges\" branch in both the Coherence Dichotomy (C6-H.5) and the Middle-Gap Dichotomy (C6-H.6), and remains a member of the final six-face alphabet $\\mathfrak B_{phys}^{(2)}$; §33 stresses that divergence is not automatically a contradiction, and its status stays OPEN.",
      "defining_relation": "\\widehat C = \\frac{C}{1+C} \\to 1",
      "notes": "Reached via both H-B1 ($COH\\to LOAD\\vee CAP^\\infty$) and H-B2 ($GAP\\to LOAD\\vee CAP^\\infty$)."
    },
    {
      "id": "ns.c6.c6h.rho_e_theta",
      "latex": "\\rho_e(\\theta)",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "邊型保留量（reserve）",
      "label_en": "Edge reserve quantity",
      "definition_zh": "屬於某個 typed transition $e:X\\to Y$ 定義域的一般 reserve 函數，是 C6-H.1（Edge-Boundary Node No-Go）命題的中心對象。該命題證明：若 $\\rho_e(\\theta_n)\\to0$ 在極限中僅意味著 $\\theta_n\\notin\\operatorname{Dom}(e)$，而未確定唯一的物理 PDE 狀態類別，則面 $\\{\\rho_e=0\\}$ 不能被當作獨立的物理動態節點使用——這是 $B_{FIELD}$、$B_{HER}$ 被剔除出物理邊界 SCC 字母表的理論依據。",
      "definition_en": "A generic reserve function belonging to the domain of some typed transition $e:X\\to Y$; the central object of Proposition C6-H.1 (Edge-Boundary Node No-Go). The proposition shows that if $\\rho_e(\\theta_n)\\to0$ only implies $\\theta_n\\notin\\operatorname{Dom}(e)$ in the limit — without pinning down a unique physical PDE state class — then the face $\\{\\rho_e=0\\}$ cannot be used as an independent physical dynamic node; this is the theoretical basis for removing $B_{FIELD}$ and $B_{HER}$ from the physical boundary SCC alphabet.",
      "notes": "Generic/schematic notation, instantiated concretely by the FIELD and HER edge-failure examples in §4."
    },
    {
      "id": "ns.c6.c6h.b_setup",
      "latex": "B_{SETUP}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "設置／合法性面",
      "label_en": "Setup / legality face",
      "definition_zh": "代表 theorem entry 失敗、legal reference scale 不可得、ancestry/provenance interface 未建立，或 remaining-time gate 不可得，本質上正是 C5/C6 的 legality class $A$。C6-H.2（Setup Quotient）證明 $B_{SETUP}\\equiv A$，因此就物理 recurrent SCC 分析而言將其自物理邊界字母表中移除（quotiented），不再視為獨立面。",
      "definition_en": "Represents theorem-entry failure, unavailability of a legal reference scale, an unestablished ancestry/provenance interface, or an unavailable remaining-time gate — essentially identical to the C5/C6 legality class $A$. C6-H.2 (Setup Quotient) proves $B_{SETUP}\\equiv A$, so for the purpose of physical recurrent SCC analysis it is removed (quotiented) from the physical boundary alphabet rather than treated as an independent face.",
      "defining_relation": "B_{SETUP} \\equiv A",
      "notes": "Classification table in §36 records its status as \"QUOTIENTED.\""
    },
    {
      "id": "ns.c6.c6h.mathfrak_b_phys_1",
      "latex": "\\mathfrak B_{phys}^{(1)}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "第一次約簡後之物理邊界字母表（七元）",
      "label_en": "First reduced physical boundary alphabet (seven-element)",
      "definition_zh": "在 C6-H.1（移除 $B_{FIELD},B_{HER}$）與 C6-H.2（將 $B_{SETUP}$ 商去為 $A$）之語意商化後得到的七元集合，將物理邊界問題自原本的十個粗粒化超類縮減為七個超類，是本輪第一個里程碑式的約簡結果。",
      "definition_en": "The seven-element set obtained after the semantic quotients of C6-H.1 (removing $B_{FIELD}, B_{HER}$) and C6-H.2 (quotienting $B_{SETUP}$ into $A$); it shrinks the physical boundary problem from the original ten coarse-grained superclasses to seven, marking the round's first reduction milestone.",
      "defining_relation": "\\mathfrak B_{phys}^{(1)} = \\{ B_{LOAD}, B_{COH}, B_{SEG}, B_{GEOM}, B_{MEAN}, B_{PROV}, B_{CAP^\\infty} \\}",
      "notes": "Superseded later in the same round by the six-element $\\mathfrak B_{phys}^{(2)}$ (§35) after the coherence and middle-gap dichotomies."
    },
    {
      "id": "ns.c6.c6h.ns_scaling_u_p",
      "latex": "u_\\lambda(x,t) = \\lambda u(\\lambda x,\\lambda^2t)",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "Navier–Stokes 拋物型重新標度（速度／壓力）",
      "label_en": "Navier–Stokes parabolic rescaling (velocity/pressure)",
      "definition_zh": "標準三維不可壓縮 Navier–Stokes 方程的拋物型自相似重新標度：速度場依 $u_\\lambda(x,t)=\\lambda u(\\lambda x,\\lambda^2t)$、壓力場依 $p_\\lambda(x,t)=\\lambda^2p(\\lambda x,\\lambda^2t)$ 標度。此標度是 §12–§13 動能與黏性耗散標度計算，以及 C6-H.3（Scaling Obstruction to Uniform Energy Coercivity）定理的基礎，用以證明 scale-invariant UV event 無法承載 uniform positive kinetic-energy debt。",
      "definition_en": "The standard parabolic self-similar rescaling of 3D incompressible Navier–Stokes: the velocity field scales as $u_\\lambda(x,t)=\\lambda u(\\lambda x,\\lambda^2t)$ and the pressure field as $p_\\lambda(x,t)=\\lambda^2p(\\lambda x,\\lambda^2t)$. This rescaling underlies the kinetic-energy and viscous-dissipation scaling computations of §12–§13 and Theorem C6-H.3 (Scaling Obstruction to Uniform Energy Coercivity), which shows that scale-invariant UV events cannot carry a uniform positive kinetic-energy debt.",
      "defining_relation": "u_\\lambda(x,t) = \\lambda u(\\lambda x,\\lambda^2t), \\quad p_\\lambda(x,t) = \\lambda^2 p(\\lambda x,\\lambda^2t)",
      "notes": "The vorticity scaling $\\omega_\\lambda=\\lambda^2\\omega(\\lambda x,\\lambda^2t)$ (§17, §40) is a derived consequence of this rescaling, not an independent primary definition."
    },
    {
      "id": "ns.c6.c6h.d_e_dissipation",
      "latex": "D_E[u;I]",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "事件視窗上的黏性耗散量",
      "label_en": "Viscous dissipation over an event window",
      "definition_zh": "對時間窗 $I=(a,b)$ 上速度場 $u$ 定義的黏性耗散量（即 $\\nu\\int_I\\|\\nabla u(s)\\|_2^2ds$）；在 N–S 重新標度下，窗口本身變為 $I_\\lambda=(\\lambda^{-2}a,\\lambda^{-2}b)$，而該量滿足 $D_E[u_\\lambda;I_\\lambda]=\\lambda^{-1}D_E[u;I]$。此標度律是 C6-H.3 定理與 §37 finite-energy coercivity audit 的關鍵工具：因 $D_E$ 隨 $\\lambda\\to\\infty$ 趨於零，scale-invariant boundary metadata 無法單獨導出 uniform positive energy debt。",
      "definition_en": "The viscous dissipation of velocity field $u$ over a time window $I=(a,b)$ (i.e. $\\nu\\int_I\\|\\nabla u(s)\\|_2^2ds$); under N–S rescaling the window itself becomes $I_\\lambda=(\\lambda^{-2}a,\\lambda^{-2}b)$, and the quantity obeys $D_E[u_\\lambda;I_\\lambda]=\\lambda^{-1}D_E[u;I]$. This scaling law is the key tool behind Theorem C6-H.3 and the finite-energy coercivity audit of §37: because $D_E\\to0$ as $\\lambda\\to\\infty$, scale-invariant boundary metadata alone cannot force a uniform positive energy debt.",
      "defining_relation": "D_E[u_\\lambda;I_\\lambda] = \\lambda^{-1} D_E[u;I]",
      "notes": "Referred to informally as $D_{energy}$ in the round's opening summary (§0); the same object underlies the event-indexed shorthand $D_E$ used in the C6-H.3 proof (§15) and the threshold $D_E\\ge\\varepsilon_0>0$ discussed throughout."
    },
    {
      "id": "ns.c6.c6h.mathcal_e_event_class",
      "latex": "\\mathcal E",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "尺度不變事件類",
      "label_en": "Scale-invariant event class",
      "definition_zh": "一個非空的 local/parabolic N–S 事件類，僅由 scale-invariant metadata 定義且在 N–S 重新標度下封閉，是定理 C6-H.3（Scaling Obstruction to Uniform Energy Coercivity）陳述與證明的中心對象。該定理證明：不存在形如 $\\nu\\int_{I_E}\\|\\nabla u\\|_2^2dt\\ge\\varepsilon_0>0$、且 $\\varepsilon_0$ 與尺度無關的估計，能單純由成員資格 $E\\in\\mathcal E$ 導出。",
      "definition_en": "A nonempty class of local/parabolic N–S events defined solely by scale-invariant metadata and closed under N–S rescaling; the central object of Theorem C6-H.3 (Scaling Obstruction to Uniform Energy Coercivity). The theorem proves that no estimate of the form $\\nu\\int_{I_E}\\|\\nabla u\\|_2^2dt\\ge\\varepsilon_0>0$, with $\\varepsilon_0$ independent of scale, can follow solely from membership $E\\in\\mathcal E$.",
      "notes": "The proof rescales a fixed $E\\in\\mathcal E$ by $\\lambda\\to\\infty$, keeping it in $\\mathcal E$ while its dissipation $D_E(\\lambda)=\\lambda^{-1}D_E\\to0$, yielding the contradiction."
    },
    {
      "id": "ns.c6.c6h.d_n_finite_budget",
      "latex": "d_n",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "有限總預算之逐項量",
      "label_en": "Finite global budget summand",
      "definition_zh": "C6-H.4（Finite Budget vs Critical Barrier Distinction）中「有限總體預算」概念的代表符號，滿足 $\\sum_n d_n<\\infty$（例如總動能耗散）。此類預算唯有在存在正下界 $d_n\\ge d_0>0$ 時才能用以排除無限多事件，但 C6-H.3 的標度論證顯示，對純粹 dimensionless UV metadata 而言此下界不可能成立，這正是 C6 boundary-cycle program 必須轉向 critical barrier 型 debt 的理由。",
      "definition_en": "The representative symbol for the \"finite global budget\" notion in C6-H.4 (Finite Budget vs Critical Barrier Distinction), satisfying $\\sum_n d_n<\\infty$ (e.g. total kinetic-energy dissipation). Such a budget is useful for excluding infinitely many events only if a positive lower bound $d_n\\ge d_0>0$ holds, but the scaling argument of C6-H.3 shows this lower bound cannot hold for purely dimensionless UV metadata — the reason the C6 boundary-cycle program must switch to critical-barrier-type debt.",
      "defining_relation": "\\sum_n d_n < \\infty",
      "notes": "Contrasted directly with $b_n/b_{crit}$ (the critical-barrier notion) in the same section, C6-H.4."
    },
    {
      "id": "ns.c6.c6h.b_n_critical_barrier",
      "latex": "b_n, b_{crit}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "臨界障礙量與其門檻值",
      "label_en": "Critical barrier quantity and threshold",
      "definition_zh": "C6-H.4 中「critical barrier」概念的代表符號：一個 scale-invariant 量 $b_n$ 具有正規性門檻 $b_n<b_{crit}\\Rightarrow\\mathrm{REG}$，因此假設性 blow-up 必迫使 $b_n\\ge b_{crit}$ 沿相關高尺度子序列成立，但這不蘊含任何 summability。此概念與 finite budget（$d_n$）形成本輪核心方法論對比，並在 §39 進一步強調 critical barrier coercivity 不等於 finite-budget cycle elimination。",
      "definition_en": "The representative symbol for the \"critical barrier\" notion in C6-H.4: a scale-invariant quantity $b_n$ carries a regularity threshold $b_n<b_{crit}\\Rightarrow\\mathrm{REG}$, so hypothetical blow-up forces $b_n\\ge b_{crit}$ along a relevant high-scale subsequence, with no summability implied. This concept is contrasted methodologically with the finite budget ($d_n$), and §39 further stresses that critical barrier coercivity is not equivalent to finite-budget cycle elimination.",
      "defining_relation": "b_n < b_{crit} \\Rightarrow \\mathrm{REG}",
      "notes": "Concretely instantiated by the Cheskidov–Dai quantity $\\mathfrak B_q^\\omega$ in §19."
    },
    {
      "id": "ns.c6.c6h.mathfrak_b_q_omega",
      "latex": "\\mathfrak B_q^\\omega",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "Cheskidov–Dai 頻殼渦量障礙量",
      "label_en": "Cheskidov–Dai shell-vorticity barrier quantity",
      "definition_zh": "依 Cheskidov–Dai frequency-localized regularity criterion schematically 定義的頻殼渦量積分 $\\mathfrak B_q^\\omega=\\int_{\\mathcal T_q}^{T^\\ast}\\|\\Delta_q\\omega(t)\\|_\\infty dt$，滿足 $\\limsup_{q\\to\\infty}\\mathfrak B_q^\\omega<c_\\nu\\Rightarrow\\mathrm{REG}$。它是 §19「critical-barrier coercivity」的典型模型，示範 critical barrier（而非 finite energy budget）才是適合 UV recurrence 的 debt 型態。",
      "definition_en": "The shell-vorticity integral $\\mathfrak B_q^\\omega=\\int_{\\mathcal T_q}^{T^\\ast}\\|\\Delta_q\\omega(t)\\|_\\infty dt$ schematically defined from the Cheskidov–Dai frequency-localized regularity criterion, satisfying $\\limsup_{q\\to\\infty}\\mathfrak B_q^\\omega<c_\\nu\\Rightarrow\\mathrm{REG}$. It is the canonical model of \"critical-barrier coercivity\" in §19, demonstrating that a critical barrier — not a finite energy budget — is the right kind of debt for UV recurrence.",
      "defining_relation": "\\mathfrak B_q^\\omega = \\int_{\\mathcal T_q}^{T^\\ast} \\|\\Delta_q\\omega(t)\\|_\\infty dt",
      "notes": "Its scaling behavior under dyadic rescaling ($\\lambda=2^m$, shell index shift $q\\mapsto q+m$) is examined in §40, showing it retains a fixed threshold across scales — unlike $D_E$."
    },
    {
      "id": "ns.c6.c6h.gamma_n_coherence",
      "latex": "\\Gamma_n",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "相干性係數",
      "label_en": "Coherence coordinate",
      "definition_zh": "一般相干性座標 $\\Gamma_n=R_n/C_n\\to0$，其中 $R_n$ 為已實現的 response/toll，$C_n$ 為可用的 source capacity；涵蓋 Duhamel coherence、operator positive-growth efficiency、local/far pressure coherence 等具體例子。它是定理 C6-H.5（Coherence Boundary Dichotomy）的中心對象，該定理證明取子序列後必有 $R_n\\to0$（即 $B_{LOAD}$）或 $C_n/R_n=\\Gamma_n^{-1}\\to\\infty$（即 $B_{CAP^\\infty}$）兩者之一成立。",
      "definition_en": "A generic coherence coordinate $\\Gamma_n=R_n/C_n\\to0$, where $R_n$ is the realized response/toll and $C_n$ the available source capacity; it covers concrete instances such as Duhamel coherence, operator positive-growth efficiency, and local/far pressure coherence. It is the central object of Theorem C6-H.5 (Coherence Boundary Dichotomy), which proves that after passing to a subsequence either $R_n\\to0$ (i.e. $B_{LOAD}$) or $C_n/R_n=\\Gamma_n^{-1}\\to\\infty$ (i.e. $B_{CAP^\\infty}$) must hold.",
      "defining_relation": "\\Gamma_n = \\frac{R_n}{C_n} \\to 0",
      "notes": "Structurally paralleled later by $R_M/C_M$ (§31, the MEAN-face analogue) and by $\\mathfrak K=C/R$ (§48, the generalized relative-capacity object)."
    },
    {
      "id": "ns.c6.c6h.b_coh",
      "latex": "B_{COH}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "相干性邊界面",
      "label_en": "Coherence boundary face",
      "definition_zh": "原始十元字母表中的一員，代表以 $\\Gamma_n=R_n/C_n\\to0$ 型態呈現的一般相干性座標崩潰的邊界。C6-H.5（Coherence Boundary Dichotomy）證明每一個此類崩潰子序列必然精煉為 $B_{LOAD}$ 或 $B_{CAP^\\infty}$ 之一，因此 $B_{COH}$ 本身在 §23 被移除，不再作為獨立終端物理節點，但其內部機制仍保留為有用的 metadata。",
      "definition_en": "A member of the original ten-face alphabet, representing collapse of a generic coherence coordinate of the form $\\Gamma_n=R_n/C_n\\to0$. The Coherence Boundary Dichotomy (C6-H.5) proves that every recurrent coherence-collapse subsequence refines to either $B_{LOAD}$ or $B_{CAP^\\infty}$; consequently $B_{COH}$ itself is removed in §23 as an independent terminal physical boundary node, though its internal mechanism remains useful metadata.",
      "defining_relation": "B_{COH} \\Longrightarrow B_{LOAD} \\vee B_{CAP^\\infty}",
      "notes": "The elimination route is certified as transition H-B1 in §42."
    },
    {
      "id": "ns.c6.c6h.vartheta_s",
      "latex": "\\vartheta(S)",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "中隙比",
      "label_en": "Middle-gap ratio",
      "definition_zh": "由 C5-E 定義的中隙比 $\\vartheta(S)=\\lambda_2^+\\lambda_3/|S|^2$（$S$ 為應變張量，$\\lambda_2^+,\\lambda_3$ 為其相關特徵值量），在 C6-H 中被重新引用作為 §24–§25「Middle-gap boundary」與定理 C6-H.6（Middle-Gap Boundary Dichotomy）的基礎座標，用以刻畫 $B_{GEOM}$ 中會被消去的 middle-gap 子面。",
      "definition_en": "The middle-gap ratio $\\vartheta(S)=\\lambda_2^+\\lambda_3/|S|^2$ ($S$ the strain tensor, $\\lambda_2^+,\\lambda_3$ its associated eigenvalue quantities), originally defined in C5-E and re-invoked in C6-H as the base coordinate for §24–§25 \"Middle-gap boundary\" and Theorem C6-H.6 (Middle-Gap Boundary Dichotomy), used to characterize the middle-gap subface of $B_{GEOM}$ that gets eliminated.",
      "defining_relation": "\\vartheta(S) = \\frac{\\lambda_2^+\\lambda_3}{|S|^2}",
      "notes": "Originates in round C5-E, not newly defined by C6-H; restated here as the foundation for the C6-H.6 dichotomy and the definition of $B_{GAP}$."
    },
    {
      "id": "ns.c6.c6h.m_delta",
      "latex": "M_\\delta",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "中隙負載積分",
      "label_en": "Middle-gap load integral",
      "definition_zh": "定義為 $M_\\delta=\\int_{\\{\\vartheta\\le\\delta\\}}\\lambda_2^+|S|^2dx$，即在中隙比低於門檻 $\\delta$ 的區域上之加權應變積分。搭配 C5-E 已證的不等式 $\\int_{\\{\\vartheta\\le\\delta\\}}|S|^3dx\\ge M_\\delta/(\\sqrt6\\,\\delta)$，是定理 C6-H.6（Middle-Gap Boundary Dichotomy）的核心量：沿 $\\delta_n\\downarrow0$ 取子序列後，$M_{\\delta_n}\\to0$（即 $B_{LOAD}$）或 $M_{\\delta_n}\\ge m_0>0$（迫使三次應變積分發散，即 $B_{CAP^\\infty}$）。",
      "definition_en": "Defined as $M_\\delta=\\int_{\\{\\vartheta\\le\\delta\\}}\\lambda_2^+|S|^2dx$, a weighted strain integral over the region where the middle-gap ratio falls below threshold $\\delta$. Combined with the C5-E inequality $\\int_{\\{\\vartheta\\le\\delta\\}}|S|^3dx\\ge M_\\delta/(\\sqrt6\\,\\delta)$, it is the central quantity of Theorem C6-H.6 (Middle-Gap Boundary Dichotomy): along a subsequence $\\delta_n\\downarrow0$, either $M_{\\delta_n}\\to0$ (i.e. $B_{LOAD}$) or $M_{\\delta_n}\\ge m_0>0$ (forcing the cubic strain integral to diverge, i.e. $B_{CAP^\\infty}$).",
      "defining_relation": "M_\\delta = \\int_{\\{\\vartheta\\le\\delta\\}} \\lambda_2^+ |S|^2dx",
      "notes": "The companion inequality driving the dichotomy, $\\int_{\\{\\vartheta\\le\\delta\\}}|S|^3dx \\ge \\frac{M_\\delta}{\\sqrt6\\,\\delta}$, was proved in C5-E."
    },
    {
      "id": "ns.c6.c6h.b_gap",
      "latex": "B_{GAP}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "中隙崩塌子面",
      "label_en": "Middle-gap collapse subface",
      "definition_zh": "$B_{GEOM}$ 之下的中隙崩塌子面，由定理 C6-H.6（Middle-Gap Boundary Dichotomy）證明必精煉為 $B_{LOAD}\\vee B_{CAP^\\infty}$（即 transition H-B2）。因此在最終物理邊界字母表中，$B_{GAP}$ 本身不再作為獨立終端節點，只有殘餘的 $B_{GEOM}^{res}$ 保留為 OPEN 物理面；分類表（§36）將其狀態記為「reducible geometry / REMOVED」。",
      "definition_en": "The middle-gap collapse subface within $B_{GEOM}$, proved by Theorem C6-H.6 (Middle-Gap Boundary Dichotomy) to refine to $B_{LOAD}\\vee B_{CAP^\\infty}$ (transition H-B2). Consequently $B_{GAP}$ itself is no longer an independent terminal node in the final physical boundary alphabet; only the residual $B_{GEOM}^{res}$ survives as an OPEN physical face. The classification table (§36) records its status as \"reducible geometry / REMOVED.\"",
      "defining_relation": "B_{GAP} \\Longrightarrow B_{LOAD} \\vee B_{CAP^\\infty}",
      "notes": "Not a member of the original ten-face $\\mathfrak B$ (which lists only $B_{GEOM}$); introduced in C6-H as the explicit gap-collapse component that $B_{GEOM}$ decomposes into."
    },
    {
      "id": "ns.c6.c6h.b_geom_res",
      "latex": "B_{GEOM}^{res}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "殘餘幾何面",
      "label_en": "Residual geometry face",
      "definition_zh": "§26 定義的殘餘幾何面，涵蓋 $B_{GEOM}$ 中扣除已被 C6-H.6 消去的中隙子面（$B_{GAP}$）後仍存留的物理幾何臨界性，包括 harmonic sign saturation、directional cone degeneration、axis-margin collapse、signature-induced geometric criticality。它取代原本的 $B_{GEOM}$ 進入最終六元字母表 $\\mathfrak B_{phys}^{(2)}$，狀態為 OPEN。",
      "definition_en": "The residual geometry face defined in §26: the physical geometric criticalities remaining within $B_{GEOM}$ after the middle-gap subface ($B_{GAP}$) is eliminated by C6-H.6, including harmonic sign saturation, directional cone degeneration, axis-margin collapse, and signature-induced geometric criticality. It replaces the original $B_{GEOM}$ in the final six-element alphabet $\\mathfrak B_{phys}^{(2)}$, with status OPEN.",
      "notes": "Conceptually $B_{GEOM} = B_{GAP} \\cup B_{GEOM}^{res}$ — only the gap subface is coercively removed; the residual is not shown to incur any uniform finite-energy cost."
    },
    {
      "id": "ns.c6.c6h.kappa_lambda_delta",
      "latex": "\\kappa_{\\lambda,\\delta}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "調和飽和下降係數",
      "label_en": "Descent coefficient (harmonic saturation)",
      "definition_zh": "C5-L 所證的下降係數 $\\kappa_{\\lambda,\\delta}=(1+\\lambda)\\delta-1>0$，用以說明當 $\\beta\\downarrow\\delta$ 自壞側逼近時，harmonic critical saturation 並非零成本。C6-H §27 引用此結果指出：即使 harmonic sign saturation 成立，它仍會導致 persistent derivative-order descent debt，但目前尚無已知的 globally finite all-order 總和，因此屬於 critical barrier/debt 而非 finite-budget elimination。",
      "definition_en": "The descent coefficient $\\kappa_{\\lambda,\\delta}=(1+\\lambda)\\delta-1>0$ proved in C5-L, showing that harmonic critical saturation is not zero-cost as $\\beta\\downarrow\\delta$ approaches from the bad side. C6-H §27 invokes this to argue that harmonic sign saturation still routes to a persistent derivative-order descent debt, for which no globally finite all-order sum is currently known — placing it in the critical-barrier/debt category rather than finite-budget elimination.",
      "defining_relation": "\\kappa_{\\lambda,\\delta} = (1+\\lambda)\\delta - 1 > 0",
      "notes": "Originates in round C5-L, not newly defined by C6-H; cited here to support §27's conclusion that harmonic saturation is not cost-free."
    },
    {
      "id": "ns.c6.c6h.b_seg",
      "latex": "B_{SEG}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "分離面",
      "label_en": "Segregation face",
      "definition_zh": "涵蓋 temporal phase segregation、spatial source segregation、target diffusion、core multiplicity、shared-source thickness collapse 等現象的物理面。儘管 C3/C5 提供了 active-worldvolume bounds、effective-volume multiplicity bounds 等工具，但因高頻權重隨尺度衰減，one-new-scale-per-generation 情境仍可存活，故 §28 指出 $B_{SEG}$ 未被 finite kinetic-energy budget coercively 消除，§29 進一步將其標明為 C6-H.3 標度障礙的直接實例，保留在最終字母表 $\\mathfrak B_{phys}^{(2)}$ 中，狀態 OPEN。",
      "definition_en": "The physical face covering temporal phase segregation, spatial source segregation, target diffusion, core multiplicity, and shared-source thickness collapse. Although C3/C5 supply active-worldvolume bounds and effective-volume multiplicity bounds among other tools, high-frequency weights decay with scale, so one-new-scale-per-generation scenarios survive; §28 shows $B_{SEG}$ is not coercively eliminated by the finite kinetic-energy budget, and §29 identifies this as a direct instance of the C6-H.3 scaling obstruction. It remains in the final alphabet $\\mathfrak B_{phys}^{(2)}$ with status OPEN.",
      "notes": "Classification table in §36 records its \"Current route\" as \"multiplicity/diffusion,\" type \"physical.\""
    },
    {
      "id": "ns.c6.c6h.b_mean",
      "latex": "B_{MEAN}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "平均旋轉面",
      "label_en": "Mean-rotation face",
      "definition_zh": "代表 coherent quadratic forcing 被 $M_\\chi'$（而非壓力）吸收的分支；大的瞬時 $|M_\\chi'|$ 可導致 mean-strain growth、rotation、oscillatory variation。§30 指出目前並無已知的、針對假設性奇異點附近 $M_\\chi$ 之 universal globally finite total-variation budget，因此 $B_{MEAN}$ 仍是物理邊界候選面，保留於最終六元字母表 $\\mathfrak B_{phys}^{(2)}$ 中，狀態 OPEN。",
      "definition_en": "Represents the branch where coherent quadratic forcing is absorbed by $M_\\chi'$ rather than by pressure; a large instantaneous $|M_\\chi'|$ can produce mean-strain growth, rotation, or oscillatory variation. §30 notes that no universal globally finite total-variation budget for $M_\\chi$ near a hypothetical singularity is currently known, so $B_{MEAN}$ remains a physical boundary candidate, retained in the final six-element alphabet $\\mathfrak B_{phys}^{(2)}$ with status OPEN.",
      "notes": "§31 develops a structural (not fully proved) coherence-type analogue for this face using $C_M, R_M$."
    },
    {
      "id": "ns.c6.c6h.m_chi",
      "latex": "M_\\chi, M_\\chi'",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "平均旋轉量與其變化率",
      "label_en": "Mean-rotation quantity and its rate",
      "definition_zh": "支撐 $B_{MEAN}$ 面的底層物理量 $M_\\chi$（近假設性奇異點的 mean/rotation 相關量），其瞬時變化率記為 $M_\\chi'$，coherent quadratic forcing 由此吸收而非由壓力吸收。本檔案未重述其完整定義式，視為承接自先前 C6 討論的既有記號；§31 以 $M_\\chi'$ 的時間積分建構 mean-variation capacity $C_M$。",
      "definition_en": "The underlying physical quantity $M_\\chi$ supporting the $B_{MEAN}$ face (a mean/rotation-related quantity near a hypothetical singularity), whose instantaneous rate is denoted $M_\\chi'$; coherent quadratic forcing is absorbed by this rate rather than by pressure. This file does not restate its full defining formula, treating it as notation inherited from earlier C6 discussion; §31 builds the mean-variation capacity $C_M$ from the time integral of $M_\\chi'$.",
      "notes": "Defining formula not given in this file; presumed defined in an earlier C6 round. Do not confuse $M_\\chi$ (mean-rotation state) with $M_\\delta$ (middle-gap load integral, §24) — unrelated despite the similar letter M."
    },
    {
      "id": "ns.c6.c6h.c_m_r_m",
      "latex": "C_M, R_M",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "平均變化容量與已實現淨變化",
      "label_en": "Mean-variation capacity and realized net change",
      "definition_zh": "§31 定義的一對量：正的 mean-variation capacity $C_M=\\int|M_\\chi'|dt$，與已實現的淨平均變化 $R_M=|M_\\chi(t_1)-M_\\chi(t_0)|$，滿足 $R_M\\le C_M$。低效率 $R_M/C_M\\to0$ 會導出「net-load collapse 或 variation-capacity inflation」，結構上類比於 $B_{COH}$ 的 $\\Gamma_n$ 機制，但文中明確指出這僅是結構類比，並非已完成的消去定理（不提供 globally finite capacity bound）。",
      "definition_en": "A pair of quantities defined in §31: the positive mean-variation capacity $C_M=\\int|M_\\chi'|dt$ and the realized net mean change $R_M=|M_\\chi(t_1)-M_\\chi(t_0)|$, satisfying $R_M\\le C_M$. Low efficiency $R_M/C_M\\to0$ yields \"net-load collapse $\\vee$ variation-capacity inflation,\" structurally analogous to the $\\Gamma_n$ mechanism of $B_{COH}$ — but the text explicitly flags this as only a structural analogue, not a completed elimination theorem (it supplies no globally finite capacity bound).",
      "defining_relation": "C_M = \\int|M_\\chi'|dt, \\quad R_M = |M_\\chi(t_1)-M_\\chi(t_0)|, \\quad R_M\\le C_M",
      "notes": "Explicitly NOT a proved dichotomy like C6-H.5; the text calls it \"a structural analogue of $B_{COH}$, not a completed elimination theorem.\""
    },
    {
      "id": "ns.c6.c6h.b_prov",
      "latex": "B_{PROV}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "壓力／溯源面",
      "label_en": "Pressure/provenance face",
      "definition_zh": "涵蓋 local-pressure takeover、far-pressure heredity loss、signature boundary（$\\det F\\to0$）、pressure-source fragmentation、local/far cancellation 等現象的物理面。§32 指出雖然某些壓力區制已被已發表的 pressure/intermittency 條件外部地 regularity-killed，但目前程序中並無定理保證每一條通往 $B_{PROV}$ 的路徑都會進入那些有利壓力區制，因此 $B_{PROV}$ 仍是物理邊界候選面，保留於最終字母表中，狀態 OPEN。",
      "definition_en": "The physical face covering local-pressure takeover, far-pressure heredity loss, the signature boundary ($\\det F\\to0$), pressure-source fragmentation, and local/far cancellation. §32 notes that although some pressure regimes are externally regularity-killed by published pressure/intermittency conditions, no theorem in the current program guarantees that every approach to $B_{PROV}$ enters those favorable regimes; hence $B_{PROV}$ remains a physical boundary candidate, retained in the final alphabet with status OPEN.",
      "notes": "The signature quantity $\\det F$ appearing in this face's description is not itself redefined in this file (inherited notation for a provenance/signature tensor $F$ from earlier rounds)."
    },
    {
      "id": "ns.c6.c6h.mathfrak_b_phys_2",
      "latex": "\\mathfrak B_{phys}^{(2)}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "第二次約簡後之物理邊界字母表（六元）",
      "label_en": "Second reduced physical boundary alphabet (six-element)",
      "definition_zh": "在邊界商化、setup 商化、coherence 二分法（C6-H.5）、middle-gap 二分法（C6-H.6）四項結果綜合之下得到的六元集合，是本輪 C6-H 的主要邊界狀態壓縮結果，也是 C6-H.8（Boundary SCC Audit）的分析對象——該六個物理邊界類之間目前無任何已 certified 的非平凡 directed cycle。",
      "definition_en": "The six-element set obtained by combining the edge-boundary quotient, setup quotient, coherence dichotomy (C6-H.5), and middle-gap dichotomy (C6-H.6); this is the main boundary-state compression result of C6-H and the object analyzed by C6-H.8 (Boundary SCC Audit), which finds no certified nontrivial closed directed cycle among these six physical boundary classes.",
      "defining_relation": "\\mathfrak B_{phys}^{(2)} = \\{ B_{LOAD}, B_{SEG}, B_{GEOM}^{res}, B_{MEAN}, B_{PROV}, B_{CAP^\\infty} \\}",
      "notes": "Restated with identical membership as $\\mathfrak B_{phys}^{C6H}$ in §57's formal ETN summary — the two notations denote the same six-element set."
    },
    {
      "id": "ns.c6.c6h.mathfrak_k",
      "latex": "\\mathfrak K",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "相對容量",
      "label_en": "Relative capacity",
      "definition_zh": "§48 對一般事件定義的相對容量 $\\mathfrak K=C/R\\ge1$，其中 $R>0$ 為已實現的 toll，$C$ 為 source/variation capacity；coherence collapse 即對應 $\\mathfrak K\\to\\infty$。此符號統一了 $\\Gamma_n^{-1}$（§22）與 $C_M/R_M$（§31）的模式，作為「realized-load / required-capacity duality」（§47）的正式表述，並被列為下一階段 C6-I 應探究 $\\mathfrak K$ 能否在與 blow-up 及有限能量相容的情況下無限發散的關鍵物件。",
      "definition_en": "The relative capacity $\\mathfrak K=C/R\\ge1$ defined in §48 for a generic event, where $R>0$ is the realized toll and $C$ the source/variation capacity; coherence collapse corresponds to $\\mathfrak K\\to\\infty$. This symbol unifies the pattern seen in $\\Gamma_n^{-1}$ (§22) and $C_M/R_M$ (§31) as the formal expression of the \"realized-load / required-capacity duality\" (§47), and is flagged as a key object for the planned next round C6-I to determine whether $\\mathfrak K$ can diverge indefinitely while remaining compatible with blow-up and finite energy.",
      "defining_relation": "\\mathfrak K = \\frac{C}{R} \\ge 1",
      "notes": "Forward-looking object; explicitly proposed in §48 as an open question for the next paper, C6-I."
    },
    {
      "id": "ns.c6.c6h.b_crit_vector",
      "latex": "\\mathbf B^{crit}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "臨界障礙向量",
      "label_en": "Critical barrier vector",
      "definition_zh": "§49 定義的一般臨界障礙向量 $\\mathbf B^{crit}=(B_\\omega,B_{\\rm harm},B_{\\rm middle},B_{\\rm op},B_{\\rm press},B_{\\rm chain})$，六個座標依序代表 frequency-localized vorticity toll、harmonic/sign geometry status、middle-eigenvalue critical toll、strain-vorticity operator toll、pressure criticality、derivative-chain root/clock geometry。與能量不同，這些座標具有 scale-critical 或 theorem-threshold 意義，是 §50「boundary recurrence should be measured in critical coordinates」方法論主張的具體物件，並構成 §57 中 $\\Theta_B^{C6H}$ 狀態元組的一個分量。",
      "definition_en": "The generic critical barrier vector $\\mathbf B^{crit}=(B_\\omega,B_{\\rm harm},B_{\\rm middle},B_{\\rm op},B_{\\rm press},B_{\\rm chain})$ defined in §49, whose six coordinates represent, in order, the frequency-localized vorticity toll, harmonic/sign geometry status, middle-eigenvalue critical toll, strain-vorticity operator toll, pressure criticality, and derivative-chain root/clock geometry. Unlike energy, these coordinates carry scale-critical or theorem-threshold meaning; it is the concrete object behind the §50 methodological claim that \"boundary recurrence should be measured in critical coordinates,\" and forms one component of the $\\Theta_B^{C6H}$ state tuple in §57.",
      "defining_relation": "\\mathbf B^{crit} = \\left( B_\\omega, B_{\\rm harm}, B_{\\rm middle}, B_{\\rm op}, B_{\\rm press}, B_{\\rm chain} \\right)",
      "notes": "Forward-looking formalization proposed for the planned next round, C6-I (see §54, proof obligation I4)."
    },
    {
      "id": "ns.c6.c6h.b_ast",
      "latex": "B_\\ast",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "邊界飽和佔位符（承自 C6-G）",
      "label_en": "Boundary-saturated placeholder (inherited from C6-G)",
      "definition_zh": "承接自 C6-G 的 minimal survivor frontier 框架 $GP_{\\rm uniform}\\vee HF_{\\rm uniform}\\vee B_\\ast\\vee A$ 中的抽象佔位符，代表任一未具體化的邊界飽和情形。§51 中 C6-H 將其正式精煉（refine）為六元具體集合 $B_\\ast\\in\\{LOAD,SEG,GEOM^{res},MEAN,PROV,CAP^\\infty\\}$，使 boundary-saturated survivor alphabet 由十個粗粒化超類縮至六個物理面，而 $GP_{\\rm uniform}$ 與 $HF_{\\rm uniform}$ 則未被本輪觸及。",
      "definition_en": "The abstract placeholder inherited from C6-G's minimal-survivor-frontier schema $GP_{\\rm uniform}\\vee HF_{\\rm uniform}\\vee B_\\ast\\vee A$, representing any unspecified boundary-saturated case. In §51, C6-H formally refines it into the concrete six-element set $B_\\ast\\in\\{LOAD,SEG,GEOM^{res},MEAN,PROV,CAP^\\infty\\}$, shrinking the boundary-saturated survivor alphabet from ten coarse-grained superclasses to six physical faces; $GP_{\\rm uniform}$ and $HF_{\\rm uniform}$ are left untouched by this round.",
      "defining_relation": "B_\\ast \\in \\{ LOAD, SEG, GEOM^{res}, MEAN, PROV, CAP^\\infty \\}",
      "notes": "The symbol itself originates in C6-G; only its refinement into a concrete six-element range is new to C6-H."
    },
    {
      "id": "ns.c6.c6h.theta_b_c6h",
      "latex": "\\Theta_B^{C6H}",
      "series": "NS",
      "first_appearance": "C6-H",
      "label_zh": "C6-H 邊界狀態元組（ETN 更新）",
      "label_en": "C6-H boundary state tuple (ETN update)",
      "definition_zh": "§57「True ETN update」定義的邊界狀態八元組 $\\Theta_B^{C6H}=\\langle$ boundary type, physical face, scaling degree, absolute load, critical barrier vector, capacity ratio, edge metadata, kill gates $\\rangle$，是本輪對邊界節點所需追蹤資訊的正式彙整，取代原先僅記錄 kinetic energy cost 的作法，呼應 §50 的方法論主張，為 C6-I 之後的 critical-coordinate 邊界圖分析提供資料結構基礎。",
      "definition_en": "The eight-component boundary state tuple $\\Theta_B^{C6H}=\\langle$ boundary type, physical face, scaling degree, absolute load, critical barrier vector, capacity ratio, edge metadata, kill gates $\\rangle$ defined in §57 (\"True ETN update\"). It formally consolidates the information that must be tracked at each boundary node, replacing the earlier practice of recording only the kinetic-energy cost, echoes the §50 methodological claim, and provides the data-structure basis for the critical-coordinate boundary-graph analysis planned for C6-I.",
      "defining_relation": "\\Theta_B^{C6H} = \\left\\langle \\text{boundary type}, \\text{physical face}, \\text{scaling degree}, \\text{absolute load}, \\text{critical barrier vector}, \\text{capacity ratio}, \\text{edge metadata}, \\text{kill gates} \\right\\rangle",
      "notes": "\"ETN\" is not expanded within this file; treated here as terminology inherited from the broader C6 framework. §57 also restates the reduced alphabet as $\\mathfrak B_{phys}^{C6H}$, identical in membership to $\\mathfrak B_{phys}^{(2)}$."
    },
    {
      "id": "ns.c6.c6i.ns_scaling_transformation",
      "latex": "u_\\lambda(x,t)=\\lambda u(\\lambda x,\\lambda^2t), \\qquad p_\\lambda(x,t)=\\lambda^2p(\\lambda x,\\lambda^2t)",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "N–S 縮放變換",
      "label_en": "N–S scaling transformation",
      "definition_zh": "標準 Navier–Stokes 尺度變換：速度依 $\\lambda$ 縮放、壓力依 $\\lambda^2$ 縮放，是本輪一切 scaling degree 計算的基準。",
      "definition_en": "The standard Navier–Stokes rescaling, with velocity scaling by $\\lambda$ and pressure by $\\lambda^2$; the baseline against which every scaling-degree computation in this round is taken."
    },
    {
      "id": "ns.c6.c6i.scaling_degree",
      "latex": "a_F \\ (\\text{pointwise}), \\qquad d_Q \\ (\\text{event})",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "逐點／事件縮放度",
      "label_en": "pointwise / event scaling degree",
      "definition_zh": "本輪縮放代數的基本記號：$a_F$ 為逐點場的縮放度，$d_Q$ 為積分型事件量的縮放度，兩者是後續所有 criticalization 的輸入。",
      "definition_en": "The round's basic scaling notation: $a_F$ is the pointwise field degree and $d_Q$ the integrated event-quantity degree, both feeding every later criticalization.",
      "defining_relation": "F_\\lambda = \\lambda^{a_F}F(\\lambda x,\\lambda^2t); \\qquad Q[u_\\lambda;E_\\lambda] = \\lambda^{d_Q}Q[u;E]"
    },
    {
      "id": "ns.c6.c6i.criticalization_operator",
      "latex": "\\mathscr C_r[Q] = r^{d_Q}Q",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界化算子",
      "label_en": "Criticalization Operator",
      "definition_zh": "C6-I.1，本輪核心算子：以事件空間尺度 $r$ 對度 $d_Q$ 的事件量做冪次修正，使其在 $\\lambda$-縮放下不變；等價記法 $Q^{crit}=r^{d_Q}Q$ 貫穿全文。",
      "definition_en": "C6-I.1, the round's central operator: rescales an event quantity of degree $d_Q$ by spatial scale $r$ so the result is invariant under $\\lambda$-rescaling; written equivalently as $Q^{crit}=r^{d_Q}Q$ throughout the paper."
    },
    {
      "id": "ns.c6.c6i.event_aspect_ratio",
      "latex": "\\theta_J = \\frac{|J|}{r^2}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "事件長寬比",
      "label_en": "event aspect ratio",
      "definition_zh": "時間窗 $J$ 與空間尺度 $r$ 的比值，本身縮放不變，用於把時空負載換算為同時刻核心負載。",
      "definition_en": "The ratio of time window $J$ to $r^2$; itself scale invariant, used to convert spacetime load into same-time core load."
    },
    {
      "id": "ns.c6.c6i.critical_middle_event_mass",
      "latex": "\\mathfrak M_J^{crit} = rM_J",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界中層事件質量",
      "label_en": "critical middle event mass",
      "definition_zh": "C6-F 中層時空質量 $M_J$（度 $1$）的臨界化版本，是本輪標舉的關鍵結果之一，縮放不變。",
      "definition_en": "The scale-invariant criticalization of the C6-F middle spacetime mass $M_J$ (degree $1$); one of the round's headline results.",
      "defining_relation": "M_J=\\int_J\\int\\lambda_2^+|S|^2\\,dxdt\\ \\ (d_{M_J}=1), \\qquad \\mathfrak M_J^{crit}=rM_J"
    },
    {
      "id": "ns.c6.c6i.critical_operator_event_mass",
      "latex": "\\mathfrak P_J^{crit} = r^3P_J",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界算子事件質量",
      "label_en": "critical operator event mass",
      "definition_zh": "C6-F 正算子事件質量 $P_J$（度 $3$）的臨界化版本，同樣是本輪標舉的關鍵結果，縮放不變。",
      "definition_en": "The scale-invariant criticalization of the C6-F positive operator event mass $P_J$ (degree $3$); likewise one of the round's headline results.",
      "defining_relation": "P_J=\\int_J[E_1'(t)]_+\\,dt\\ \\ (d_{P_J}=3), \\qquad \\mathfrak P_J^{crit}=r^3P_J"
    },
    {
      "id": "ns.c6.c6i.critical_shared_core_junction",
      "latex": "\\mathfrak m_\\ast^{crit} \\ge \\frac{\\Omega_{ST}q_0}{\\theta_J}\\mathfrak M_J^{crit}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界共享核心接合定理",
      "label_en": "Critical Shared-Core Junction Theorem",
      "definition_zh": "C6-I.2：將 C6-F 的同時刻中層核心負載下界不等式乘上 $r^3$ 並以 $\\theta_J$ 改寫後得到的完全縮放不變版本。",
      "definition_en": "C6-I.2: the fully scale-invariant rewriting of C6-F's same-time middle core-load lower bound, obtained by multiplying through by $r^3$ and expressing it via $\\theta_J$."
    },
    {
      "id": "ns.c6.c6i.critical_operator_derivative_junction",
      "latex": "\\mathfrak Q_{E_\\ast}^{crit}\\,\\mathfrak D_{E_\\ast}^{crit} \\ge \\frac{\\Omega_{ST}q_0}{\\theta_J}\\mathfrak P_J^{crit}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界算子—導數接合定理",
      "label_en": "Critical Operator–Derivative Junction",
      "definition_zh": "C6-I.3：以臨界化的算子範數與二階導數範數乘積下界 $\\mathfrak P_J^{crit}$，證明 C6-F 的算子/高階導數橋接在臨界尺度下無縮放損失。",
      "definition_en": "C6-I.3: bounds the product of criticalized operator and second-derivative norms below by $\\mathfrak P_J^{crit}$, showing the C6-F operator/high-derivative bridge survives criticalization with no scaling loss.",
      "defining_relation": "\\mathfrak Q_E^{crit}=r^{5/2}\\|\\mathcal Q_{SV}\\|_{L^2(E)}, \\qquad \\mathfrak D_E^{crit}=r^{5/2}\\|\\Delta S\\|_{L^2(E)}"
    },
    {
      "id": "ns.c6.c6i.miller_operator_criticality_identity",
      "latex": "d_{\\int_0^{T^\\ast}\\left(\\|\\mathcal Q_{SV}\\|_{\\dot H^\\alpha}/\\|S\\|_{\\dot H^1}\\right)^{2/(1+\\alpha)}dt} = 0",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "Miller 算子臨界性恆等式",
      "label_en": "Miller Operator Criticality Identity",
      "definition_zh": "C6-I.4：證明 Miller operator blow-up 判準積分（$p=2/(1+\\alpha)$）的縮放度恆為 $0$，故此判準本身即是道地的臨界 ledger 成員。",
      "definition_en": "C6-I.4: proves the Miller operator blow-up criterion integral (with $p=2/(1+\\alpha)$) has scaling degree exactly $0$, so it belongs natively to the critical ledger."
    },
    {
      "id": "ns.c6.c6i.critical_ledger_vector",
      "latex": "\\mathbf L_E^{crit} = \\left(E_{CKN}, B_\\omega, B_{\\rm middle}, B_{\\rm op}, B_{\\rm pressure}, \\mathfrak M_J^{crit}, \\mathfrak P_J^{crit}, \\mathfrak C_S^{crit}, \\mathfrak Q^{crit}, \\mathfrak D^{crit}, \\widehat C^{Duh}, \\widehat{\\mathcal R}_k, \\widehat\\tau_k, \\Gamma, \\Omega, \\ldots\\right)",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界帳本向量",
      "label_en": "Critical Ledger Vector",
      "definition_zh": "C6-I.5：把所有主要 C6 量統一收進同一個在 N–S 縮放下彼此可比較的向量，取代單一 scalar cycle currency，是本輪核心產物。",
      "definition_en": "C6-I.5: collects all the main C6 quantities into one vector whose coordinates are mutually comparable under N–S scaling, replacing a single scalar cycle currency — the round's central product."
    },
    {
      "id": "ns.c6.c6i.critical_capacity_infinity_boundary",
      "latex": "B_{CAP^{crit,\\infty}}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界無窮容量邊界",
      "label_en": "critical capacity-at-infinity boundary",
      "definition_zh": "C6-I.6（Raw-Infinity No-Go）證明正度量原始容量 $C\\to\\infty$ 只是 UV 縮放假象，並非真正物理邊界；取代 C6-H 的 $B_{CAP^\\infty}$，成為正式修正後的容量邊界。",
      "definition_en": "Introduced by C6-I.6 (Raw-Infinity No-Go), which shows raw divergence of a positive-degree capacity is merely UV rescaling, not a genuine physical boundary; replaces C6-H's raw $B_{CAP^\\infty}$ as the corrected capacity boundary.",
      "defining_relation": "r^{d_C}C\\to\\infty \\quad\\text{or}\\quad C/R\\to\\infty\\ \\ (d_C=d_R)",
      "notes": "Directly supersedes C6-H's uncriticalized $B_{CAP^\\infty}$."
    },
    {
      "id": "ns.c6.c6i.critical_middle_gap_dichotomy",
      "latex": "\\mathfrak C_{\\delta}^{S} \\ge \\frac{\\mathfrak M_{\\delta}}{\\sqrt6\\,\\delta}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界中層缺口二分定理",
      "label_en": "Critical Middle-Gap Dichotomy",
      "definition_zh": "C6-I.7：criticalize 後的 C5-E 中層缺口不等式；當 $\\delta_n\\to0$ 時，導出臨界負載塌陷（I-GAP-L）或臨界立方項發散為真正 $CAP^{crit,\\infty}$（I-GAP-C）的二分。",
      "definition_en": "C6-I.7: the criticalized C5-E middle-gap inequality; as $\\delta_n\\to0$ it forces a dichotomy between critical load collapse (I-GAP-L) and the critical cubic term diverging to genuine $CAP^{crit,\\infty}$ (I-GAP-C)."
    },
    {
      "id": "ns.c6.c6i.critical_load_boundary",
      "latex": "B_{LOAD^{crit}}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界負載邊界",
      "label_en": "critical load boundary",
      "definition_zh": "對度 $d_R$ 的事件量定義已實現臨界負載 $R^{crit}=r^{d_R}R$，邊界表示 $R_n^{crit}\\to0$，取代 C6-H 未臨界化的 $LOAD$ 終端。",
      "definition_en": "For an event quantity of degree $d_R$, the realized critical load is $R^{crit}=r^{d_R}R$, and the boundary means $R_n^{crit}\\to0$; replaces C6-H's uncriticalized $LOAD$ terminal.",
      "defining_relation": "R^{crit} = r^{d_R}R; \\qquad B_{LOAD^{crit}}:\\ R_n^{crit}\\to0"
    },
    {
      "id": "ns.c6.c6i.critical_boundary_alphabet",
      "latex": "\\mathfrak B_{crit} = \\{LOAD^{crit},\\, SEG,\\, GEOM^{res},\\, MEAN,\\, PROV,\\, CAP^{crit,\\infty}\\}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界邊界字母表",
      "label_en": "critical boundary alphabet",
      "definition_zh": "修正 C6-H 邊界字母表後得到的物理邊界集合，將 $LOAD$、$CAP^\\infty$ 替換為臨界化版本，使全部座標皆縮放一致。",
      "definition_en": "The corrected C6-H boundary alphabet, with $LOAD$ and $CAP^\\infty$ replaced by their criticalized forms so every member has a scale-consistent interpretation."
    },
    {
      "id": "ns.c6.c6i.geometric_scale_ladder",
      "latex": "r_n = r_0a^{-n}, \\qquad a>1",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "幾何尺度階梯",
      "label_en": "geometric scale ladder",
      "definition_zh": "事件空間尺度以幾何速率遞減的序列，用來構造無限 UV 事件序列同時滿足有限總物理時間與有限總能量代價的架構。",
      "definition_en": "A sequence of event spatial scales decaying geometrically, used to construct an infinite UV event sequence compatible with finite total physical time and finite total energy cost."
    },
    {
      "id": "ns.c6.c6i.critical_zeno_compatibility_lemma",
      "latex": "\\textbf{fixed nonzero critical toll per scale} \\not\\Rightarrow \\textbf{finite-time contradiction}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界 Zeno 相容性引理",
      "label_en": "Critical Zeno Compatibility Lemma",
      "definition_zh": "C6-I.8：在幾何尺度階梯上，每代固定正的無因次臨界代價可與有限總物理時間及有限總原始能量代價同時成立；純屬縮放層次 no-go，不構造真實奇異解。",
      "definition_en": "C6-I.8: on a geometric scale ladder, a fixed positive dimensionless critical toll per generation is compatible with finite total physical time and finite total raw energy cost; a purely scaling-level no-go, not a construction of an actual singular solution.",
      "defining_relation": "D_n^{crit}=r_n^{-1}D_n\\ge d_0>0 \\ \\Rightarrow\\ D_n\\ge d_0r_n, \\qquad \\sum_n r_n<\\infty"
    },
    {
      "id": "ns.c6.c6i.log_scale_variable",
      "latex": "s = -\\log r",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "對數尺度變數",
      "label_en": "log-scale variable",
      "definition_zh": "把 UV 極限 $r\\downarrow0$ 轉為 $s\\to+\\infty$，使幾何尺度階梯化為等差數列，讓跨代 recurrence 問題成為對數尺度時間中的動力系統。",
      "definition_en": "Turns the UV limit $r\\downarrow0$ into $s\\to+\\infty$, converting the geometric scale ladder into an arithmetic progression and recasting cross-generation recurrence as a dynamical system in logarithmic scale time."
    },
    {
      "id": "ns.c6.c6i.renormalized_cycle_reframing",
      "latex": "\\widehat\\Theta(s_n)",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "重整化循環重構",
      "label_en": "Renormalized-Cycle Reframing",
      "definition_zh": "C6-I.9：主張無限 UV 循環 $E_1,E_2,\\ldots$（$r_n\\downarrow0$）應表示為對數尺度軌道 $\\widehat\\Theta(s_n)$，把 recurrence 問題改寫為此軌道是否有不動點、週期軌道或向 REG 邊界漂移。",
      "definition_en": "C6-I.9: proposes representing an infinite UV cycle $E_1,E_2,\\ldots$ ($r_n\\downarrow0$) as a log-scale orbit $\\widehat\\Theta(s_n)$, reframing recurrence as whether this orbit has a fixed point, a periodic orbit, or drifts to a REG boundary.",
      "defining_relation": "E_1,E_2,\\ldots\\ (r_n\\downarrow0) \\ \\longmapsto\\ \\widehat\\Theta(s_n)"
    },
    {
      "id": "ns.c6.c6i.barrier_zeno_no_go",
      "latex": "\\textbf{a successful C6 cycle proof needs cross-scale structure, not merely per-scale critical non-smallness}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "障壁—Zeno 不可能定理",
      "label_en": "Barrier-Zeno No-Go",
      "definition_zh": "C6-I.10：僅由每代固定正臨界障壁代價、有限總動能、有限剩餘時間三者，無法單獨導出矛盾；成功的 C6 循環證明需要額外的跨尺度結構。",
      "definition_en": "C6-I.10: no contradiction follows solely from a fixed positive critical barrier toll per generation, finite global kinetic energy, and finite remaining time; a successful C6 cycle proof needs additional cross-scale structure.",
      "defining_relation": "\\sum_n|J_n|<\\infty, \\qquad \\sum_nD_n<\\infty \\ \\ \\text{while criticalized descriptors remain } O(1)"
    },
    {
      "id": "ns.c6.c6i.load_capacity_compactification",
      "latex": "\\widehat L,\\ \\widehat{\\mathfrak K}",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "負載—容量緊化",
      "label_en": "load–capacity compactification",
      "definition_zh": "將臨界負載 $L^{crit}$ 與臨界相對容量 $\\mathfrak K^{crit}$ 各以 $x/(1+x)$ 映入緊區間，使 $LOAD^{crit}$ 對應 $\\widehat L=0$、$CAP^{crit,\\infty}$ 對應 $\\widehat{\\mathfrak K}=1$。",
      "definition_en": "Maps the critical load $L^{crit}$ and critical relative capacity $\\mathfrak K^{crit}$ via $x/(1+x)$ into compact intervals, so $LOAD^{crit}$ is $\\widehat L=0$ and $CAP^{crit,\\infty}$ is $\\widehat{\\mathfrak K}=1$.",
      "defining_relation": "\\widehat L = \\frac{L^{crit}}{1+L^{crit}}, \\qquad \\widehat{\\mathfrak K} = \\frac{\\mathfrak K^{crit}}{1+\\mathfrak K^{crit}}"
    },
    {
      "id": "ns.c6.c6i.critical_event_state",
      "latex": "\\Theta_{crit}^{C6I} = \\left\\langle r,\\theta_J,\\mathbf L^{crit},\\mathbf B^{crit},\\widehat L,\\widehat{\\mathfrak K},\\text{joint node},\\text{boundary face},\\text{barrier polarity},\\text{provenance}\\right\\rangle",
      "series": "NS",
      "first_appearance": "C6-I",
      "label_zh": "臨界事件狀態（C6-I ETN）",
      "label_en": "critical event state (C6-I ETN update)",
      "definition_zh": "本輪對 Event-Type-Node 的最終更新，把尺度、長寬比、臨界帳本／障壁向量、緊化負載與容量、障壁極性等打包成單一狀態，其對數尺度版本為 $\\widehat\\Theta(s)$。",
      "definition_en": "The round's final Event-Type-Node update, packaging scale, aspect ratio, the critical ledger/barrier vectors, compactified load and capacity, and barrier polarity into one state, with log-scale-time version $\\widehat\\Theta(s)$.",
      "defining_relation": "\\widehat\\Theta(s) = \\Theta_{crit}\\left(r=e^{-s}\\right)"
    },
    {
      "id": "ns.c6.c6j.tau",
      "latex": "\\tau",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "剩餘時間",
      "label_en": "remaining time",
      "definition_zh": "τ = T*−t，即假設有限時間爆破時刻 T* 與當前物理時間 t 之間的剩餘時間差，是建立 backward Leray 重整化座標系統的第一步。它把物理時間趨近奇異時刻的極限 t↑T* 轉換為 τ→0+，為定義 backward log-scale 時間 s=-log τ 提供基礎變量。",
      "definition_en": "τ = T*−t is the remaining physical time to a hypothetical finite-time blow-up instant T*, the first step in constructing the backward-Leray renormalized coordinate system. It converts the approach t↑T* into τ→0+ and underlies the subsequent definition of backward log-scale time s = -log τ.",
      "defining_relation": "\\tau=T^\\ast-t",
      "notes": "T* and t are pre-existing physical blow-up-analysis notation; τ is the auxiliary variable C6-J introduces to streamline the rescaling."
    },
    {
      "id": "ns.c6.c6j.s_backward_log_time",
      "latex": "s",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "反向對數尺度時間",
      "label_en": "backward logarithmic scale-time",
      "definition_zh": "s=-log τ=-log(T*-t)，本輪正式採用的 backward parabolic/Leray log-scale 時間變量，把假設有限時間爆破的終點 t↑T* 映成 s→+∞。C6-J 在此變量下證明重整化 N–S 方程（C6-J.1）為 autonomous，s 從而作為對應動力系統的演化時間貫穿全篇（fixed point、periodic orbit、各能量恒等式皆以 s 為自變量）。",
      "definition_en": "s=-log τ=-log(T*-t) is the backward parabolic/Leray log-scale time formally adopted here, mapping the hypothetical blow-up endpoint t↑T* to s→+∞. Under this variable C6-J proves the renormalized N–S equation (C6-J.1) is autonomous, and s serves as the evolution time of the resulting dynamical system throughout (fixed points, periodic orbits, and the energy identities are all posed in s).",
      "defining_relation": "s=-\\log\\tau",
      "notes": "C6-I's scale-time was s_r=-log r; C6-J shows s=2s_r, so the two differ only by a factor of 2. The §0 preamble loosely writes the upgrade as 's=-log r' before the rigorous §1 definition via τ supersedes it."
    },
    {
      "id": "ns.c6.c6j.r_of_s",
      "latex": "r(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "拋物尺度",
      "label_en": "parabolic scale",
      "definition_zh": "r(s)=√τ=e^{-s/2}，backward Leray 重整化所用的空間/速度伸縮因子，將物理座標 (x,u,p) 映為 renormalized 座標 (y,U,P)。它同時是聯繫 renormalized 與 physical 量的橋樑：e^{-s/2}=r 正是物理能量 telescoping 恆等式所帶的 subcritical weight（§24–25），並重現 C6-I 的 Zeno 求和條件。",
      "definition_en": "r(s)=√τ=e^{-s/2} is the spatial/velocity rescaling factor of the backward Leray renormalization, mapping physical variables (x,u,p) to renormalized ones (y,U,P). It also bridges renormalized and physical quantities: e^{-s/2}=r is exactly the subcritical weight carried by the physical-energy telescoping identity (§24-25), reproducing C6-I's geometric Zeno summability.",
      "defining_relation": "r(s)=\\sqrt{\\tau}=e^{-s/2}",
      "notes": "Same letter r as C6-I's discrete geometric-ladder radius r_n=r_0 a^{-n}; r(s) is its continuous scale-time analogue."
    },
    {
      "id": "ns.c6.c6j.y_self_similar_variable",
      "latex": "y",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "自相似空間變量",
      "label_en": "self-similar spatial variable",
      "definition_zh": "y=(x-x*)/√(T*-t)=(x-x*)/r(s)，以假設奇異中心 x* 為原點、以 r(s) 為尺度無量綱化的空間座標。它是把物理速度場 u(x,t) 轉換成 renormalized 場 U(y,s) 的座標基礎，並在 backward Leray 方程（C6-J.1）的伸縮/對流項 (y·∇)U 中扮演核心角色。",
      "definition_en": "y=(x-x*)/√(T*-t)=(x-x*)/r(s) is the spatial coordinate recentered at the hypothetical singular center x* and nondimensionalized by r(s). It underlies the passage from the physical velocity field u(x,t) to the renormalized field U(y,s), and appears centrally in the dilation/convection term (y·∇)U of the backward Leray equation (C6-J.1).",
      "defining_relation": "y=\\frac{x-x^\\ast}{\\sqrt{T^\\ast-t}}=\\frac{x-x^\\ast}{r(s)}"
    },
    {
      "id": "ns.c6.c6j.u_renormalized_velocity",
      "latex": "U(y,s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化速度場",
      "label_en": "renormalized velocity field",
      "definition_zh": "U(y,s)=√(T*-t)·u(x,t)=r(s)u(x,t)，本輪 backward Leray 重整化流的中心對象。在 s 時間下 U 滿足 autonomous 方程（C6-J.1）；U 的 fixed point 對應 backward self-similar blow-up profile，periodic orbit 對應 backward discretely self-similar（DSS）profile。其臨界範數 ‖U‖_{L^3}、‖U‖_{\\dot H^{1/2}} 在重整化下不變，且假設爆破要求兩者同時發散（C6-J.2）。",
      "definition_en": "U(y,s)=√(T*-t)·u(x,t)=r(s)u(x,t) is the central object of this round's backward-Leray renormalized flow. Under s it solves the autonomous equation (C6-J.1); its fixed points correspond to backward self-similar blow-up profiles and its periodic orbits to backward discretely self-similar (DSS) profiles. Its critical norms ‖U‖_{L^3} and ‖U‖_{\\dot H^{1/2}} are invariant under the rescaling, and hypothetical blow-up forces both to diverge (C6-J.2).",
      "defining_relation": "U(y,s)=\\sqrt{T^\\ast-t}\\,u(x,t)=r(s)u(x,t)"
    },
    {
      "id": "ns.c6.c6j.p_renormalized_pressure",
      "latex": "P(y,s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化壓力場",
      "label_en": "renormalized pressure field",
      "definition_zh": "P(y,s)=(T*-t)p(x,t)=r(s)^2 p(x,t)，與 U 配對的重整化壓力，使 backward Leray 方程（C6-J.1）中壓力梯度項 ∇P 與其餘各項具有一致尺度。在 L² 能量恒等式（C6-J.3）的推導中，不可壓縮性給出 ∫U·∇P=0，確保壓力對能量平衡無貢獻。",
      "definition_en": "P(y,s)=(T*-t)p(x,t)=r(s)^2 p(x,t) is the renormalized pressure paired with U, giving the pressure-gradient term ∇P in the backward Leray equation (C6-J.1) consistent scaling with the other terms. In deriving the renormalized L² balance (C6-J.3), incompressibility gives ∫U·∇P=0, so pressure contributes nothing to the energy balance.",
      "defining_relation": "P(y,s)=(T^\\ast-t)p(x,t)=r(s)^2p(x,t)"
    },
    {
      "id": "ns.c6.c6j.backward_leray_flow_equation",
      "latex": "\\partial_sU+\\frac12U+\\frac12(y\\cdot\\nabla)U+(U\\cdot\\nabla)U+\\nabla P=\\nu\\Delta U,\\qquad \\nabla\\cdot U=0",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "反向 Leray 流方程（C6-J.1）",
      "label_en": "Backward Leray flow equation (C6-J.1)",
      "definition_zh": "本輪首個主定理：由標準變量代換直接導出，重整化速度場 U 在 scale-time s 下滿足自治（autonomous）方程 ∂_sU+½U+½(y·∇)U+(U·∇)U+∇P=νΔU，搭配不可壓縮條件 ∇·U=0。因方程不顯含 s，有限時間物理終點 t↑T* 被轉成無限 scale-time 動力系統問題 s→∞，使 fixed point/periodic orbit 等動力系統語言得以嚴格套用於 N–S 爆破分析。",
      "definition_en": "The round's first main theorem, obtained by a direct change of variables: the renormalized velocity U obeys the autonomous equation ∂_sU+½U+½(y·∇)U+(U·∇)U+∇P=νΔU together with the incompressibility condition ∇·U=0, in scale-time s. Because the equation has no explicit s-dependence, the finite physical-time endpoint t↑T* becomes an infinite-time dynamical-systems problem s→∞, licensing fixed-point/periodic-orbit language for N–S blow-up analysis.",
      "notes": "Status \"PROVED\" per the round's formal status table (§65); the autonomy is what makes \"fixed point = backward self-similar\" and \"periodic orbit = backward DSS\" precise later in the round."
    },
    {
      "id": "ns.c6.c6j.u_star_fixed_point",
      "latex": "U_\\ast(y)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "反向自相似輪廓（固定點）",
      "label_en": "backward self-similar profile (fixed point)",
      "definition_zh": "當 U(y,s)=U_*(y) 與 s 無關時的解，滿足 stationary backward Leray 方程 ½U_*+½(y·∇)U_*+(U_*·∇)U_*+∇P_*=νΔU_*，物理上對應 backward self-similar blow-up profile。已知 Liouville 型定理（如 §43 的 Chae–Wolf 結果）在適當可積性假設下排除大類非平凡 U_*，但這些定理只適用於 field-level 輪廓，不自動適用於僅涉及 C6 defect metadata 的週期軌道（§6 guard）。",
      "definition_en": "The s-independent solution U(y,s)=U_*(y), solving the stationary backward Leray equation ½U_*+½(y·∇)U_*+(U_*·∇)U_*+∇P_*=νΔU_*, corresponding physically to a backward self-similar blow-up profile. Known Liouville-type theorems (e.g. the Chae-Wolf result of §43) exclude broad classes of nontrivial U_* under suitable integrability, but only at the field level -- they do not automatically apply to a periodic orbit of only the C6 defect metadata (§6 guard).",
      "defining_relation": "\\frac12U_\\ast+\\frac12(y\\cdot\\nabla)U_\\ast+(U_\\ast\\cdot\\nabla)U_\\ast+\\nabla P_\\ast=\\nu\\Delta U_\\ast"
    },
    {
      "id": "ns.c6.c6j.period_l",
      "latex": "L",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化流週期",
      "label_en": "period of the renormalized flow",
      "definition_zh": "由 U(y,s+L)=U(y,s) 定義的 scale-time 週期，刻畫 backward Leray 流的 periodic orbit，物理上對應 backward discretely self-similar（DSS）情景，尺度比為 λ=e^{L/2}。L 亦設定週期 L² balance（§18）的積分區間，並在後續 skew-product 模型中控制 defect metadata θ 的近似週期性（θ(s+L)≈θ(s)，§39）。",
      "definition_en": "The scale-time period defined by U(y,s+L)=U(y,s), characterizing a periodic orbit of the backward Leray flow, physically a backward discretely self-similar (DSS) scenario with dilation ratio λ=e^{L/2}. L also sets the integration window of the periodic L² balance (§18) and later governs the approximate periodicity of the defect metadata θ in the skew-product model (θ(s+L)≈θ(s), §39).",
      "defining_relation": "U(y,s+L)=U(y,s)"
    },
    {
      "id": "ns.c6.c6j.lambda_dss_ratio",
      "latex": "\\lambda",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "反向 DSS 伸縮比",
      "label_en": "backward DSS dilation ratio",
      "definition_zh": "λ=e^{L/2}>1，由週期 L 導出，刻畫物理解在相隔 λ 倍尺度之間所滿足的 backward discrete self-similarity（DSS）關係。它把 scale-time 中的週期軌道語言翻譯回物理自相似伸縮語言，是「periodic orbit = backward DSS」（本輪主要結果之一）這一對應的具體係數。",
      "definition_en": "λ=e^{L/2}>1, derived from the period L, quantifies the backward discrete self-similarity (DSS) relation the physical solution obeys between scales separated by a factor λ. It translates the scale-time periodic-orbit language back into physical self-similar dilation language, the concrete coefficient realizing \"periodic orbit = backward DSS\" (one of the round's main results).",
      "defining_relation": "\\lambda=e^{L/2}>1"
    },
    {
      "id": "ns.c6.c6j.e_renormalized_energy",
      "latex": "E(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化 L² 能量",
      "label_en": "renormalized L² energy",
      "definition_zh": "E(s)=‖U(s)‖_2^2，重整化速度場的 L² 範數平方。C6-J.3（renormalized L² balance）證明其恰滿足 ½E'+ν‖∇U‖_2^2-¼E=0，即 E'=½E-2ν‖∇U‖_2^2——伸縮項貢獻 anti-dissipative 的 +½E，使 E 本身並非普遍單調量（本輪核心負面結果之一，亦見 NG-J2）。E 是後續加權族 V_α=e^{-αs}E 的基底。",
      "definition_en": "E(s)=‖U(s)‖_2^2, the squared L² norm of the renormalized velocity. C6-J.3 (the renormalized L² balance) shows it obeys exactly ½E'+ν‖∇U‖_2^2-¼E=0, i.e. E'=½E-2ν‖∇U‖_2^2 -- the dilation term contributes an anti-dissipative +½E, so E itself is not a universal monotone quantity (one of the round's core negative results, cf. NG-J2). E is the base of the later weighted family V_α=e^{-αs}E.",
      "defining_relation": "E(s)=\\|U(s)\\|_2^2",
      "notes": "Not to be confused with the physical kinetic energy ‖u(t)‖_2^2; the two are related by E(s)=r(s)^{-1}‖u(t)‖_2^2 (§22). The physical energy is instead identified with V_{1/2}, not with E itself."
    },
    {
      "id": "ns.c6.c6j.v_alpha_weighted_potential",
      "latex": "V_\\alpha(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "加權 L² 位能族",
      "label_en": "weighted L² potential family",
      "definition_zh": "對 α∈ℝ 定義 V_α(s)=e^{-αs}E(s)。C6-J.4 給出恆等式 V_α'=e^{-αs}[(½-α)‖U‖_2^2-2ν‖∇U‖_2^2]。核心結果 C6-J.5（Criticality–Monotonicity Tradeoff）證明此族中普遍單調性（V_α'≤0）恰於 α≥½ 開始成立，而每個這樣的位能都帶有明確衰減因子 e^{-αs}，故臨界尺度敏感性與普遍單調性在此自然指數族中無法共存。",
      "definition_en": "For α∈ℝ, define V_α(s)=e^{-αs}E(s). C6-J.4 gives the identity V_α'=e^{-αs}[(½-α)‖U‖_2^2-2ν‖∇U‖_2^2]. The central result C6-J.5 (Criticality-Monotonicity Tradeoff) shows universal monotonicity (V_α'≤0) within this family begins exactly at α≥½, and every such potential carries an explicit decaying factor e^{-αs} -- so critical scale-sensitivity and universal monotonicity cannot coexist in this natural exponential family.",
      "defining_relation": "V_\\alpha(s)=e^{-\\alpha s}E(s)"
    },
    {
      "id": "ns.c6.c6j.v_half_physical_energy",
      "latex": "V_{1/2}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "物理能量（α=1/2 情形）",
      "label_en": "physical energy (the α=1/2 case)",
      "definition_zh": "V_α 家族在最弱單調權重 α=½ 處的特例；§22 的恆等式 e^{-s/2}‖U(s)‖_2^2=r‖U‖_2^2=‖u(t)‖_2^2 證明它恰為物理動能 ‖u(t)‖_2^2。這是本輪的重要正面結果：physical-energy telescoping 確實存在，其權重 e^{-s/2}=r 正好重現 C6-I 的 Zeno 求和條件（§24），但正因帶有次臨界權重，無法對尺度不變的 recurrent event 分配固定正代價（§25）。",
      "definition_en": "The V_α family member at the weakest monotone weight α=½; the §22 identity e^{-s/2}‖U(s)‖_2^2=r‖U‖_2^2=‖u(t)‖_2^2 shows it equals exactly the physical kinetic energy ‖u(t)‖_2^2. This is an important positive result of the round: physical-energy telescoping does exist, with weight e^{-s/2}=r exactly reproducing C6-I's Zeno summability (§24) -- but precisely because the weight is subcritical, it cannot assign a fixed positive price to a scale-invariant recurrent event (§25).",
      "defining_relation": "e^{-s/2}\\|U(s)\\|_2^2=\\|u(t)\\|_2^2"
    },
    {
      "id": "ns.c6.c6j.d_crit_dissipation_density",
      "latex": "D_{crit}(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化臨界耗散密度",
      "label_en": "renormalized critical dissipation density",
      "definition_zh": "D_crit(s)=ν‖∇U(s)‖_2^2，重整化黏性耗散率。它使物理能量 telescoping 恆等式可寫成 V_{1/2}(s_1)-V_{1/2}(s_2)=2∫_{s_1}^{s_2} e^{-s/2}D_crit(s)ds（§24），說明單調位能是以權重 e^{-s/2}=r 對臨界耗散積分而得，即 C6-I geometric Zeno lemma 的連續版本。§26 的 simple critical Lyapunov test 指出，任何 E'≤0 的證明都需要額外不等式 D_crit≥¼E，但目前無此類全空間普遍不等式。",
      "definition_en": "D_crit(s)=ν‖∇U(s)‖_2^2, the renormalized viscous dissipation rate. It lets the physical-energy telescoping identity be written V_{1/2}(s_1)-V_{1/2}(s_2)=2∫_{s_1}^{s_2} e^{-s/2}D_crit(s)ds (§24): the monotone potential integrates critical dissipation with weight e^{-s/2}=r, the continuous-s analogue of C6-I's geometric Zeno lemma. The §26 \"simple critical Lyapunov test\" notes that any proof of E'≤0 would require the additional inequality D_crit≥¼E, for which no universal whole-space form is currently known.",
      "defining_relation": "D_{crit}(s)=\\nu\\|\\nabla U(s)\\|_2^2",
      "notes": "Distinct from the bare \"D\" used only informally in the §0 summary's compressed statement of C6-J.3 (\"½E'+νD-¼E=0\"), where D stands for ‖∇U‖_2^2 without the ν factor; D_crit is the formally boxed, ν-weighted quantity used throughout the technical sections."
    },
    {
      "id": "ns.c6.c6j.criticality_monotonicity_tradeoff",
      "latex": "\\textbf{Criticality–Monotonicity Tradeoff}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "臨界性—單調性取捨（C6-J.5）",
      "label_en": "Criticality-Monotonicity Tradeoff (C6-J.5)",
      "definition_zh": "本輪第二個核心定理：在自然指數加權 L² 位能族 V_α=e^{-αs}E 中，普遍單調性（α≥½，§21）與不帶衰減因子的臨界尺度敏感性無法同時成立——唯一同時逼近兩者邊界的 α=½ 情形，其位能已帶有物理上有意義但次臨界的權重 e^{-s/2}。此定理說明為何單純「尋找單調位能」的策略無法直接排除 C6 defect cycle，並促使 §27 起轉向真正尺度不變的臨界場範數 ‖U‖_{L^3}、‖U‖_{\\dot H^{1/2}}。",
      "definition_en": "The round's second central theorem: within the natural exponentially-weighted L² potential family V_α=e^{-αs}E, universal monotonicity (α≥½, §21) and (unweighted) critical scale-sensitivity cannot both hold -- the unique boundary case α=½ that achieves monotonicity already carries the physically meaningful but subcritical weight e^{-s/2}. This explains why simply \"finding a monotone potential\" cannot by itself rule out C6 defect cycles, and motivates the pivot from §27 onward to the genuinely scale-invariant critical field norms ‖U‖_{L^3}, ‖U‖_{\\dot H^{1/2}}.",
      "defining_relation": "\\alpha\\ge\\frac12 \\Rightarrow V_\\alpha'(s)\\le0"
    },
    {
      "id": "ns.c6.c6j.f_crit_field_capacity",
      "latex": "\\mathfrak F_{crit}(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化臨界場容量",
      "label_en": "renormalized critical field capacity",
      "definition_zh": "𝔉_crit(s)=‖U(s)‖_{L^3}+‖U(s)‖_{\\dot H^{1/2}}，合併兩個在 backward Leray 重整化下嚴格不變（§8–9）的臨界拓撲範數。假設爆破要求 𝔉_crit(s)→∞（§10）。C6-J.2（§11）由此證明：任何假設爆破軌道的尾段 {U(s):s≥s_0} 都不能在 L^3 或 \\dot H^{1/2} 中 precompact。𝔉_crit 亦是後續 critical fiber radius 𝔯_fiber、fiber 座標 κ(s)、以及 ETN 狀態 Θ_U^{C6J} 的核心構件。",
      "definition_en": "𝔉_crit(s)=‖U(s)‖_{L^3}+‖U(s)‖_{\\dot H^{1/2}} combines the two critical topological norms shown strictly invariant under the backward Leray rescaling (§8-9). Hypothetical blow-up forces 𝔉_crit(s)→∞ (§10). From this, C6-J.2 (§11) proves that no tail {U(s):s≥s_0} of a hypothetical blow-up orbit can be precompact in L^3 or \\dot H^{1/2}. 𝔉_crit is also the core building block of the later critical fiber radius 𝔯_fiber, the fiber coordinate κ(s), and the ETN state Θ_U^{C6J}.",
      "defining_relation": "\\mathfrak F_{crit}(s)=\\|U(s)\\|_{L^3}+\\|U(s)\\|_{\\dot H^{1/2}}"
    },
    {
      "id": "ns.c6.c6j.f_crit_hat_compactified",
      "latex": "\\widehat{\\mathfrak F}_{crit}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "緊化臨界場容量",
      "label_en": "compactified critical field capacity",
      "definition_zh": "𝔉_crit 經 x↦x/(1+x) 緊化後所得的 [0,1) 值量：𝔉̂_crit=𝔉_crit/(1+𝔉_crit)。它把假設爆破判準 𝔉_crit→∞ 重寫成有界形式 𝔉̂_crit(s)→1（§27）。§28 強調此極限是 genuine critical infinity 而非單純尺度化人工產物（不同於未臨界化的 A_k→∞），因為 ‖U‖_3、‖U‖_{\\dot H^{1/2}} 本身已是尺度不變量。",
      "definition_en": "The [0,1)-valued compactification of 𝔉_crit via x↦x/(1+x): 𝔉̂_crit=𝔉_crit/(1+𝔉_crit). It rewrites the blow-up criterion 𝔉_crit→∞ into the bounded form 𝔉̂_crit(s)→1 (§27). §28 stresses this limit is a genuine critical infinity rather than a mere scaling artifact (unlike an uncriticalized A_k→∞), precisely because ‖U‖_3 and ‖U‖_{\\dot H^{1/2}} are themselves already scale-invariant.",
      "defining_relation": "\\widehat{\\mathfrak F}_{crit}=\\frac{\\mathfrak F_{crit}}{1+\\mathfrak F_{crit}}\\in[0,1)"
    },
    {
      "id": "ns.c6.c6j.x_crit_state_space",
      "latex": "\\mathcal X_{crit}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "重整化臨界場狀態空間",
      "label_en": "renormalized critical field state space",
      "definition_zh": "承載至少奇異時刻前的局部光滑性、臨界場範數、以及 C6 所需空間/來源資料的合適重整化場狀態空間（§29）。它是 defect projection π 的定義域，也是「full field U(s)」與「僅其 defect metadata 投影 π(U(s))」這一本輪核心區分（§30）的載體——C6-A 到 C6-I 主要只研究了 π(U(s))，而非完整位於 X_crit 中的 U(s)。",
      "definition_en": "A suitable renormalized field state space carrying at least local smoothness prior to the singular time, critical field norms, and the spatial/provenance data C6 needs (§29). It is the domain of the defect projection π, and the carrier of this round's core distinction between the full field U(s) and its defect-metadata projection π(U(s)) (§30) -- C6-A through C6-I had studied only π(U(s)), not the full U(s) living in X_crit.",
      "notes": "Defined by listed structural properties in the source rather than by a single formula."
    },
    {
      "id": "ns.c6.c6j.k_def_defect_space",
      "latex": "\\mathcal K_{def}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "緊化缺陷狀態空間",
      "label_en": "compactified defect state space",
      "definition_zh": "由 TS、GP、HF、臨界邊界面、composition reserves 等組成的緊化 C6 defect metadata 狀態空間（§29），是 defect projection π 的值域。C6 defect recurrence（π(U(s_n))→θ* 或返回 K⊂K_def 附近）發生於此空間，但 K_def 中的緊致性並不蘊含 X_crit 中對應纖維的緊致性（§30），此落差正是 Critical Fiber Escape 現象的來源。",
      "definition_en": "The compactified C6 defect-metadata state space (§29), built from TS, GP, HF, the critical boundary faces, and composition reserves -- the codomain of the defect projection π. C6 defect recurrence (π(U(s_n))→θ* or return near a compact K⊂K_def) takes place here, but compactness in K_def does not imply compactness of the corresponding fiber in X_crit (§30); this gap is exactly the source of the Critical Fiber Escape phenomenon.",
      "notes": "The file is not fully consistent in spelling: the §0 summary writes π:X_crit→K_defect (point 21), while the formal definition from §29 onward uses K_def. The sequel C6-K continues with the spelled-out \\mathcal K_{\\rm defect}. Treat K_def and K_defect as the same object."
    },
    {
      "id": "ns.c6.c6j.pi_defect_projection",
      "latex": "\\pi",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "缺陷投影",
      "label_en": "defect projection",
      "definition_zh": "π:X_crit→K_def（§29），把完整重整化場狀態 U(s) 投影到其 C6 defect metadata θ=π(U(s))。本輪的關鍵觀察（§30）是：defect recurrence（π(U(s_n)) 收斂或返回緊集）不蘊含 U(s_n) 在 X_crit 中 precompact，因為纖維 π^{-1}(θ) 可能非緊。這把整個 C6 cycle 問題重新表述為 skew-product 問題：緊致 recurrent base（於 K_def 中）加上可能逃逸的 critical fiber（於 X_crit 中）。",
      "definition_en": "π: X_crit → K_def (§29), projecting the full renormalized field state U(s) onto its C6 defect metadata θ=π(U(s)). The round's key observation (§30) is that defect recurrence (π(U(s_n)) converging or returning to a compact set) does not imply U(s_n) is precompact in X_crit, because the fiber π^{-1}(θ) may be noncompact. This reframes the entire C6 cycle problem as a skew-product problem: a compact recurrent base (in K_def) plus a possibly escaping critical fiber (in X_crit).",
      "defining_relation": "\\pi:\\mathcal X_{crit}\\to\\mathcal K_{def}"
    },
    {
      "id": "ns.c6.c6j.r_fiber_radius",
      "latex": "\\mathfrak R_{fiber}(K)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "臨界纖維半徑",
      "label_en": "critical fiber radius",
      "definition_zh": "對緊集 K⊂K_def，定義 𝔯_fiber(K)=sup{𝔉_crit(U):π(U)∈K}∈[0,∞]（§31）。若 𝔯_fiber(K)<∞，defect set K 一致控制其纖維上的臨界場容量。此量是 C6-J.6（Bounded-Fiber Recurrence No-Go，§32）與 C6-J.7（Critical Fiber Escape Theorem，§34）的中心測度：前者證明有限 𝔯_fiber 的緊 defect trap 不能支持假設爆破，後者證明任何被無窮多次訪問的緊 defect set 必有 𝔯_fiber(K)=∞。",
      "definition_en": "For compact K⊂K_def, define 𝔯_fiber(K)=sup{𝔉_crit(U):π(U)∈K}∈[0,∞] (§31). If 𝔯_fiber(K)<∞, the defect set K uniformly controls the critical field capacity over its fiber. This quantity is the central measure of both C6-J.6 (Bounded-Fiber Recurrence No-Go, §32), showing a compact defect trap with finite 𝔯_fiber cannot support hypothetical blow-up, and C6-J.7 (Critical Fiber Escape Theorem, §34), showing any compact defect set visited infinitely often must have 𝔯_fiber(K)=∞.",
      "defining_relation": "\\mathfrak R_{fiber}(K)=\\sup\\{\\mathfrak F_{crit}(U):\\pi(U)\\in K\\}\\in[0,\\infty]",
      "notes": "Same symbol as ns.c6.Rfiber, which files it under first_appearance 'C6-K' even though that entry's own text attributes the defining theorem to C6-J -- this entry, extracted directly from the C6-J source, is the more precise first_appearance attribution."
    },
    {
      "id": "ns.c6.c6j.critical_fiber_escape",
      "latex": "\\textbf{Critical Fiber Escape}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "臨界纖維逃逸（C6-J.7）",
      "label_en": "Critical Fiber Escape (C6-J.7)",
      "definition_zh": "本輪標題定理與核心結論：任何被假設爆破重整化軌道無窮多次訪問的緊 defect set K，必滿足 𝔯_fiber(K)=∞，等價地說，每個倖存的緊 defect recurrence 都要求其臨界纖維非緊。這確立了 C6 cycle 問題的最終形式——緊致 recurrent defect base 加上必然逃逸的 noncompact critical fiber 之 skew-product 問題（§34；亦見結論 §66）。它並非矛盾，而是假設爆破下的必要狀態性質（§65 formal status：「PROVED AS NECESSARY STATE PROPERTY」；亦見 NG-J8）。C6-K 直接以此定理為起點，將 J-F1–J-F6（amplitude/multiplicity/secondary-scale/translation-tail/frequency escape/profile splitting，§54）候選逃逸機制提升為嚴謹的 profile-decomposition 論證。",
      "definition_en": "The round's titular theorem and central conclusion: any compact defect set K visited infinitely often by a hypothetical blow-up renormalized orbit must satisfy 𝔯_fiber(K)=∞ -- equivalently, every surviving compact defect recurrence requires noncompactness in its critical fiber. This fixes the final shape of the C6 cycle problem as a skew-product problem: a compact recurrent defect base plus a critical fiber that must escape (§34; see also the conclusion, §66). It is not a contradiction but a necessary state property under hypothetical blow-up (§65 formal status: \"PROVED AS NECESSARY STATE PROPERTY\"; cf. NG-J8). The sequel C6-K takes this theorem as its starting point, upgrading the candidate escape mechanisms J-F1-F6 (amplitude/multiplicity/secondary-scale/translation-tail/frequency escape/profile splitting, §54) into rigorous profile-decomposition arguments.",
      "defining_relation": "\\mathfrak R_{fiber}(K)=\\infty",
      "notes": "Confirmed by direct inspection of the sequel file NS_C6K_CriticalFiber_ProfileSplitting_v0.1.md, which opens by restating this theorem and the same six escape-mechanism candidates before formalizing them."
    },
    {
      "id": "ns.c6.c6j.theta_defect_base",
      "latex": "\\theta(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "缺陷元資料座標",
      "label_en": "defect metadata coordinate (base)",
      "definition_zh": "θ(s)=π(U(s))∈K_def，即重整化場沿投影 π 的像，代表「compact defect metadata」（§37–38）。在 §38 的 skew-product 模型 (θ(s),κ(s)) 中，θ 是緊致的 base 座標；C6-J.8（Projected-Cycle Reframing，§39）指出，一個與假設爆破相容的 C6 recurrent defect cycle 不是完整臨界場狀態空間中的閉軌道，而必須是 skew-product 軌道：θ(s+L)≈θ(s) 而 κ(s+L)>κ(s)（平均或沿子序列）。",
      "definition_en": "θ(s)=π(U(s))∈K_def, the image of the renormalized field under the projection π, representing \"compact defect metadata\" (§37-38). In the §38 skew-product model (θ(s),κ(s)), θ is the compact base coordinate; C6-J.8 (Projected-Cycle Reframing, §39) states that a C6 recurrent defect cycle compatible with hypothetical blow-up is not a closed orbit in the full critical field state space, but must be a skew-product orbit: θ(s+L)≈θ(s) while κ(s+L)>κ(s) (on average or along a subsequence).",
      "defining_relation": "\\theta(s)=\\pi(U(s))",
      "notes": "Relabeled θ_def in the §64 ETN tuple Θ_skew^{C6J}=(θ_def,κ_fiber); same object."
    },
    {
      "id": "ns.c6.c6j.kappa_fiber_coordinate",
      "latex": "\\kappa(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "臨界纖維座標",
      "label_en": "critical fiber coordinate",
      "definition_zh": "κ(s)=log(1+𝔉_crit(s))，skew-product 模型 (θ(s),κ(s)) 中代表無界臨界纖維方向的座標（§38）。假設爆破要求 κ(s)→∞，而 base θ(s) 可保持 recurrent。§57–58 進一步把 boundary graph 的轉移關係從單純 B_i→B_j 升級為帶 κ 座標的 (B_i,κ)→(B_j,κ')，並要求任何與爆破相容的 recurrent base cycle 滿足 κ_n→∞——這是 C6-G/H 所缺的方向性座標。",
      "definition_en": "κ(s)=log(1+𝔉_crit(s)), the coordinate representing the unbounded critical-fiber direction in the skew-product model (θ(s),κ(s)) (§38). Hypothetical blow-up requires κ(s)→∞ while the base θ(s) may remain recurrent. §57-58 further upgrades the boundary-graph transition from a plain B_i→B_j to one carrying the κ coordinate, (B_i,κ)→(B_j,κ'), requiring any blow-up-compatible recurrent base cycle to satisfy κ_n→∞ -- the directional coordinate missing from C6-G/H.",
      "defining_relation": "\\kappa(s)=\\log\\left(1+\\mathfrak F_{crit}(s)\\right)",
      "notes": "Relabeled κ_fiber in the §64 ETN tuple with the equivalent formula log[1+‖U‖_3+‖U‖_{\\dot H^{1/2}}]. In the sequel C6-K, the symbol κ_n is reused for a related but distinct concentration-scale quantity in the profile-decomposition analysis -- do not conflate the two across rounds."
    },
    {
      "id": "ns.c6.c6j.field_cap_infinity",
      "latex": "CAP_{field}^{crit,\\infty}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "場容量無窮大（FIELD-CAP∞）",
      "label_en": "field capacity infinity (FIELD-CAP∞)",
      "definition_zh": "C6-J 在 §36 對 C6-I 的 CAP^{crit,∞} 邊界重新細分出的兩型之一：field capacity infinity，即 ‖U‖_3+‖U‖_{\\dot H^{1/2}}→∞（等價於 𝔉_crit→∞）。§56 強調對真正的假設爆破而言，field capacity infinity（記作 CAP_field^{crit,∞}）並非可有可無的邊界選項之一，而是必要的 fiber 方向——與另一型 edge capacity infinity（EDGE-CAP∞）含義本質不同。",
      "definition_en": "One of two refined subtypes C6-J splits out of C6-I's CAP^{crit,∞} boundary (§36): field capacity infinity, i.e. ‖U‖_3+‖U‖_{\\dot H^{1/2}}→∞ (equivalently 𝔉_crit→∞). §56 stresses that for genuine hypothetical blow-up, field capacity infinity (denoted CAP_field^{crit,∞}) is not merely one optional boundary among others but a required fiber direction -- with a meaning essentially different from the other subtype, edge capacity infinity (EDGE-CAP∞).",
      "defining_relation": "\\mathfrak F_{crit}\\to\\infty",
      "notes": "CAP^{crit,∞} itself (unqualified) is C6-I notation, treated there as a single critical boundary; C6-J's contribution here is this field/edge refinement."
    },
    {
      "id": "ns.c6.c6j.edge_cap_infinity",
      "latex": "\\Gamma^{-1}\\to\\infty",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "邊容量無窮大（EDGE-CAP∞）",
      "label_en": "edge capacity infinity (EDGE-CAP∞)",
      "definition_zh": "C6-J 在 §36／§56 細分出的另一型 capacity infinity：edge capacity infinity，以 Γ^{-1}→∞ 為例（Γ 為先前 C6 輪次的邊/來源容量記號，本檔並未重新定義）。與 field capacity infinity 不同，它描述的是 required source capacity inflation，而非 field-critical norm infinity；本輪強調兩者「都是 critical，但含義不同」。",
      "definition_en": "The other subtype of capacity infinity C6-J distinguishes in §36/§56: edge capacity infinity, exemplified by Γ^{-1}→∞ (Γ is an edge/source-capacity notation from an earlier C6 round, not redefined in this file). Unlike field capacity infinity, it describes required source-capacity inflation rather than field-critical-norm infinity; the round stresses both are critical but \"have different meanings.\"",
      "notes": "Γ itself is not defined within C6-J -- it is invoked only as an \"e.g.\" contrast example against FIELD-CAP∞/𝔉_crit. Consult the earlier C6 round that introduces Γ (edge/boundary-face capacity) for its precise definition."
    },
    {
      "id": "ns.c6.c6j.etn_state_theta_u",
      "latex": "\\Theta_U^{C6J}(s)",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "反向重整化場狀態（ETN 更新）",
      "label_en": "backward renormalized field state (ETN update)",
      "definition_zh": "§64「True ETN update」給出的本輪正式狀態元組：Θ_U^{C6J}(s)=⟨U(s),P(s),‖U‖_3,‖U‖_{\\dot H^{1/2}},E(s),D_crit(s),π(U(s))⟩，將本輪引入的主要場量（重整化速度/壓力、兩個臨界範數、重整化能量、臨界耗散密度、defect 投影）統一封裝成單一狀態記錄，供後續輪次接續。",
      "definition_en": "The round's formal state tuple from the §64 \"True ETN update\": Θ_U^{C6J}(s)=⟨U(s),P(s),‖U‖_3,‖U‖_{\\dot H^{1/2}},E(s),D_crit(s),π(U(s))⟩, packaging this round's principal field quantities (renormalized velocity/pressure, the two critical norms, renormalized energy, critical dissipation density, and the defect projection) into a single state record for continuity into subsequent rounds.",
      "defining_relation": "\\Theta_U^{C6J}(s)=\\left\\langle U(s),P(s),\\|U\\|_3,\\|U\\|_{\\dot H^{1/2}},E(s),D_{crit}(s),\\pi(U(s))\\right\\rangle"
    },
    {
      "id": "ns.c6.c6j.etn_skew_state",
      "latex": "\\Theta_{\\rm skew}^{C6J}",
      "series": "NS",
      "first_appearance": "C6-J",
      "label_zh": "斜積狀態（ETN 更新）",
      "label_en": "skew-product state (ETN update)",
      "definition_zh": "§64 給出的第二個正式狀態，把本輪的 skew-product 重述壓縮成一對：Θ_skew^{C6J}=(θ_def,κ_fiber)，其中 κ_fiber=log[1+‖U‖_3+‖U‖_{\\dot H^{1/2}}]（即 §38 的 κ(s)）。假設爆破要求 κ_fiber→∞，這是本輪「compact defect base + noncompact critical fiber」核心圖像的正式狀態封裝，也是下一輪 C6-K 的起點。",
      "definition_en": "The second formal state given in §64, compressing the round's skew-product reframing into a pair: Θ_skew^{C6J}=(θ_def,κ_fiber), where κ_fiber=log[1+‖U‖_3+‖U‖_{\\dot H^{1/2}}] (i.e. the κ(s) of §38). Hypothetical blow-up requires κ_fiber→∞; this is the formal state-packaging of the round's core \"compact defect base + noncompact critical fiber\" picture, and the starting point of the sequel C6-K.",
      "defining_relation": "\\Theta_{\\rm skew}^{C6J}=(\\theta_{def},\\kappa_{fiber}),\\ \\kappa_{fiber}=\\log\\left[1+\\|U\\|_3+\\|U\\|_{\\dot H^{1/2}}\\right]"
    },
    {
      "id": "ns.c1.c1a",
      "latex": "\\mathrm{C1a}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "高頻尾端逃逸（C1a）",
      "label_en": "High-Frequency Tail Escape (C1a)",
      "definition_zh": "本輪三分解的第一層（第4節定理4.1）：對任意固定頻率截斷 $J$，若 $T_\\ast$ 為有限 blow-up time，則高通尾端 $\\|P_{>J}u(t)\\|_3$ 在 $t\\uparrow T_\\ast$ 時 limsup 必發散至無窮；本輪已閉合（CLOSED）。",
      "definition_en": "The first layer of this round's three-part decomposition (Section 4, Theorem 4.1): for every fixed frequency cutoff $J$, if $T_\\ast$ is a finite blow-up time then $\\limsup_{t\\uparrow T_\\ast}\\|P_{>J}u(t)\\|_3=\\infty$; closed in this round.",
      "defining_relation": "\\mathrm{Blowup}\\Rightarrow\\forall J,\\ \\limsup_{t\\uparrow T_\\ast}\\|P_{>J}u(t)\\|_3=\\infty"
    },
    {
      "id": "ns.c1.c1b",
      "latex": "\\mathrm{C1b}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "非線性UV補給鏈（C1b）",
      "label_en": "Nonlinear UV Replenishment Chain (C1b)",
      "definition_zh": "三分解的第二層（第11節定理11.1）：存在遞迴選取的 $J_n\\to\\infty,\\ t_n\\to T_\\ast$，使區間 $[t_{n-1},t_n]$ 上的非線性來源項 $\\mathcal N_n$ 滿足 $\\|\\mathcal N_n\\|_3\\ge A_n-\\varepsilon_n\\to\\infty$；本輪已閉合（CLOSED）。",
      "definition_en": "The second layer (Section 11, Theorem 11.1): there exist recursively chosen $J_n\\to\\infty$, $t_n\\to T_\\ast$ such that the nonlinear source term $\\mathcal N_n$ on $[t_{n-1},t_n]$ satisfies $\\|\\mathcal N_n\\|_3\\ge A_n-\\varepsilon_n\\to\\infty$; closed in this round.",
      "defining_relation": "\\mathrm{Blowup}\\Rightarrow\\exists(J_n,t_n,\\mathcal N_n):J_n\\to\\infty,\\ t_n\\to T_\\ast,\\ \\|\\mathcal N_n\\|_3\\to\\infty"
    },
    {
      "id": "ns.c1.c1c",
      "latex": "\\mathrm{C1c}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "持續三元譜系（C1c，未解）",
      "label_en": "Persistent Triadic Genealogy (C1c, open)",
      "definition_zh": "三分解的第三層（第15節）：是否存在單一持續、來源保持的三元交互譜系 $(j_1,t_1)\\prec(j_2,t_2)\\prec\\cdots$ 一路延伸到 $j\\to\\infty$；本輪明確保持 OPEN。",
      "definition_en": "The third layer (Section 15): whether a single, source-preserving triadic interaction genealogy $(j_1,t_1)\\prec(j_2,t_2)\\prec\\cdots$ extends all the way to $j\\to\\infty$; explicitly left OPEN in this round.",
      "defining_relation": "\\mathrm{Blowup}\\Rightarrow\\text{one persistent source-preserving triadic genealogy to }j=\\infty"
    },
    {
      "id": "ns.c1.blowup_pred",
      "latex": "\\mathrm{Blowup}(T_\\ast)",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "Blowup 述詞",
      "label_en": "Blowup predicate",
      "definition_zh": "表示「$T_\\ast$ 為有限時間 blow-up time」此一假設的簡寫述詞，貫穿全篇作為 C1a/C1b/C1c 各條件式的前件（第0、16、19節）。",
      "definition_en": "Shorthand predicate asserting that $T_\\ast$ is a finite-time blow-up time; used throughout as the antecedent of the C1a/C1b/C1c implications (Sections 0, 16, 19)."
    },
    {
      "id": "ns.c1.xlegaluvchain",
      "latex": "\\mathrm{XLegalUVChain}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "X-合法UV鏈（前版目標）",
      "label_en": "XLegalUVChain (prior-round target)",
      "definition_zh": "前一版整合框架的結論物件（$\\mathrm{Blowup}(T_\\ast)\\Rightarrow\\mathrm{XLegalUVChain}$）；本輪第0節將它拆解為 C1a+C1b+C1c 三個更精確的子命題。",
      "definition_en": "The conclusion object from the previous version's framework ($\\mathrm{Blowup}(T_\\ast)\\Rightarrow\\mathrm{XLegalUVChain}$); Section 0 of this round decomposes it into the sharper sub-statements C1a+C1b+C1c.",
      "notes": "Legacy notation carried over from a prior round rather than newly defined here; included because this round's structure is framed entirely as its decomposition."
    },
    {
      "id": "ns.c1.p_le_j",
      "latex": "P_{\\le J}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "低通投影算子",
      "label_en": "Low-pass projector",
      "definition_zh": "第3節定義的平滑 Littlewood–Paley low-pass projector，頻率支撐在 $|\\xi|\\lesssim2^J$，滿足 Bernstein 不等式 $\\|P_{\\le J}f\\|_3\\le C_{\\mathrm{LP}}2^{J/2}\\|f\\|_2$。",
      "definition_en": "The smooth Littlewood–Paley low-pass projector defined in Section 3, frequency-supported on $|\\xi|\\lesssim2^J$, satisfying the Bernstein inequality $\\|P_{\\le J}f\\|_3\\le C_{\\mathrm{LP}}2^{J/2}\\|f\\|_2$."
    },
    {
      "id": "ns.c1.p_gt_j",
      "latex": "P_{>J}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "高通（UV尾端）投影算子",
      "label_en": "High-pass (UV tail) projector",
      "definition_zh": "第3節定義為 $P_{>J}=I-P_{\\le J}$，即高於截斷 $2^J$ 的頻率尾端投影；$\\|P_{>J}u(t)\\|_3$ 是定理4.1（C1a）與整個補給鏈論證的核心量。",
      "definition_en": "Defined in Section 3 as $P_{>J}=I-P_{\\le J}$, the projector onto frequencies above the cutoff $2^J$; $\\|P_{>J}u(t)\\|_3$ is the central quantity in Theorem 4.1 (C1a) and the replenishment-chain argument.",
      "defining_relation": "P_{>J}=I-P_{\\le J}"
    },
    {
      "id": "ns.c1.c_lp",
      "latex": "C_{\\mathrm{LP}}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "Bernstein不等式常數",
      "label_en": "Bernstein inequality constant",
      "definition_zh": "第3節 Bernstein 不等式中的 universal 常數，用以推出 $\\|P_{\\le J}u(t)\\|_3\\le C_{\\mathrm{LP}}2^{J/2}\\|u_0\\|_2$，即對固定 $J$ 一致有限的界，是定理4.1證明的關鍵。",
      "definition_en": "The universal constant in the Section 3 Bernstein inequality, used to derive the uniform-in-time finite bound $\\|P_{\\le J}u(t)\\|_3\\le C_{\\mathrm{LP}}2^{J/2}\\|u_0\\|_2$ for fixed $J$, key to the proof of Theorem 4.1."
    },
    {
      "id": "ns.c1.x_jt",
      "latex": "X(J,t)",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "X-積分觀測證書",
      "label_en": "X-Integral observation certificate",
      "definition_zh": "第6節定義的觀測型證書元組 $X(J,t)=\\langle u(t),P_{\\le J}u(t),P_{>J}u(t),\\|P_{>J}u(t)\\|_3\\rangle$，把定理4.1的逃逸結論包裝成可重播但尚非因果譜系的 multiscale escape certificate。",
      "definition_en": "The observational certificate tuple defined in Section 6, $X(J,t)=\\langle u(t),P_{\\le J}u(t),P_{>J}u(t),\\|P_{>J}u(t)\\|_3\\rangle$, packaging Theorem 4.1's escape conclusion into a replayable but not-yet-causal multiscale escape certificate.",
      "defining_relation": "X(J,t)=\\left\\langle u(t),P_{\\le J}u(t),P_{>J}u(t),\\|P_{>J}u(t)\\|_3\\right\\rangle"
    },
    {
      "id": "ns.c1.prov_x",
      "latex": "\\operatorname{Prov}X(J,t)",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "X 的來源記錄",
      "label_en": "Provenance record of X",
      "definition_zh": "第6節定義 $\\operatorname{Prov}X(J,t)=\\langle u_0,\\text{同一N–S解},t,J,P_{>J}\\rangle$，記錄證書 $X(J,t)$ 所依附的初值、軌跡與投影，使其可追溯而非憑空聲稱。",
      "definition_en": "Defined in Section 6 as $\\operatorname{Prov}X(J,t)=\\langle u_0,\\text{same N–S solution},t,J,P_{>J}\\rangle$, recording the initial data, trajectory and projector underlying the certificate $X(J,t)$ so it remains traceable.",
      "defining_relation": "\\operatorname{Prov}X(J,t)=\\left\\langle u_0,\\text{same N--S solution},t,J,P_{>J}\\right\\rangle"
    },
    {
      "id": "ns.c1.a_n",
      "latex": "A_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "目標發散序列",
      "label_en": "Target divergence sequence",
      "definition_zh": "第8節引入的任意遞增序列 $A_n\\uparrow\\infty$，作為第 $n$ 步高頻尾端 $\\|P_{>J_n}u(t_n)\\|_3$ 須達到的下界目標，也是定理11.1（C1b）與 $\\operatorname{XUVRepCert}_n$ 的參數。",
      "definition_en": "An arbitrary sequence $A_n\\uparrow\\infty$ introduced in Section 8, the lower-bound target that the high-frequency tail $\\|P_{>J_n}u(t_n)\\|_3$ must reach at step $n$; also a parameter of Theorem 11.1 (C1b) and of $\\operatorname{XUVRepCert}_n$."
    },
    {
      "id": "ns.c1.eps_n",
      "latex": "\\varepsilon_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "目標趨零序列",
      "label_en": "Target smallness sequence",
      "definition_zh": "第8節引入的任意遞減序列 $\\varepsilon_n\\downarrow0$，作為第 $n$ 步高頻尾端 $\\|P_{>J_n}u(t_{n-1})\\|_3$ 的上界，用以在遞迴選取中製造供 $\\mathcal N_n$ 補上的「先小後大」落差。",
      "definition_en": "An arbitrary sequence $\\varepsilon_n\\downarrow0$ introduced in Section 8, the upper bound on $\\|P_{>J_n}u(t_{n-1})\\|_3$ at step $n$, creating the small-then-large gap across $[t_{n-1},t_n]$ that $\\mathcal N_n$ must supply."
    },
    {
      "id": "ns.c1.t_n",
      "latex": "t_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "遞迴選取時間序列",
      "label_en": "Recursively selected time sequence",
      "definition_zh": "第8節交替使用「固定時刻高頻趨零」與定理4.1遞迴選出的時間序列 $t_0<t_1<t_2<\\cdots\\uparrow T_\\ast$，供第9節 Duhamel 分解與第13節 $\\operatorname{XUVRepCert}_n$ 的區間端點使用。",
      "definition_en": "The time sequence $t_0<t_1<t_2<\\cdots\\uparrow T_\\ast$ recursively selected in Section 8 by alternating the fixed-time tail-vanishing fact with Theorem 4.1; supplies the interval endpoints used in the Section 9 Duhamel decomposition and the Section 13 $\\operatorname{XUVRepCert}_n$."
    },
    {
      "id": "ns.c1.j_n",
      "latex": "J_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "遞迴選取頻率尺度序列",
      "label_en": "Recursively selected frequency-scale sequence",
      "definition_zh": "第8節與 $t_n$ 同步遞迴選出的截斷尺度序列 $J_1<J_2<\\cdots\\uparrow\\infty$，滿足 $\\|P_{>J_n}u(t_{n-1})\\|_3\\le\\varepsilon_n$ 且 $\\|P_{>J_n}u(t_n)\\|_3\\ge A_n$，是定理11.1與 $\\mathcal N_n$ 定義所用的頻率截斷。",
      "definition_en": "The cutoff-scale sequence $J_1<J_2<\\cdots\\uparrow\\infty$ selected in Section 8 in lockstep with $t_n$, satisfying $\\|P_{>J_n}u(t_{n-1})\\|_3\\le\\varepsilon_n$ and $\\|P_{>J_n}u(t_n)\\|_3\\ge A_n$; the frequency cutoff used in Theorem 11.1 and in defining $\\mathcal N_n$."
    },
    {
      "id": "ns.c1.n_script_n",
      "latex": "\\mathcal N_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "非線性UV來源項",
      "label_en": "Nonlinear UV source term",
      "definition_zh": "第9節由 Duhamel 恆等式投影到 $>J_n$ 後定義：$\\mathcal N_n=\\int_{t_{n-1}}^{t_n}e^{\\nu(t_n-s)\\Delta}P_{>J_n}\\mathbb P\\nabla\\cdot(u\\otimes u)(s)\\,ds$，代表區間 $[t_{n-1},t_n]$ 內純由非線性項注入高頻尾端的部分；第10–11節證明其範數 $\\ge A_n-\\varepsilon_n$，是C1b核心。",
      "definition_en": "Defined in Section 9 by projecting the Duhamel identity onto $>J_n$: $\\mathcal N_n=\\int_{t_{n-1}}^{t_n}e^{\\nu(t_n-s)\\Delta}P_{>J_n}\\mathbb P\\nabla\\cdot(u\\otimes u)(s)\\,ds$, the high-frequency content injected purely by the nonlinear term over $[t_{n-1},t_n]$; Sections 10–11 show $\\|\\mathcal N_n\\|_3\\ge A_n-\\varepsilon_n$, the core of C1b.",
      "defining_relation": "\\mathcal N_n=\\int_{t_{n-1}}^{t_n}e^{\\nu(t_n-s)\\Delta}P_{>J_n}\\mathbb P\\nabla\\cdot(u\\otimes u)(s)\\,ds"
    },
    {
      "id": "ns.c1.xuvrepcert",
      "latex": "\\operatorname{XUVRepCert}_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "X-積分補給證書",
      "label_en": "X-Integration replenishment certificate",
      "definition_zh": "第13節定義的第 $n$ 步證書元組 $\\langle t_{n-1},t_n,J_n,\\varepsilon_n,A_n,S_n,N_n,G_n\\rangle$，搭配7條守衛條件，記錄補給鏈每一步的來源、尺度與時間可重播性。",
      "definition_en": "The step-$n$ certificate tuple defined in Section 13, $\\langle t_{n-1},t_n,J_n,\\varepsilon_n,A_n,S_n,N_n,G_n\\rangle$, paired with 7 guard conditions, recording the provenance, scale and time-replayability of each step of the chain.",
      "defining_relation": "\\operatorname{XUVRepCert}_n=\\left\\langle t_{n-1},t_n,J_n,\\varepsilon_n,A_n,S_n,N_n,G_n\\right\\rangle"
    },
    {
      "id": "ns.c1.s_n",
      "latex": "S_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "前段尾端範數",
      "label_en": "Pre-interval tail norm",
      "definition_zh": "第13節定義 $S_n=\\|P_{>J_n}u(t_{n-1})\\|_3$，即區間起點 $t_{n-1}$ 時、尺度 $J_n$ 以上尾端的 $L^3$ 範數，依構造滿足 $S_n\\le\\varepsilon_n$。",
      "definition_en": "Defined in Section 13 as $S_n=\\|P_{>J_n}u(t_{n-1})\\|_3$, the $L^3$ norm of the $>J_n$ tail at the interval's starting time $t_{n-1}$; by construction $S_n\\le\\varepsilon_n$.",
      "defining_relation": "S_n=\\|P_{>J_n}u(t_{n-1})\\|_3"
    },
    {
      "id": "ns.c1.n_n",
      "latex": "N_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "補給量範數",
      "label_en": "Replenishment magnitude",
      "definition_zh": "第13節定義 $N_n=\\|\\mathcal N_n\\|_3$，即非線性UV來源項的 $L^3$ 範數；注意 $N_n$（純量範數）與 $\\mathcal N_n$（積分/算子物件本身）是不同符號。",
      "definition_en": "Defined in Section 13 as $N_n=\\|\\mathcal N_n\\|_3$, the $L^3$ norm of the nonlinear source term; note $N_n$ (scalar norm) is distinct from $\\mathcal N_n$ (the integral/operator object itself).",
      "defining_relation": "N_n=\\|\\mathcal N_n\\|_3"
    },
    {
      "id": "ns.c1.g_n",
      "latex": "G_n",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "守衛記錄",
      "label_en": "Guard record",
      "definition_zh": "第13節 $\\operatorname{XUVRepCert}_n$ 元組的第八分量；正文未逐字說明其內容，但緊接列出的7條守衛條件（如 $t_{n-1}<t_n<T_\\ast$、$S_n\\le\\varepsilon_n$、heat evolution 不放大 $L^3$ 尾端）應即為 $G_n$ 所記錄者。",
      "definition_en": "The eighth component of the $\\operatorname{XUVRepCert}_n$ tuple in Section 13; the prose does not gloss it verbatim, but the 7 guard conditions listed immediately after (e.g. $t_{n-1}<t_n<T_\\ast$, $S_n\\le\\varepsilon_n$, heat evolution not amplifying the $L^3$ tail) are presumably what it records.",
      "notes": "Confidence caveat: G_n's precise content is inferred from adjacent context, not explicitly glossed in the source text."
    },
    {
      "id": "ns.c1.delta_j",
      "latex": "\\Delta_j",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "二進殼層",
      "label_en": "Dyadic shell",
      "definition_zh": "第14節為證明 Fourier parent-scale lemma 引入的 dyadic Littlewood–Paley shell；與第3節累積型的 $P_{\\le J}/P_{>J}$ 不同，$\\Delta_j$ 恰好落在尺度 $j$，用於分析三元交互 $\\Delta_j\\mathbb P\\nabla\\cdot(\\Delta_{j_1}u\\otimes\\Delta_{j_2}u)$ 的 Fourier 支撐。",
      "definition_en": "The dyadic Littlewood–Paley shell introduced in Section 14 for the Fourier parent-scale lemma; unlike the cumulative $P_{\\le J}/P_{>J}$ of Section 3, $\\Delta_j$ isolates exactly scale $j$, used to analyze the Fourier support of triadic interactions $\\Delta_j\\mathbb P\\nabla\\cdot(\\Delta_{j_1}u\\otimes\\Delta_{j_2}u)$."
    },
    {
      "id": "ns.c1.c_0",
      "latex": "C_0",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "parent-scale 通用常數",
      "label_en": "Parent-scale universal constant",
      "definition_zh": "第14節 Fourier parent-scale lemma 中的 universal integer：若輸入殼層 $j_1,j_2<j-C_0$，輸出殼層 $j$ 的三元非線性項恆為零，故任何生成殼層 $j$ 的交互至少一 parent 滿足 $\\max\\{j_1,j_2\\}\\ge j-C_0$；第18節強調此僅為 support 事實，非 parent 振幅下界。",
      "definition_en": "The universal integer in the Section 14 Fourier parent-scale lemma: if both input shells satisfy $j_1,j_2<j-C_0$, the triadic term at output shell $j$ vanishes identically, so any interaction producing shell $j$ must have a parent with $\\max\\{j_1,j_2\\}\\ge j-C_0$; Section 18 stresses this is a pure support fact, not a parent-amplitude lower bound.",
      "defining_relation": "j_1,j_2<j-C_0\\ \\Rightarrow\\ \\Delta_j\\mathbb P\\nabla\\cdot(\\Delta_{j_1}u\\otimes\\Delta_{j_2}u)=0"
    },
    {
      "id": "ns.c1.prec_genealogy",
      "latex": "(j_1,t_1)\\prec(j_2,t_2)\\prec\\cdots",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "譜系先後序關係",
      "label_en": "Genealogy precedence ordering",
      "definition_zh": "第15節為描述仍未證的 C1c 引入的（尺度,時間）配對先後序記號，設想一條沿此序關係、使 $j_n\\to\\infty$ 的 nested causal branch；本輪只列出所需性質，未構造它，故 C1c 保持 OPEN。",
      "definition_en": "An ordering notation on (scale, time) pairs introduced in Section 15 to describe the still-open C1c: a hypothetical nested causal branch following this order with $j_n\\to\\infty$; this round only lists the required properties without constructing it, leaving C1c OPEN.",
      "notes": "Notation for an as-yet-unconstructed object, introduced to state what C1c would require rather than to define something that exists in this round."
    },
    {
      "id": "ns.c1.n_triad",
      "latex": "\\mathcal N_{j;j_1,j_2}^{(n)}",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "三元殼層分量",
      "label_en": "Triadic shell component",
      "definition_zh": "第17節分解 $\\mathcal N_n=\\sum_{j>J_n}\\sum_{j_1,j_2}\\mathcal N_{j;j_1,j_2}^{(n)}$ 中的分量，代表補給項 $\\mathcal N_n$ 中由輸入殼層 $j_1,j_2$ 生成輸出殼層 $j$ 的部分；Route A（C1c genealogy extraction）即嘗試從中抽出持續 parent branch。",
      "definition_en": "The component in the Section 17 decomposition $\\mathcal N_n=\\sum_{j>J_n}\\sum_{j_1,j_2}\\mathcal N_{j;j_1,j_2}^{(n)}$, representing the part of $\\mathcal N_n$ generated by input shells $j_1,j_2$ into output shell $j$; Route A (C1c genealogy extraction) attempts to extract a persistent parent branch from these.",
      "defining_relation": "\\mathcal N_n=\\sum_{j>J_n}\\sum_{j_1,j_2}\\mathcal N_{j;j_1,j_2}^{(n)}"
    },
    {
      "id": "ns.c1.cost",
      "latex": "\\operatorname{Cost}(\\mathcal N_n)",
      "series": "NS",
      "first_appearance": "C1",
      "label_zh": "強制性補給成本（提案）",
      "label_en": "Coercive replenishment cost (proposed)",
      "definition_zh": "第17節 Route B 提出但未定義的量：設想每次 UV 補給 $\\mathcal N_n$ 須付出某正的、可加總的 coercive cost；若 $\\sum_n\\operatorname{Cost}(\\mathcal N_n)=\\infty$ 卻與有限全域 budget 矛盾，將直接導向 C2（Finite Obstruction）。",
      "definition_en": "A quantity proposed but not defined in this round (Section 17, Route B): each UV replenishment $\\mathcal N_n$ is hypothesized to pay a positive, summable coercive cost; if $\\sum_n\\operatorname{Cost}(\\mathcal N_n)=\\infty$ could be shown to contradict a finite global budget, it would lead directly to C2 (Finite Obstruction).",
      "notes": "Placeholder/forward-looking notation; given a precise definition (if at all) only in a future C2 round, per the source text."
    },
    {
      "id": "ns.c2.u_q_dyadic",
      "latex": "u_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "第 q 層 dyadic 分量",
      "label_en": "dyadic (Littlewood–Paley) shell of u",
      "definition_zh": "第 1 節引入的 Littlewood–Paley dyadic 分解分量，$u_q=\\Delta_q u$ 表示 $u$ 在頻率尺度 $\\lambda_q$ 附近的部分，是全篇 dyadic shell 分析的基礎記號。",
      "definition_en": "The Littlewood–Paley dyadic block of u introduced in §1; u_q=Δ_q u isolates the component of u near frequency scale λ_q, underlying all dyadic-shell analysis in this round.",
      "defining_relation": "u_q=\\Delta_qu",
      "notes": "與 $\\lambda_q=2^q$ 配對使用，可能延續自前一輪 C1 的 dyadic 框架。"
    },
    {
      "id": "ns.c2.lambda_q",
      "latex": "\\lambda_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "dyadic 頻率尺度",
      "label_en": "dyadic frequency scale",
      "definition_zh": "第 1 節定義的標準 dyadic 頻率尺度，貫穿全文用於索引 shell、定義 $\\Lambda(t)$、$K_q$ 等量。",
      "definition_en": "The standard dyadic frequency scale defined in §1, used throughout to index shells and define Λ(t), K_q, and related quantities.",
      "defining_relation": "\\lambda_q=2^q",
      "notes": "抽象 ledger 模型（§14 起）重用同一公式但改記為 $\\lambda_n=2^n$，代表 toy model 而非真實解。"
    },
    {
      "id": "ns.c2.dissipation_wavenumber",
      "latex": "\\Lambda(t)",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "耗散波數",
      "label_en": "dissipation wavenumber",
      "definition_zh": "第 2 節引入（Cheskidov–Shvydkoy）的 time-dependent dissipation wavenumber：使所有 $p>q$ 的 shell amplitude 已被黏性吸收的最小 $\\lambda_q$。本輪核心外部輸入，定理 3.1 證明假想 blow-up 下 $\\Lambda\\in L^1(0,T_\\ast)\\setminus L^{5/2}(0,T_\\ast)$。",
      "definition_en": "Time-dependent dissipation wavenumber (Cheskidov–Shvydkoy, §2): the smallest λ_q beyond which shell amplitudes are already absorbed by viscosity. Central external input; Theorem 3.1 shows hypothetical blow-up forces Λ∈L¹(0,T*)\\L^{5/2}(0,T*).",
      "defining_relation": "\\Lambda(t)=\\min\\left\\{\\lambda_q:\\lambda_p^{-1}\\|u_p(t)\\|_\\infty<c_0\\nu,\\quad\\forall p>q\\right\\}",
      "notes": "$c_0$ 為定義中固定的 viscosity-normalized 小常數，僅在此處出現。"
    },
    {
      "id": "ns.c2.q_t",
      "latex": "Q(t)",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "耗散波數指標",
      "label_en": "dissipation wavenumber index",
      "definition_zh": "第 2 節由 $\\Lambda(t)=\\lambda_{Q(t)}$ 定義的整數指標；$q\\le Q(t)$ 表示該 shell 仍可能有 nonlinear/inertial 動力，也用於 §7 $K_q$ 定義中的指示函數 $1_{\\{q\\le Q(t)\\}}$。",
      "definition_en": "Integer index defined by Λ(t)=λ_{Q(t)} in §2; q≤Q(t) marks shells still potentially inertial/nonlinear, and appears in the indicator 1_{q≤Q(t)} inside K_q's definition (§7).",
      "defining_relation": "\\Lambda(t)=\\lambda_{Q(t)}"
    },
    {
      "id": "ns.c2.e_q",
      "latex": "E_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "耗散波數第 q 層佔用集",
      "label_en": "dyadic occupancy set of Λ",
      "definition_zh": "第 5 節定義：使 $\\Lambda(t)$ 落在第 $q$ 層 $[2^q,2^{q+1})$ 的時間集合，用以把定理 3.1 的積分條件轉譯為 dyadic spike-packing law（定理 5.1）。",
      "definition_en": "The set of times (§5) where Λ(t) lies in the q-th dyadic band [2^q,2^{q+1}); used to translate Theorem 3.1's integral conditions into the dyadic spike-packing law (Theorem 5.1).",
      "defining_relation": "E_q=\\left\\{t\\in(0,T_\\ast):2^q\\le\\Lambda(t)<2^{q+1}\\right\\}"
    },
    {
      "id": "ns.c2.m_q",
      "latex": "m_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "第 q 層佔用測度",
      "label_en": "occupancy measure",
      "definition_zh": "$E_q$ 的 Lebesgue 測度，第 5 節核心量；定理 5.1 證明 $(m_q)\\in\\ell^1(2^q)\\setminus\\ell^1(2^{5q/2})$，即 blow-up 必須滿足的 dyadic spike-packing law。",
      "definition_en": "Lebesgue measure of E_q (§5); Theorem 5.1's central quantity, showing (m_q)∈ℓ¹(2^q)\\ℓ¹(2^{5q/2}) is the packing law any hypothetical blow-up occupancy must obey.",
      "defining_relation": "m_q=|E_q|",
      "notes": "$\\ell^1(2^{aq})$ 指帶權重 $2^{aq}$ 的加權 $\\ell^1$ 空間，即 $\\sum_q2^{aq}m_q<\\infty$。"
    },
    {
      "id": "ns.c2.alpha_exponent",
      "latex": "\\alpha",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "占用測度冪次指數",
      "label_en": "occupancy power-law exponent",
      "definition_zh": "第 6 節作為 asymptotic diagnostic 引入，假設 $m_q\\sim2^{-\\alpha q}$；推出 hypothetical blow-up 的純冪次占用指數須落於 $1<\\alpha\\le5/2$，自然 parabolic window（$\\alpha=2$）恰落在允許域內。",
      "definition_en": "Diagnostic power-law exponent (§6) via m_q∼2^{-αq}; forces any pure power-law blow-up occupancy into 1<α≤5/2, with the natural parabolic window (α=2) landing inside this allowed band.",
      "defining_relation": "m_q\\sim2^{-\\alpha q}"
    },
    {
      "id": "ns.c2.parabolic_window_tau",
      "latex": "\\tau_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "自然拋物窗口時間尺度",
      "label_en": "natural parabolic time window",
      "definition_zh": "第 6 節提到的自然 parabolic 時間尺度 $\\tau_q\\sim\\lambda_q^{-2}$（對應 $\\alpha=2$），第 14 節在抽象 ledger 模型中被具體化為 $\\tau_n=\\lambda_n^{-2}=2^{-2n}$。",
      "definition_en": "The natural parabolic timescale τ_q∼λ_q^{-2} noted in §6 (corresponding to α=2); concretely instantiated in the abstract cascade ledger (§14) as τ_n=λ_n^{-2}=2^{-2n}.",
      "defining_relation": "\\tau_n=\\lambda_n^{-2}=2^{-2n}",
      "notes": "§6 用指標 q 作一般性討論；§14 起改用指標 n 作為抽象 toy-model ledger 的精確定義，該模型明言「不是 N–S solution」。"
    },
    {
      "id": "ns.c2.k_q",
      "latex": "K_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "高頻殼層臨界通行費",
      "label_en": "high-shell critical toll",
      "definition_zh": "第 7 節依 Cheskidov–Dai frequency-localized regularity criterion 定義的 dimensionless 量；定理 7.1（外部逆否）證明若 $T_\\ast$ 為 singular time，則沿無限多高頻殼層 $K_q\\gtrsim1$，是「每個高尺度須支付臨界通行費」的嚴格版本。",
      "definition_en": "Dimensionless high-shell quantity (§7) from the Cheskidov–Dai frequency-localized criterion; Theorem 7.1 (external contrapositive) shows a singular T* forces K_q≳1 along infinitely many high shells — the rigorous form of 'every high scale must pay a critical toll.'",
      "defining_relation": "K_q=\\int_{T/2}^{T}1_{\\{q\\le Q(t)\\}}\\lambda_q\\|u_q(t)\\|_\\infty\\,dt",
      "notes": "與抽象 ledger 中的類比量 $K_n$（§18）不同：$K_q$ 是對真實解成立的下界結果，$K_n$ 只是玩具模型中展示相容性的構造量。"
    },
    {
      "id": "ns.c2.c_star",
      "latex": "c_\\ast",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "臨界正則性門檻常數",
      "label_en": "critical regularity threshold constant",
      "definition_zh": "定理 7.1 中 Cheskidov–Dai 判準所允許的正則性門檻常數，$c_\\ast>0$；singular time 蘊含 $\\limsup_{q\\to\\infty}K_q>c_\\ast$。",
      "definition_en": "The regularity threshold constant c*>0 permitted by the Cheskidov–Dai criterion in Theorem 7.1; a singular time forces limsup_{q→∞} K_q > c*.",
      "defining_relation": "\\limsup_{q\\to\\infty}K_q>c_\\ast"
    },
    {
      "id": "ns.c2.u_lambda_scaling",
      "latex": "u_\\lambda",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "N–S 標準縮放",
      "label_en": "Navier–Stokes scaling transformation",
      "definition_zh": "第 9 節的標準 N–S scaling：若 $u$ 解方程，則 $u_\\lambda$ 仍解同一 viscosity-normalized 方程；驅動第 10–13 節 quadratic Sobolev cost 的 scale-blindness no-go。",
      "definition_en": "Standard N–S scaling (§9): if u solves the equation, so does u_λ under the same viscosity-normalized form; drives the scale-blindness no-go for quadratic Sobolev costs in §10–13.",
      "defining_relation": "u_\\lambda(x,t)=\\lambda u(\\lambda x,\\lambda^2t)"
    },
    {
      "id": "ns.c2.d_s_functional",
      "latex": "\\mathcal D_s",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "二次 Sobolev 代價泛函",
      "label_en": "quadratic Sobolev cost functional",
      "definition_zh": "第 10 節定義的候選 cost 泛函；在 N–S scaling 下滿足 $\\mathcal D_s[u_\\lambda;I_\\lambda]=\\lambda^{2s-3}\\mathcal D_s[u;I]$，是定理 11.1（$s<3/2$ no-go）與 §13 臨界 $s=3/2$ 討論的核心對象。",
      "definition_en": "Candidate cost functional (§10); under N–S scaling it obeys D_s[u_λ;I_λ]=λ^{2s-3}D_s[u;I], the central object behind Theorem 11.1's no-go (s<3/2) and the critical s=3/2 discussion in §13.",
      "defining_relation": "\\mathcal D_s[u;I]=\\int_I\\|u(t)\\|_{\\dot H^s}^2\\,dt",
      "notes": "$s=1$ 特例即 $\\mathcal D_{\\mathrm{energy}}$（§12），$s=3/2$ 特例即臨界 $\\mathcal D_{3/2}$（§13）。"
    },
    {
      "id": "ns.c2.i_lambda",
      "latex": "I_\\lambda",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "縮放後時間區間",
      "label_en": "rescaled time interval",
      "definition_zh": "第 10 節配合 N–S scaling 對時間區間 $I$ 的縮放定義，使 $\\mathcal D_s[u_\\lambda;I_\\lambda]$ 的 scaling law 得以成立。",
      "definition_en": "The rescaling of a time interval I accompanying N–S scaling (§10), needed for the scaling law of D_s[u_λ;I_λ] to hold.",
      "defining_relation": "I_\\lambda=\\lambda^{-2}I"
    },
    {
      "id": "ns.c2.event_class",
      "latex": "\\mathcal E",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "尺度不變事件類",
      "label_en": "scale-invariant event class",
      "definition_zh": "第 11 節定理 11.1 所考慮的抽象事件類，對 N–S scaling 不變；定理證明當 $s<3/2$ 時，不存在僅由 $\\mathcal E$ 本身推出的 scale-independent 正下界常數 $c$ 使 $\\mathcal D_s\\ge c$。",
      "definition_en": "The abstract N–S-scaling-invariant event class considered in Theorem 11.1 (§11); the theorem shows no scale-independent positive lower bound c with D_s≥c can be derived from E alone when s<3/2."
    },
    {
      "id": "ns.c2.d_energy",
      "latex": "\\mathcal D_{\\mathrm{energy}}",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "標準能量耗散代價",
      "label_en": "standard energy dissipation cost",
      "definition_zh": "第 12 節即 $\\mathcal D_s$ 在 $s=1$ 之特例，滿足 $\\mathcal D_{\\mathrm{energy}}[u_\\lambda;I_\\lambda]=\\lambda^{-1}\\mathcal D_{\\mathrm{energy}}[u;I]\\to0$（$\\lambda\\to\\infty$），正式淘汰「固定正 energy-dissipation cost」的通行費策略。",
      "definition_en": "The s=1 special case of D_s (§12); D_energy[u_λ;I_λ]=λ^{-1}D_energy[u;I]→0 as λ→∞, formally ruling out any fixed-positive energy-dissipation toll strategy.",
      "defining_relation": "\\mathcal D_{\\mathrm{energy}}=\\nu\\int_I\\|\\nabla u(t)\\|_2^2dt"
    },
    {
      "id": "ns.c2.cost_n",
      "latex": "\\operatorname{Cost}_n",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "逐事件加法代價（前輪策略）",
      "label_en": "per-event additive cost (prior-round strategy)",
      "definition_zh": "第 12 節回顧上一輪最自然的策略：$\\sum_n\\operatorname{Cost}_n\\le\\frac12\\|u_0\\|_2^2$ 且 $\\operatorname{Cost}_n\\ge c>0$；本輪證明高頻下 $\\operatorname{Cost}_n$ 可如 $2^{-J_n}$ 般縮小，策略不成立。",
      "definition_en": "The additive per-event cost from the prior round's most natural strategy (§12): Σ Cost_n ≤ ½‖u_0‖²₂ with Cost_n≥c>0; this round shows Cost_n can shrink like 2^{-J_n} at high frequency, defeating the strategy.",
      "defining_relation": "\\sum_n\\operatorname{Cost}_n\\le\\frac12\\|u_0\\|_2^2",
      "notes": "$J_n$ 記號延續自 C1 輪。"
    },
    {
      "id": "ns.c2.d_3_2",
      "latex": "\\mathcal D_{3/2}",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "臨界二次 Sobolev 代價",
      "label_en": "critical quadratic Sobolev cost",
      "definition_zh": "第 13 節即 $\\mathcal D_s$ 在 $s=3/2$ 之特例，是唯一 scale-independent 的二次代價，但 standard energy inequality 並不控制它，形成 criticality wall。",
      "definition_en": "The s=3/2 special case of D_s (§13), the unique scale-independent quadratic cost; the standard energy inequality does not control it, producing a criticality wall.",
      "defining_relation": "\\mathcal D_{3/2}[u_\\lambda;I_\\lambda]=\\mathcal D_{3/2}[u;I]"
    },
    {
      "id": "ns.c2.ledger_lambda_n",
      "latex": "\\lambda_n",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "抽象 ledger 尺度",
      "label_en": "abstract ledger scale",
      "definition_zh": "第 14 節抽象 geometric cascade ledger（純 scalar toy model，明言「不是 N–S solution」）所用的 dyadic 尺度記號，與 §1 的 $\\lambda_q$ 公式相同但索引改為 $n$，代表獨立於真實解的構造。",
      "definition_en": "The dyadic scale used in the abstract geometric cascade ledger (§14) — a pure scalar toy model explicitly flagged as 'not an N–S solution' — formally identical to λ_q (§1) but reindexed by n.",
      "defining_relation": "\\lambda_n=2^n"
    },
    {
      "id": "ns.c2.ledger_u_n",
      "latex": "U_n",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "ledger 特徵速度振幅",
      "label_en": "ledger characteristic velocity amplitude",
      "definition_zh": "第 14 節 ledger 模型賦予每個尺度的特徵速度振幅，符合 N–S critical scaling，用於構造 §18 的 $K_n$。",
      "definition_en": "Characteristic velocity amplitude assigned to each scale in the ledger model (§14), consistent with N–S critical scaling; used to build K_n (§18).",
      "defining_relation": "U_n\\sim\\lambda_n"
    },
    {
      "id": "ns.c2.ledger_k_n",
      "latex": "K_n",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "ledger 臨界通行費類比量",
      "label_en": "ledger critical-toll analog",
      "definition_zh": "第 18 節（Ledger D）在 toy model 中構造的 $K_q$ 類比量，代入 $U_n\\sim\\lambda_n,\\tau_n=\\lambda_n^{-2}$ 得 $K_n\\asymp1$，顯示每個尺度都可支付 order-one 通行費而不與其餘 ledger 條件矛盾。",
      "definition_en": "The K_q-analog constructed in the toy model (§18, Ledger D); substituting U_n∼λ_n, τ_n=λ_n^{-2} gives K_n≍1, showing every scale can pay an order-one toll without contradicting the other ledger constraints.",
      "defining_relation": "K_n\\sim\\lambda_n^2\\lambda_n^{-2}=1"
    },
    {
      "id": "ns.c2.ledger_d_n",
      "latex": "D_n",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "ledger 能量耗散類比量",
      "label_en": "ledger energy-dissipation analog",
      "definition_zh": "第 19 節（Ledger E）依 scaling theorem 賦予每個尺度的 ordinary energy-dissipation cost，使 $\\sum_nD_n<\\infty$ 與 $K_n\\asymp1$、$\\Lambda\\notin L^{5/2}$ 等條件同時相容，構成 C2e no-go 的核心構造。",
      "definition_en": "The ordinary energy-dissipation cost per scale in the ledger (§19, Ledger E), assigned via the scaling theorem so Σ D_n<∞ holds simultaneously with K_n≍1 and Λ∉L^{5/2} — the core construction behind the C2e no-go.",
      "defining_relation": "D_n\\sim\\lambda_n^{-1}",
      "notes": "不要與 §10 的 $\\mathcal D_s$（花體 D）或 §23 ETN 中的 $D_q$ 混淆，三者記號相近但脈絡不同。"
    },
    {
      "id": "ns.c2.tao_averaged_b",
      "latex": "\\widetilde B(u,u)",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "Tao 平均化雙線性算子",
      "label_en": "Tao's averaged bilinear operator",
      "definition_zh": "第 21 節引用 Tao 構造之 averaged bilinear operator，滿足 $\\langle\\widetilde B(u,u),u\\rangle=0$（保留能量消去），但對應的 averaged 3D N–S 方程可產生 finite-time blow-up，說明「energy identity + 泛型調和分析結構」不足以證明真正 N–S regularity，與本輪 C2 no-go 同向。",
      "definition_en": "Tao's averaged bilinear operator (§21), satisfying ⟨B̃(u,u),u⟩=0 (preserving energy cancellation) yet admitting finite-time blow-up in the corresponding averaged 3D N–S equation — evidence that energy identity plus generic harmonic-analysis structure cannot suffice, echoing this round's C2 no-go.",
      "defining_relation": "\\langle\\widetilde B(u,u),u\\rangle=0"
    },
    {
      "id": "ns.c2.g_n_gate",
      "latex": "\\mathsf G_n",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "跨尺度資格審核關係",
      "label_en": "cross-scale qualification (gating) relation",
      "definition_zh": "第 22 節重新定位 X 積分角色所提出的關係：檢驗事件 $n$ 是否有資格把來源結構傳給事件 $n+1$，取代原先「逐事件付費、預算耗盡」的加法圖像，開啟 C3 cross-scale coupling rigidity 的研究方向。",
      "definition_en": "The relation proposed in §22 repositioning the role of the X-integral: it checks whether event n qualifies to pass source structure to event n+1, replacing the earlier additive 'pay-per-event, exhaust the budget' picture and opening the C3 direction.",
      "defining_relation": "\\mathsf G_n\\left(X_n,\\rho_{n\\to n+1},X_{n+1}\\right)",
      "notes": "$X_n$ 為 X 積分狀態（定義見 X_Integral 系列文件，本輪僅重新使用/定位），$\\rho_{n\\to n+1}$ 為本輪新提出的尺度間傳遞關係。"
    },
    {
      "id": "ns.c2.rho_transfer",
      "latex": "\\rho_{n\\to n+1}",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "尺度間傳遞係數",
      "label_en": "inter-scale transfer coefficient",
      "definition_zh": "第 22 節作為 $\\mathsf G_n$ 的一個變元引入，代表事件 $n$ 到事件 $n+1$ 之間的傳遞/耦合關係，須整合 parent provenance、frequency support、triad geometry 等結構性條件而非單純 scalar amplitude。",
      "definition_en": "Introduced in §22 as an argument of G_n, representing the transfer/coupling relation from event n to event n+1; meant to encode structural conditions (parent provenance, frequency support, triad geometry, etc.) rather than a bare scalar amplitude."
    },
    {
      "id": "ns.c2.theta_q_typed",
      "latex": "\\Theta_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "升級版 typed 張力狀態",
      "label_en": "upgraded typed tension state",
      "definition_zh": "第 23 節將 True ETN 每殼層狀態從純 scalar pair $(T_q,D_q)$ 升級為六元組 typed tension relation，明言真正的 tension 不再是 scalar 而是 typed multiscale relation。",
      "definition_en": "Section 23 upgrades True ETN's per-shell state from the coarse scalar pair (T_q,D_q) into a six-component typed tension relation, asserting that genuine tension is no longer a scalar but a typed multiscale relation.",
      "defining_relation": "\\Theta_q=\\left\\langle A_q,D_q,R_q,\\mathcal P_q,\\mathcal G_q,\\mathcal S_q\\right\\rangle",
      "notes": "舊版 $\\Theta_q=(T_q,D_q)$ 來自先前 ETN 文件，被本節判定「仍太粗」而升級。"
    },
    {
      "id": "ns.c2.a_q",
      "latex": "A_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "振幅狀態",
      "label_en": "amplitude state",
      "definition_zh": "第 23 節 $\\Theta_q$ 六元組的第一分量，代表殼層 $q$ 的振幅狀態。",
      "definition_en": "First component of the Θ_q sextuple (§23), the amplitude state of shell q."
    },
    {
      "id": "ns.c2.d_q_etn",
      "latex": "D_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "黏性耗散（ETN 分量）",
      "label_en": "viscous dissipation (ETN component)",
      "definition_zh": "第 23 節 $\\Theta_q$ 六元組分量，代表殼層 $q$ 的黏性耗散；亦是舊版 $\\Theta_q=(T_q,D_q)$ 中留存的分量。",
      "definition_en": "A component of the Θ_q sextuple (§23), the viscous dissipation of shell q; also the surviving component from the old pair Θ_q=(T_q,D_q).",
      "notes": "與 §10 的 $\\mathcal D_s$、§19 的 $D_n$ 記號相近但脈絡不同，勿混淆。"
    },
    {
      "id": "ns.c2.r_q",
      "latex": "R_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "非線性補充",
      "label_en": "nonlinear replenishment",
      "definition_zh": "第 23 節 $\\Theta_q$ 六元組分量，代表殼層 $q$ 的 nonlinear replenishment，概念上呼應 C1 輪的 UV replenishment。",
      "definition_en": "A component of the Θ_q sextuple (§23), the nonlinear replenishment at shell q — conceptually echoing the UV replenishment of round C1."
    },
    {
      "id": "ns.c2.p_q",
      "latex": "\\mathcal P_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "來源結構分量",
      "label_en": "provenance / parent-structure component",
      "definition_zh": "第 23 節 $\\Theta_q$ 六元組分量，記錄殼層 $q$ 的 provenance / parent structure，對應 §22 $\\mathsf G_n$ 所要審核的 parent provenance 條件。",
      "definition_en": "A component of the Θ_q sextuple (§23) recording provenance / parent structure of shell q, corresponding to the parent-provenance check envisioned for G_n in §22."
    },
    {
      "id": "ns.c2.g_q_triad",
      "latex": "\\mathcal G_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "triad 幾何分量",
      "label_en": "triad geometry component",
      "definition_zh": "第 23 節 $\\Theta_q$ 六元組分量，記錄殼層 $q$ 的 triad geometry；字體與 §22 的關係函數 $\\mathsf G_n$ 不同，不應混淆。",
      "definition_en": "A component of the Θ_q sextuple (§23) recording triad geometry of shell q; a different typeface from the gating relation G_n of §22 — not to be confused with it."
    },
    {
      "id": "ns.c2.s_q",
      "latex": "\\mathcal S_q",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "空間集中度分量",
      "label_en": "spatial concentration / support-state component",
      "definition_zh": "第 23 節 $\\Theta_q$ 六元組分量，記錄殼層 $q$ 的空間集中度／support state。",
      "definition_en": "A component of the Θ_q sextuple (§23), recording the spatial concentration / support state of shell q."
    },
    {
      "id": "ns.c2.leray_nonlinear",
      "latex": "\\mathbb P\\nabla\\cdot(u\\otimes u)",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "真實 N–S（Leray 投影）非線性項",
      "label_en": "true Leray-projected N–S nonlinear operator",
      "definition_zh": "第 24 節 C3-C 分支所指的「true N–S symbol」，即 Leray 投影 $\\mathbb P$ 作用下的 $\\nabla\\cdot(u\\otimes u)$；用以與 Tao 的 averaged bilinear operator（§21）對比，追問其 incompressibility 結構是否禁止 averaged-model 式的完美 cascade wiring。",
      "definition_en": "The 'true N–S symbol' referenced in the C3-C branch (§24): the Leray-projected nonlinear term P∇·(u⊗u), contrasted with Tao's averaged bilinear operator (§21) to ask whether its incompressibility structure forbids averaged-model-style perfect cascade wiring."
    },
    {
      "id": "ns.c2.c3_label",
      "latex": "\\mathrm{C3}",
      "series": "NS",
      "first_appearance": "C2",
      "label_zh": "C3：跨尺度耦合剛性",
      "label_en": "C3: Cross-Scale Coupling Rigidity",
      "definition_zh": "第 24 節正式命名的下一輪研究主題：目標不再是證明 $\\sum_n\\text{cost}_n=\\infty$，而是證明任意 hypothetical blow-up chain 必須滿足某些不相容的 cross-scale constraints；本輪 C2 的 scalar additive 路線正式降級為此。",
      "definition_en": "The next research round formally named in §24: instead of proving Σcost_n=∞, C3 aims to show any hypothetical blow-up chain must satisfy mutually incompatible cross-scale constraints. This round's (C2) scalar-additive route is formally downgraded in favor of C3.",
      "defining_relation": "\\mathrm{C3}=\\textbf{Cross-Scale Coupling Rigidity}"
    },
    {
      "id": "ns.c3.c3a.u_lambda_scaling",
      "latex": "u_\\lambda(x,t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "拋物型重新縮放",
      "label_en": "Parabolic rescaling",
      "definition_zh": "第1節重述標準 Navier–Stokes 拋物縮放，用以推導使量綱平衡的臨界 Sobolev 指數 $s=1/2$，是本輪 critical $\\dot H^{1/2}$ 論證的出發點。",
      "definition_en": "Section 1 restates the standard NS parabolic rescaling of the velocity field, used to derive the scaling-critical Sobolev exponent $s=1/2$ that anchors this round's $\\dot H^{1/2}$ argument.",
      "defining_relation": "u_\\lambda(x,t) = \\lambda u(\\lambda x,\\lambda^2 t)",
      "notes": "屬標準 NS scaling，非 C3-A 原創，此處重述以自成一體地推出 critical exponent；同一縮放也用於第7節 helicity 不變性 $H[u_\\lambda]=H[u]$ 及第17節 $\\mathcal R_\\lambda$ 的 scaling 性質。"
    },
    {
      "id": "ns.c3.c3a.energy_e",
      "latex": "E(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "動能",
      "label_en": "Kinetic energy",
      "definition_zh": "第2節定義的標準動能，滿足 $\\frac{d}{dt}E+\\nu\\|\\nabla u\\|_2^2=0$；本輪中的角色是作為第8節 Trilemma 三個 quantity 之一，代表 positive + nonlinearly conserved 但 subcritical 的一角。",
      "definition_en": "The standard kinetic energy defined in Section 2, obeying $\\frac{d}{dt}E+\\nu\\|\\nabla u\\|_2^2=0$; its role here is as one vertex of the Section 8 Trilemma, exemplifying positive + nonlinearly conserved but subcritical.",
      "defining_relation": "E(t) = \\frac12\\|u(t)\\|_2^2",
      "notes": "與 u, p, nu 等同屬系列既有基礎量，此輪新意在於把它明確放入三難表格中與 helicity、$\\dot H^{1/2}$ 比較。"
    },
    {
      "id": "ns.c3.c3a.freq_cutoff_proj",
      "latex": "P_{\\le K}, P_{>K}",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "頻率截斷投影",
      "label_en": "Frequency cutoff projectors",
      "definition_zh": "第3節引入的 $L^2$-正交 Fourier 截斷投影算子，在頻率 $K$ 處把 $u$ 分成低頻與高頻部分，是本輪 exact flux antisymmetry 論證的基礎工具。",
      "definition_en": "The $L^2$-orthogonal Fourier cutoff projectors introduced in Section 3, splitting $u$ at frequency $K$ into low- and high-frequency parts; the basic tool behind this round's exact flux-antisymmetry argument.",
      "notes": "與第10節螺旋投影 $P^\\pm$、貫穿全文的 Leray 投影 $\\mathbb P$ 是三種不同的投影記號，勿混淆。"
    },
    {
      "id": "ns.c3.c3a.u_low_high",
      "latex": "u_L, u_H",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "低頻／高頻速度場",
      "label_en": "Low/high-frequency velocity parts",
      "definition_zh": "第3節由截斷投影定義的 $u$ 之低頻分量 $u_L$ 與高頻分量 $u_H$，用以建構 $E_L,E_H$ 與 flux $\\Pi_K$。",
      "definition_en": "The low- and high-frequency components of $u$ defined in Section 3 via the cutoff projectors, used to build $E_L, E_H$, and the flux $\\Pi_K$.",
      "defining_relation": "u_L=P_{\\le K}u, \\qquad u_H=P_{>K}u"
    },
    {
      "id": "ns.c3.c3a.energy_low_high",
      "latex": "E_L, E_H",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "低頻／高頻能量",
      "label_en": "Low/high-frequency energy",
      "definition_zh": "第3節定義，將動能依頻率 $K$ 分成低頻能量 $E_L$ 與高頻能量 $E_H$，兩者透過 flux $\\Pi_K$ 精確反向耦合。",
      "definition_en": "Defined in Section 3 by splitting kinetic energy at frequency $K$; $E_L$ and $E_H$ are exactly, oppositely coupled through the flux $\\Pi_K$.",
      "defining_relation": "E_L=\\frac12\\|u_L\\|_2^2, \\qquad E_H=\\frac12\\|u_H\\|_2^2"
    },
    {
      "id": "ns.c3.c3a.nonlinear_n",
      "latex": "N(u)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "投影非線性項",
      "label_en": "Projected nonlinear (inertial) term",
      "definition_zh": "第3節為 Leray 投影後的對流非線性項 $\\mathbb P(u\\cdot\\nabla u)$ 引入的簡寫，用於定義 flux $\\Pi_K=-\\langle N(u),u_H\\rangle$。",
      "definition_en": "A shorthand introduced in Section 3 for the Leray-projected convective nonlinear term $\\mathbb P(u\\cdot\\nabla u)$, used to define the flux $\\Pi_K=-\\langle N(u),u_H\\rangle$.",
      "defining_relation": "N(u)=\\mathbb P(u\\cdot\\nabla u)",
      "notes": "此簡寫僅在第3節使用；第6節的 $\\mathcal P_{\\mathrm{crit}}$ 與第10節的 $\\mathcal R_\\pm$ 雖用到同一 $\\mathbb P(u\\cdot\\nabla u)$ 表達式，卻未再沿用 $N(u)$ 記號而是重新展開書寫。"
    },
    {
      "id": "ns.c3.c3a.flux_pi_k",
      "latex": "\\Pi_K(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "跨截斷頻率能量通量",
      "label_en": "Flux into high frequencies across cutoff K",
      "definition_zh": "第3節定義的跨頻率 $K$ 能量通量，滿足 $\\frac{dE_H}{dt}+\\nu\\|\\nabla u_H\\|_2^2=\\Pi_K$ 且 $\\frac{dE_L}{dt}+\\nu\\|\\nabla u_L\\|_2^2=-\\Pi_K$。",
      "definition_en": "The energy flux across cutoff frequency $K$ defined in Section 3, satisfying $\\frac{dE_H}{dt}+\\nu\\|\\nabla u_H\\|_2^2=\\Pi_K$ and $\\frac{dE_L}{dt}+\\nu\\|\\nabla u_L\\|_2^2=-\\Pi_K$.",
      "defining_relation": "\\Pi_K(t) = -\\langle N(u),u_H\\rangle",
      "notes": "第15節守衛清單第7點明確指出 pair-production rate $\\mathcal R$ 不能被 $\\Pi_K$ 取代，兩者性質不同（$\\mathcal R$ 不是 energy flux）。"
    },
    {
      "id": "ns.c3.c3a.flux_antisymmetry",
      "latex": "\\text{high-side gain}=\\text{low-side nonlinear loss}",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "低高頻能量通量精確反對稱恆等式",
      "label_en": "Exact low/high energy flux antisymmetry identity",
      "definition_zh": "第3節 boxed 的結構性結論：跨任意截斷 $K$ 的非線性能量轉移在 $L^2$ energy 層級精確反對稱，文中稱為 C3 的第一個 exact parent-depletion identity；第19節狀態表列為 PROVED。",
      "definition_en": "The boxed structural conclusion of Section 3: nonlinear energy transfer across any cutoff $K$ is exactly antisymmetric at the $L^2$ energy level, termed this round's first exact parent-depletion identity; listed PROVED in the Section 19 status table.",
      "defining_relation": "\\frac{dE_H}{dt}+\\nu\\|\\nabla u_H\\|_2^2=\\Pi_K, \\qquad \\frac{dE_L}{dt}+\\nu\\|\\nabla u_L\\|_2^2=-\\Pi_K",
      "notes": "第4節隨即證明此恆等式雖真，仍不足以對 C1 的 critical UV replenishment 構成約束。"
    },
    {
      "id": "ns.c3.c3a.v_lambda_a",
      "latex": "v_{\\lambda,A}(x)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "雙參數重縮放測試場",
      "label_en": "Two-parameter rescaled test field",
      "definition_zh": "第4節由固定、Fourier support 位於 unit annulus 的 divergence-free Schwartz 場 $v$ 構造的雙參數（振幅 $A$、尺度 $\\lambda$）測試場族，證明可同時令 $\\|v_{\\lambda,A}\\|_3=M$ 任意大而 $\\|v_{\\lambda,A}\\|_2^2<\\varepsilon$ 任意小；命題4.1並排除任何形如 $\\|P_{>K}u\\|_3\\ge M\\Rightarrow\\|P_{>K}u\\|_2^2\\ge c(M)$ 的 uniform lower bound。",
      "definition_en": "A two-parameter (amplitude $A$, scale $\\lambda$) rescaled family in Section 4, built from a fixed divergence-free Schwartz field $v$ supported in the unit annulus; shows $\\|v_{\\lambda,A}\\|_3=M$ can be made arbitrarily large while $\\|v_{\\lambda,A}\\|_2^2<\\varepsilon$ stays arbitrarily small, and Proposition 4.1 rules out any uniform bound $\\|P_{>K}u\\|_3\\ge M\\Rightarrow\\|P_{>K}u\\|_2^2\\ge c(M)$.",
      "defining_relation": "v_{\\lambda,A}(x) = A\\lambda v(\\lambda x)",
      "notes": "此處振幅參數 $A$ 與第5節起代表 critical quadratic size 的 $A(t)$ 是同一字母的不同用途，務必區分。第19節狀態表對應行「$L^3$ large with arbitrarily small $L^2$ energy: PROVED」。"
    },
    {
      "id": "ns.c3.c3a.critical_size_a",
      "latex": "A(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "臨界正定二次量",
      "label_en": "Critical positive quadratic size",
      "definition_zh": "第5節定義的 scaling-critical 正定量，經 Sobolev embedding $\\dot H^{1/2}\\hookrightarrow L^3$ 把 C1 的 $L^3$ blow-up 判準轉譯成 $\\limsup_{t\\uparrow T_\\ast}A(t)=\\infty$；第9節進一步證明 $A=H_++H_-$。",
      "definition_en": "The scaling-critical positive quantity defined in Section 5; via $\\dot H^{1/2}\\hookrightarrow L^3$ it translates C1's $L^3$ blow-up criterion into $\\limsup_{t\\uparrow T_\\ast}A(t)=\\infty$, and Section 9 shows $A=H_++H_-$.",
      "defining_relation": "A(t) = \\|u(t)\\|_{\\dot H^{1/2}}^2 = H_+(t)+H_-(t)",
      "notes": "與第4節 $v_{\\lambda,A}$ 中的振幅 $A$ 為同名不同義的獨立記號。"
    },
    {
      "id": "ns.c3.c3a.p_crit",
      "latex": "\\mathcal P_{\\mathrm{crit}}(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "臨界非線性生成項",
      "label_en": "Critical nonlinear production term",
      "definition_zh": "第6節定義，出現在 $\\frac12\\frac{d}{dt}\\|u\\|_{\\dot H^{1/2}}^2+\\nu\\|u\\|_{\\dot H^{3/2}}^2=\\mathcal P_{\\mathrm{crit}}$ 中，一般不為零，顯示 $\\dot H^{1/2}$ 雖 positive+critical 卻失去 exact nonlinear conservation。",
      "definition_en": "Defined in Section 6, appearing in $\\frac12\\frac{d}{dt}\\|u\\|_{\\dot H^{1/2}}^2+\\nu\\|u\\|_{\\dot H^{3/2}}^2=\\mathcal P_{\\mathrm{crit}}$; generically nonzero, showing $\\dot H^{1/2}$ is positive+critical but lacks exact nonlinear conservation.",
      "defining_relation": "\\mathcal P_{\\mathrm{crit}} = -\\left\\langle |D|u, \\mathbb P(u\\cdot\\nabla u) \\right\\rangle",
      "notes": "是第10–11節 sector-resolved $\\mathcal R_\\pm$／$\\mathcal R$ 的整場（未分 helicity sector）前身；本文未明示兩者精確代數關係，但扮演類似角色（皆為伴隨耗散項出現的非線性生成項）。"
    },
    {
      "id": "ns.c3.c3a.helicity_h",
      "latex": "H(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "螺度",
      "label_en": "Helicity",
      "definition_zh": "第7節定義的標準螺度泛函，非線性貢獻恆為零（$\\frac{dH}{dt}=-2\\nu\\int\\omega\\cdot(\\nabla\\times\\omega)dx$）且在 $u_\\lambda$ 縮放下不變，但不具正定性；是 Trilemma 三量之一。",
      "definition_en": "The standard helicity functional defined in Section 7; its nonlinear contribution vanishes identically ($\\frac{dH}{dt}=-2\\nu\\int\\omega\\cdot(\\nabla\\times\\omega)dx$) and it is scale-invariant under $u_\\lambda$, but it is sign-indefinite — one vertex of the Trilemma.",
      "defining_relation": "H(t) = \\int_{\\mathbb R^3} u\\cdot\\omega\\,dx",
      "notes": "第9節進一步分解為 $H=H_+-H_-$。"
    },
    {
      "id": "ns.c3.c3a.trilemma",
      "latex": "\\text{positive}+\\text{critical}+\\text{nonlinearly conserved}",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "守恆—臨界—正定三難",
      "label_en": "Conservation–Criticality–Positivity Trilemma",
      "definition_zh": "第8節本輪核心結構性結果：$\\|u\\|_2^2$、helicity $H$、$\\|u\\|_{\\dot H^{1/2}}^2$ 三個最自然的 quadratic quantity 各自只具備 positive／critical／conserved 三性質中的兩個，沒有一個同時具備三者，迫使研究從 scalar invariant 轉向 signed paired（helical sector）structure。",
      "definition_en": "This round's central structural result (Section 8): the three most natural quadratic quantities — $\\|u\\|_2^2$, helicity $H$, and $\\|u\\|_{\\dot H^{1/2}}^2$ — each have only two of the three properties (positive, scaling-critical, nonlinearly conserved); none has all three, forcing the shift from scalar invariants to signed paired (helical-sector) structure developed from Section 9 onward.",
      "notes": "第19節狀態表列為 PROVED/STRUCTURAL；為全篇標題概念，本質是三量比較表的結論而非單一公式，此處以其自身 boxed 文字作為 latex 記號。"
    },
    {
      "id": "ns.c3.c3a.helical_eigenbasis",
      "latex": "h^\\pm(\\xi)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "螺旋本徵基底",
      "label_en": "Helical eigenbasis",
      "definition_zh": "第9節引入的 Fourier space curl 算子本徵向量場，滿足 $i\\xi\\times h^\\pm=\\pm|\\xi|h^\\pm$，是 helical decomposition（分解 $u=u^++u^-$）的基底。",
      "definition_en": "The Fourier-space eigenvectors of the curl operator introduced in Section 9, satisfying $i\\xi\\times h^\\pm=\\pm|\\xi|h^\\pm$; the basis underlying the helical decomposition $u=u^++u^-$.",
      "defining_relation": "i\\xi\\times h^\\pm(\\xi) = \\pm|\\xi|h^\\pm(\\xi)"
    },
    {
      "id": "ns.c3.c3a.helical_components",
      "latex": "u^\\pm",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "正／負螺旋分量",
      "label_en": "Positive/negative helical components",
      "definition_zh": "第9節將 $u$ 依 $h^\\pm(\\xi)$ 基底分解出的兩個分量，各自為 curl 算子的正／負本徵場，是全篇 sector-resolved 論證（$H_\\pm,D_\\pm,\\mathcal R_\\pm$）的基礎物件。",
      "definition_en": "The two components into which Section 9 splits $u$ via the $h^\\pm(\\xi)$ basis, each a positive/negative eigenfield of the curl operator; the foundational objects for all sector-resolved quantities ($H_\\pm,D_\\pm,\\mathcal R_\\pm$) in the paper.",
      "defining_relation": "u=u^++u^-, \\qquad \\nabla\\times u^\\pm=\\pm|D|u^\\pm"
    },
    {
      "id": "ns.c3.c3a.sector_helicity_pm",
      "latex": "H_+(t), H_-(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "正／負螺旋扇區臨界量",
      "label_en": "Positive/negative-sector critical sizes (sector helicities)",
      "definition_zh": "第9節定義為各螺旋分量的 $\\dot H^{1/2}$ 範數平方，滿足 $H=H_+-H_-$（重建螺度）與 $A=H_++H_-$（重建臨界正定量），是本輪把 scalar invariant 拆成 signed paired structure 的關鍵記號。",
      "definition_en": "Defined in Section 9 as the $\\dot H^{1/2}$-norm-squared of each helical component; satisfy $H=H_+-H_-$ (recovering helicity) and $A=H_++H_-$ (recovering the critical positive size) — the key notation by which this round splits a scalar invariant into signed paired structure.",
      "defining_relation": "H_+(t) = \\|u^+(t)\\|_{\\dot H^{1/2}}^2, \\quad H_-(t) = \\|u^-(t)\\|_{\\dot H^{1/2}}^2, \\quad H=H_+-H_-, \\quad A=H_++H_-",
      "notes": "文中稱為「positive sector helicities」，但其定義本身是 $\\dot H^{1/2}$ 範數而非螺度積分，命名與第7節的 $H$ 相關但構造不同，讀者需留意此處的雙重意涵。"
    },
    {
      "id": "ns.c3.c3a.helical_projectors",
      "latex": "P^\\pm",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "螺旋投影算子",
      "label_en": "Helical projectors",
      "definition_zh": "第10節引入，對應 $h^\\pm(\\xi)$ 基底的投影算子，用於從非線性項 $\\mathbb P(u\\cdot\\nabla u)$ 中取出各 sector 的分量以定義 $\\mathcal R_\\pm$。",
      "definition_en": "Introduced in Section 10 as the projectors associated with the $h^\\pm(\\xi)$ basis, used to extract each sector's component from the nonlinear term $\\mathbb P(u\\cdot\\nabla u)$ in defining $\\mathcal R_\\pm$.",
      "notes": "與第3節的頻率截斷投影 $P_{\\le K}/P_{>K}$、Leray 投影 $\\mathbb P$ 為三種不同投影記號，本文未給出 $P^\\pm$ 的顯式公式。"
    },
    {
      "id": "ns.c3.c3a.sector_dissipation_pm",
      "latex": "D_\\pm",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "扇區耗散量",
      "label_en": "Sector dissipation",
      "definition_zh": "第10節定義，各螺旋 sector 在 $\\dot H^{3/2}$ 層級的耗散量，出現於 $\\frac12H_\\pm'+\\nu D_\\pm=\\mathcal R_\\pm$ 及第11節 pair-production identity 的 $\\nu(D_++D_-)$ 項。",
      "definition_en": "Defined in Section 10 as each helical sector's dissipation at the $\\dot H^{3/2}$ level, appearing in $\\frac12H_\\pm'+\\nu D_\\pm=\\mathcal R_\\pm$ and in the $\\nu(D_++D_-)$ term of the Section 11 pair-production identity.",
      "defining_relation": "D_\\pm = \\|u^\\pm\\|_{\\dot H^{3/2}}^2"
    },
    {
      "id": "ns.c3.c3a.sector_production_pm",
      "latex": "\\mathcal R_\\pm",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "扇區非線性生成項",
      "label_en": "Sector nonlinear production terms",
      "definition_zh": "第10節定義的各 sector 非線性生成率，出現在 $\\frac12H_\\pm'+\\nu D_\\pm=\\mathcal R_\\pm$ 中；因非線性螺度守恆推出 $\\mathcal R_+=\\mathcal R_-$（第19節列 PROVED），此共同值即定義為 $\\mathcal R$。",
      "definition_en": "The per-sector nonlinear production rates defined in Section 10, appearing in $\\frac12H_\\pm'+\\nu D_\\pm=\\mathcal R_\\pm$; nonlinear helicity conservation forces $\\mathcal R_+=\\mathcal R_-$ (PROVED per Section 19), and this common value is what defines $\\mathcal R$.",
      "defining_relation": "\\mathcal R_\\pm = -\\left\\langle |D|u^\\pm, P^\\pm\\mathbb P(u\\cdot\\nabla u)\\right\\rangle, \\qquad \\mathcal R_+=\\mathcal R_-"
    },
    {
      "id": "ns.c3.c3a.pair_production_rate",
      "latex": "\\mathcal R(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "臨界螺旋成對生成率",
      "label_en": "Critical helical pair-production rate",
      "definition_zh": "第10–11節定義為兩 sector 生成項的共同值，第11節將其命名為 critical helical pair-production rate，是全篇最核心的新量，明言不是 energy flux（區別於 $\\Pi_K$）；在縮放 $u_\\lambda$ 下 $\\mathcal R_\\lambda(t)=\\lambda^2\\mathcal R(\\lambda^2t)$，本身也是 scaling-critical rate（第17節）。",
      "definition_en": "Defined in Sections 10–11 as the common value of the two sector-production terms and named the critical helical pair-production rate in Section 11 — the paper's central new quantity, explicitly not an energy flux (unlike $\\Pi_K$). Under the rescaling $u_\\lambda$ it obeys $\\mathcal R_\\lambda(t)=\\lambda^2\\mathcal R(\\lambda^2t)$, i.e. it is itself scaling-critical (Section 17), which is why Theorem 12.1 does not yet yield a contradiction.",
      "defining_relation": "\\mathcal R(t) = \\mathcal R_+(t) = \\mathcal R_-(t)"
    },
    {
      "id": "ns.c3.c3a.pair_production_identity",
      "latex": "\\frac12A'(t) + \\nu\\left(D_+(t)+D_-(t)\\right) = 2\\mathcal R(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "精確臨界成對生成恆等式",
      "label_en": "Exact critical pair-production identity",
      "definition_zh": "第11節 boxed 的核心恆等式，由兩 sector 演化方程相加得到，顯示全 N–S 臨界正定量 $A$ 的非線性成長並非任意 scalar source，而是兩個 signed helical sectors 的等量非線性生成；為定理12.1證明的直接依據。",
      "definition_en": "The boxed core identity of Section 11, obtained by summing the two sector evolution equations; shows the nonlinear growth of the full critical positive size $A$ is not an arbitrary scalar source but equal nonlinear production in the two signed helical sectors — the direct basis for the proof of Theorem 12.1."
    },
    {
      "id": "ns.c3.c3a.thm_pair_production_divergence",
      "latex": "\\int_0^{T_\\ast}[\\mathcal R(t)]_+\\,dt=\\infty",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "定理 12.1（成對生成發散）",
      "label_en": "Theorem 12.1 (Pair-Production Divergence)",
      "definition_zh": "本輪主定理（第12節）：若 $T_\\ast<\\infty$ 為 maximal finite blow-up time，則正部生成率的時間積分必發散；證明依賴 $L^3$ endpoint regularity criterion（Escauriaza–Seregin–Šverák）與第11節的 pair-production identity。第19節列為 PROVED given standard endpoint regularity。",
      "definition_en": "This round's main theorem (Section 12): if $T_\\ast<\\infty$ is the maximal finite blow-up time, the time integral of the positive part of the production rate must diverge; the proof relies on the $L^3$ endpoint regularity criterion (Escauriaza–Seregin–Šverák) and the Section 11 pair-production identity. Listed in Section 19 as PROVED given standard endpoint regularity.",
      "defining_relation": "[x]_+=\\max\\{x,0\\}"
    },
    {
      "id": "ns.c3.c3a.xhelpaircert",
      "latex": "\\operatorname{XHelPairCert}_n",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "X 螺旋成對證書",
      "label_en": "X Helical Pair Certificate",
      "definition_zh": "第15節為 X 積分框架新增的每區間（$I_n=[t_{n-1},t_n]$）證書，在既有（前一輪的）$\\operatorname{XUVRepCert}_n$ 之外，記錄本輪的 sector helicities、dissipations、pair-production rate 與 helical provenance $\\operatorname{Prov}_{\\rm hel}$，並附 8 項守衛檢查條件。",
      "definition_en": "A per-interval ($I_n=[t_{n-1},t_n]$) certificate added to the X-integral framework in Section 15, alongside the prior round's $\\operatorname{XUVRepCert}_n$; records this round's sector helicities, dissipations, pair-production rate, and helical provenance $\\operatorname{Prov}_{\\rm hel}$, subject to 8 listed guard checks.",
      "defining_relation": "\\operatorname{XHelPairCert}_n = \\left\\langle H_+,H_-,D_+,D_-,\\mathcal R,\\operatorname{Prov}_{\\rm hel}\\right\\rangle_{I_n}",
      "notes": "$\\operatorname{Prov}_{\\rm hel}$ 本身未在文中進一步展開定義，僅作為證書內一個 provenance 欄位出現。$I_n$、$\\operatorname{XUVRepCert}_n$ 為承襲自前一輪（C1）的既有記號，非本輪新定義，此處僅標註以利區分。"
    },
    {
      "id": "ns.c3.c3a.theta_crit",
      "latex": "\\Theta_{\\rm crit}(t)",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "臨界張力座標元組",
      "label_en": "Critical tension coordinate tuple",
      "definition_zh": "第16節提議加入 True ETN state 的成對臨界座標元組，整合 sector helicities、其和差（即 $H$ 與 $A$）、pair-production rate 與 dissipations，取代舊有單一 shell 記號 $E_j,T_j,D_j$，文中稱其更接近真正的「張力」。",
      "definition_en": "The paired critical-coordinate tuple Section 16 proposes adding to the True ETN state, bundling sector helicities, their sum/difference (recovering $H$ and $A$), the pair-production rate, and the dissipations; proposed to supersede the older single-shell notation $E_j,T_j,D_j$ as a closer approximation to genuine tension.",
      "defining_relation": "\\Theta_{\\rm crit}(t) = \\left\\langle H_+(t),H_-(t),H_+(t)-H_-(t),H_+(t)+H_-(t),\\mathcal R(t),D_+(t),D_-(t)\\right\\rangle",
      "notes": "$E_j,T_j,D_j$ 是既有 ETN 框架（外部前作）的舊記號，此處僅作對比，非本輪定義。"
    },
    {
      "id": "ns.c3.c3a.minority_factor_estimate",
      "latex": "|\\mathcal R| \\le C \\min\\left\\{\\|u^+\\|_{\\dot H^{1/2}},\\|u^-\\|_{\\dot H^{1/2}}\\right\\}\\|u\\|_{\\dot H^{3/2}}^2",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "H1／少數方生成率估計（候選引理）",
      "label_en": "H1 — Minority-factor estimate (candidate lemma)",
      "definition_zh": "第18節提出、明確標記為 CANDIDATE LEMMA（非定理）的待驗證不等式，猜測 $|\\mathcal R|$ 受兩 sector 中較小的 $\\dot H^{1/2}$ 範數（少數方）與 $\\|u\\|_{\\dot H^{3/2}}^2$ 所控制；若成立則任何 blow-up 都不能讓兩個 helicity sectors 同時永久保持小 critical size。第19節列為 OPEN。",
      "definition_en": "An unproven inequality proposed in Section 18 and explicitly marked CANDIDATE LEMMA (not a theorem), conjecturing that $|\\mathcal R|$ is controlled by the smaller (minority) of the two sectors' $\\dot H^{1/2}$ norms times $\\|u\\|_{\\dot H^{3/2}}^2$; if true, no blow-up could keep both helicity sectors permanently small in critical size. Listed OPEN in Section 19."
    },
    {
      "id": "ns.c3.c3a.triad_sign_classification",
      "latex": "(s_1,s_2,s_3)\\in\\{+,-\\}^3",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "H2／三元交互作用符號分類",
      "label_en": "H2 — Triad sign classification",
      "definition_zh": "第18節提出的下一步工作 H2：把每個 helicity triad 交互作用依三個因子的正負號窮舉為 8 類，逐一判定哪些對 $\\mathcal R$ 恰好抵消、哪些只是 redistribute、哪些真正 pair-produce critical size；本文尚未展開此分類。",
      "definition_en": "The H2 task proposed in Section 18: exhaustively classify every helicity-triad interaction into 8 classes by the signs of its three factors, to determine which exactly cancel in $\\mathcal R$, which merely redistribute, and which genuinely pair-produce critical size; not yet carried out in this document."
    },
    {
      "id": "ns.c3.c3a.persistent_triadic_obstruction",
      "latex": "q\\to q+1\\to q+2\\to\\cdots",
      "series": "NS",
      "first_appearance": "C3-A",
      "label_zh": "持續三元阻礙（未決）",
      "label_en": "Persistent triadic obstruction (open)",
      "definition_zh": "第19節狀態表中列為 OPEN 的項目，對應第18節 H2（triad 符號分類）與 H3（跨尺度 pair-production congestion，即同一 mixed-helicity parent 是否能無限次支援 $q\\to q+1\\to\\cdots$ 而不出現 depletion、alignment loss、back-transfer、viscous penalty 或 branch multiplicity explosion），是 C3-B 待處理的核心未決問題。",
      "definition_en": "The item listed OPEN in the Section 19 status table, corresponding to Section 18's H2 (triad sign classification) and H3 (cross-scale pair-production congestion — whether the same mixed-helicity parent structure can indefinitely support $q\\to q+1\\to\\cdots$ without depletion, alignment loss, back-transfer, viscous penalty, or branch-multiplicity explosion); the central open question deferred to C3-B.",
      "notes": "H3 本身未引入獨立新記號，$q$ 沿用既有 dyadic shell index 慣例；此條目主要對應狀態表用語與 H3 描述，而非單一公式。"
    },
    {
      "id": "ns.c3.c3b.d_operator",
      "latex": "D",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "分數階拉普拉斯算子",
      "label_en": "Fractional Laplacian operator",
      "definition_zh": "第0節重申的基礎算子 $D=\\sqrt{-\\Delta}$，是全篇所有 $H_\\pm,D_\\pm,\\mathcal E_\\pm$ 等 critical Sobolev 範數量的構造基礎。",
      "definition_en": "Base operator $D=\\sqrt{-\\Delta}$ restated in Sec. 0; used to build every critical Sobolev-norm quantity ($H_\\pm$, $D_\\pm$, $\\mathcal E_\\pm$) in the paper.",
      "defining_relation": "D=\\sqrt{-\\Delta}",
      "notes": "承襲自更早輪次；本輪僅重申作為記號基礎，注意與純量 $D_\\pm(t)$（第0節，另條目）字母相同但意義不同。"
    },
    {
      "id": "ns.c3.c3b.u_pm",
      "latex": "u^\\pm",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "螺旋（手性）投影場",
      "label_en": "Helical (chirality) projection fields",
      "definition_zh": "第1節正式定義：divergence-free $u$ 分解出的正/負螺旋分量，滿足 $\\nabla\\times u^\\pm=\\pm Du^\\pm$，且在所有 $D$-生成的 Sobolev 內積下彼此正交。",
      "definition_en": "Sec. 1 formal definition: the positive/negative helical components of divergence-free $u$, satisfying $\\nabla\\times u^\\pm=\\pm Du^\\pm$ and mutually orthogonal in every $D$-generated Sobolev inner product.",
      "defining_relation": "u^\\pm=\\frac12\\left(u\\pm D^{-1}\\nabla\\times u\\right)",
      "notes": "概念源自上一輪 C3-A 的 helical decomposition；本輪第1節給出正式算子公式並推導正交性，為全篇基礎。"
    },
    {
      "id": "ns.c3.c3b.h_pm",
      "latex": "H_\\pm(t)",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "螺旋扇區臨界能量（瞬時）",
      "label_en": "Helical-sector critical energy (instantaneous)",
      "definition_zh": "第0節回顧 C3-A 的定義：正/負螺旋分量的 $\\dot H^{1/2}$ 範數平方，是累積量 $\\mathcal E_\\pm$ 的瞬時部分，並在定理5.1證明中用於構造 $S_n\\to\\infty$。",
      "definition_en": "Recapped from C3-A in Sec. 0: squared $\\dot H^{1/2}$ norm of each helical component, the instantaneous part of $\\mathcal E_\\pm$; used in the proof of Thm 5.1 to build $S_n\\to\\infty$.",
      "defining_relation": "H_\\pm(t)=\\|D^{1/2}u^\\pm(t)\\|_2^2"
    },
    {
      "id": "ns.c3.c3b.d_pm",
      "latex": "D_\\pm(t)",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "螺旋扇區耗散量",
      "label_en": "Helical-sector dissipation quantity",
      "definition_zh": "第0節回顧 C3-A 定義：正/負螺旋分量的 $\\dot H^{3/2}$ 範數平方，出現於 sector balance 方程 $\\frac12H_\\pm'+\\nu D_\\pm=\\mathcal R_\\pm$，其時間積分構成 $\\mathcal E_\\pm$ 的累積耗散部分。",
      "definition_en": "Recapped from C3-A in Sec. 0: squared $\\dot H^{3/2}$ norm of each helical component in the sector balance $\\frac12H_\\pm'+\\nu D_\\pm=\\mathcal R_\\pm$; its time integral forms the accumulated-dissipation part of $\\mathcal E_\\pm$.",
      "defining_relation": "D_\\pm(t)=\\|D^{3/2}u^\\pm(t)\\|_2^2",
      "notes": "此為純量耗散量，與算子 $D=\\sqrt{-\\Delta}$ 僅共用字母，意義不同。"
    },
    {
      "id": "ns.c3.c3b.r_common",
      "latex": "\\mathcal R(t)",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "共同臨界配對產生率",
      "label_en": "Common critical pair-production rate",
      "definition_zh": "第3節定義：由 C3-A 的 $\\mathcal R_+=\\mathcal R_-$ 結果導出的單一共同量，等於 $\\mathcal E_\\pm$ 的導數，是全篇（fixed-low bound、UV escape 定理）核心追蹤對象。",
      "definition_en": "Sec. 3 definition: since $\\mathcal R_+=\\mathcal R_-$ (a C3-A result), a single common rate equal to the derivative of either $\\mathcal E_\\pm$; the central quantity tracked throughout (fixed-low bound, UV-escape theorem).",
      "defining_relation": "\\mathcal R(t)=\\mathcal E_+'(t)=\\mathcal E_-'(t)",
      "notes": "sector-wise 版本 $\\mathcal R_\\pm(t)$ 源自 C3-A 的 sector balance；本輪確立兩者相等並統一為 $\\mathcal R(t)$。"
    },
    {
      "id": "ns.c3.c3b.e_pm",
      "latex": "\\mathcal E_\\pm(t)",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "累積臨界螺旋能量",
      "label_en": "Cumulative critical helical energy",
      "definition_zh": "第2節定義：瞬時臨界能量 $\\frac12H_\\pm(t)$ 加上耗散時間積分，即 Lei–Lin–Zhou critical-helicity identity 中的量，滿足 $\\mathcal E_+(t)=\\mathcal E_-(t)+c_0$。",
      "definition_en": "Sec. 2 definition: instantaneous critical energy $\\frac12H_\\pm(t)$ plus time-integrated dissipation; the quantity in the Lei–Lin–Zhou critical-helicity identity, satisfying $\\mathcal E_+(t)=\\mathcal E_-(t)+c_0$.",
      "defining_relation": "\\mathcal E_\\pm(t)=\\frac12\\|D^{1/2}u^\\pm(t)\\|_2^2+\\nu\\int_0^t\\|D^{3/2}u^\\pm(s)\\|_2^2\\,ds",
      "notes": "本輪最核心的新量之一；定理5.1（C3-B.1）與推論6.1（C3-B.2）皆以此為主角。"
    },
    {
      "id": "ns.c3.c3b.c0",
      "latex": "c_0",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "初始螺旋偏差常數",
      "label_en": "Initial helicity-bias constant",
      "definition_zh": "第2節定義：由初始資料 $u_0^\\pm$ 決定的守恆常數，滿足 $\\mathcal E_+(t)-\\mathcal E_-(t)=c_0$ 對所有存在時間成立（Lei–Lin–Zhou 外部定理）。",
      "definition_en": "Sec. 2 definition: a conserved constant fixed by initial data $u_0^\\pm$, satisfying $\\mathcal E_+(t)-\\mathcal E_-(t)=c_0$ for all times of smooth existence (external Lei–Lin–Zhou theorem).",
      "defining_relation": "c_0=\\frac12\\left(\\|D^{1/2}u_0^+\\|_2^2-\\|D^{1/2}u_0^-\\|_2^2\\right)",
      "notes": "第7節指出：無論 $c_0$ 多大，blow-up 時 $\\mathcal E_\\pm\\to\\infty$ 會使其漸近可忽略。"
    },
    {
      "id": "ns.c3.c3b.s_n",
      "latex": "S_n",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "雙扇區能量和序列",
      "label_en": "Bi-sector energy sum sequence",
      "definition_zh": "定理5.1證明中定義：沿 $t_n\\uparrow T_\\ast$ 的 $\\mathcal E_++\\mathcal E_-$ 序列，用以推出 $\\mathcal E_\\pm(t_n)=\\frac{S_n\\pm c_0}{2}\\to\\infty$ 及推論6.1的比值極限1。",
      "definition_en": "Defined in the proof of Thm 5.1: the sequence $\\mathcal E_++\\mathcal E_-$ along $t_n\\uparrow T_\\ast$, used to derive $\\mathcal E_\\pm(t_n)=\\frac{S_n\\pm c_0}{2}\\to\\infty$ and Cor. 6.1's ratio limit of 1.",
      "defining_relation": "S_n:=\\mathcal E_+(t_n)+\\mathcal E_-(t_n)"
    },
    {
      "id": "ns.c3.c3b.triad_wavenumbers",
      "latex": "k,p,q",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "傅立葉三元組波數",
      "label_en": "Fourier triad wavenumbers",
      "definition_zh": "第9節定義：滿足 closure $\\mathbf k+\\mathbf p+\\mathbf q=0$ 的三個波向量之模，按大小排序 $0<k\\le p\\le q$，是第9–17、25–26節 triad algebra 的基本座標。",
      "definition_en": "Sec. 9 definition: magnitudes of three wavevectors satisfying the closure $\\mathbf k+\\mathbf p+\\mathbf q=0$, ordered $0<k\\le p\\le q$; the basic coordinates of the triad algebra in Secs. 9-17, 25-26.",
      "defining_relation": "\\mathbf k+\\mathbf p+\\mathbf q=0, 0<k\\le p\\le q"
    },
    {
      "id": "ns.c3.c3b.helical_signs",
      "latex": "s_k,s_p,s_q",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "三元組螺旋號",
      "label_en": "Triad helical signs",
      "definition_zh": "第9節定義：三元組中每個模態各自所屬的螺旋（手性）正負號，決定該 triad 屬於同手性（Class I）或異手性（Class II–IV）。",
      "definition_en": "Sec. 9 definition: the helicity sign of each triad mode, determining whether the triad is homochiral (Class I) or heterochiral (Classes II-IV).",
      "defining_relation": "s_k,s_p,s_q\\in\\{+1,-1\\}"
    },
    {
      "id": "ns.c3.c3b.mode_energy",
      "latex": "e_k,e_p,e_q",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "螺旋模態能量密度",
      "label_en": "Helical mode energy density",
      "definition_zh": "第9節定義（$e_p,e_q$ 同理）：波數 $\\mathbf k$ 處、其自身螺旋號分量的能量密度，滿足 triad-wise 守恆 $\\dot e_k+\\dot e_p+\\dot e_q=0$ 及 $s_kk\\dot e_k+s_pp\\dot e_p+s_qq\\dot e_q=0$。",
      "definition_en": "Sec. 9 definition (with $e_p,e_q$ analogous): energy density of the mode at $\\mathbf k$ in its own helical sign, obeying triad-wise conservation $\\dot e_k+\\dot e_p+\\dot e_q=0$ and $s_kk\\dot e_k+s_pp\\dot e_p+s_qq\\dot e_q=0$.",
      "defining_relation": "e_k=\\frac12|u^{s_k}(\\mathbf k)|^2"
    },
    {
      "id": "ns.c3.c3b.theta_tau",
      "latex": "\\Theta_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "三元組轉移零空間純量",
      "label_en": "Triad transfer nullspace scalar",
      "definition_zh": "引理10.1定義：因 $(\\dot e_k,\\dot e_p,\\dot e_q)$ 同時正交於 $(1,1,1)$ 與 $(s_kk,s_pp,s_qq)$（一維零空間），所得的比例純量，決定 triad 的實際轉移速率與正負號。",
      "definition_en": "Lemma 10.1 definition: since $(\\dot e_k,\\dot e_p,\\dot e_q)$ lies in the (one-dimensional) joint orthogonal complement of $(1,1,1)$ and $(s_kk,s_pp,s_qq)$, this proportionality scalar fixes the triad's actual transfer rate and sign.",
      "defining_relation": "(\\dot e_k,\\dot e_p,\\dot e_q)=\\Theta_\\tau(s_p p-s_q q,s_q q-s_k k,s_k k-s_p p)",
      "notes": "第31.5節（C3-C.5）指出 $\\Theta_\\tau$ 尚包含 triad phase/amplitude/geometric coefficient，其持續正號性列為未解問題。"
    },
    {
      "id": "ns.c3.c3b.helicity_classes",
      "latex": "\\mathrm{I},\\mathrm{II},\\mathrm{III},\\mathrm{IV}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "四類螺旋號組態",
      "label_en": "Four helicity-sign configuration classes",
      "definition_zh": "第12節定義（固定最小波數號為 $+$ 後）：四種獨立 helicity 組態，I 為同手性，II–IV 為異手性，依 unique sign 落在最小/中間/最大波數區分。",
      "definition_en": "Sec. 12 definition (after fixing the smallest-wavenumber sign to $+$): the four independent helicity configurations; I is homochiral, II-IV heterochiral, classified by whether the unique sign sits at the smallest/middle/largest wavenumber.",
      "defining_relation": "\\mathrm{I}:(+++),\\mathrm{II}:(+--),\\mathrm{III}:(+-+),\\mathrm{IV}:(++-)"
    },
    {
      "id": "ns.c3.c3b.script_a_tau",
      "latex": "\\mathscr A_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "三元組絕對臨界內容",
      "label_en": "Triad absolute critical content",
      "definition_zh": "第11節定義：三元組對 $\\frac12\\|D^{1/2}u\\|_2^2$ 的貢獻量；同手性三元組滿足 $\\dot{\\mathscr A}_\\tau=0$（第13節），異手性則 $\\dot{\\mathscr A}_\\tau=2\\mathcal R_\\tau$（第14節）。",
      "definition_en": "Sec. 11 definition: a triad's contribution to $\\frac12\\|D^{1/2}u\\|_2^2$; vanishes identically for homochiral triads (Sec. 13, $\\dot{\\mathscr A}_\\tau=0$), while $\\dot{\\mathscr A}_\\tau=2\\mathcal R_\\tau$ for heterochiral ones (Sec. 14).",
      "defining_relation": "\\mathscr A_\\tau=k e_k+p e_p+q e_q"
    },
    {
      "id": "ns.c3.c3b.script_h_tau",
      "latex": "\\mathscr H_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "帶號螺旋度半密度",
      "label_en": "Signed helicity half-density",
      "definition_zh": "第11節定義：三元組的帶號螺旋度貢獻，在 nonlinear dynamics 下恆為守恆量 $\\dot{\\mathscr H}_\\tau=0$；Class I 時等於 $\\mathscr A_\\tau$（第13節）。",
      "definition_en": "Sec. 11 definition: the triad's signed-helicity contribution, exactly conserved under the nonlinear dynamics ($\\dot{\\mathscr H}_\\tau=0$); equals $\\mathscr A_\\tau$ for Class I (Sec. 13).",
      "defining_relation": "\\mathscr H_\\tau=s_k k e_k+s_p p e_p+s_q q e_q"
    },
    {
      "id": "ns.c3.c3b.r_tau",
      "latex": "r_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "唯一螺旋號模態波數",
      "label_en": "Unique-helicity-sign mode wavenumber",
      "definition_zh": "第14節定義：異手性三元組中與另兩者號不同的「唯一號模態」之波數，是 UV escape（第23、25節）與 X-guard 框架的核心座標，依 Class II/III/IV 分別等於 $k,p,q$。",
      "definition_en": "Sec. 14 definition: the wavenumber of the single mode in a heterochiral triad whose sign differs from the other two; the central coordinate of the UV-escape argument (Secs. 23, 25) and X-guard framework, equal to $k,p,q$ in Class II/III/IV respectively."
    },
    {
      "id": "ns.c3.c3b.e_uniq",
      "latex": "e_{\\rm uniq}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "唯一號模態能量",
      "label_en": "Unique-sign mode energy",
      "definition_zh": "第14節定義：波數為 $r_\\tau$ 的唯一螺旋號模態之能量，滿足 $\\dot{\\mathscr A}_\\tau=2r_\\tau\\dot e_{\\rm uniq}$，將 critical pair production 連結到單一模態的能量變化率。",
      "definition_en": "Sec. 14 definition: the energy of the unique-sign mode at wavenumber $r_\\tau$, satisfying $\\dot{\\mathscr A}_\\tau=2r_\\tau\\dot e_{\\rm uniq}$, linking critical pair production to a single mode's rate of energy change.",
      "defining_relation": "\\dot{\\mathscr A}_\\tau=2r_\\tau\\dot e_{\\rm uniq}"
    },
    {
      "id": "ns.c3.c3b.r_tau_rate",
      "latex": "\\mathcal R_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "三元組配對產生率貢獻",
      "label_en": "Triad's pair-production rate contribution",
      "definition_zh": "第14節定義：單一三元組對共同產生率 $\\mathcal R$ 的貢獻，把「critical pair production」重新解讀為波數加權的、流向唯一螺旋號模態的能量轉移。",
      "definition_en": "Sec. 14 definition: a single triad's contribution to the common production rate $\\mathcal R$, reinterpreting critical pair production as wavenumber-weighted energy transfer into the unique-sign mode.",
      "defining_relation": "\\mathcal R_\\tau=r_\\tau\\dot e_{\\rm uniq}",
      "notes": "第16節給出絕對值公式 $|\\mathcal R_\\tau|=r_\\tau\\Delta_\\tau|\\Theta_\\tau|$；第17節給出上界 $|\\mathcal R_\\tau|\\le r_\\tau^2|\\Theta_\\tau|$。"
    },
    {
      "id": "ns.c3.c3b.r_class_formulas",
      "latex": "\\mathcal R_{\\mathrm{II}},\\mathcal R_{\\mathrm{III}},\\mathcal R_{\\mathrm{IV}}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "分類三元組產生率精確公式",
      "label_en": "Class-specific exact triad production-rate formulas",
      "definition_zh": "第15節（Exact triad table）：依引理10.1，Class II/III/IV 各自 unique-sign 模態在最小/中間/最大波數時 $\\mathcal R_\\tau$ 的顯式表達式。",
      "definition_en": "Sec. 15 (exact triad table): explicit formulas for $\\mathcal R_\\tau$ in Classes II/III/IV, where the unique-sign mode sits at the smallest/middle/largest wavenumber respectively.",
      "defining_relation": "\\mathcal R_{\\mathrm{II}}=k(q-p)\\Theta_\\tau, \\mathcal R_{\\mathrm{III}}=p(q-k)\\Theta_\\tau, \\mathcal R_{\\mathrm{IV}}=q(k-p)\\Theta_\\tau"
    },
    {
      "id": "ns.c3.c3b.delta_tau",
      "latex": "\\Delta_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "同號徑向波數差",
      "label_en": "Same-sign radial wavenumber gap",
      "definition_zh": "第16節定義：異手性三元組中另兩個同螺旋號模態的波數差，統一寫出 $|\\mathcal R_\\tau|=r_\\tau\\Delta_\\tau|\\Theta_\\tau|$；$\\Delta_\\tau=0$ 時即使 heterochiral 仍有 $\\mathcal R_\\tau=0$。",
      "definition_en": "Sec. 16 definition: the wavenumber gap between the two same-sign modes in a heterochiral triad, giving $|\\mathcal R_\\tau|=r_\\tau\\Delta_\\tau|\\Theta_\\tau|$; when $\\Delta_\\tau=0$, $\\mathcal R_\\tau=0$ even though the triad is heterochiral.",
      "defining_relation": "\\Delta_\\tau=\\left|a_\\tau-b_\\tau\\right|",
      "notes": "第17節由 triangle inequalities 證明 $0\\le\\Delta_\\tau\\le r_\\tau$；$a_\\tau,b_\\tau$ 泛指該 triad 中除 $r_\\tau$ 外的另兩個波數。"
    },
    {
      "id": "ns.c3.c3b.x_guards",
      "latex": "G-H, G-U, G-\\Delta, G-GEO, G-PHASE",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "X-Guard 最低形成資格五守衛",
      "label_en": "X-Guard five formation-eligibility conditions",
      "definition_zh": "第18節定義：一個 triad 要對 $\\mathcal R$ 有非零貢獻須同時通過的五個守衛——異手性、唯一號模態不退化、徑向差非零（$\\Delta_\\tau>0$）、幾何耦合非退化、相位轉移非零（$\\Theta_\\tau\\ne0$），防止把「mixed sign」直接等同「危險 transfer」。",
      "definition_en": "Sec. 18 definition: five guards a triad must simultaneously pass for nonzero contribution to $\\mathcal R$ — heterochiral, nondegenerate unique-sign participation, nonzero radial gap ($\\Delta_\\tau>0$), nondegenerate geometric coupling, and nonzero phase transfer ($\\Theta_\\tau\\ne0$); prevents conflating 'mixed sign' with 'dangerous transfer.'",
      "notes": "五守衛併入第29節的 $\\operatorname{XHelUV}_n$ 證書結構前四項。"
    },
    {
      "id": "ns.c3.c3b.r_uniq_pm",
      "latex": "\\mathcal R_{\\mathrm{uniq}+},\\mathcal R_{\\mathrm{uniq}-}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "全域唯一號產生率分支",
      "label_en": "Global unique-sign production channels",
      "definition_zh": "第19節定義：把 $\\mathcal R=\\mathcal R_{\\mathrm{uniq}+}+\\mathcal R_{\\mathrm{uniq}-}$ 依 physical-space bilinear 形式（含螺旋投影 $\\mathbb P^\\pm$）寫出，代表每個 heterochiral triad 只由其唯一號模態計帳一次。",
      "definition_en": "Sec. 19 definition: splits $\\mathcal R=\\mathcal R_{\\mathrm{uniq}+}+\\mathcal R_{\\mathrm{uniq}-}$ in physical-space bilinear form (via helical projectors $\\mathbb P^\\pm$), each heterochiral triad counted exactly once by its unique-sign mode.",
      "defining_relation": "\\mathcal R_{\\mathrm{uniq}+}=-\\langle Du^+,\\mathbb P^+((u^-\\cdot\\nabla)u^-)\\rangle, \\mathcal R_{\\mathrm{uniq}-}=-\\langle Du^-,\\mathbb P^-((u^+\\cdot\\nabla)u^+)\\rangle"
    },
    {
      "id": "ns.c3.c3b.p_leq_k",
      "latex": "P_{\\le K}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "低通傅立葉截斷投影",
      "label_en": "Low-pass Fourier cutoff projector",
      "definition_zh": "第20節定義：平滑 Fourier low-pass 投影至波數 $\\le K$，用以定義固定低頻的唯一號產生率 $\\mathcal R_{\\le K}^{\\mathrm{uniq}}$。",
      "definition_en": "Sec. 20 definition: a smooth Fourier low-pass projector onto wavenumbers $\\le K$, used to define the fixed-low-frequency unique-sign production rate $\\mathcal R_{\\le K}^{\\mathrm{uniq}}$."
    },
    {
      "id": "ns.c3.c3b.r_leq_k_uniq",
      "latex": "\\mathcal R_{\\le K}^{\\mathrm{uniq}}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "固定低頻唯一號產生率",
      "label_en": "Fixed-low unique-sign production rate",
      "definition_zh": "第20節定義，定理21.1（C3-B.3）證明其滿足 $|\\mathcal R_{\\le K}^{\\mathrm{uniq}}(t)|\\le CK^{7/2}\\|u_0\\|_2^3$；第22節證明其時間積分有限，不能承擔 divergent cumulative production。",
      "definition_en": "Sec. 20 definition, bounded in Thm 21.1 (C3-B.3) by $|\\mathcal R_{\\le K}^{\\mathrm{uniq}}(t)|\\le CK^{7/2}\\|u_0\\|_2^3$; Sec. 22 shows its time integral is finite, so it cannot sustain divergent cumulative production.",
      "defining_relation": "\\mathcal R_{\\le K}^{\\mathrm{uniq}}=-\\langle DP_{\\le K}u^+,(u^-\\cdot\\nabla)u^-\\rangle-\\langle DP_{\\le K}u^-,(u^+\\cdot\\nabla)u^+\\rangle"
    },
    {
      "id": "ns.c3.c3b.r_gt_k_uniq",
      "latex": "\\mathcal R_{>K}^{\\mathrm{uniq}}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "高頻唯一號產生率尾項",
      "label_en": "High-frequency unique-sign production tail",
      "definition_zh": "第23節定義（定理23.1／C3-B.4核心對象）：總產生率扣除固定低頻部分的餘項；證明若 $T_\\ast<\\infty$，則對任意 fixed $K$ 其正部時間積分必發散——即 Unique-Sign UV Escape。",
      "definition_en": "Sec. 23 definition (central object of Thm 23.1 / C3-B.4): the remainder of the total production rate after subtracting the fixed-low part; if $T_\\ast<\\infty$, its positive-part time integral diverges for every fixed $K$ — the Unique-Sign UV Escape.",
      "defining_relation": "\\mathcal R_{>K}^{\\mathrm{uniq}}=\\mathcal R-\\mathcal R_{\\le K}^{\\mathrm{uniq}}"
    },
    {
      "id": "ns.c3.c3b.xheluv_n",
      "latex": "\\operatorname{XHelUV}_n",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "奇異鏈 X-積分證書",
      "label_en": "Singular-chain X-Integration certificate",
      "definition_zh": "第29節定義：本輪更新的 X-Integration 證書元組，逐尺度須保存唯一號波數、徑向差、號、相位轉移、產生率與雙扇區累積臨界能量，並列出九項守衛。",
      "definition_en": "Sec. 29 definition: this round's updated X-Integration certificate tuple, preserved scale-by-scale, packaging the unique-sign wavenumber, radial gap, sign, phase transfer, production rate and both sectors' cumulative critical energy, with nine listed guards.",
      "defining_relation": "\\operatorname{XHelUV}_n=\\langle r_n,\\Delta_n,s_n,\\Theta_n,\\mathcal R_n,\\mathcal E_n^+,\\mathcal E_n^-,\\operatorname{Prov}_n\\rangle",
      "notes": "元組末項 $\\operatorname{Prov}_n$（provenance）僅作為元組成分列出，未給出獨立公式，對應第0節『unique-sign mode provenance』新工作項目，信心度較低。"
    },
    {
      "id": "ns.c3.c3b.r_q_uniq",
      "latex": "\\mathcal R_q^{\\mathrm{uniq}}",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "殼層 q 唯一號產生率",
      "label_en": "Shell-q unique-sign production rate",
      "definition_zh": "第31節（C3-C.1，下一步 proof obligations）定義：唯一號模態位於 dyadic shell $q$ 的配對產生量，由 C3-B.4 導出 tail law $\\forall Q,\\int[\\sum_{q>Q}\\mathcal R_q^{\\mathrm{uniq}}]_+dt=\\infty$。",
      "definition_en": "Defined in Sec. 31 (C3-C.1, next-step proof obligations): the pair production carried by the unique-sign mode in dyadic shell $q$; C3-B.4 yields the tail law $\\forall Q,\\int[\\sum_{q>Q}\\mathcal R_q^{\\mathrm{uniq}}]_+dt=\\infty$.",
      "defining_relation": "\\forall Q,\\int[\\sum_{q>Q}\\mathcal R_q^{\\mathrm{uniq}}]_+dt=\\infty",
      "notes": "前瞻性定義，銜接下一輪 C3-C，但文字寫在本文件第31節內。"
    },
    {
      "id": "ns.c3.c3b.eta_tau",
      "latex": "\\eta_\\tau",
      "series": "NS",
      "first_appearance": "C3-B",
      "label_zh": "相對徑向差比",
      "label_en": "Relative radial-gap ratio",
      "definition_zh": "第31節（C3-C.2）定義：徑向差 $\\Delta_\\tau$ 相對唯一號波數 $r_\\tau$ 的比值，用於研究 $\\eta_\\tau\\ll1$ 的近退化三元組是否可被微擾吸收。",
      "definition_en": "Defined in Sec. 31 (C3-C.2): the ratio of radial gap $\\Delta_\\tau$ to unique-sign wavenumber $r_\\tau$, used to study whether near-radially-degenerate triads ($\\eta_\\tau\\ll1$) can be perturbatively absorbed.",
      "defining_relation": "\\eta_\\tau=\\frac{\\Delta_\\tau}{r_\\tau}",
      "notes": "前瞻性定義，屬第31節『下一步 proof obligations』（銜接 C3-C），形式上寫在本文件內。"
    },
    {
      "id": "ns.c3.c3c.theta_tau",
      "latex": "\\Theta_\\tau",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "三元組轉移純量參數",
      "label_en": "Triad transfer scalar parameter",
      "definition_zh": "定義於第1節（回顧自先前輪次）：唯一純量 $\\Theta_\\tau$ 使 $(\\dot e_k,\\dot e_p,\\dot e_q)$ 正比於 $(s_pp-s_qq,\\,s_qq-s_kk,\\,s_kk-s_pp)$，同時滿足能量與帶號螺度守恆；本輪 Class II/III/IV 所有 $\\dot e$ 與 $\\mathcal R$ 公式皆以它為共同因子。",
      "definition_en": "Defined (reviewed) in §1: the unique scalar $\\Theta_\\tau$ such that $(\\dot e_k,\\dot e_p,\\dot e_q)$ is proportional to $(s_pp-s_qq,\\,s_qq-s_kk,\\,s_kk-s_pp)$, consistent with triadwise energy and signed-helicity conservation; every Class II/III/IV transfer formula in this round factors through it.",
      "defining_relation": "\\begin{pmatrix}\\dot e_k\\\\\\dot e_p\\\\\\dot e_q\\end{pmatrix}=\\Theta_\\tau\\begin{pmatrix}s_pp-s_qq\\\\s_qq-s_kk\\\\s_kk-s_pp\\end{pmatrix}",
      "notes": "Section title is explicitly '回顧' (review), so this restates rather than newly defines the parameter; included because every formula in this round is built on it."
    },
    {
      "id": "ns.c3.c3c.class_ii_transfer",
      "latex": "\\dot e_k=(q-p)\\Theta_\\tau,\\ \\dot e_p=-(q+k)\\Theta_\\tau,\\ \\dot e_q=(p+k)\\Theta_\\tau",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class II 精確轉移方程",
      "label_en": "Class II exact transfer equations",
      "definition_zh": "第2節：對 helicity 符號組態 $(s_k,s_p,s_q)=(+,-,-)$（$0<k\\le p\\le q$）代入三元組轉移代數得到的三個精確方程，是本輪推導 $\\mathcal R_{\\mathrm{II}}$ 及後續所有結果的起點。",
      "definition_en": "§2: the three exact equations from substituting sign pattern $(s_k,s_p,s_q)=(+,-,-)$, $0<k\\le p\\le q$, into the triad transfer algebra; the starting point for $\\mathcal R_{\\mathrm{II}}$ and everything downstream in this round.",
      "defining_relation": "\\dot e_k=(q-p)\\Theta_\\tau,\\quad\\dot e_p=-(q+k)\\Theta_\\tau,\\quad\\dot e_q=(p+k)\\Theta_\\tau",
      "notes": "Listed by name in the §30 status table as 'Class-II exact transfer equations: PROVED'."
    },
    {
      "id": "ns.c3.c3c.r_ii",
      "latex": "\\mathcal R_{\\mathrm{II}}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class II 臨界配對生成量",
      "label_en": "Class II critical pair-production contribution",
      "definition_zh": "第2節定義：$\\mathcal R_{\\mathrm{II}}=k(q-p)\\Theta_\\tau$，即 Class II triad 中最小波數 $k$（唯一符號模）的能量變化率，是本輪 quadratic tax 與 congestion 論證的核心對象。",
      "definition_en": "Defined in §2 as $\\mathcal R_{\\mathrm{II}}=k(q-p)\\Theta_\\tau$, the energy-transfer rate into the unique-sign smallest mode $k$ of a Class II triad; the central quantity behind this round's quadratic-tax and congestion arguments.",
      "defining_relation": "\\mathcal R_{\\mathrm{II}}=k(q-p)\\Theta_\\tau"
    },
    {
      "id": "ns.c3.c3c.x_p_x_q",
      "latex": "X_p=p|\\dot e_p|,\\quad X_q=q|\\dot e_q|",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "高頻模臨界交換量",
      "label_en": "High-mode critical exchange magnitudes",
      "definition_zh": "第4節定義：$X_p=p|\\dot e_p|$、$X_q=q|\\dot e_q|$；在 Class II 中分別等於 $p(q+k)|\\Theta_\\tau|$ 與 $q(p+k)|\\Theta_\\tau|$，度量兩個高頻模 $p,q$ 各自的絕對能量交換規模。",
      "definition_en": "Defined in §4 as $X_p=p|\\dot e_p|$, $X_q=q|\\dot e_q|$ — equal to $p(q+k)|\\Theta_\\tau|$ and $q(p+k)|\\Theta_\\tau|$ respectively for Class II — measuring the absolute exchange scale carried by each high mode $p,q$.",
      "defining_relation": "X_p=p|\\dot e_p|=p(q+k)|\\Theta_\\tau|,\\quad X_q=q|\\dot e_q|=q(p+k)|\\Theta_\\tau|"
    },
    {
      "id": "ns.c3.c3c.x_hi",
      "latex": "X_{\\mathrm{hi}}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "隱藏高頻交換規模",
      "label_en": "Hidden high-frequency exchange scale",
      "definition_zh": "第4節定義 $X_{\\mathrm{hi}}=\\min\\{X_p,X_q\\}\\ge p^2|\\Theta_\\tau|$，代表 Class II triad 中兩個高頻模背後、不被 $\\mathcal R_{\\mathrm{II}}$ 淨值反映的最小絕對交換規模，是 quadratic tax 定理的比較基準。",
      "definition_en": "Defined in §4 as $X_{\\mathrm{hi}}=\\min\\{X_p,X_q\\}\\ge p^2|\\Theta_\\tau|$, the smaller of the two high-mode exchange magnitudes; the reference scale against which the quadratic nonlocality tax measures $\\mathcal R_{\\mathrm{II}}$'s smallness.",
      "defining_relation": "X_{\\mathrm{hi}}=\\min\\{X_p,X_q\\}\\ge p^2|\\Theta_\\tau|"
    },
    {
      "id": "ns.c3.c3c.quadratic_nonlocality_tax",
      "latex": "|\\mathcal R_{\\mathrm{II}}|\\le\\left(\\frac{k}{p}\\right)^2X_{\\mathrm{hi}}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class II 二次非局部稅（定理 C3-C.1）",
      "label_en": "Class-II Quadratic Nonlocality Tax (Theorem C3-C.1)",
      "definition_zh": "定理5.1（C3-C.1）：$|\\mathcal R_{\\mathrm{II}}|\\le(k/p)^2X_{\\mathrm{hi}}$，由 $|\\mathcal R_{\\mathrm{II}}|\\le k^2|\\Theta_\\tau|$ 與 $X_{\\mathrm{hi}}\\ge p^2|\\Theta_\\tau|$ 直接合併而得；dyadic 形式（第6節）：若 $p\\ge2^Nk$ 則 $|\\mathcal R_{\\mathrm{II}}|\\le2^{-2N}X_{\\mathrm{hi}}$，每多一個 dyadic 分離，效率再降4倍。",
      "definition_en": "Theorem 5.1 (C3-C.1): $|\\mathcal R_{\\mathrm{II}}|\\le(k/p)^2X_{\\mathrm{hi}}$, combining $|\\mathcal R_{\\mathrm{II}}|\\le k^2|\\Theta_\\tau|$ (§3) with $X_{\\mathrm{hi}}\\ge p^2|\\Theta_\\tau|$ (§4); the dyadic form (§6) states that if $p\\ge2^Nk$ then $|\\mathcal R_{\\mathrm{II}}|\\le2^{-2N}X_{\\mathrm{hi}}$ — each extra dyadic separation costs another factor of 4.",
      "defining_relation": "|\\mathcal R_{\\mathrm{II}}|\\le\\left(\\frac{k}{p}\\right)^2X_{\\mathrm{hi}};\\quad p\\ge2^Nk\\Rightarrow|\\mathcal R_{\\mathrm{II}}|\\le2^{-2N}X_{\\mathrm{hi}}",
      "notes": "Restated in §7 as the 'hidden-exchange debt' certificate $X_{\\mathrm{hi}}\\ge(p/k)^2|\\mathcal R_{\\mathrm{II}}|$; registered as checklist item G-tax in §23."
    },
    {
      "id": "ns.c3.c3c.chi_tau",
      "latex": "\\chi_\\tau=\\frac{k}{p}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "非局部性比值",
      "label_en": "Nonlocality ratio",
      "definition_zh": "第7、11節定義 $\\chi_\\tau=k/p$（世系中記為 $\\chi_n=k_n/p_n$），量化 triad 的 strongly-nonlocal 程度；第23節登記為 guard G-$\\chi$。",
      "definition_en": "Defined in §7/§11 as $\\chi_\\tau=k/p$ (written $\\chi_n=k_n/p_n$ along a genealogy in §12/§23), quantifying how strongly nonlocal a triad is; registered as guard G-$\\chi$ in §23.",
      "defining_relation": "\\chi_\\tau=\\frac{k}{p}"
    },
    {
      "id": "ns.c3.c3c.delta_tau",
      "latex": "\\delta_\\tau=\\frac{q-p}{p}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "徑向漂移比值",
      "label_en": "Radial drift ratio",
      "definition_zh": "第11節「Radial step bound」定義 $\\delta_\\tau=(q-p)/p$（世系中為 $\\delta_n$），並證明 $0\\le\\delta_\\tau\\le\\chi_\\tau\\le1$；第23節登記為 guard G-$\\delta$。",
      "definition_en": "Defined in §11 ('Radial step bound') as $\\delta_\\tau=(q-p)/p$ (written $\\delta_n$ along the genealogy), with the proven bound $0\\le\\delta_\\tau\\le\\chi_\\tau\\le1$; registered as guard G-$\\delta$ in §23.",
      "defining_relation": "0\\le\\delta_\\tau=\\frac{q-p}{p}\\le\\chi_\\tau\\le1"
    },
    {
      "id": "ns.c3.c3c.delta_rad",
      "latex": "\\Delta_{\\mathrm{rad}}=q-p",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "高頻端徑向前進量",
      "label_en": "High-end radial advancement",
      "definition_zh": "第11節定義的未正規化徑向步長 $\\Delta_{\\mathrm{rad}}=q-p$（high receiver $q$ 與 high donor $p$ 之差），三角不等式給出 $\\Delta_{\\mathrm{rad}}\\le k$，正規化後即為 $\\delta_\\tau$。",
      "definition_en": "The un-normalized radial step defined in §11, $\\Delta_{\\mathrm{rad}}=q-p$ (gap between high receiver $q$ and high donor $p$); bounded by the triangle inequality as $\\Delta_{\\mathrm{rad}}\\le k$, normalizing to $\\delta_\\tau$.",
      "defining_relation": "\\Delta_{\\mathrm{rad}}=q-p\\le k"
    },
    {
      "id": "ns.c3.c3c.radial_genealogy",
      "latex": "p_n=p_0\\prod_{j=0}^{n-1}(1+\\delta_j)",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class II 徑向世系",
      "label_en": "Class-II radial genealogy",
      "definition_zh": "第12節構造的理想化 high-end 世系 $p_0\\to q_0=p_1\\to q_1=p_2\\to\\cdots$，遞迴式 $p_{n+1}=p_n(1+\\delta_n)$，通解 $p_n=p_0\\prod_{j<n}(1+\\delta_j)$，是第13節 congestion lemma 證明的基礎。",
      "definition_en": "The idealized source-preserving high-end genealogy $p_0\\to q_0=p_1\\to q_1=p_2\\to\\cdots$ built in §12, with recursion $p_{n+1}=p_n(1+\\delta_n)$ and closed form $p_n=p_0\\prod_{j<n}(1+\\delta_j)$; the basis for the §13 congestion lemma's proof.",
      "defining_relation": "p_{n+1}=p_n(1+\\delta_n)\\ \\Longrightarrow\\ p_n=p_0\\prod_{j=0}^{n-1}(1+\\delta_j)"
    },
    {
      "id": "ns.c3.c3c.radial_drift_congestion_lemma",
      "latex": "\\sum_{n=0}^{\\infty}\\delta_n=\\infty",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "徑向漂移擁塞引理（定理 C3-C.2）",
      "label_en": "Radial-Drift Congestion Lemma (Theorem C3-C.2)",
      "definition_zh": "定理13.1（C3-C.2）：若 Class II high-end genealogy 滿足 $p_n\\to\\infty$，則必有 $\\sum_n\\delta_n=\\infty$（進而 $\\sum_n\\chi_n=\\infty$），否則乘積 $\\prod(1+\\delta_n)$ 收斂使 $p_n$ 有界，矛盾；第14節推出均勻 $\\chi_n\\le\\varepsilon$ 下跨一個 dyadic scale 至少需 $m\\gtrsim1/\\varepsilon$ 步。",
      "definition_en": "Theorem 13.1 (C3-C.2): if a Class-II high-end genealogy has $p_n\\to\\infty$, then $\\sum_n\\delta_n=\\infty$ (hence $\\sum_n\\chi_n=\\infty$), proved by contradiction via convergence of $\\prod(1+\\delta_n)$; §14 derives that under uniform $\\chi_n\\le\\varepsilon$, crossing one dyadic scale costs at least $m\\gtrsim1/\\varepsilon$ steps.",
      "defining_relation": "p_n\\to\\infty\\Rightarrow\\sum_{n=0}^{\\infty}\\delta_n=\\infty\\Rightarrow\\sum_{n=0}^{\\infty}\\chi_n=\\infty;\\qquad \\chi_n\\le\\varepsilon\\Rightarrow m\\gtrsim\\varepsilon^{-1}",
      "notes": "The step-count bound (§14) is registered as guards G-drift/G-congestion in §23."
    },
    {
      "id": "ns.c3.c3c.cancellation_ratio",
      "latex": "\\frac{|\\dot e_p+\\dot e_q|}{\\min\\{|\\dot e_p|,|\\dot e_q|\\}}\\le\\frac{k}{p}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "高頻交換相消比（定理 8.1）",
      "label_en": "High-exchange cancellation ratio (Theorem 8.1)",
      "definition_zh": "定理8.1：Class II 的 $\\dot e_p+\\dot e_q=-\\dot e_k$，兩高頻模轉移殘餘量相對其較小者之比 $\\le k/p$；當 $k/p\\to0$，兩個高頻轉移趨於大量相反、幾乎完全相消，對應 Waleffe 對強非局部 reverse-type 交互作用 pair cancellation 的精確代數版本。",
      "definition_en": "Theorem 8.1: since $\\dot e_p+\\dot e_q=-\\dot e_k$ for Class II, the ratio of the residual to the smaller of the two high-mode transfers is $\\le k/p$; as $k/p\\to0$ the two high-mode transfers become large, nearly-canceling opposites — the exact algebraic counterpart of Waleffe's pair-cancellation description for strongly nonlocal reverse-type interactions.",
      "defining_relation": "\\frac{|\\dot e_p+\\dot e_q|}{\\min\\{|\\dot e_p|,|\\dot e_q|\\}}\\le\\frac{k}{p}"
    },
    {
      "id": "ns.c3.c3c.donor_receiver_arrow",
      "latex": "p\\longrightarrow\\{k,q\\}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "施者—受者流向記號",
      "label_en": "Donor-to-receivers flow notation",
      "definition_zh": "本輪反覆使用的記號 $X\\longrightarrow\\{Y,Z\\}$，表示能量施者 $X$（$\\dot e_X<0$）餵給受者 $Y,Z$（$\\dot e>0$）；Class II positive pair production 為 $p\\to\\{k,q\\}$（第9節），Class III／IV 則為 $k\\to\\{p,q\\}$（第16、18節）。",
      "definition_en": "Recurring notation $X\\longrightarrow\\{Y,Z\\}$ meaning donor $X$ ($\\dot e_X<0$) feeds receivers $Y,Z$ ($\\dot e>0$); Class II's positive pair production is $p\\to\\{k,q\\}$ (§9), while Class III and IV are both $k\\to\\{p,q\\}$ (§16, §18)."
    },
    {
      "id": "ns.c3.c3c.r_iii",
      "latex": "\\mathcal R_{\\mathrm{III}}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class III 臨界配對生成量",
      "label_en": "Class III critical pair-production contribution",
      "definition_zh": "第16節：對符號組態 $(+,-,+)$（unique sign 在中頻 $p$），$\\mathcal R_{\\mathrm{III}}=p(q-k)\\Theta_\\tau$；正production時 $k$ 為 donor（$k\\to\\{p,q\\}$），第17節證明強非局部時 $|\\mathcal R_{\\mathrm{III}}|\\sim p^2|\\Theta_\\tau|$，沒有 $(k/p)^2$ 稅。",
      "definition_en": "§16: for sign pattern $(+,-,+)$ (unique sign at middle mode $p$), $\\mathcal R_{\\mathrm{III}}=p(q-k)\\Theta_\\tau$; when positive, $k$ is the donor ($k\\to\\{p,q\\}$), and §17 shows that in the strongly nonlocal regime $|\\mathcal R_{\\mathrm{III}}|\\sim p^2|\\Theta_\\tau|$ — no $(k/p)^2$ tax.",
      "defining_relation": "\\mathcal R_{\\mathrm{III}}=p(q-k)\\Theta_\\tau",
      "notes": "Classified a SURVIVOR in the §27 survivor map: forward-compatible and untaxed by the radial-gap suppression."
    },
    {
      "id": "ns.c3.c3c.r_iv",
      "latex": "\\mathcal R_{\\mathrm{IV}}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class IV 臨界配對生成量",
      "label_en": "Class IV critical pair-production contribution",
      "definition_zh": "第18節：對符號組態 $(+,+,-)$（unique sign 在最高頻 $q$），$\\mathcal R_{\\mathrm{IV}}=q(k-p)\\Theta_\\tau$；正production時同樣 $k$ 為 donor（$k\\to\\{p,q\\}$）。第19節證明 Class IV 是三個 heterochiral classes 中唯一 unique-sign 模位於 triad 最大波數者。",
      "definition_en": "§18: for sign pattern $(+,+,-)$ (unique sign at largest mode $q$), $\\mathcal R_{\\mathrm{IV}}=q(k-p)\\Theta_\\tau$; when positive, $k$ is again the donor ($k\\to\\{p,q\\}$). §19 proves Class IV is the only heterochiral class whose unique-sign mode sits at the triad's largest wavenumber.",
      "defining_relation": "\\mathcal R_{\\mathrm{IV}}=q(k-p)\\Theta_\\tau",
      "notes": "Classified PRIMARY FRONTIER SURVIVOR in the §27 survivor map — top-priority class for C3-D."
    },
    {
      "id": "ns.c3.c3c.class_iv_top_unique",
      "latex": "\\text{Class IV: unique-sign mode}=q\\ (\\text{largest})",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "Class IV 頂端唯一性分類",
      "label_en": "Class-IV top-unique classification",
      "definition_zh": "第19節表格比較三個 heterochiral classes 的 unique-sign 波數位置：Class II 在最小 $k$，Class III 在中間 $p$，Class IV 在最大 $q$；故若追蹤 unique-sign critical pair-production frontier，Class IV 是唯一直接 frontier-capable 的 class（不表示 global blow-up 只能由 Class IV 構成）。",
      "definition_en": "The §19 table compares the unique-sign wavenumber position across the three heterochiral classes: Class II at smallest $k$, Class III at middle $p$, Class IV at largest $q$; hence Class IV is the only class directly capable of extending the unique-sign pair-production frontier (not implying blow-up must be built from Class IV alone)."
    },
    {
      "id": "ns.c3.c3c.forward_compatible",
      "latex": "\\text{forward-compatible}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "前向相容分類（定理 C3-C.3）",
      "label_en": "Forward-Compatible Survivor Classification (Theorem C3-C.3)",
      "definition_zh": "第21節定義：positive pair-production event 稱為 forward-compatible，若最低波數 $k$ 是 energy donor。由精確符號關係，Class III、IV 均 forward-compatible；Class II 的 donor 是 $p$ 而非 $k$，故在此精確意義下不是 forward-compatible，此分類為純代數推導，不依賴 Waleffe 統計不穩定性假設。",
      "definition_en": "§21 defines a positive pair-production event as forward-compatible if the lowest wavenumber $k$ is the energy donor. By exact sign relations, Class III and IV are both forward-compatible; Class II's donor is $p$, not $k$, so it is not forward-compatible in this precise sense — a classification following purely from exact conservation algebra, not Waleffe's statistical instability assumption."
    },
    {
      "id": "ns.c3.c3c.c_ii_n",
      "latex": "\\mathfrak C_{II}^{(N)}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "第 N 層 Class II 三元組集合",
      "label_en": "N-th shell Class-II triad set",
      "definition_zh": "第25節（定理25.1）定義：滿足 $p\\ge2^Nk$ 的所有 Class-II triads 集合，用以界定 strongly-nonlocal（dyadic separation $\\ge N$）子族，是 $P_{II}^{(N)}$、$V_{II}^{(N)}$ 的求和範圍。",
      "definition_en": "Defined in §25 (Theorem 25.1) as the set of all Class-II triads satisfying $p\\ge2^Nk$, delimiting the strongly-nonlocal (dyadic separation $\\ge N$) subfamily over which $P_{II}^{(N)}$ and $V_{II}^{(N)}$ are summed.",
      "defining_relation": "\\mathfrak C_{II}^{(N)}=\\{\\tau:\\ p\\ge2^Nk\\}"
    },
    {
      "id": "ns.c3.c3c.p_ii_n",
      "latex": "P_{II}^{(N)}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "累積配對生成量",
      "label_en": "Cumulative pair production",
      "definition_zh": "第25節定義 $P_{II}^{(N)}=\\int\\sum_{\\tau\\in\\mathfrak C_{II}^{(N)}}|\\mathcal R_\\tau(t)|\\,dt$，即 $N$ 層以上 strongly-nonlocal Class II triads 的臨界配對生成絕對值對時間與 triad 求和積分的總量，定理25.1證明其 $\\le2^{-2N}V_{II}^{(N)}$。",
      "definition_en": "Defined in §25 as $P_{II}^{(N)}=\\int\\sum_{\\tau\\in\\mathfrak C_{II}^{(N)}}|\\mathcal R_\\tau(t)|\\,dt$, the total (time-integrated, triad-summed) absolute critical pair production from strongly-nonlocal Class-II triads at dyadic level $\\ge N$; Theorem 25.1 shows it is $\\le2^{-2N}V_{II}^{(N)}$.",
      "defining_relation": "P_{II}^{(N)}=\\int\\sum_{\\tau\\in\\mathfrak C_{II}^{(N)}}|\\mathcal R_\\tau(t)|\\,dt\\ \\le\\ 2^{-2N}V_{II}^{(N)}"
    },
    {
      "id": "ns.c3.c3c.v_ii_n",
      "latex": "V_{II}^{(N)}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "隱藏高頻交換變差量",
      "label_en": "Hidden high-exchange variation",
      "definition_zh": "第25節定義 $V_{II}^{(N)}=\\int\\sum_{\\tau\\in\\mathfrak C_{II}^{(N)}}X_{\\mathrm{hi},\\tau}(t)\\,dt$；第26節指出若能證 $V_{II}^{(N)}=o(2^{2N})$（$N\\to\\infty$）則 $P_{II}^{(N)}\\to0$，此為第30節狀態表列為 OPEN 的具體後續 proof target；第24節強調它是 absolute turnover，energy conservation 不控制它，故目前無已證全域有限界。",
      "definition_en": "Defined in §25 as $V_{II}^{(N)}=\\int\\sum_{\\tau\\in\\mathfrak C_{II}^{(N)}}X_{\\mathrm{hi},\\tau}(t)\\,dt$; §26 notes that proving $V_{II}^{(N)}=o(2^{2N})$ as $N\\to\\infty$ would force $P_{II}^{(N)}\\to0$ — listed OPEN in §30; §24 stresses it is absolute exchange turnover, unbounded by energy conservation alone, so no finite global bound is currently proved.",
      "defining_relation": "V_{II}^{(N)}=\\int\\sum_{\\tau\\in\\mathfrak C_{II}^{(N)}}X_{\\mathrm{hi},\\tau}(t)\\,dt",
      "notes": "The paper's key concrete forward proof target: 'control the growth rate of $V_{II}^{(N)}$' (§26), central to C3-D obligation D3."
    },
    {
      "id": "ns.c3.c3c.r_iv_q",
      "latex": "\\mathcal R_{IV,q}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "第 q 殼層 Class IV frontier 生成量",
      "label_en": "Shell-q Class-IV frontier production",
      "definition_zh": "第29節 D1 定義：unique-sign highest mode 位於殼層 $q$ 的 Class-IV positive pair production；C3-D 的 proof obligation 之一是研究 $\\sum_{q>Q}\\int[\\mathcal R_{IV,q}]_+\\,dt$ 在 blow-up 情境下是否必有 nontrivial lower envelope。",
      "definition_en": "Defined in §29 (obligation D1) as the Class-IV positive pair production whose unique-sign highest mode sits at shell $q$; a C3-D proof obligation is whether $\\sum_{q>Q}\\int[\\mathcal R_{IV,q}]_+\\,dt$ must have a nontrivial lower envelope in a blow-up scenario.",
      "defining_relation": "\\sum_{q>Q}\\int[\\mathcal R_{IV,q}]_+\\,dt",
      "notes": "Forward-looking definition for the next round (C3-D), not analyzed within C3-C itself."
    },
    {
      "id": "ns.c3.c3c.x_integration_guards",
      "latex": "\\text{G-}\\chi,\\ \\text{G-}\\delta,\\ \\text{G-tax},\\ \\text{G-drift},\\ \\text{G-congestion}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "C3-C X-Integration 守衛清單",
      "label_en": "C3-C X-Integration guard checklist",
      "definition_zh": "第23節將本輪結果整理為五個守衛條件：G-$\\chi$（$\\chi_n=k_n/p_n$）、G-$\\delta$（$\\delta_n\\le\\chi_n$）、G-tax（$|\\mathcal R_n|/X_{\\mathrm{hi},n}\\le\\chi_n^2$）、G-drift（$p_n\\to\\infty\\Rightarrow\\sum\\delta_n=\\infty$）、G-congestion（$\\chi_n\\le\\varepsilon\\Rightarrow$每 dyadic scale 至少 $O(\\varepsilon^{-1})$ 步）；Class-II singular certificate 不能只宣稱「all interactions legal」，還須攜帶這些 debt。",
      "definition_en": "§23 packages this round's results into five guard conditions a singular certificate must carry: G-$\\chi$ ($\\chi_n=k_n/p_n$), G-$\\delta$ ($\\delta_n\\le\\chi_n$), G-tax ($|\\mathcal R_n|/X_{\\mathrm{hi},n}\\le\\chi_n^2$), G-drift ($p_n\\to\\infty\\Rightarrow\\sum\\delta_n=\\infty$), G-congestion ($\\chi_n\\le\\varepsilon\\Rightarrow$ at least $O(\\varepsilon^{-1})$ steps per dyadic scale) — a Class-II singular certificate cannot just assert 'all interactions legal'.",
      "notes": "Bundles ns.c3.c3c.chi_tau, delta_tau, quadratic_nonlocality_tax, radial_drift_congestion_lemma under the paper's 'X-Integral Unified Program' (see Internal dependencies)."
    },
    {
      "id": "ns.c3.c3c.c3d_frontier_rigidity",
      "latex": "\\textbf{C3-D --- Forward Heterochiral Frontier Rigidity}",
      "series": "NS",
      "first_appearance": "C3-C",
      "label_zh": "C3-D：前向異手性前緣剛性（下輪主線）",
      "label_en": "C3-D: Forward Heterochiral Frontier Rigidity (next round's program)",
      "definition_zh": "第28節於本輪末提出的下一主線名稱：核心 survivor 為 Classes III/IV 加上 non-negligibly local 的 Class II，Class IV 優先度最高；第29節列出四項 proof obligations（D1 Class-IV dyadic frontier production、D2 分離 III/IV ancestry、D3 控制 $V_{II}^{(N)}$、D4 wave-space advection formulation）。",
      "definition_en": "The name coined at the end of this round (§28) for the next research program: core survivors are Classes III/IV plus non-negligibly-local Class II, with Class IV top priority; §29 lists four proof obligations (D1 Class-IV dyadic frontier production, D2 separating III/IV ancestry, D3 controlling $V_{II}^{(N)}$, D4 a wave-space advection formulation).",
      "notes": "Listed OPEN in the §30 status table; a forward pointer to the next paper in the series, not a result proved in C3-C."
    },
    {
      "id": "ns.c3.c3d.cutoff_k",
      "latex": "K",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "銳截止波數",
      "label_en": "sharp cutoff wavenumber",
      "definition_zh": "第3節引入的正實數閾值 K>0，用以將單一 triad 的三個模 k,p,q 分成高於／低於截止的兩側，是本輪所有 cutoff-flux 分析的自變量。",
      "definition_en": "A positive threshold K>0 introduced in Sec.3 that splits a triad's three modes k,p,q into sides above/below the spectral cutoff; the independent variable of all cutoff-flux results in this round.",
      "notes": "與波數 k,p,q 本身不同，K 是外加的觀測截止，可自由掃過整個譜。"
    },
    {
      "id": "ns.c3.c3d.e_highside",
      "latex": "E_{>K}^{(\\tau)}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Triad高側能量",
      "label_en": "triad high-side energy",
      "definition_zh": "第3節定義，單一 triad τ 中模長大於 K 的所有模能量之和，用來定義 cutoff flux。",
      "definition_en": "Defined in Sec.3 as the sum of modal energies of a single triad τ whose wavenumber exceeds K; the quantity whose time-derivative defines the cutoff flux.",
      "defining_relation": "E_{>K}^{(\\tau)}=\\sum_{r\\in\\{k,p,q\\},\\,r>K}e_r"
    },
    {
      "id": "ns.c3.c3d.phi_tau",
      "latex": "\\Phi_\\tau(K)",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Triad截止能量流",
      "label_en": "triad cutoff flux",
      "definition_zh": "第3節核心定義：單一 triad 高側能量 $E_{>K}^{(\\tau)}$ 的非線性時間導數，符號約定 $\\Phi_\\tau(K)>0$ 表示能量經此 triad 向截止以上的高波數流動。",
      "definition_en": "The central Sec.3 definition: the nonlinear time-derivative of the triad's high-side energy; sign convention $\\Phi_\\tau(K)>0$ means energy flows through this triad toward wavenumbers above K.",
      "defining_relation": "\\Phi_\\tau(K)=\\left(\\frac{d}{dt}E_{>K}^{(\\tau)}\\right)_{\\mathrm{nonlinear}}"
    },
    {
      "id": "ns.c3.c3d.phi_ii",
      "latex": "\\Phi_{II}(K)",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class II分段截止流",
      "label_en": "Class-II piecewise cutoff flux",
      "definition_zh": "定理4.1（C3-D.1）給出的分段公式，顯示 positive Class-II triad 在 $k<K<p$ 為負（reverse），在 $p<K<q$ 為正（forward），是本輪 sign-reversal 結果的核心對象。",
      "definition_en": "The piecewise formula from Theorem 4.1 (C3-D.1): a positive Class-II triad's flux is negative on $k<K<p$ (reverse) and positive on $p<K<q$ (forward) — the central object behind this round's sign-reversal result.",
      "defining_relation": "\\Phi_{II}(K)=\\begin{cases}0,&0<K<k\\\\-(q-p)\\Theta_\\tau,&k<K<p\\\\(p+k)\\Theta_\\tau,&p<K<q\\\\0,&K>q\\end{cases}",
      "notes": "第7節另證明 forward window 必落在截止的 O(k) boundary layer 內：K-k<p<K<q<K+k。"
    },
    {
      "id": "ns.c3.c3d.phi_iii_iv",
      "latex": "\\Phi_{III}(K),\\ \\Phi_{IV}(K)",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class III/IV均勻正向流",
      "label_en": "Class III/IV uniform forward flux",
      "definition_zh": "定理8.1（C3-D.2）證明：positive Class III 或 IV triad 的 $\\Phi_\\tau(K)$ 對所有中間截止 $K\\in(k,q)$ 恆正，與 Class II 的 reversal 形成對比。",
      "definition_en": "Theorem 8.1 (C3-D.2): for a positive Class III or IV triad, $\\Phi_\\tau(K)>0$ holds for every intermediate cutoff $K\\in(k,q)$, contrasting with Class II's sign reversal.",
      "defining_relation": "\\Phi_\\tau(K)>0\\quad\\forall K\\in(k,q)"
    },
    {
      "id": "ns.c3.c3d.frakf_tau",
      "latex": "\\mathfrak F_\\tau",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "對數截止積分流",
      "label_en": "log-cutoff integrated flux",
      "definition_zh": "第10節定義，將 $\\Phi_\\tau(K)$ 以對數尺度測度 $dK/K$ 對所有截止積分，得到單一 triad 對「跨尺度總淨方向」的量度。",
      "definition_en": "Defined in Sec.10 as the integral of $\\Phi_\\tau(K)$ against the logarithmic scale measure $dK/K$ over all cutoffs, giving a single triad's net cross-scale flux direction.",
      "defining_relation": "\\mathfrak F_\\tau=\\int_0^\\infty\\Phi_\\tau(K)\\,\\frac{dK}{K}"
    },
    {
      "id": "ns.c3.c3d.frakf_ii",
      "latex": "\\mathfrak F_{II}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class II對數積分流",
      "label_en": "Class-II log-integrated flux",
      "definition_zh": "第10–12節導出的 Class II 顯式 $\\mathfrak F_{II}$ 公式；定理12.1（C3-D.3）證明當 $k/p<\\chi_\\ast\\approx0.27846$ 時 $\\mathfrak F_{II}<0$，即 reverse 貢獻主導。",
      "definition_en": "The explicit Class-II formula derived in Sec.10–12; Theorem 12.1 (C3-D.3) proves $\\mathfrak F_{II}<0$ whenever $k/p<\\chi_\\ast\\approx0.27846$, i.e. the reverse contribution dominates.",
      "defining_relation": "\\mathfrak F_{II}=-(q-p)\\Theta_\\tau\\log\\frac{p}{k}+(p+k)\\Theta_\\tau\\log\\frac{q}{p}"
    },
    {
      "id": "ns.c3.c3d.frakf_iii_iv",
      "latex": "\\mathfrak F_{III},\\ \\mathfrak F_{IV}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class III/IV對數積分流",
      "label_en": "Class III/IV log-integrated flux",
      "definition_zh": "第14節由定理8.1直接推得：positive Class III、IV 的 $\\mathfrak F_\\tau$ 恆正，即在對數尺度上均勻正向，與 Class II 相對。",
      "definition_en": "Sec.14 derives directly from Theorem 8.1 that positive Class III and IV always have $\\mathfrak F_\\tau>0$ — uniformly forward on the logarithmic scale, opposite to Class II.",
      "defining_relation": "\\mathfrak F_{III}>0,\\qquad\\mathfrak F_{IV}>0"
    },
    {
      "id": "ns.c3.c3d.chi",
      "latex": "\\chi",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "非局部性比值",
      "label_en": "nonlocality ratio",
      "definition_zh": "$\\chi=k/p$，衡量 triad 非局部程度的比值；本輪（第11節起）系統性用其界定 forward-window 厚度、log-flux 反轉閾值 $\\chi_\\ast$ 與 kernel/amplitude 指數分類。",
      "definition_en": "$\\chi=k/p$, the ratio measuring a triad's nonlocality; this round (from Sec.11 on) systematically uses it to bound the forward-window thickness, the log-flux reversal threshold $\\chi_\\ast$, and the kernel/amplitude exponent classification.",
      "defining_relation": "\\chi=\\frac{k}{p}",
      "notes": "符號承接自 C3-C（本輪 Sec.0 已引用），但 C3-D 賦予其新的定量角色。Carried over from C3-C but given new quantitative roles here."
    },
    {
      "id": "ns.c3.c3d.delta",
      "latex": "\\delta",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "高頻對相對展寬",
      "label_en": "high-pair relative spread",
      "definition_zh": "第11節定義 $\\delta=(q-p)/p$，度量 $p,q$ 間的相對間距；triangle geometry 給出 $0<\\delta\\le\\chi\\le1$，是改寫 $\\mathfrak F_{II}$ 公式的關鍵變量。",
      "definition_en": "Defined in Sec.11 as $\\delta=(q-p)/p$, measuring the relative spacing between $p$ and $q$; triangle geometry gives $0<\\delta\\le\\chi\\le1$, the key variable used to rewrite the $\\mathfrak F_{II}$ formula.",
      "defining_relation": "\\delta=\\frac{q-p}{p}"
    },
    {
      "id": "ns.c3.c3d.chi_star",
      "latex": "\\chi_\\ast",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "對數反轉臨界值",
      "label_en": "log-cutoff reversal threshold",
      "definition_zh": "第12節（C3-D.3）定義為 $\\log(1/\\chi_\\ast)=1+\\chi_\\ast$ 的唯一正根，等於 Lambert $W$ 函數值 $W(e^{-1})\\approx0.278464542761$；當 $\\chi<\\chi_\\ast$ 時 positive Class II 必有 $\\mathfrak F_{II}<0$。",
      "definition_en": "Defined in Sec.12 (Theorem C3-D.3) as the unique positive root of $\\log(1/\\chi_\\ast)=1+\\chi_\\ast$, equal to the Lambert-$W$ value $W(e^{-1})\\approx0.278464542761$; whenever $\\chi<\\chi_\\ast$, a positive Class-II triad must have $\\mathfrak F_{II}<0$.",
      "defining_relation": "\\log\\frac{1}{\\chi_\\ast}=1+\\chi_\\ast \\Longleftrightarrow \\chi_\\ast=W(e^{-1})\\approx0.278464542761"
    },
    {
      "id": "ns.c3.c3d.g_helical",
      "latex": "|g_{s_ks_ps_q}(k,p,q)|",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Waleffe耦合係數",
      "label_en": "Waleffe coupling coefficient",
      "definition_zh": "第16節引入的標準 Waleffe helical 三波耦合係數量值，以三角形面積相關量 $Q$ 表示，是第17–20節所有 nonlocality bound 的幾何來源。",
      "definition_en": "The standard Waleffe helical triad-coupling coefficient magnitude introduced in Sec.16, expressed via the triangle-area quantity $Q$; the geometric source of every nonlocality bound in Sec.17–20.",
      "defining_relation": "|g_{s_ks_ps_q}(k,p,q)|=\\frac{Q}{4kpq}\\left|s_kk+s_pp+s_qq\\right|"
    },
    {
      "id": "ns.c3.c3d.q_factor",
      "latex": "Q",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "三角形面積因子",
      "label_en": "triangle-area factor",
      "definition_zh": "第16節定義 $Q^2=2(k^2p^2+p^2q^2+q^2k^2)-k^4-p^4-q^4$，等於波數三角形面積的固定倍數，滿足 $Q\\le2kp$，用於構成 helical 耦合係數。",
      "definition_en": "Defined in Sec.16 by $Q^2=2(k^2p^2+p^2q^2+q^2k^2)-k^4-p^4-q^4$, equal to a fixed multiple of the wavenumber-triangle area, satisfying $Q\\le2kp$; the building block of the helical coupling coefficient.",
      "defining_relation": "Q^2=2(k^2p^2+p^2q^2+q^2k^2)-k^4-p^4-q^4"
    },
    {
      "id": "ns.c3.c3d.g_ii",
      "latex": "|g_{II}|",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class II係數上界",
      "label_en": "Class-II coefficient bound",
      "definition_zh": "第17節證明 Class II 的 $|k-p-q|\\le2q$，故 $|g_{II}|\\le C$ 為 universal 常數，即非局部極限下 $g_{II}=O(1)$、無自動衰減（第18節）。",
      "definition_en": "Sec.17 shows Class II's $|k-p-q|\\le2q$ gives $|g_{II}|\\le C$ for a universal constant, i.e. $g_{II}=O(1)$ in the strong-nonlocal limit with no automatic suppression (Sec.18).",
      "defining_relation": "|g_{II}|\\le C"
    },
    {
      "id": "ns.c3.c3d.g_iii_iv",
      "latex": "|g_{III}|,\\ |g_{IV}|",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class III/IV係數上界",
      "label_en": "Class III/IV coefficient bound",
      "definition_zh": "第17節分別證明 $|g_{III}|,|g_{IV}|\\le Ck/q\\le Ck/p$，即在非局部極限下 $g_{III},g_{IV}=O(\\chi)$，比 Class II 多一階線性抑制（第18節）。",
      "definition_en": "Sec.17 proves $|g_{III}|,|g_{IV}|\\le Ck/q\\le Ck/p$, so in the strong-nonlocal limit $g_{III},g_{IV}=O(\\chi)$ — one extra order of linear suppression relative to Class II (Sec.18).",
      "defining_relation": "|g_{III}|\\le C\\frac{k}{q}\\le C\\frac{k}{p},\\qquad|g_{IV}|\\le C\\frac{k}{q}\\le C\\frac{k}{p}"
    },
    {
      "id": "ns.c3.c3d.amplitudes_akpq",
      "latex": "a_k,\\ a_p,\\ a_q",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "helical模振幅",
      "label_en": "helical mode amplitudes",
      "definition_zh": "第19節定義 $a_k=|u^{s_k}(\\mathbf k)|$ 等，為三個 helical Fourier 模的振幅量值，透過 $|\\Theta_\\tau|\\le C|g_\\tau|a_ka_pa_q$ 將 transfer 與 kernel、振幅連結。",
      "definition_en": "Defined in Sec.19 as $a_k=|u^{s_k}(\\mathbf k)|$ etc., the amplitude magnitudes of the three helical Fourier modes; linked to the transfer scalar via $|\\Theta_\\tau|\\le C|g_\\tau|a_ka_pa_q$.",
      "defining_relation": "a_k=|u^{s_k}(\\mathbf k)|,\\quad a_p=|u^{s_p}(\\mathbf p)|,\\quad a_q=|u^{s_q}(\\mathbf q)|",
      "notes": "第21節 Amplitude Compensation Debt：Class II 需 $a_{k_n}a_{p_n}a_{q_n}\\gtrsim\\chi_n^{-2}$，Class III/IV 需 $\\gtrsim\\chi_n^{-1}$，才能在 $\\chi_n\\to0$ 下維持不消失的 pair production。"
    },
    {
      "id": "ns.c3.c3d.theta_tau",
      "latex": "\\Theta_\\tau",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "純量transfer參數",
      "label_en": "scalar transfer parameter",
      "definition_zh": "第2節「回顧」引入的純量參數，決定三模能量變化率的共同比例因子（承接自更早的 C3 分輪）；本輪據此推出每一 class 的 $\\Phi_\\tau(K)$ 分段公式與 $\\mathfrak F_\\tau$。",
      "definition_en": "The scalar parameter recapped in Sec.2 that sets the common proportionality of the three modal energy rates (carried over from an earlier C3 sub-round); this round uses it to derive each class's piecewise $\\Phi_\\tau(K)$ and $\\mathfrak F_\\tau$.",
      "defining_relation": "(\\dot e_k,\\dot e_p,\\dot e_q)=\\Theta_\\tau(s_pp-s_qq, s_qq-s_kk, s_kk-s_pp)",
      "notes": "標記為本輪「回顧」對象，非 C3-D 首次定義，但本輪為其賦予截止流/log-flux 的具體新結果。"
    },
    {
      "id": "ns.c3.c3d.psi_tau",
      "latex": "\\Psi_\\tau",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class IV正向transfer參數",
      "label_en": "Class-IV positive transfer parameter",
      "definition_zh": "第2節針對 Class IV 定義 $\\Psi_\\tau=-\\Theta_\\tau>0$，使 positive pair production 情形下的 transfer 參數符號一致為正，供第8、19節推導使用。",
      "definition_en": "Defined in Sec.2 for Class IV as $\\Psi_\\tau=-\\Theta_\\tau>0$, so the transfer parameter has a uniformly positive sign under positive pair production; used throughout Sec.8, 19.",
      "defining_relation": "\\Psi_\\tau=-\\Theta_\\tau>0"
    },
    {
      "id": "ns.c3.c3d.r_ii",
      "latex": "\\mathcal R_{II}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class II臨界對產生",
      "label_en": "Class-II critical pair production",
      "definition_zh": "第2節定義 $\\mathcal R_{II}=k(q-p)\\Theta_\\tau$；第19–20節導出量值界 $|\\mathcal R_{II}|\\le Ck^2a_ka_pa_q$，相對 $p^2a_ka_pa_q$ 呈二次非局部指數 $\\chi^2$。",
      "definition_en": "Defined in Sec.2 as $\\mathcal R_{II}=k(q-p)\\Theta_\\tau$; Sec.19–20 derive the bound $|\\mathcal R_{II}|\\le Ck^2a_ka_pa_q$, giving a quadratic nonlocality exponent $\\chi^2$ relative to $p^2a_ka_pa_q$.",
      "defining_relation": "\\mathcal R_{II}=k(q-p)\\Theta_\\tau,\\qquad|\\mathcal R_{II}|\\le Ck^2a_ka_pa_q"
    },
    {
      "id": "ns.c3.c3d.r_iii",
      "latex": "\\mathcal R_{III}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class III臨界對產生",
      "label_en": "Class-III critical pair production",
      "definition_zh": "第2節定義 $\\mathcal R_{III}=p(q-k)\\Theta_\\tau$；第19–20節導出 $|\\mathcal R_{III}|\\le Ckpa_ka_pa_q$，呈線性非局部指數 $\\chi$。",
      "definition_en": "Defined in Sec.2 as $\\mathcal R_{III}=p(q-k)\\Theta_\\tau$; Sec.19–20 give $|\\mathcal R_{III}|\\le Ckpa_ka_pa_q$, a linear nonlocality exponent $\\chi$.",
      "defining_relation": "\\mathcal R_{III}=p(q-k)\\Theta_\\tau,\\qquad|\\mathcal R_{III}|\\le Ckpa_ka_pa_q"
    },
    {
      "id": "ns.c3.c3d.r_iv",
      "latex": "\\mathcal R_{IV}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Class IV臨界對產生",
      "label_en": "Class-IV critical pair production",
      "definition_zh": "第2節定義 $\\mathcal R_{IV}=q(k-p)\\Theta_\\tau$；第19–20節導出 $|\\mathcal R_{IV}|\\le Ckpa_ka_pa_q$，同樣呈線性非局部指數 $\\chi$。",
      "definition_en": "Defined in Sec.2 as $\\mathcal R_{IV}=q(k-p)\\Theta_\\tau$; Sec.19–20 give $|\\mathcal R_{IV}|\\le Ckpa_ka_pa_q$, also a linear nonlocality exponent $\\chi$.",
      "defining_relation": "\\mathcal R_{IV}=q(k-p)\\Theta_\\tau,\\qquad|\\mathcal R_{IV}|\\le Ckpa_ka_pa_q"
    },
    {
      "id": "ns.c3.c3d.aluie_eyink_sigma",
      "latex": "\\sigma_p,\\ \\sigma_3,\\ P",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "Aluie–Eyink尺度指數",
      "label_en": "Aluie–Eyink scaling exponent",
      "definition_zh": "第22節引用 Aluie–Eyink sharp spectral filter 定理的符號：正的慣性域 scaling 指數 $0<\\sigma_p<1$，以及非局部 band 對比波數 $P\\gg K$，衰減因子例如 $(K/P)^{2\\sigma_3}$；屬外部文獻符號，非本文原創定義。",
      "definition_en": "Symbols cited in Sec.22 from the Aluie–Eyink sharp-spectral-filter theorem: a positive inertial-range scaling exponent $0<\\sigma_p<1$, and an outer comparison wavenumber $P\\gg K$ with decay factor e.g. $(K/P)^{2\\sigma_3}$; imported from external literature, not defined by this paper.",
      "notes": "用於第23節「不能偷用 locality theorem 解 Clay 問題」的討論，屬引用符號而非本輪原創定義。"
    },
    {
      "id": "ns.c3.c3d.guard_fsig",
      "latex": "G-FSIG",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "截止流簽章guard",
      "label_en": "cutoff-flux signature guard",
      "definition_zh": "第24節新增的 X-Integration guard，規定必須完整保存映射 $K\\mapsto\\Phi_\\tau(K)$，不能只記錄單一 transfer 振幅。",
      "definition_en": "An X-Integration guard added in Sec.24 requiring the full map $K\\mapsto\\Phi_\\tau(K)$ to be stored, not merely a single transfer amplitude.",
      "defining_relation": "K\\mapsto\\Phi_\\tau(K)"
    },
    {
      "id": "ns.c3.c3d.guard_fwin",
      "latex": "G-FWIN",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "正向窗寬guard",
      "label_en": "forward-window width guard",
      "definition_zh": "第24節定義 Class II 的對數正向窗寬 $\\operatorname{width}_{\\log}(p,q)=\\log(q/p)\\le\\chi$，作為 X-Integration 必存的幾何量。",
      "definition_en": "Defined in Sec.24 as Class II's logarithmic forward-window width $\\operatorname{width}_{\\log}(p,q)=\\log(q/p)\\le\\chi$, a geometric quantity X-Integration must retain.",
      "defining_relation": "\\operatorname{width}_{\\log}(p,q)=\\log(q/p)\\le\\chi"
    },
    {
      "id": "ns.c3.c3d.guard_log",
      "latex": "G-LOG",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "對數截止符號guard",
      "label_en": "log-cutoff sign guard",
      "definition_zh": "第24節規定：若 $\\chi<\\chi_\\ast$，positive Class II 事件必須標記 $\\mathfrak F_{II}<0$，將定理12.1直接嵌入 X-Integration 記帳規則。",
      "definition_en": "Sec.24 requires that whenever $\\chi<\\chi_\\ast$, a positive Class-II event must be flagged $\\mathfrak F_{II}<0$, embedding Theorem 12.1 directly into the X-Integration bookkeeping rule.",
      "defining_relation": "\\chi<\\chi_\\ast\\Rightarrow\\mathfrak F_{II}<0"
    },
    {
      "id": "ns.c3.c3d.guard_kernel",
      "latex": "G-KERNEL",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "幾何抑制guard",
      "label_en": "geometry suppression guard",
      "definition_zh": "第24節規定必須保存 $g_{II}=O(1)$ 與 $g_{III/IV}=O(\\chi)$ 的幾何抑制分類，作為 X-Integration 的必存記錄。",
      "definition_en": "Sec.24 requires storing the geometric-suppression classification $g_{II}=O(1)$ versus $g_{III/IV}=O(\\chi)$ as mandatory X-Integration bookkeeping.",
      "defining_relation": "g_{II}=O(1),\\qquad g_{III/IV}=O(\\chi)"
    },
    {
      "id": "ns.c3.c3d.guard_amp",
      "latex": "G-AMP",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "振幅補償guard",
      "label_en": "amplitude compensation guard",
      "definition_zh": "第24節規定：若 $\\chi\\to0$ 但 pair production 不消失，必須顯式記錄振幅增長的來源，對應第21節的 Amplitude Compensation Debt。",
      "definition_en": "Sec.24 requires that if $\\chi\\to0$ while pair production does not vanish, the source of amplitude growth must be explicitly recorded, corresponding to the Amplitude Compensation Debt of Sec.21."
    },
    {
      "id": "ns.c3.c3d.c3e_local_ratio_c",
      "latex": "c",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "局部比值下界(C3-E預告)",
      "label_en": "local ratio lower bound (C3-E preview)",
      "definition_zh": "第28節為下一輪 C3-E 預告的固定常數 $c>0$，限制 $c\\le k/p\\le1$ 以界定「局部」heterochiral triad 的範圍。",
      "definition_en": "A fixed constant $c>0$ previewed in Sec.28 for the next round C3-E, constraining $c\\le k/p\\le1$ to delimit \"local\" heterochiral triads.",
      "defining_relation": "c\\le\\frac{k}{p}\\le1",
      "notes": "屬本文末段為 C3-E 鋪陳的符號，非 C3-D 本輪定理內容。"
    },
    {
      "id": "ns.c3.c3d.c3e_r_loc",
      "latex": "\\mathcal R_q^{\\rm loc}",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "局部殼層對產生(C3-E預告)",
      "label_en": "local shell pair production (C3-E preview)",
      "definition_zh": "第28節 E1 預告的 dyadic shell 局部 pair-production 量，用於研究 $\\sum_q\\int[\\mathcal R_q^{\\rm loc}]_+dt=\\infty$ 是否為 blow-up 必要條件。",
      "definition_en": "The dyadic-shell local pair-production quantity previewed in Sec.28 (E1), used to ask whether $\\sum_q\\int[\\mathcal R_q^{\\rm loc}]_+dt=\\infty$ is necessary for blow-up.",
      "notes": "C3-E 前瞻符號，本文未給出其顯式公式。"
    },
    {
      "id": "ns.c3.c3d.c3e_lambda",
      "latex": "\\lambda",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "局部共同尺度(C3-E預告)",
      "label_en": "common local scale (C3-E preview)",
      "definition_zh": "第28節 E3 預告，表示局部 triad 的共同尺度 $k\\sim p\\sim q\\sim\\lambda$，用於定義黏性時間 $\\tau_\\nu$。",
      "definition_en": "Previewed in Sec.28 (E3) as the common scale of a local triad, $k\\sim p\\sim q\\sim\\lambda$, used to define the viscous time $\\tau_\\nu$.",
      "defining_relation": "k\\sim p\\sim q\\sim\\lambda",
      "notes": "C3-E 前瞻符號。"
    },
    {
      "id": "ns.c3.c3d.c3e_tau_nu",
      "latex": "\\tau_\\nu",
      "series": "NS",
      "first_appearance": "C3-D",
      "label_zh": "黏性時間尺度(C3-E預告)",
      "label_en": "viscous time scale (C3-E preview)",
      "definition_zh": "第28節 E3 預告，定義 $\\tau_\\nu\\sim(\\nu\\lambda^2)^{-1}$，用於研究 positive pair-production 相位相干是否須在 $O(\\lambda^{-2})$ 窗內完成。",
      "definition_en": "Previewed in Sec.28 (E3) as $\\tau_\\nu\\sim(\\nu\\lambda^2)^{-1}$, used to ask whether positive pair-production coherence must complete within an $O(\\lambda^{-2})$ window.",
      "defining_relation": "\\tau_\\nu\\sim(\\nu\\lambda^2)^{-1}",
      "notes": "C3-E 前瞻符號。"
    },
    {
      "id": "ns.c3.c3e.p_high_pass",
      "latex": "P_{>J}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "高頻投影算子",
      "label_en": "High-pass projector",
      "definition_zh": "第1節定義的光滑 Littlewood–Paley 高頻投影算子，投影到頻率 $|\\xi|\\gtrsim\\lambda_J$ 的部分；熱半群作用其上滿足指數譜隙衰減 $\\|e^{\\nu\\tau\\Delta}P_{>J}f\\|_{L^p}\\le Ce^{-c\\nu\\lambda_J^2\\tau}\\|P_{>J}f\\|_{L^p}$。",
      "definition_en": "The smooth Littlewood–Paley high-pass projector onto frequencies $|\\xi|\\gtrsim\\lambda_J$, introduced in Section 1; the heat semigroup restricted to its range obeys the exponential spectral-gap decay $\\|e^{\\nu\\tau\\Delta}P_{>J}f\\|_{L^p}\\le Ce^{-c\\nu\\lambda_J^2\\tau}\\|P_{>J}f\\|_{L^p}$.",
      "notes": "$C\\ge1,c>0$ 為 universal constants；其後固定 $p=3$。"
    },
    {
      "id": "ns.c3.c3e.lambda_dyadic",
      "latex": "\\lambda_J=2^J",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "截止頻率（dyadic 尺度）",
      "label_en": "Dyadic cutoff frequency",
      "definition_zh": "第1節定義的 dyadic 截止頻率 $\\lambda_J=2^J$；同一記法自第11節起以 shell 指標重寫為 $\\lambda_q=2^q$，用於局部 block 分析。",
      "definition_en": "The dyadic cutoff/shell frequency $\\lambda_J=2^J$ defined in Section 1; the same pattern is reused from Section 11 onward as $\\lambda_q=2^q$ for local shell-indexed analysis.",
      "defining_relation": "\\lambda_J=2^J"
    },
    {
      "id": "ns.c3.c3e.duhamel_hightail",
      "latex": "\\mathcal N_J[s,t]",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "高頻尾端 Duhamel 非線性項",
      "label_en": "High-tail Duhamel nonlinear term",
      "definition_zh": "第2節定義，將 Duhamel 公式投影到 $>J$ 頻段後得到的非線性來源項，是黏性窗口更新定理的核心驅動項。",
      "definition_en": "Defined in Section 2 as the high-frequency projection of the Duhamel nonlinear term; the driving source in the viscous-window renewal theorem.",
      "defining_relation": "\\mathcal N_J[s,t]=\\int_s^t e^{\\nu(t-r)\\Delta}P_{>J}\\mathbb P\\nabla\\cdot(u\\otimes u)(r)\\,dr"
    },
    {
      "id": "ns.c3.c3e.h_hightail_norm",
      "latex": "H_J(t)=\\|P_{>J}u(t)\\|_3",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "高頻尾端範數",
      "label_en": "High-tail norm",
      "definition_zh": "第2節定義的 $L^3$ 高頻尾端範數；第3節在窗口端點離散化為 $H_m=H_J(t_m)$，第7節 XViscRenew 證書中再標記為窗口起訖值 $H_{\\rm in},H_{\\rm out}$。",
      "definition_en": "The $L^3$ high-tail norm defined in Section 2; discretized at window endpoints as $H_m=H_J(t_m)$ in Section 3, and relabeled as window-boundary values $H_{\\rm in},H_{\\rm out}$ inside the XViscRenew certificate of Section 7.",
      "defining_relation": "H_J(t)=\\|P_{>J}u(t)\\|_3",
      "notes": "$H_{\\rm in},H_{\\rm out}$ 文中未明寫等式，依上下文即窗口首尾的 $H_m$ 值。"
    },
    {
      "id": "ns.c3.c3e.theta_window_param",
      "latex": "\\theta>0",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "窗口尺度參數",
      "label_en": "Window-scale parameter",
      "definition_zh": "第3節固定的正參數，用來設定黏性窗口長度 $\\tau_J=\\theta/(\\nu\\lambda_J^2)$；選 $\\theta$ 足夠大使收縮比 $\\rho=Ce^{-c\\theta}<1$。",
      "definition_en": "A fixed positive parameter from Section 3 that sets the viscous window length $\\tau_J=\\theta/(\\nu\\lambda_J^2)$; chosen large enough that the contraction ratio $\\rho=Ce^{-c\\theta}<1$."
    },
    {
      "id": "ns.c3.c3e.tau_viscous_window",
      "latex": "\\tau_J=\\dfrac{\\theta}{\\nu\\lambda_J^2}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "黏性窗口長度",
      "label_en": "Viscous window duration",
      "definition_zh": "第3節定義的單一黏性窗口時間長度，本輪核心尺度；後續以同一 $O((\\nu\\lambda^2)^{-1})$ 形式重複出現為 $\\tau_q$（第18節）與 $\\tau_{q_n},\\tau_n$（第19節 Zeno 定理），皆為同一概念在不同指標下的重寫。",
      "definition_en": "The single viscous-window time length defined in Section 3, the round's central time scale; the same $O((\\nu\\lambda^2)^{-1})$ form recurs re-indexed as $\\tau_q$ (Section 18) and $\\tau_{q_n},\\tau_n$ (Zeno Theorem 19.1).",
      "defining_relation": "\\tau_J=\\frac{\\theta}{\\nu\\lambda_J^2}"
    },
    {
      "id": "ns.c3.c3e.rho_contraction",
      "latex": "\\rho:=Ce^{-c\\theta}<1",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "窗口收縮比",
      "label_en": "Window contraction ratio",
      "definition_zh": "第3節定義的線性繼承收縮比（範例取 $\\rho\\le1/4$），滿足遞迴 $H_m\\le\\rho H_{m-1}+S_m$；為 Viscous-Window Renewal Theorem（定理4.1）與 X 證書 G-INHERIT 守衛的核心常數。",
      "definition_en": "The linear-inheritance contraction ratio (fixed e.g. at $\\rho\\le1/4$) from Section 3, satisfying the recursion $H_m\\le\\rho H_{m-1}+S_m$; the key constant in the Viscous-Window Renewal Theorem (Thm 4.1) and the G-INHERIT guard.",
      "defining_relation": "\\rho:=Ce^{-c\\theta}<1"
    },
    {
      "id": "ns.c3.c3e.s_window_source",
      "latex": "S_m=\\|\\mathcal N_J[t_{m-1},t_m]\\|_3",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "窗口累積非線性來源量",
      "label_en": "Window-integrated source magnitude",
      "definition_zh": "第3節定義，單一黏性窗口內非線性來源項的 $L^3$ 範數；定理4.1證明中取 $S_\\ast=\\max_jS_j$，第7、21節分別以 $S_n$ 出現在 XViscRenew 與 XLocalHet 證書中。",
      "definition_en": "The $L^3$ norm of the nonlinear source over one viscous window, from Section 3; the proof of Theorem 4.1 uses $S_\\ast=\\max_jS_j$, and it reappears as $S_n$ inside both the XViscRenew (Sec.7) and XLocalHet (Sec.21) certificates.",
      "defining_relation": "S_m=\\|\\mathcal N_J[t_{m-1},t_m]\\|_3"
    },
    {
      "id": "ns.c3.c3e.i_star_renewal_window",
      "latex": "I_n^\\star",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "已認證的更新窗口",
      "label_en": "Certified renewal window",
      "definition_zh": "第6節結合 C1 序列與定理4.1所得的具體窗口：對每個 $n$ 存在 $I_n^\\star\\subset[t_{n-1},t_n]$ 使來源量達到 $A_n$ 量級且窗長受控（推論6.1），是 UV renewal chain 的存在性見證。",
      "definition_en": "The concrete window produced in Section 6 by combining C1's sequences with Theorem 4.1: for each $n$ there is $I_n^\\star\\subset[t_{n-1},t_n]$ whose source reaches scale $A_n$ while its length stays controlled (Corollary 6.1); it witnesses the UV renewal chain's existence.",
      "defining_relation": "\\|\\mathcal N_{J_n}[I_n^\\star]\\|_3\\gtrsim A_n,\\quad |I_n^\\star|\\lesssim(\\nu2^{2J_n})^{-1}"
    },
    {
      "id": "ns.c3.c3e.xviscrenew",
      "latex": "\\operatorname{XViscRenew}_n=\\left\\langle J_n,I_n^\\star,\\tau_{J_n},H_{\\rm in},H_{\\rm out},S_n,\\rho,\\operatorname{Prov}_n\\right\\rangle",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "黏性窗口更新證書",
      "label_en": "Viscous-window renewal certificate",
      "definition_zh": "第7節定義的 X-Integration 證書元組，打包每一代高頻更新事件的尺度、窗口、起訖範數、來源量、收縮比與親緣資訊，須通過 G-TIME/G-SOURCE/G-INHERIT/G-PROV/G-RENEW 五個守衛。",
      "definition_en": "The X-Integration certificate tuple defined in Section 7, packaging each generation's scale, window, boundary norms, source magnitude, contraction ratio, and provenance; must pass the five guards G-TIME/G-SOURCE/G-INHERIT/G-PROV/G-RENEW.",
      "defining_relation": "\\operatorname{XViscRenew}_n=\\left\\langle J_n,I_n^\\star,\\tau_{J_n},H_{\\rm in},H_{\\rm out},S_n,\\rho,\\operatorname{Prov}_n\\right\\rangle"
    },
    {
      "id": "ns.c3.c3e.g_time",
      "latex": "\\text{G-TIME}:\\ |I_n^\\star|\\lesssim(\\nu2^{2J_n})^{-1}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "G-TIME 守衛",
      "label_en": "G-TIME guard",
      "definition_zh": "第7節 XViscRenew 證書的守衛之一，要求更新窗口長度不超過黏性時間尺度 $(\\nu2^{2J_n})^{-1}$。",
      "definition_en": "One of the guards on the XViscRenew certificate (Sec.7), requiring the renewal window's length to stay within the viscous time scale $(\\nu2^{2J_n})^{-1}$.",
      "defining_relation": "|I_n^\\star|\\lesssim(\\nu2^{2J_n})^{-1}"
    },
    {
      "id": "ns.c3.c3e.g_source",
      "latex": "\\text{G-SOURCE}:\\ S_n=\\|\\mathcal N_{J_n}[I_n^\\star]\\|_3\\gtrsim A_n",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "G-SOURCE 守衛",
      "label_en": "G-SOURCE guard",
      "definition_zh": "第7節守衛，要求該窗口內實際非線性來源量達到目標量級 $A_n$。",
      "definition_en": "The Section 7 guard requiring the actual nonlinear source magnitude in the window to reach the target scale $A_n$.",
      "defining_relation": "S_n=\\|\\mathcal N_{J_n}[I_n^\\star]\\|_3\\gtrsim A_n"
    },
    {
      "id": "ns.c3.c3e.g_inherit",
      "latex": "\\text{G-INHERIT}:\\ \\rho<1",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "G-INHERIT 守衛",
      "label_en": "G-INHERIT guard",
      "definition_zh": "第7節守衛，限制線性繼承在一個黏性窗口後最多保留 fraction $\\rho<1$，其餘必須來自新的非線性來源。",
      "definition_en": "The Section 7 guard capping linear inheritance across one viscous window at fraction $\\rho<1$; the remainder must come from fresh nonlinear source.",
      "defining_relation": "\\rho<1"
    },
    {
      "id": "ns.c3.c3e.g_prov",
      "latex": "\\text{G-PROV}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "G-PROV 守衛",
      "label_en": "G-PROV guard",
      "definition_zh": "第7節守衛，要求來源必須來自原始 $\\mathbb P\\nabla\\cdot(u\\otimes u)$ 項，不得為外加 forcing。",
      "definition_en": "The Section 7 guard requiring the source to originate from the genuine $\\mathbb P\\nabla\\cdot(u\\otimes u)$ term rather than any external forcing."
    },
    {
      "id": "ns.c3.c3e.g_renew",
      "latex": "\\text{G-RENEW}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "G-RENEW 守衛",
      "label_en": "G-RENEW guard",
      "definition_zh": "第7節守衛，聲明上一代證書合法不代表下一代自動合法，每個更高 $J_n$ 都必須在更短窗口內重新取得非線性來源證書。",
      "definition_en": "The Section 7 guard stating that a valid certificate at one generation does not automatically transfer; each higher $J_n$ must re-earn its nonlinear-source certificate in a shorter window."
    },
    {
      "id": "ns.c3.c3e.prov_n",
      "latex": "\\operatorname{Prov}_n",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "親子世系來源標記",
      "label_en": "Parent-child provenance",
      "definition_zh": "在 XViscRenew（第7節）與 XLocalHet（第21節）證書中都出現的分量，記錄該代來源與上一代 parent 之間的合法承接關係，防止不合法的來源重複計數。",
      "definition_en": "A component appearing in both the XViscRenew (Sec.7) and XLocalHet (Sec.21) certificates, recording the legitimate parent-child lineage of a generation's source so as to prevent illegitimate double-counting."
    },
    {
      "id": "ns.c3.c3e.n_loc_het",
      "latex": "\\mathcal N_J^{\\rm loc,het}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "局部異手性來源分量",
      "label_en": "Local heterochiral source component",
      "definition_zh": "第8節將高頻來源分解為 $\\mathcal N_J=\\mathcal N_J^{\\rm loc,het}+\\mathcal N_J^{\\rm rem}$ 中的第一項，只含 bounded scale-ratio（$c_1\\lambda\\le k,p,q\\le c_2\\lambda$）的異手性 triad 交互作用，是本輪聚焦分析的對象。",
      "definition_en": "The first term in the decomposition $\\mathcal N_J=\\mathcal N_J^{\\rm loc,het}+\\mathcal N_J^{\\rm rem}$ (Sec.8), containing only heterochiral triad interactions with bounded scale ratio ($c_1\\lambda\\le k,p,q\\le c_2\\lambda$); this round's main object of study.",
      "defining_relation": "\\mathcal N_J=\\mathcal N_J^{\\rm loc,het}+\\mathcal N_J^{\\rm rem}"
    },
    {
      "id": "ns.c3.c3e.n_rem",
      "latex": "\\mathcal N_J^{\\rm rem}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "剩餘來源分量",
      "label_en": "Remainder source component",
      "definition_zh": "第8節分解中的第二項，收納 strongly nonlocal heterochiral、homochiral、boundary/cross-band 及其他尚未被 reduction 排除的來源；本輪尚未證明其全域可忽略。",
      "definition_en": "The second term in Section 8's decomposition, collecting strongly-nonlocal heterochiral, homochiral, boundary/cross-band, and other not-yet-eliminated sources; this round does not prove it is globally negligible.",
      "defining_relation": "\\mathcal N_J=\\mathcal N_J^{\\rm loc,het}+\\mathcal N_J^{\\rm rem}"
    },
    {
      "id": "ns.c3.c3e.triad_tau",
      "latex": "\\tau=(\\mathbf k,\\mathbf p,\\mathbf q;s_k,s_p,s_q)",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "螺旋三元組標記",
      "label_en": "Helical triad label",
      "definition_zh": "第9節記法，標記一個由波數 $\\mathbf k,\\mathbf p,\\mathbf q$ 與各自螺旋號 $s_k,s_p,s_q$ 構成的 helical triad，是本輪相位—振幅分析的基本單位。",
      "definition_en": "The Section 9 notation labeling one helical triad by wavevectors $\\mathbf k,\\mathbf p,\\mathbf q$ together with their helicity signs $s_k,s_p,s_q$; the basic unit of this round's phase-amplitude analysis.",
      "notes": "此三元組記法可能承接自更早的 C3-A/B 輪（helical pair production），本輪在其上新增相位、振幅權重與效率的具體結構。"
    },
    {
      "id": "ns.c3.c3e.mode_amp_phase",
      "latex": "u^{s_k}(\\mathbf k)=a_ke^{i\\phi_k}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "模態振幅—相位分解",
      "label_en": "Mode amplitude-phase decomposition",
      "definition_zh": "第9節將 helical 模態係數寫成振幅 $a_k$ 與相位 $\\phi_k$ 的極式；$p,q$ 分量同理定義 $a_p,\\phi_p,a_q,\\phi_q$。",
      "definition_en": "Section 9 writes each helical mode coefficient in polar form as amplitude $a_k$ times a phase factor; the $p,q$ components $a_p,\\phi_p,a_q,\\phi_q$ are defined analogously.",
      "defining_relation": "u^{s_k}(\\mathbf k)=a_ke^{i\\phi_k}"
    },
    {
      "id": "ns.c3.c3e.gamma_tau",
      "latex": "\\gamma_\\tau",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "幾何/基底相位",
      "label_en": "Geometric/basis phase",
      "definition_zh": "第9節記法，吸收 helical 基底選取所帶來的所有幾何相位，併入 effective triad phase $\\Phi_\\tau$ 的定義中。",
      "definition_en": "The Section 9 term absorbing all geometric phase arising from the choice of helical basis, folded into the effective triad phase $\\Phi_\\tau$."
    },
    {
      "id": "ns.c3.c3e.phi_triad",
      "latex": "\\Phi_\\tau=\\phi_k+\\phi_p+\\phi_q+\\gamma_\\tau",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "有效三元組相位",
      "label_en": "Effective triad phase",
      "definition_zh": "第9節定義的 effective triad phase，決定 triad 生成是正是負；符號慣例（$\\sin$或$\\cos$）不影響本文結論。",
      "definition_en": "The effective triad phase defined in Section 9, which determines whether a triad's production is positive or negative; the sign convention ($\\sin$ vs $\\cos$) does not affect the paper's conclusions.",
      "defining_relation": "\\Phi_\\tau=\\phi_k+\\phi_p+\\phi_q+\\gamma_\\tau"
    },
    {
      "id": "ns.c3.c3e.r_tau_production",
      "latex": "\\mathcal R_\\tau=W_\\tau a_ka_pa_q\\sigma_\\tau",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "單三元組帶號產生量",
      "label_en": "Signed triad production",
      "definition_zh": "第9節定義的單一 helical triad 帶號產生率，由振幅權重 $W_\\tau$ 與相位效率 $\\sigma_\\tau$ 相乘而成，是後續 $\\mathcal M_\\lambda,\\mathcal P_\\lambda,\\eta_\\lambda$ 的基本構件。",
      "definition_en": "The signed production rate of a single helical triad, defined in Section 9 as amplitude weight $W_\\tau$ times phase efficiency $\\sigma_\\tau$; the basic building block for the later $\\mathcal M_\\lambda,\\mathcal P_\\lambda,\\eta_\\lambda$.",
      "defining_relation": "\\mathcal R_\\tau=W_\\tau a_ka_pa_q\\sigma_\\tau"
    },
    {
      "id": "ns.c3.c3e.w_tau_weight",
      "latex": "W_\\tau\\ge0",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "三元組振幅權重",
      "label_en": "Triad amplitude weight",
      "definition_zh": "第9節定義，由波數與螺旋幾何決定的非負振幅權重，出現在 $\\mathcal R_\\tau$ 與 $\\mathcal M_\\lambda$ 的定義中。",
      "definition_en": "The nonnegative amplitude weight determined by wavenumber and helical geometry, defined in Section 9 and used in both $\\mathcal R_\\tau$ and $\\mathcal M_\\lambda$."
    },
    {
      "id": "ns.c3.c3e.sigma_tau",
      "latex": "-1\\le\\sigma_\\tau\\le1",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "相位效率（單三元組）",
      "label_en": "Phase efficiency (single triad)",
      "definition_zh": "第9節定義，單一 triad 的相位效率，數值上是 $\\sin\\Phi_\\tau$ 或 $\\cos\\Phi_\\tau$ 之一（依 convention），量測相位對齊帶來的產生效率折損。",
      "definition_en": "The phase efficiency of a single triad, defined in Section 9, numerically equal to $\\sin\\Phi_\\tau$ or $\\cos\\Phi_\\tau$ depending on convention; measures how phase alignment discounts the production rate.",
      "defining_relation": "-1\\le\\sigma_\\tau\\le1"
    },
    {
      "id": "ns.c3.c3e.triad_family_loc_het",
      "latex": "\\mathfrak T_\\lambda^{\\rm loc,het}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "局部異手性三元組族",
      "label_en": "Local heterochiral triad family",
      "definition_zh": "第10節定義，尺度 $\\lambda$ 上所有 bounded scale-ratio 異手性 helical triads 所構成的集合，是 $\\mathcal M_\\lambda,\\mathcal P_\\lambda$ 求和範圍；第22節簡寫為 $\\mathfrak T_q$。",
      "definition_en": "The set of all bounded-scale-ratio heterochiral helical triads at scale $\\lambda$, defined in Section 10 as the summation range for $\\mathcal M_\\lambda,\\mathcal P_\\lambda$; abbreviated $\\mathfrak T_q$ in Section 22.",
      "defining_relation": "c_1\\lambda\\le k,p,q\\le c_2\\lambda"
    },
    {
      "id": "ns.c3.c3e.m_lambda_capacity",
      "latex": "\\mathcal M_\\lambda(t)=\\sum_{\\tau\\in\\mathfrak T_\\lambda^{\\rm loc,het}}W_\\tau a_ka_pa_q",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "瞬時最大振幅容量",
      "label_en": "Instantaneous maximal amplitude capacity",
      "definition_zh": "第10節定義的瞬時 amplitude capacity 上界，把所有局部異手性 triads 的振幅權重相加而不計相位；第11節重寫為 $\\mathcal M_q$ 並給出估計 $\\mathcal M_q\\le C\\lambda_q^{7/2}U_q^3$。",
      "definition_en": "The instantaneous amplitude-capacity bound defined in Section 10, summing amplitude weights over all local heterochiral triads while ignoring phase; rewritten as $\\mathcal M_q$ in Section 11, where it is bounded by $\\mathcal M_q\\le C\\lambda_q^{7/2}U_q^3$.",
      "defining_relation": "\\mathcal M_\\lambda(t)=\\sum_{\\tau\\in\\mathfrak T_\\lambda^{\\rm loc,het}}W_\\tau a_ka_pa_q"
    },
    {
      "id": "ns.c3.c3e.p_lambda_production",
      "latex": "\\mathcal P_\\lambda(t)=\\left[\\sum_{\\tau\\in\\mathfrak T_\\lambda^{\\rm loc,het}}\\mathcal R_\\tau\\right]_+",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "實際正產生量",
      "label_en": "Actual positive production",
      "definition_zh": "第10節定義，局部異手性 triads 帶號產生量加總後取正部，滿足 $0\\le\\mathcal P_\\lambda\\le\\mathcal M_\\lambda$；自第11節起重寫為 $\\mathcal P_q$。",
      "definition_en": "Defined in Section 10 as the positive part of the summed signed production over local heterochiral triads, satisfying $0\\le\\mathcal P_\\lambda\\le\\mathcal M_\\lambda$; rewritten as $\\mathcal P_q$ from Section 11 onward.",
      "defining_relation": "\\mathcal P_\\lambda(t)=\\left[\\sum_{\\tau\\in\\mathfrak T_\\lambda^{\\rm loc,het}}\\mathcal R_\\tau\\right]_+"
    },
    {
      "id": "ns.c3.c3e.eta_lambda_efficiency",
      "latex": "\\eta_\\lambda(t)=\\mathcal P_\\lambda(t)/\\mathcal M_\\lambda(t)",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "相位—同調效率",
      "label_en": "Phase-coherence efficiency",
      "definition_zh": "第10節定義的本輪核心診斷量，$\\eta_\\lambda\\in[0,1]$（分母為零時取0），量測局部異手性 triads 相位對齊程度；文中強調這是精確的 normalized diagnostic 而非 turbulence closure。",
      "definition_en": "This round's central diagnostic quantity, defined in Section 10, with $\\eta_\\lambda\\in[0,1]$ (0 when the denominator vanishes), measuring how phase-aligned the local heterochiral triads are; the paper stresses this is an exact normalized diagnostic, not a turbulence closure.",
      "defining_relation": "\\eta_\\lambda(t)=\\begin{cases}\\dfrac{\\mathcal P_\\lambda(t)}{\\mathcal M_\\lambda(t)},&\\mathcal M_\\lambda(t)>0,\\\\0,&\\mathcal M_\\lambda(t)=0.\\end{cases}"
    },
    {
      "id": "ns.c3.c3e.u_q_shell_norm",
      "latex": "U_q=\\left(\\sum_{|r-q|\\le C_0}\\|u_r\\|_2^2\\right)^{1/2}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "局部 shell L² 範數",
      "label_en": "Local shell L² norm",
      "definition_zh": "第11節定義，鄰近 dyadic shells（$|r-q|\\le C_0$）速度分量的合併 $L^2$ 範數，用於界定局部 critical trilinear 上界 $\\mathcal M_q\\le C\\lambda_q^{7/2}U_q^3$ 與臨界振幅 $A_q^{\\rm crit}$。",
      "definition_en": "Defined in Section 11 as the combined $L^2$ norm of velocity over neighboring dyadic shells ($|r-q|\\le C_0$); used to bound the local critical trilinear term, $\\mathcal M_q\\le C\\lambda_q^{7/2}U_q^3$, and to define the critical amplitude $A_q^{\\rm crit}$.",
      "defining_relation": "U_q=\\left(\\sum_{|r-q|\\le C_0}\\|u_r\\|_2^2\\right)^{1/2}"
    },
    {
      "id": "ns.c3.c3e.d_q_crit_dissipation",
      "latex": "\\mathcal D_q^{\\rm crit}\\asymp\\nu\\lambda_q^3U_q^2",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "局部臨界黏性耗散尺度",
      "label_en": "Local critical viscous dissipation scale",
      "definition_zh": "第12節定義的同一 local block 之臨界黏性耗散量級，與 $\\mathcal P_q$ 相除給出決定 $\\eta_q,A_q^{\\rm crit}$ 的核心比值。",
      "definition_en": "The critical viscous dissipation scale of the same local block, defined in Section 12; dividing $\\mathcal P_q$ by it yields the ratio central to $\\eta_q$ and $A_q^{\\rm crit}$.",
      "defining_relation": "\\mathcal D_q^{\\rm crit}\\asymp\\nu\\lambda_q^3U_q^2"
    },
    {
      "id": "ns.c3.c3e.a_q_crit_amplitude",
      "latex": "A_q^{\\rm crit}=\\lambda_q^{1/2}U_q",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "局部臨界振幅",
      "label_en": "Local critical amplitude",
      "definition_zh": "第12節定義，在 N–S scaling 下 dimensionless/critical 的局部振幅量；定理13.1（Coherence–Amplitude Tradeoff）給出核心不等式 $\\eta_qA_q^{\\rm crit}\\gtrsim\\nu$，是本輪第二個核心量。",
      "definition_en": "The local amplitude quantity that is dimensionless/critical under N–S scaling, defined in Section 12; Theorem 13.1 (the Coherence–Amplitude Tradeoff) gives the key inequality $\\eta_qA_q^{\\rm crit}\\gtrsim\\nu$, making this the round's second central quantity.",
      "defining_relation": "A_q^{\\rm crit}=\\lambda_q^{1/2}U_q"
    },
    {
      "id": "ns.c3.c3e.i_q_window",
      "latex": "I_q",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "更新窗口（通用記法）",
      "label_en": "Renewal window (generic)",
      "definition_zh": "第16節起使用的通用黏性窗口記法，用來定義窗口內累積量 $M_q(I_q),P_q(I_q),\\bar\\eta_q(I_q)$；同一角色在第21節寫作 $I_n$，第24節相空間 cell 中寫作 $I_\\lambda$。",
      "definition_en": "The generic viscous-window notation used from Section 16 onward to define window-integrated quantities $M_q(I_q),P_q(I_q),\\bar\\eta_q(I_q)$; the same role is written $I_n$ in Section 21 and $I_\\lambda$ in the Section 24 phase-space cell.",
      "defining_relation": "|I_\\lambda|\\sim(\\nu\\lambda^2)^{-1}"
    },
    {
      "id": "ns.c3.c3e.m_q_integrated_capacity",
      "latex": "M_q(I_q)=\\int_{I_q}\\mathcal M_q(t)\\,dt",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "窗口累積振幅容量",
      "label_en": "Window-integrated amplitude capacity",
      "definition_zh": "第16節定義，將瞬時容量 $\\mathcal M_q(t)$ 對整個更新窗口 $I_q$ 積分所得的（非花體）$M_q(I_q)$，與瞬時版本是不同的記法層級。",
      "definition_en": "Defined in Section 16 as the time-integral of the instantaneous capacity $\\mathcal M_q(t)$ over the whole renewal window $I_q$; the plain-letter $M_q(I_q)$ is a distinct notational layer from the script $\\mathcal M_q(t)$.",
      "defining_relation": "M_q(I_q)=\\int_{I_q}\\mathcal M_q(t)\\,dt"
    },
    {
      "id": "ns.c3.c3e.p_q_integrated_production",
      "latex": "P_q(I_q)=\\int_{I_q}\\mathcal P_q(t)\\,dt",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "窗口累積實際產生量",
      "label_en": "Window-integrated actual production",
      "definition_zh": "第16節定義，瞬時正產生量 $\\mathcal P_q(t)$ 對窗口 $I_q$ 的時間積分。",
      "definition_en": "Defined in Section 16 as the time-integral of the instantaneous positive production $\\mathcal P_q(t)$ over the window $I_q$.",
      "defining_relation": "P_q(I_q)=\\int_{I_q}\\mathcal P_q(t)\\,dt"
    },
    {
      "id": "ns.c3.c3e.eta_bar_q_weighted",
      "latex": "\\bar\\eta_q(I_q)=P_q(I_q)/M_q(I_q)",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "窗口加權相位效率",
      "label_en": "Window-weighted phase efficiency",
      "definition_zh": "第16節定義的窗口平均相位效率，$\\bar\\eta_q(I_q)\\in[0,1]$；若某更新事件需要 $P_q(I_q)\\ge B_q$，則必須 $\\bar\\eta_q(I_q)\\ge B_q/M_q(I_q)$。",
      "definition_en": "The window-averaged phase efficiency defined in Section 16, with $\\bar\\eta_q(I_q)\\in[0,1]$; a renewal event requiring $P_q(I_q)\\ge B_q$ forces $\\bar\\eta_q(I_q)\\ge B_q/M_q(I_q)$.",
      "defining_relation": "\\bar\\eta_q(I_q)=\\frac{P_q(I_q)}{M_q(I_q)}"
    },
    {
      "id": "ns.c3.c3e.xlocalhet",
      "latex": "\\operatorname{XLocalHet}_n=\\left\\langle q_n,I_n,\\mathcal P_n,\\mathcal M_n,\\bar\\eta_n,A_n^{\\rm crit},\\mathcal G_n,\\mathcal S_n,\\operatorname{Prov}_n\\right\\rangle",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "局部異手性 Zeno 鏈證書",
      "label_en": "Local-heterochiral Zeno-chain certificate",
      "definition_zh": "第21節定義的第二個 X-Integration 證書元組，用來檢驗一條 hypothetical local singular chain 在每個 genealogy 步驟是否同時具備尺度、窗口、產生量、容量、相位效率、臨界振幅、triad 幾何、空間支撐與親緣資訊。",
      "definition_en": "The second X-Integration certificate tuple, defined in Section 21, used to check whether a hypothetical local singular chain carries scale, window, production, capacity, phase efficiency, critical amplitude, triad geometry, spatial support, and provenance at every genealogy step.",
      "defining_relation": "\\operatorname{XLocalHet}_n=\\left\\langle q_n,I_n,\\mathcal P_n,\\mathcal M_n,\\bar\\eta_n,A_n^{\\rm crit},\\mathcal G_n,\\mathcal S_n,\\operatorname{Prov}_n\\right\\rangle"
    },
    {
      "id": "ns.c3.c3e.g_n_triad_geometry",
      "latex": "\\mathcal G_n",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "螺旋三元組幾何",
      "label_en": "Helical triad geometry",
      "definition_zh": "第21節 XLocalHet 證書的分量之一，記錄第 $n$ 代 helical triad 的幾何資訊；本輪僅列為證書欄位，具體結構留待 C3-F 發展。",
      "definition_en": "One component of the XLocalHet certificate (Sec.21), recording the helical triad geometry at generation $n$; this round only lists it as a certificate field, leaving its concrete structure to C3-F.",
      "notes": "本輪未給出 $\\mathcal G_n$ 的具體公式，屬 placeholder。"
    },
    {
      "id": "ns.c3.c3e.s_n_spatial_support",
      "latex": "\\mathcal S_n",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "空間支撐/集中資訊",
      "label_en": "Spatial support/concentration information",
      "definition_zh": "第21節 XLocalHet 證書的分量之一，記錄第 $n$ 代的空間支撐或集中資訊，呼應第23–24節的空間集中接口，但本輪未給出具體公式，留待 C3-F。",
      "definition_en": "Another XLocalHet certificate component (Sec.21) recording generation-$n$ spatial support/concentration data, echoing the Sec.23–24 spatial-concentration interface; left without an explicit formula in this round, deferred to C3-F.",
      "notes": "屬 placeholder，尚未展開。"
    },
    {
      "id": "ns.c3.c3e.r_t_concentration_radius",
      "latex": "R(t)\\sim\\sqrt{T_\\ast-t}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "集中球半徑",
      "label_en": "Concentration ball radius",
      "definition_zh": "第23節引用 Barker–Prange 結果，Type-I-like 情境下 critical $L^3$ mass 集中所在的 shrinking ball 半徑量級，與 parabolic 頻率尺度互為倒數：$R(t)\\lambda(t)\\sim1$；屬 conditional spatial interface，非本文自證結果。",
      "definition_en": "Cited from Barker–Prange in Section 23: under Type-I-like scenarios, the shrinking-ball radius at which critical $L^3$ mass concentrates, reciprocal to the parabolic frequency scale via $R(t)\\lambda(t)\\sim1$; introduced only as a conditional spatial interface, not proved in this paper.",
      "defining_relation": "R(t)\\sim\\sqrt{T_\\ast-t}"
    },
    {
      "id": "ns.c3.c3e.c_lambda_phase_space_cell",
      "latex": "\\mathcal C_\\lambda=B(x_\\lambda,c\\lambda^{-1})\\times\\{\\xi:|\\xi|\\sim\\lambda\\}\\times I_\\lambda",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "聯合空間—頻率相空間胞元",
      "label_en": "Joint space-frequency phase-space cell",
      "definition_zh": "第24節定義，把空間球、頻率殼與時間窗口組合成單一相空間胞元；singular genealogy 若存在，需要一串 $\\mathcal C_{\\lambda_1}\\rightsquigarrow\\mathcal C_{\\lambda_2}\\rightsquigarrow\\cdots$ 在 space、frequency、time、helicity、phase 五個方向保持合法關聯，是通往下一輪 C3-F 的橋接構造。",
      "definition_en": "Defined in Section 24 by combining a spatial ball, a frequency shell, and a time window into a single phase-space cell; a hypothetical singular genealogy would need a chain $\\mathcal C_{\\lambda_1}\\rightsquigarrow\\mathcal C_{\\lambda_2}\\rightsquigarrow\\cdots$ staying legally linked across space, frequency, time, helicity, and phase — the bridge construct toward the next round, C3-F.",
      "defining_relation": "\\mathcal C_\\lambda=B(x_\\lambda,c\\lambda^{-1})\\times\\{\\xi:|\\xi|\\sim\\lambda\\}\\times I_\\lambda"
    },
    {
      "id": "ns.c3.c3e.u_q_alpha_wavepacket",
      "latex": "u_q=\\sum_\\alpha u_{q,\\alpha}",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "空間局部化波包分解",
      "label_en": "Spatially localized wave-packet decomposition",
      "definition_zh": "第29節（C3-F proof obligations, F1）預告的記法，把 dyadic shell $u_q$ 再分解成空間局部化波包 $u_{q,\\alpha}$，用以建立 $(q,\\alpha,s)$ 來源圖；本輪僅作為下一輪任務的前導定義。",
      "definition_en": "Notation previewed in Section 29 (C3-F proof obligation F1), further decomposing the dyadic shell $u_q$ into spatially localized wave packets $u_{q,\\alpha}$ to build a $(q,\\alpha,s)$ provenance graph; introduced here only as a lead-in to the next round's task.",
      "defining_relation": "u_q=\\sum_\\alpha u_{q,\\alpha}",
      "notes": "屬 C3-F 前導定義，非本輪已證結果。"
    },
    {
      "id": "ns.c3.c3e.overlap_operator",
      "latex": "\\operatorname{Overlap}(u_{k,\\alpha},u_{p,\\beta},u_{q,\\gamma})",
      "series": "NS",
      "first_appearance": "C3-E",
      "label_zh": "空間重疊算子",
      "label_en": "Spatial overlap operator",
      "definition_zh": "第29節（F3）預告的記法，用來檢驗三個 Fourier shell 波包是否在物理空間真正重疊，而非只有代數 triad 關係；為 C3-F 的 Spatial overlap guard 任務鋪墊。",
      "definition_en": "Notation previewed in Section 29 (F3) to test whether three Fourier-shell wave packets physically overlap in space, rather than merely satisfying an algebraic triad relation; sets up C3-F's Spatial Overlap Guard task.",
      "defining_relation": "\\operatorname{Overlap}(u_{k,\\alpha},u_{p,\\beta},u_{q,\\gamma})",
      "notes": "屬 C3-F 前導定義，非本輪已證結果。"
    },
    {
      "id": "ns.c3.c3f.t_q_operator",
      "latex": "\\mathcal T_q",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "環形 Leray 臨界算子",
      "label_en": "Annular Leray critical operator",
      "definition_zh": "第1節定義的核心新算子 $\\mathcal T_qF=D\\Delta_q\\mathbb P\\nabla\\cdot F$，作用於張量場 $F$，總微分階數為2，其 Fourier multiplier 支撐在 annulus $|\\xi|\\sim\\lambda_q$。",
      "definition_en": "The round's central new operator (Sec. 1), $\\mathcal T_qF=D\\Delta_q\\mathbb P\\nabla\\cdot F$ acting on a tensor field $F$; total differential order 2, multiplier supported on the annulus $|\\xi|\\sim\\lambda_q$.",
      "defining_relation": "\\mathcal T_qF=D\\Delta_q\\mathbb P\\nabla\\cdot F"
    },
    {
      "id": "ns.c3.c3f.d_op",
      "latex": "D=\\sqrt{-\\Delta}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "半階微分算子",
      "label_en": "Square-root-Laplacian operator",
      "definition_zh": "第1節引入，與 $\\Delta_q\\mathbb P\\nabla\\cdot$ 組合構成 $\\mathcal T_q$，貢獻1階微分，使 $\\mathcal T_q$ 總階數為2。",
      "definition_en": "Introduced in Sec. 1; combined with $\\Delta_q\\mathbb P\\nabla\\cdot$ to build $\\mathcal T_q$, contributing order 1 so that $\\mathcal T_q$ has total differential order 2."
    },
    {
      "id": "ns.c3.c3f.k_q_kernel",
      "latex": "K_q=\\mathcal F^{-1}m_q",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "算子核及其縮放",
      "label_en": "Kernel of 𝒯_q and its dyadic scaling",
      "definition_zh": "第2節定義 $K_q=\\mathcal F^{-1}m_q$，滿足縮放 $K_q(x)=\\lambda_q^5K(\\lambda_qx)$，其中 $K\\in\\mathcal S(\\mathbb R^3)$ 為固定 Schwartz profile；其 rapid tail decay 是 physical-space quasi-locality 的來源。",
      "definition_en": "Sec. 2 defines $K_q=\\mathcal F^{-1}m_q$ with scaling $K_q(x)=\\lambda_q^5K(\\lambda_qx)$ for a fixed Schwartz profile $K\\in\\mathcal S(\\mathbb R^3)$; its rapid off-support decay is the source of physical-space quasi-locality.",
      "defining_relation": "K_q(x)=\\lambda_q^5K(\\lambda_qx)"
    },
    {
      "id": "ns.c3.c3f.dist_ab",
      "latex": "d=\\operatorname{dist}(A,B)",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "支撐集間距離",
      "label_en": "Support separation distance",
      "definition_zh": "定理3.1中，$f\\otimes g$ 支撐於 $A$、$h$ 支撐於 $B$ 時兩者的距離，用以量化 off-diagonal interaction 隨 $\\lambda_qd$ 的 rapid decay。",
      "definition_en": "In Thm 3.1, the distance between support $A$ of $f\\otimes g$ and support $B$ of $h$; controls the rapid decay of the off-diagonal interaction in $\\lambda_qd$."
    },
    {
      "id": "ns.c3.c3f.q_qm_cube",
      "latex": "Q_{q,m},\\quad \\ell_q=c\\lambda_q^{-1}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "尺度 q 空間分割立方體",
      "label_en": "Scale-q spatial packet cube",
      "definition_zh": "第5節在尺度 $\\lambda_q$ 上取的空間立方體格點，$m\\in\\mathbb Z^3$，邊長 $\\ell_q=c\\lambda_q^{-1}$，搭配 smooth bounded-overlap partition $\\sum_m\\chi_{q,m}=1$ 構成 admissible dyadic packet 格架。",
      "definition_en": "Sec. 5's spatial grid at scale $\\lambda_q$, indexed by $m\\in\\mathbb Z^3$ with side length $\\ell_q=c\\lambda_q^{-1}$, paired with a smooth bounded-overlap partition $\\sum_m\\chi_{q,m}=1$ defining the admissible dyadic packet grid."
    },
    {
      "id": "ns.c3.c3f.u_qm_packet",
      "latex": "u_{q,m}=\\chi_{q,m}u_q",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "局域化 shell packet",
      "label_en": "Spatially localized shell packet",
      "definition_zh": "第5節定義，將 shell $u_q$ 乘上分割函數 $\\chi_{q,m}$ 局域化到立方體 $Q_{q,m}$，稱為 admissible dyadic packet；後續 $U_q$ 與 ancestry chain 均建立於此。",
      "definition_en": "Sec. 5's localization of shell $u_q$ to cube $Q_{q,m}$ via $\\chi_{q,m}$, termed an admissible dyadic packet; the basic unit underlying $U_q$ and the later ancestry chain.",
      "defining_relation": "u_{q,m}=\\chi_{q,m}u_q",
      "notes": "Section 5 also allows an optional enlarged projector $\\widetilde\\Delta_q$ to restore strict annular localization if needed."
    },
    {
      "id": "ns.c3.c3f.core_radius_r",
      "latex": "R",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "核心半徑參數",
      "label_en": "Core radius parameter",
      "definition_zh": "第6節引入的無量綱參數 $R\\ge1$，定義 parent pair 是否屬於 $R$-core（重疊區域距 output cube 不超過 $R\\lambda_q^{-1}$）或 $R$-tail；是全文 $R_q,R_n,R_\\ast$ 等符號的基底。",
      "definition_en": "Sec. 6's dimensionless parameter $R\\ge1$ separating parent pairs into $R$-core (overlap region within $R\\lambda_q^{-1}$ of the output cube) versus $R$-tail; the base symbol behind $R_q$, $R_n$, $R_\\ast$ later in the paper."
    },
    {
      "id": "ns.c3.c3f.m_r_branching",
      "latex": "M_R",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "有效核心親代組數上界",
      "label_en": "Effective core parent-tuple bound",
      "definition_zh": "命題7.1，固定 $R$ 下每個 output packet 的 effective core parent tuples 數目上界，與 $q$ 無關，$M_R=O(R^3)$ 乘上有限 helicity/角扇區組合因子；是 finite branching 的定量陳述。",
      "definition_en": "Prop. 7.1's bound on the number of effective core parent tuples per output packet at fixed $R$, independent of $q$, scaling as $M_R=O(R^3)$ times a finite helicity/angular-sector combinatorial factor; the quantitative statement of finite branching.",
      "defining_relation": "M_R=O(R^3)"
    },
    {
      "id": "ns.c3.c3f.u_q_norm",
      "latex": "U_q^2=\\sum_m\\|u_{q,m}\\|_2^2",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "聚合 packet 範數",
      "label_en": "Aggregate packet norm",
      "definition_zh": "第8節定義 $U_q^2=\\sum_m\\|u_{q,m}\\|_2^2$，由 bounded overlap 得 $U_q\\asymp\\|u_q\\|_2$；用於建構 $\\mathcal M_q^{\\rm scale}$ 與第29節 Zeno 論證中的臨界振幅 $A_n^{\\rm crit}$。",
      "definition_en": "Sec. 8's aggregate norm $U_q^2=\\sum_m\\|u_{q,m}\\|_2^2$, comparable to $\\|u_q\\|_2$ by bounded overlap; feeds into $\\mathcal M_q^{\\rm scale}$ and the critical amplitude $A_n^{\\rm crit}$ in the Sec. 29 Zeno argument.",
      "defining_relation": "U_q^2=\\sum_m\\|u_{q,m}\\|_2^2"
    },
    {
      "id": "ns.c3.c3f.tail_bound",
      "latex": "\\mathcal R_q^{\\rm tail}(R)",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "Packet 尾端聚合貢獻",
      "label_en": "Aggregate packet tail contribution",
      "definition_zh": "定理8.1中，所有距 output 超過 $R\\lambda_q^{-1}$ 的 packet interactions 總和，滿足 $|\\mathcal R_q^{\\rm tail}(R)|\\le C_NR^{-N}\\lambda_q^{7/2}U_q^3$，是 locality–coherence tradeoff（定理10.1）的核心估計量。",
      "definition_en": "Thm 8.1's sum of all packet interactions farther than $R\\lambda_q^{-1}$ from the output, bounded by $C_NR^{-N}\\lambda_q^{7/2}U_q^3$; the key estimate feeding the locality–coherence tradeoff (Thm 10.1).",
      "defining_relation": "|\\mathcal R_q^{\\rm tail}(R)|\\le C_NR^{-N}\\lambda_q^{7/2}U_q^3"
    },
    {
      "id": "ns.c3.c3f.m_q_scale",
      "latex": "\\mathcal M_q^{\\rm scale}=\\lambda_q^{7/2}U_q^3",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "尺度化振幅容量",
      "label_en": "Scaling proxy for amplitude capacity",
      "definition_zh": "第8節定義 $\\mathcal M_q^{\\rm scale}=\\lambda_q^{7/2}U_q^3$，與 C3-E 的 maximal local amplitude capacity $\\mathcal M_q$ 同 scaling（$\\mathcal M_q\\le C_0\\mathcal M_q^{\\rm scale}$），用以在 tail bound 中取代 $\\mathcal M_q$。",
      "definition_en": "Sec. 8's $\\mathcal M_q^{\\rm scale}=\\lambda_q^{7/2}U_q^3$, sharing scaling with C3-E's amplitude capacity $\\mathcal M_q$ (via $\\mathcal M_q\\le C_0\\mathcal M_q^{\\rm scale}$); used in place of $\\mathcal M_q$ inside the tail bound.",
      "defining_relation": "\\mathcal M_q^{\\rm scale}=\\lambda_q^{7/2}U_q^3"
    },
    {
      "id": "ns.c3.c3f.eta_q",
      "latex": "\\eta_q=\\mathcal P_q/\\mathcal M_q",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "相位效率",
      "label_en": "Phase efficiency",
      "definition_zh": "承 C3-E 定義 $\\eta_q=\\mathcal P_q/\\mathcal M_q\\in[0,1]$（第9節回顧），本輪賦予其新角色：透過定理10.1，$\\eta_q$ 直接決定維持一半 production 所需的 ancestry core 半徑 $R_q\\gtrsim\\eta_q^{-1/N}$。",
      "definition_en": "Carried over from C3-E, $\\eta_q=\\mathcal P_q/\\mathcal M_q\\in[0,1]$ (reviewed Sec. 9); this round gives it a new role via Thm 10.1, where $\\eta_q$ directly sets the ancestry core radius $R_q\\gtrsim\\eta_q^{-1/N}$ needed to retain half the production.",
      "defining_relation": "\\eta_q=\\mathcal P_q/\\mathcal M_q",
      "notes": "$\\mathcal P_q$ (actual positive local pair production) is likewise carried from C3-E, used here only as η_q's numerator."
    },
    {
      "id": "ns.c3.c3f.r_q_radius",
      "latex": "R_q\\ge C_N'\\eta_q^{-1/N}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "相干性所需核心半徑",
      "label_en": "Coherence-required core radius",
      "definition_zh": "定理10.1，使 $C_NR_q^{-N}\\mathcal M_q^{\\rm scale}\\le\\frac12\\mathcal P_q$ 成立的半徑選擇，充分條件為 $R_q\\ge C_N'\\eta_q^{-1/N}$；沿 ancestry chain 成為第 $n$ 步的位移上界 $R_n$。",
      "definition_en": "Thm 10.1's radius choice ensuring $C_NR_q^{-N}\\mathcal M_q^{\\rm scale}\\le\\frac12\\mathcal P_q$, sufficient once $R_q\\ge C_N'\\eta_q^{-1/N}$; becomes $R_n$ bounding the spatial displacement at ancestry step $n$.",
      "defining_relation": "R_q\\ge C_N'\\eta_q^{-1/N}"
    },
    {
      "id": "ns.c3.c3f.b_v",
      "latex": "B_v",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "Output packet 產生量",
      "label_en": "Output packet production magnitude",
      "definition_zh": "第12節，output packet $v$ 的 signed/positive production magnitude；core source-selection lemma（引理12.1）即圍繞 $B_v$ 如何被有限多個 parent tuple 貢獻主導而展開。",
      "definition_en": "Sec. 12's signed/positive production magnitude of output packet $v$; the core source-selection lemma (12.1) shows $B_v$ must be dominated by one of finitely many parent-tuple contributions."
    },
    {
      "id": "ns.c3.c3f.core_parent_set",
      "latex": "\\mathcal A(v),\\quad \\#\\mathcal A(v)\\le M",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "核心親代組集合",
      "label_en": "Core parent-tuple set",
      "definition_zh": "第12節，packet $v$ 的 core parent tuple 集合，基數 $\\#\\mathcal A(v)\\le M$；引理12.1證明其中存在 $\\alpha_\\ast$ 使 $|T_{v,\\alpha_\\ast}|\\ge\\frac{1-\\varepsilon}{M}B_v$。",
      "definition_en": "Sec. 12's set of core parent tuples for packet $v$, with $\\#\\mathcal A(v)\\le M$; Lemma 12.1 shows some $\\alpha_\\ast\\in\\mathcal A(v)$ satisfies $|T_{v,\\alpha_\\ast}|\\ge\\frac{1-\\varepsilon}{M}B_v$."
    },
    {
      "id": "ns.c3.c3f.t_v_alpha",
      "latex": "T_{v,\\alpha},\\quad T_v^{\\rm tail}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "親代組貢獻量與尾端項",
      "label_en": "Parent-tuple contribution and tail term",
      "definition_zh": "第12節，$T_{v,\\alpha}$ 為 core parent tuple $\\alpha\\in\\mathcal A(v)$ 對 $B_v$ 的貢獻，$T_v^{\\rm tail}$（滿足 $|T_v^{\\rm tail}|\\le\\varepsilon B_v$）為其餘尾端貢獻，合成 $B_v\\le|\\sum_\\alpha T_{v,\\alpha}+T_v^{\\rm tail}|$。",
      "definition_en": "Sec. 12's contribution $T_{v,\\alpha}$ of core parent tuple $\\alpha\\in\\mathcal A(v)$ to $B_v$, plus residual tail $T_v^{\\rm tail}$ (with $|T_v^{\\rm tail}|\\le\\varepsilon B_v$), combining as $B_v\\le|\\sum_\\alpha T_{v,\\alpha}+T_v^{\\rm tail}|$."
    },
    {
      "id": "ns.c3.c3f.ancestry_chain",
      "latex": "v_0\\rightsquigarrow v_1\\rightsquigarrow v_2\\rightsquigarrow\\cdots",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "Packet 世系鏈",
      "label_en": "Packet ancestry chain",
      "definition_zh": "第14節起假設的 source-certified packet genealogy 序列，帶特徵尺度 $\\lambda_0<\\lambda_1<\\cdots$、中心 $x_0,x_1,\\ldots$、時刻 $t_0<t_1<\\cdots$；是第16–19節所有收斂/cone定理的研究對象。",
      "definition_en": "Sec. 14's assumed source-certified packet genealogy, carrying scales $\\lambda_0<\\lambda_1<\\cdots$, centers $x_0,x_1,\\ldots$, and times $t_0<t_1<\\cdots$; the object studied by all convergence/cone theorems in Secs. 16–19."
    },
    {
      "id": "ns.c3.c3f.scale_ratio_bounds",
      "latex": "r_-\\lambda_n\\le\\lambda_{n+1}\\le r_+\\lambda_n",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "尺度跳躍界限",
      "label_en": "Bounded local scale-jump ratios",
      "definition_zh": "第14節假設的固定常數 $1<r_-\\le r_+<\\infty$，限制世系鏈相鄰尺度比，確保 $\\lambda_n\\ge\\lambda_0r_-^n$ 幾何增長，供第17、18節加總使用。",
      "definition_en": "Sec. 14's fixed constants $1<r_-\\le r_+<\\infty$ bounding the successive scale ratio $\\lambda_{n+1}/\\lambda_n$, giving geometric growth $\\lambda_n\\ge\\lambda_0r_-^n$ used in the summations of Secs. 17–18.",
      "defining_relation": "r_-\\lambda_n\\le\\lambda_{n+1}\\le r_+\\lambda_n"
    },
    {
      "id": "ns.c3.c3f.x_star",
      "latex": "x_\\ast",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "世系空間收斂中心",
      "label_en": "Ancestry spatial limit center",
      "definition_zh": "定理16.1，在 $\\sum_nR_n/\\lambda_n<\\infty$ 下 ancestry chain centers $x_n$ 的 Cauchy 極限點，滿足 $|x_n-x_\\ast|\\le C\\sum_{m\\ge n}R_m/\\lambda_m$；coherent route 下進一步得 $|x_n-x_\\ast|\\le C'\\lambda_n^{-1}$（推論17.1）。",
      "definition_en": "Thm 16.1's Cauchy limit point of ancestry centers $x_n$ under $\\sum_nR_n/\\lambda_n<\\infty$, with $|x_n-x_\\ast|\\le C\\sum_{m\\ge n}R_m/\\lambda_m$; under the coherent route this sharpens to $|x_n-x_\\ast|\\le C'\\lambda_n^{-1}$ (Cor. 17.1)."
    },
    {
      "id": "ns.c3.c3f.t_star_ancestry",
      "latex": "T_\\ast,\\quad T_\\infty",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "世系時間收斂極限",
      "label_en": "Ancestry temporal limit",
      "definition_zh": "第18節，由 viscous-window renewal $0<t_{n+1}-t_n\\le C_t(\\nu\\lambda_n^2)^{-1}$ 與 $\\lambda_n$ 幾何增長得 $\\sum_n(\\nu\\lambda_n^2)^{-1}<\\infty$，故 $t_n\\to T_\\infty$；若此鏈代表假設性 terminal singular cascade，則 $T_\\infty=T_\\ast$。",
      "definition_en": "Sec. 18: viscous-window renewal $0<t_{n+1}-t_n\\le C_t(\\nu\\lambda_n^2)^{-1}$ plus geometric $\\lambda_n$ growth gives $\\sum_n(\\nu\\lambda_n^2)^{-1}<\\infty$, so $t_n\\to T_\\infty$; if the chain represents a hypothetical terminal singular cascade, $T_\\infty=T_\\ast$.",
      "notes": "T* itself is series-standard (blow-up time); the new content is its identification with T_∞, the quantitative limit of this round's explicit ancestry-chain times."
    },
    {
      "id": "ns.c3.c3f.parabolic_cone",
      "latex": "\\lambda_n|x_n-x_\\ast|\\le C_x,\\quad \\nu\\lambda_n^2(T_\\ast-t_n)\\le C_t'",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "拋物型世系錐",
      "label_en": "Parabolic ancestry cone",
      "definition_zh": "定理19.1（C3-F.4）結論：coherent genealogy 被迫滿足 $\\lambda_n|x_n-x_\\ast|\\le C_x$ 且 $\\nu\\lambda_n^2(T_\\ast-t_n)\\le C_t'$；第20節指出其幾何 $\\lambda_n^{-1}\\sim\\sqrt{\\nu(T_\\ast-t_n)}$ 與 Barker–Prange 的 concentration radius $R(t)=O(\\sqrt{T_\\ast-t})$ 相容，但兩者為不同假設下的獨立定理，不能互相取代。",
      "definition_en": "Thm 19.1 (C3-F.4)'s conclusion that a coherent genealogy is forced into $\\lambda_n|x_n-x_\\ast|\\le C_x$ and $\\nu\\lambda_n^2(T_\\ast-t_n)\\le C_t'$; Sec. 20 notes the resulting geometry $\\lambda_n^{-1}\\sim\\sqrt{\\nu(T_\\ast-t_n)}$ matches Barker–Prange's concentration radius $R(t)=O(\\sqrt{T_\\ast-t})$, though the two remain independent theorems under different hypotheses.",
      "defining_relation": "\\lambda_n|x_n-x_\\ast|\\le C_x,\\qquad \\nu\\lambda_n^2(T_\\ast-t_n)\\le C_t'"
    },
    {
      "id": "ns.c3.c3f.rooted_tree",
      "latex": "\\mathcal T",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "根樹（與 𝒯_q 記號衝突）",
      "label_en": "Rooted tree (notation clash with 𝒯_q)",
      "definition_zh": "第22節為陳述 Kőnig infinity lemma 反例引入的 rooted tree 記號 $\\mathcal T$，與第1節算子 $\\mathcal T_q$ 共用花體 T；兩者為完全不同物件，須依上下文（有無下標 $q$）區分。",
      "definition_en": "Sec. 22's rooted-tree symbol $\\mathcal T$, used to state the Kőnig's-lemma counterexample to \"finite branching blocks infinite cascades\"; it reuses the same calligraphic T as the operator $\\mathcal T_q$ from Sec. 1, an unrelated object disambiguated only by the $q$ subscript.",
      "notes": "Confirmed genuine notation overload in the source text, not an artifact of extraction."
    },
    {
      "id": "ns.c3.c3f.c1c_split",
      "latex": "\\text{C1c-a},\\ \\text{C1c-b},\\ \\text{C1c-c}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "C1c 的三段拆解",
      "label_en": "Three-way split of C1c",
      "definition_zh": "第24節將原本單一猜想 C1c（Blowup ⇒ persistent source-preserving genealogy）拆成三段：C1c-a（static packet source selection，本輪已證）、C1c-b（infinite path extraction，離散組合定理）、C1c-c（causal orientation，即目前最關鍵的 open gap）。",
      "definition_en": "Sec. 24 splits the single conjecture C1c (Blowup ⇒ persistent source-preserving genealogy) into three parts: C1c-a (static packet source selection, proved this round), C1c-b (infinite path extraction, a discrete combinatorial theorem), and C1c-c (causal orientation — the currently critical open gap)."
    },
    {
      "id": "ns.c3.c3f.duhamel_source",
      "latex": "\\operatorname{Source}[u(r),u(r)]",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "Duhamel 非線性源項",
      "label_en": "Duhamel nonlinear source term",
      "definition_zh": "第26節，output packet 的 Duhamel 表示 $v(t)=\\text{linear inheritance}+\\int_s^t\\operatorname{Source}[u(r),u(r)]\\,dr$ 中的瞬時非線性源；用以論證 nonlinear integral 大時必存在某 $r<t$ 使瞬時 source 不可忽略，給出 strictly earlier source time。",
      "definition_en": "Sec. 26's instantaneous nonlinear source in the Duhamel representation $v(t)=\\text{linear inheritance}+\\int_s^t\\operatorname{Source}[u(r),u(r)]\\,dr$; a large nonlinear integral forces some $r<t$ with non-negligible instantaneous source, giving a strictly earlier source time.",
      "defining_relation": "v(t)=\\text{linear inheritance}+\\int_s^t\\operatorname{Source}[u(r),u(r)]\\,dr"
    },
    {
      "id": "ns.c3.c3f.tau_v_crossing",
      "latex": "\\tau_v=\\inf\\{t:A_v(t)\\ge a_v\\}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "首次跨閾值時刻",
      "label_en": "First-crossing time",
      "definition_zh": "第27節，對 packet amplitude functional $A_v(t)$ 與閾值 $a_v>0$ 定義的首次跨越時刻；causal parenthood 候選判準為 $\\tau_p<\\tau_c$，但此策略目前僅是 proof program，尚缺 amplitude 微分不等式支撐。",
      "definition_en": "Sec. 27's first time the packet amplitude functional $A_v(t)$ crosses threshold $a_v>0$; the proposed causal criterion is $\\tau_p<\\tau_c$ for parent/child, though this remains only a proof program pending a supporting amplitude differential inequality.",
      "defining_relation": "\\tau_v=\\inf\\{t:A_v(t)\\ge a_v\\}"
    },
    {
      "id": "ns.c3.c3f.use_reuse",
      "latex": "\\operatorname{Use}_p,\\ \\operatorname{Reuse}(v_p)",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "親代使用量／重複使用量",
      "label_en": "Parent-use ledger / reuse capacity",
      "definition_zh": "第28、34節（G2）引入的量，衡量單一 parent packet $p$ 可支持多少 cumulative child production；目標是證 $\\operatorname{Use}_p$ 受 $p$ 的 local energy/helicity/strain budget 控制，以避免對同一 source 的 double counting。",
      "definition_en": "Introduced in Secs. 28 and 34 (G2), measuring how much cumulative child production a single parent packet $p$ can support; the goal is bounding $\\operatorname{Use}_p$ by $p$'s local energy/helicity/strain budget to prevent double-counting the same source.",
      "defining_relation": "\\operatorname{Use}_p=\\sum_c\\text{source contribution }p\\to c"
    },
    {
      "id": "ns.c3.c3f.a_n_crit",
      "latex": "A_n^{\\rm crit}=\\lambda_n^{1/2}U_n\\sim1",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "臨界 packet 振幅",
      "label_en": "Critical packet amplitude",
      "definition_zh": "第29節 Zeno no-go 論證中設定的臨界振幅假設 $A_n^{\\rm crit}=\\lambda_n^{1/2}U_n\\sim1$，導出 $U_n^2\\sim\\lambda_n^{-1}$，用以檢驗 perfect parabolic ancestry 是否違反 finite energy budget。",
      "definition_en": "Sec. 29's critical-amplitude ansatz in the Zeno no-go argument, $A_n^{\\rm crit}=\\lambda_n^{1/2}U_n\\sim1$, giving $U_n^2\\sim\\lambda_n^{-1}$; used to test whether perfect parabolic ancestry violates the finite energy budget.",
      "defining_relation": "A_n^{\\rm crit}=\\lambda_n^{1/2}U_n\\sim1"
    },
    {
      "id": "ns.c3.c3f.d_n_dissipation",
      "latex": "D_n\\sim\\lambda_n^{-1}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "每代黏性耗散量",
      "label_en": "Per-generation viscous dissipation",
      "definition_zh": "第29節，第 $n$ 代 packet 在一個 viscous window 內的 ordinary energy dissipation，$D_n\\sim\\nu\\lambda_n^2U_n^2\\cdot(\\nu\\lambda_n^2)^{-1}\\sim\\lambda_n^{-1}$；因 $\\lambda_n$ 幾何增長，$\\sum_nD_n<\\infty$，證明完美時空局域化仍與 finite energy dissipation 相容（No-Go 29.1）。",
      "definition_en": "Sec. 29's ordinary energy dissipation of generation $n$ over one viscous window, $D_n\\sim\\nu\\lambda_n^2U_n^2\\cdot(\\nu\\lambda_n^2)^{-1}\\sim\\lambda_n^{-1}$; since $\\lambda_n$ grows geometrically, $\\sum_nD_n<\\infty$, so perfect space-time localization remains compatible with finite energy dissipation bookkeeping (No-Go 29.1).",
      "defining_relation": "D_n\\sim\\nu\\lambda_n^2U_n^2\\cdot(\\nu\\lambda_n^2)^{-1}\\sim\\lambda_n^{-1}"
    },
    {
      "id": "ns.c3.c3f.xedge",
      "latex": "\\operatorname{XEdge}=\\langle q_p,m_p,s_p,t_p;q_c,m_c,s_c,t_c;\\mathcal T;\\eta;R;\\operatorname{Prov}\\rangle",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "相空間世系候選邊",
      "label_en": "Phase-space ancestry candidate edge",
      "definition_zh": "第31節定義的候選 parent-child edge 資料結構，攜帶親/子代的 frequency-space-helicity-time 索引 $(q,m,s,t)$、算子 $\\mathcal T$、效率 $\\eta$、半徑 $R$ 與 provenance $\\operatorname{Prov}$；須通過 G-FREQ 等八個守衛才成立。",
      "definition_en": "Sec. 31's candidate parent-child edge structure, carrying parent/child frequency-space-helicity-time indices $(q,m,s,t)$, the operator $\\mathcal T$, efficiency $\\eta$, radius $R$, and provenance $\\operatorname{Prov}$; must pass eight named guards to be admitted.",
      "defining_relation": "\\operatorname{XEdge}=\\langle q_p,m_p,s_p,t_p;q_c,m_c,s_c,t_c;\\mathcal T;\\eta;R;\\operatorname{Prov}\\rangle"
    },
    {
      "id": "ns.c3.c3f.xedge_guards",
      "latex": "\\text{G-FREQ, G-SPACE, G-TIME, G-TAIL, G-CORE, G-HEL, G-PHASE, G-REUSE}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "XEdge 八守衛",
      "label_en": "XEdge's eight guards",
      "definition_zh": "第31節為 XEdge 設下的八項檢核：G-FREQ（$q_c-q_p=O(1)$）、G-SPACE（$|x_c-x_p|\\lesssim R\\lambda_p^{-1}$）、G-TIME（$t_p<t_c$）、G-TAIL（尾端受 $R^{-N}$ 控制）、G-CORE、G-HEL、G-PHASE、G-REUSE；本輪指出最未閉合者為 G-TIME 與 G-REUSE。",
      "definition_en": "Sec. 31's eight admissibility checks for an XEdge: G-FREQ ($q_c-q_p=O(1)$), G-SPACE ($|x_c-x_p|\\lesssim R\\lambda_p^{-1}$), G-TIME ($t_p<t_c$), G-TAIL ($R^{-N}$-controlled), G-CORE, G-HEL, G-PHASE, G-REUSE; this round flags G-TIME and G-REUSE as least closed.",
      "defining_relation": "\\text{G-FREQ: }q_c-q_p=O(1);\\ \\text{G-SPACE: }|x_c-x_p|\\lesssim R\\lambda_p^{-1};\\ \\text{G-TIME: }t_p<t_c"
    },
    {
      "id": "ns.c3.c3f.theta_qms",
      "latex": "\\Theta_{q,m,s}(t),\\quad \\Theta_{q_1,m_1,s_1}\\bowtie\\Theta_{q_2,m_2,s_2}\\longrightarrow\\Theta_{q_3,m_3,s_3}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "細化 ETN 狀態與交互關係",
      "label_en": "Refined ETN state and interaction relation",
      "definition_zh": "第32節將 True ETN 狀態記法由僅記頻率的 $\\Theta_q(t)$ 升級為同時記空間格 $m$ 與 helicity/sign $s$ 的 $\\Theta_{q,m,s}(t)$，並用 $\\bowtie$ 表示一對親代狀態交互產生子代狀態，使無限維張力場升級成 time-oriented phase-space tension hypergraph。",
      "definition_en": "Sec. 32 upgrades the True-ETN state from the frequency-only $\\Theta_q(t)$ to $\\Theta_{q,m,s}(t)$, also indexed by spatial cell $m$ and helicity/sign $s$, with $\\bowtie$ denoting two parent states interacting to produce a child state — upgrading the \"infinite-dimensional tension field\" into a time-oriented phase-space tension hypergraph.",
      "defining_relation": "\\Theta_{q_1,m_1,s_1}\\bowtie\\Theta_{q_2,m_2,s_2}\\longrightarrow\\Theta_{q_3,m_3,s_3}"
    },
    {
      "id": "ns.c3.c3f.c3g_obligations",
      "latex": "\\text{G1, G2, G3, G4, G5}",
      "series": "NS",
      "first_appearance": "C3-F",
      "label_zh": "C3-G 的五項證明義務",
      "label_en": "C3-G's five proof obligations",
      "definition_zh": "第34節為下一輪 C3-G 開出的五項任務：G1 first-crossing causal lemma、G2 parent-use ledger（即 $\\operatorname{Use}_p$ 受 budget 控制）、G3 no-double-counting theorem、G4 time-oriented finite-branching tree、G5 depletion along ray。",
      "definition_en": "Sec. 34's five tasks opening the next round C3-G: G1 first-crossing causal lemma, G2 parent-use ledger (bounding $\\operatorname{Use}_p$ by budget), G3 no-double-counting theorem, G4 time-oriented finite-branching tree (to extract an infinite causal ray), and G5 depletion along ray."
    },
    {
      "id": "ns.c3.c3g.a_q_sigma",
      "latex": "a_q^\\sigma(t) = \\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "臨界 dyadic shell 振幅",
      "label_en": "Critical dyadic shell amplitude",
      "definition_zh": "第 1 節定義的無因次臨界殼層振幅：以黏性尺度 $\\nu\\lambda_q$ 對 helical dyadic shell 速度的 $L^\\infty$ 範數正規化，取代前一輪 packet $L^2$ 振幅，是貫穿全文的核心度量。",
      "definition_en": "The dimensionless critical shell amplitude defined in Section 1, normalizing the $L^\\infty$ norm of the helical dyadic shell velocity by the viscous scale $\\nu\\lambda_q$; replaces the previous round's packet $L^2$ amplitude and is the central quantity used throughout this round.",
      "defining_relation": "a_q^\\sigma(t) = \\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}",
      "notes": "N–S scaling 不變、無因次；下標 $q$ 為 dyadic shell 指標，上標 $\\sigma\\in\\{+,-\\}$ 為 helicity sign。"
    },
    {
      "id": "ns.c3.c3g.u_q_sigma",
      "latex": "u_q^\\sigma = \\Delta_qP^\\sigma u",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Helical dyadic shell 速度",
      "label_en": "Helical dyadic shell velocity",
      "definition_zh": "第 1、3 節使用的量，為速度場先經 helicity 投影 $P^\\sigma$ 再做 Littlewood–Paley dyadic 投影 $\\Delta_q$ 所得之分量，滿足 $u_p=u_p^++u_p^-$。",
      "definition_en": "The velocity component obtained by applying the helicity projector $P^\\sigma$ and then the Littlewood–Paley dyadic projector $\\Delta_q$ to $u$; satisfies $u_p=u_p^++u_p^-$.",
      "defining_relation": "u_q^\\sigma = \\Delta_qP^\\sigma u",
      "notes": "$\\Delta_q$、$P^\\sigma$ 可能承襲自更早輪次的 helicity-classified 框架，本輪重新用於定義 $a_q^\\sigma$。"
    },
    {
      "id": "ns.c3.c3g.lambda_q",
      "latex": "\\lambda_q=2^q",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Dyadic 頻率尺度",
      "label_en": "Dyadic frequency scale",
      "definition_zh": "第 1 節重申的 dyadic 頻率定義，$q$ 為 Littlewood–Paley shell 整數指標，用於正規化振幅 $a_q^\\sigma$ 及黏性視窗長度。",
      "definition_en": "The dyadic frequency scale with integer Littlewood–Paley shell index $q$; used to normalize the amplitude $a_q^\\sigma$ and viscous window lengths throughout.",
      "defining_relation": "\\lambda_q=2^q"
    },
    {
      "id": "ns.c3.c3g.helicity_sign",
      "latex": "\\sigma\\in\\{+,-\\}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Helicity 符號指標",
      "label_en": "Helicity sign label",
      "definition_zh": "標記正、負 helical 分量的符號指標，貫穿全文用於 shell-sign node $(q,\\sigma)$ 及各 helical 分解。",
      "definition_en": "The sign index labeling positive/negative helical components, used throughout for shell-sign nodes $(q,\\sigma)$ and helical decompositions.",
      "defining_relation": "\\sigma\\in\\{+,-\\}"
    },
    {
      "id": "ns.c3.c3g.dissipation_wavenumber",
      "latex": "\\Lambda(t) = \\lambda_{Q(t)}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "耗散波數",
      "label_en": "Dissipation wavenumber",
      "definition_zh": "第 2 節引入的 Cheskidov–Shvydkoy 型耗散波數，定義為 $\\lambda_{Q(t)}$；C2 已證 hypothetical blow-up 要求 $\\Lambda\\notin L^{5/2}(0,T_\\ast)$，故必為 unbounded。",
      "definition_en": "The Cheskidov–Shvydkoy-type dissipation wavenumber introduced in Section 2, defined as $\\lambda_{Q(t)}$; C2 shows hypothetical blow-up forces $\\Lambda\\notin L^{5/2}(0,T_\\ast)$, hence $\\Lambda$ must be unbounded.",
      "defining_relation": "\\Lambda(t)=\\lambda_{Q(t)}"
    },
    {
      "id": "ns.c3.c3g.cutoff_index",
      "latex": "Q(t)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "耗散截止指標",
      "label_en": "Dissipation cutoff index",
      "definition_zh": "第 2 節定義：使所有 $p\\ge Q(t)$ 的 shells 皆滿足 $\\lambda_p^{-1}\\|u_p(t)\\|_\\infty<c_0\\nu$ 的最低 cutoff 整數指標，決定 $\\Lambda(t)=\\lambda_{Q(t)}$。",
      "definition_en": "The lowest cutoff integer index such that every shell $p\\ge Q(t)$ satisfies $\\lambda_p^{-1}\\|u_p(t)\\|_\\infty<c_0\\nu$; determines $\\Lambda(t)=\\lambda_{Q(t)}$.",
      "notes": "與第 14 節起用作 frontier level 的自由整數參數 $Q$ 為不同物件（後者非時間相依），勿混淆。"
    },
    {
      "id": "ns.c3.c3g.c0_threshold",
      "latex": "c_0",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "耗散門檻常數",
      "label_en": "Dissipation threshold constant",
      "definition_zh": "第 2 節定義 $Q(t)$ 所用的固定小常數，並推出 $Q(t)$ 附近存在 active shell 滿足 $\\|u_p(t)\\|_\\infty/(\\nu\\lambda_p)\\gtrsim c_0$，故至少一 helicity sign 滿足 $a_p^\\sigma\\gtrsim c_0/2$。",
      "definition_en": "The fixed small constant used to define $Q(t)$ in Section 2; yields an active shell near $Q(t)$ with $\\|u_p(t)\\|_\\infty/(\\nu\\lambda_p)\\gtrsim c_0$, hence some helicity sign satisfies $a_p^\\sigma\\gtrsim c_0/2$."
    },
    {
      "id": "ns.c3.c3g.c_dagger",
      "latex": "c_\\dagger",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "恆常活躍門檻常數",
      "label_en": "Persistent-activity constant",
      "definition_zh": "External/derived interface 2.1（第 2 節）給出的固定正常數：hypothetical $T_\\ast<\\infty$ 下，存在任意大 $q$、時刻 $t<T_\\ast$ 與符號 $\\sigma$ 使 $a_q^\\sigma(t)\\ge c_\\dagger$。",
      "definition_en": "The fixed positive constant from External/derived interface 2.1 (Section 2): under hypothetical $T_\\ast<\\infty$, arbitrarily large $q$, times $t<T_\\ast$, and signs $\\sigma$ satisfy $a_q^\\sigma(t)\\ge c_\\dagger$.",
      "defining_relation": "a_q^\\sigma(t)\\ge c_\\dagger",
      "notes": "第 13 節取 $\\beta_\\ast<c_\\dagger$ 以確保任意高頻 shells 皆會 first-cross。"
    },
    {
      "id": "ns.c3.c3g.local_operator",
      "latex": "\\mathcal L_q^\\sigma",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "局部來源算子",
      "label_en": "Local (bounded scale-ratio) source operator",
      "definition_zh": "第 3 節將非線性來源分解為 $\\mathcal L_q^\\sigma+\\mathcal R_q^\\sigma$ 中保留 bounded scale-ratio 局部交互作用的部分，明確定義為對 $|p-q|,|r-q|\\le C_L$ 的求和。",
      "definition_en": "In the Section 3 decomposition $\\mathcal L_q^\\sigma+\\mathcal R_q^\\sigma$ of the nonlinear source, the part retaining bounded-scale-ratio local interactions, explicitly the sum over $|p-q|,|r-q|\\le C_L$.",
      "defining_relation": "\\mathcal L_q^\\sigma=\\sum_{|p-q|\\le C_L,\\,|r-q|\\le C_L,\\,\\sigma_1,\\sigma_2}\\Delta_qP^\\sigma\\mathbb P\\nabla\\cdot(u_p^{\\sigma_1}\\otimes u_r^{\\sigma_2})"
    },
    {
      "id": "ns.c3.c3g.remainder_operator",
      "latex": "\\mathcal R_q^\\sigma",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "非局部殘餘算子",
      "label_en": "Nonlocal remainder operator",
      "definition_zh": "第 3 節非線性來源分解中的非局部／未解析餘項，其在一個 crossing window 上的正規化 Duhamel 貢獻記為 $\\operatorname{Rem}_q^\\sigma(I)$，是 eventual local-source dominance 假設的控制對象。",
      "definition_en": "The nonlocal/unresolved remainder part of the Section 3 nonlinear-source decomposition; its normalized Duhamel contribution over a window is denoted $\\operatorname{Rem}_q^\\sigma(I)$ and is what the eventual local-source dominance hypothesis controls.",
      "notes": "與第 30 節引自 C3-B 的量 $\\mathcal R(t)$（helicity pair-production 判準）為不同符號，僅字母偶合，勿混淆。"
    },
    {
      "id": "ns.c3.c3g.C_L",
      "latex": "C_L",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "局部交互作用尺度半徑",
      "label_en": "Local-interaction scale radius",
      "definition_zh": "第 3 節固定的常數，界定「局部」parent 的尺度範圍 $|p-q|\\le C_L$，貫穿全文用於 Critical Activation DAG 邊的尺度跳躍界。",
      "definition_en": "The fixed constant from Section 3 bounding the scale range $|p-q|\\le C_L$ of a \"local\" parent; used throughout for the bounded shell-jump of edges in the Critical Activation DAG.",
      "defining_relation": "|p-q|\\le C_L"
    },
    {
      "id": "ns.c3.c3g.M_L",
      "latex": "M_L<\\infty",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "局部 parent 類型數",
      "label_en": "Number of local parent types",
      "definition_zh": "第 3 節：固定 $C_L$ 後，局部 parent 的類型數 $M_L<\\infty$，用於定理 9.1 中界定 $\\beta_\\ast$ 的上界公式。",
      "definition_en": "With $C_L$ fixed, the finite number $M_L<\\infty$ of local parent types (Section 3), entering the upper bound formula for $\\beta_\\ast$ in Theorem 9.1."
    },
    {
      "id": "ns.c3.c3g.viscous_ds",
      "latex": "ds=\\nu\\lambda_q^2\\,dt",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "無因次黏性時間",
      "label_en": "Dimensionless viscous time",
      "definition_zh": "第 5 節定義的重新標度時間變數，使局部來源在一個視窗內的正規化貢獻不再殘留 $\\lambda_q$，是本輪 criticality 的關鍵觀察。",
      "definition_en": "The rescaled time variable defined in Section 5, ensuring the normalized local-source contribution over one window carries no residual $\\lambda_q$ — the key criticality observation of this round.",
      "defining_relation": "ds=\\nu\\lambda_q^2\\,dt"
    },
    {
      "id": "ns.c3.c3g.theta_window",
      "latex": "\\theta",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "無因次視窗長度",
      "label_en": "Dimensionless window length",
      "definition_zh": "第 5–6 節固定的正參數，為局部黏性視窗 $|I_q|=\\theta/(\\nu\\lambda_q^2)$ 在無因次時間中的長度，並決定第 6 節收縮因子 $\\rho=C_he^{-c_h\\theta}$。",
      "definition_en": "The fixed positive parameter (Sections 5–6) giving the dimensionless length of the local viscous window $|I_q|=\\theta/(\\nu\\lambda_q^2)$, and determining the contraction factor $\\rho=C_he^{-c_h\\theta}$ in Section 6."
    },
    {
      "id": "ns.c3.c3g.I_q",
      "latex": "|I_q|=\\frac{\\theta}{\\nu\\lambda_q^2}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "局部黏性視窗",
      "label_en": "Local viscous window",
      "definition_zh": "第 5 節定義的與 shell $q$ 相關之時間視窗長度；第 7 節起具體化為 crossing window $I=[t-\\theta(\\nu\\lambda_q^2)^{-1},t]$，是 $\\operatorname{Rem}_q^\\sigma(I)$ 與定理 9.1 證明的作用域。",
      "definition_en": "The window length associated to shell $q$ defined in Section 5; from Section 7 onward instantiated as the concrete crossing window $I=[t-\\theta(\\nu\\lambda_q^2)^{-1},t]$, the domain on which $\\operatorname{Rem}_q^\\sigma(I)$ and the Theorem 9.1 proof operate.",
      "defining_relation": "|I_q|=\\theta/(\\nu\\lambda_q^2)"
    },
    {
      "id": "ns.c3.c3g.heat_decay_const",
      "latex": "C_h,\\ c_h",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "熱核衰減常數",
      "label_en": "Heat-kernel decay constants",
      "definition_zh": "第 6 節 annular heat decay 估計 $\\|e^{\\nu\\tau\\Delta}u_q^\\sigma\\|_\\infty\\le C_he^{-c_h\\nu\\lambda_q^2\\tau}\\|u_q^\\sigma\\|_\\infty$ 中的常數，用以定義收縮因子 $\\rho=C_he^{-c_h\\theta}$。",
      "definition_en": "The constants in the Section 6 annular heat-decay estimate $\\|e^{\\nu\\tau\\Delta}u_q^\\sigma\\|_\\infty\\le C_he^{-c_h\\nu\\lambda_q^2\\tau}\\|u_q^\\sigma\\|_\\infty$, used to define the contraction factor $\\rho=C_he^{-c_h\\theta}$.",
      "defining_relation": "\\|e^{\\nu\\tau\\Delta}u_q^\\sigma\\|_\\infty\\le C_he^{-c_h\\nu\\lambda_q^2\\tau}\\|u_q^\\sigma\\|_\\infty",
      "notes": "此處 $\\tau$ 為熱半群時間差變數，與 $\\tau_{q,\\sigma}$（first-crossing 時刻）及第 26 節 triad 指標 $\\tau$ 為三種不同用法。"
    },
    {
      "id": "ns.c3.c3g.rho_contraction",
      "latex": "\\rho=C_he^{-c_h\\theta}<1",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "單視窗收縮因子",
      "label_en": "One-window contraction factor",
      "definition_zh": "第 6 節由固定 $\\theta$ 使熱核衰減達到的收縮率 $\\rho<1$，貫穿定理 9.1 的 Duhamel 不等式與第 24 節 persistence 遞迴 $a_m\\le\\rho a_{m-1}$。",
      "definition_en": "The contraction ratio $\\rho<1$ obtained in Section 6 by fixing $\\theta$ so heat-kernel decay dominates; used throughout Theorem 9.1's Duhamel inequality and the Section 24 persistence recurrence $a_m\\le\\rho a_{m-1}$.",
      "defining_relation": "\\rho=C_he^{-c_h\\theta}<1"
    },
    {
      "id": "ns.c3.c3g.rem_q_sigma",
      "latex": "\\operatorname{Rem}_q^\\sigma(I)\\le\\varepsilon\\beta",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "正規化非局部餘項",
      "label_en": "Normalized nonlocal remainder",
      "definition_zh": "第 7 節定義：非局部殘餘 $\\mathcal R_q^\\sigma$ 在 child crossing window $I=[t-\\theta(\\nu\\lambda_q^2)^{-1},t]$ 上的正規化 Duhamel 貢獻，eventual local-source dominance 假設要求其 $\\le\\varepsilon\\beta$。",
      "definition_en": "Defined in Section 7 as the normalized Duhamel contribution of the nonlocal remainder $\\mathcal R_q^\\sigma$ over the child crossing window $I=[t-\\theta(\\nu\\lambda_q^2)^{-1},t]$; the eventual local-source dominance hypothesis requires it to be $\\le\\varepsilon\\beta$.",
      "defining_relation": "\\operatorname{Rem}_q^\\sigma(I)\\le\\varepsilon\\beta"
    },
    {
      "id": "ns.c3.c3g.epsilon_bound",
      "latex": "0\\le\\varepsilon<1-\\rho",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "非局部餘項容忍度",
      "label_en": "Nonlocal remainder tolerance",
      "definition_zh": "第 7 節參數，滿足 $0\\le\\varepsilon<1-\\rho$，界定 eventual local-source dominance 假設中非局部項可容忍的大小。",
      "definition_en": "The Section 7 parameter satisfying $0\\le\\varepsilon<1-\\rho$, bounding how large the nonlocal contribution may be under the eventual local-source dominance hypothesis.",
      "defining_relation": "0\\le\\varepsilon<1-\\rho"
    },
    {
      "id": "ns.c3.c3g.local_dominance",
      "latex": "\\textbf{eventual local-source dominance}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "最終局部來源主導假設",
      "label_en": "Eventual local-source dominance (hypothesis)",
      "definition_zh": "第 7 節命名的核心路線假設：非局部殘餘的正規化 Duhamel 貢獻滿足 $\\operatorname{Rem}_q^\\sigma(I)\\le\\varepsilon\\beta$（$\\varepsilon<1-\\rho$）；本輪幾乎所有定理（9.1、15.1、18.1）皆以此為條件。",
      "definition_en": "The named route hypothesis of Section 7: the nonlocal remainder's normalized Duhamel contribution satisfies $\\operatorname{Rem}_q^\\sigma(I)\\le\\varepsilon\\beta$ with $\\varepsilon<1-\\rho$; nearly every theorem in this round (9.1, 15.1, 18.1) is conditional on it.",
      "notes": "C3-C/D 已建立 suppression/compensation debt，但尚未無條件證明此假設對所有 hypothetical blow-up 成立。"
    },
    {
      "id": "ns.c3.c3g.beta_threshold",
      "latex": "\\beta>0",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "一般 first-crossing 門檻",
      "label_en": "Generic first-crossing threshold",
      "definition_zh": "第 8 節固定的一般性門檻值，用於定義 $\\tau_{q,\\sigma}=\\inf\\{t>0:a_q^\\sigma(t)\\ge\\beta\\}$；定理 9.1 進一步構造出滿足額外條件的具體值 $\\beta_\\ast$。",
      "definition_en": "The generic fixed threshold from Section 8 used to define $\\tau_{q,\\sigma}=\\inf\\{t>0:a_q^\\sigma(t)\\ge\\beta\\}$; Theorem 9.1 later constructs a specific value $\\beta_\\ast$ satisfying additional requirements."
    },
    {
      "id": "ns.c3.c3g.shell_sign_node",
      "latex": "(q,\\sigma)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Shell-sign 節點",
      "label_en": "Shell-sign node",
      "definition_zh": "第 8 節起將 dyadic shell 指標 $q$ 與 helicity sign $\\sigma$ 併為一節點，作為 first-crossing 時間 $\\tau_{q,\\sigma}$ 及 Critical Activation DAG 頂點集 $\\mathcal V_\\beta$ 的基本單位。",
      "definition_en": "From Section 8 onward, the pairing of dyadic shell index $q$ with helicity sign $\\sigma$ into a single node, the basic unit for first-crossing times $\\tau_{q,\\sigma}$ and the vertex set $\\mathcal V_\\beta$ of the Critical Activation DAG.",
      "defining_relation": "(q,\\sigma)"
    },
    {
      "id": "ns.c3.c3g.tau_q_sigma",
      "latex": "\\tau_{q,\\sigma}=\\inf\\{t>0:a_q^\\sigma(t)\\ge\\beta\\}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "First-crossing 時刻",
      "label_en": "First-crossing time",
      "definition_zh": "第 8 節核心定義：shell-sign node $(q,\\sigma)$ 首次使 $a_q^\\sigma(t)\\ge\\beta$ 的時刻，未達標則設為 $\\infty$。定理 9.1 證明 child 與其 local parent 有嚴格排序 $\\tau_{p,\\sigma_p}<\\tau_{q,\\sigma}$。",
      "definition_en": "The core Section 8 definition: the first time a shell-sign node $(q,\\sigma)$ satisfies $a_q^\\sigma(t)\\ge\\beta$; set to $\\infty$ if never crossed. Theorem 9.1 proves a strict ordering $\\tau_{p,\\sigma_p}<\\tau_{q,\\sigma}$ between a child and its local parent.",
      "defining_relation": "\\tau_{q,\\sigma}=\\inf\\{t>0:a_q^\\sigma(t)\\ge\\beta\\}",
      "notes": "同一字母 $\\tau$ 亦在第 6 節作熱半群時間差、在第 26–27 節作 triad 指標使用，三者意義不同。"
    },
    {
      "id": "ns.c3.c3g.beta_star",
      "latex": "\\beta_\\ast>0",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "臨界 first-crossing 門檻值",
      "label_en": "Critical first-crossing threshold value",
      "definition_zh": "定理 9.1（C3-G.1）證明存在的具體門檻，只依賴 $\\theta,\\rho,\\varepsilon,C,M_L$，滿足 $0<\\beta_\\ast<(1-\\rho-\\varepsilon)/(CM_L\\theta)$，保證 child first crossing 必有更早的 local parent first crossing。",
      "definition_en": "The specific threshold whose existence Theorem 9.1 (C3-G.1) proves, depending only on $\\theta,\\rho,\\varepsilon,C,M_L$, satisfying $0<\\beta_\\ast<(1-\\rho-\\varepsilon)/(CM_L\\theta)$; guarantees a child's first crossing forces an earlier local parent first crossing.",
      "defining_relation": "0<\\beta_\\ast<\\dfrac{1-\\rho-\\varepsilon}{CM_L\\theta}"
    },
    {
      "id": "ns.c3.c3g.V_beta",
      "latex": "\\mathcal V_\\beta=\\{(q,\\sigma):\\tau_{q,\\sigma}<T_\\ast\\}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "活化節點集",
      "label_en": "Activation node set",
      "definition_zh": "第 12 節定義的 Critical Activation DAG 頂點集，收集所有在 blow-up 時間前 first-cross 門檻的 shell-sign nodes。",
      "definition_en": "The vertex set of the Critical Activation DAG defined in Section 12, collecting all shell-sign nodes that first-cross the threshold before the blow-up time.",
      "defining_relation": "\\mathcal V_\\beta=\\{(q,\\sigma):\\tau_{q,\\sigma}<T_\\ast\\}"
    },
    {
      "id": "ns.c3.c3g.activation_dag",
      "latex": "\\textbf{Critical Activation DAG}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "臨界活化 DAG",
      "label_en": "Critical Activation DAG",
      "definition_zh": "第 12 節命名的有向無環圖：頂點集 $\\mathcal V_\\beta$，邊為定理 9.1 選出的 parent-child 關係 $(p,\\sigma_p)\\to(q,\\sigma)$，因 $\\tau_p<\\tau_q$ 保證無迴圈，並受 $|p-q|\\le C_L$ 局部尺度限制。",
      "definition_en": "The directed acyclic graph named in Section 12: vertex set $\\mathcal V_\\beta$, edges given by the parent-child relation $(p,\\sigma_p)\\to(q,\\sigma)$ selected by Theorem 9.1, acyclic because $\\tau_p<\\tau_q$, further constrained by the local scale bound $|p-q|\\le C_L$.",
      "defining_relation": "(p,\\sigma_p)\\longrightarrow(q,\\sigma),\\quad \\tau_p<\\tau_q,\\quad |p-q|\\le C_L"
    },
    {
      "id": "ns.c3.c3g.T_Q",
      "latex": "T_Q=\\inf\\{\\tau_{q,\\sigma}:q\\ge Q,\\ \\sigma\\in\\{+,-\\}\\}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "前沿穿越時刻",
      "label_en": "Frontier crossing time",
      "definition_zh": "第 14 節定義：頻率 $\\ge Q$ 的所有 nodes 中最早的 first-crossing 時刻；hypothetical blow-up 下 $T_Q<T_\\ast$ 且 $T_Q\\uparrow T_\\ast$（$Q\\to\\infty$）。定理 15.1 以此建立跨頻率邊界的 parent 存在性。",
      "definition_en": "Defined in Section 14 as the earliest first-crossing time among all nodes with frequency $\\ge Q$; under hypothetical blow-up, $T_Q<T_\\ast$ and $T_Q\\uparrow T_\\ast$ as $Q\\to\\infty$. Theorem 15.1 uses it to establish parent existence across the frequency boundary.",
      "defining_relation": "T_Q=\\inf\\{\\tau_{q,\\sigma}:q\\ge Q,\\ \\sigma\\in\\{+,-\\}\\}",
      "notes": "此處整數 $Q$ 為自由 frontier 參數，與第 2 節時間相依的 $Q(t)$ 為不同物件，僅共用字母。"
    },
    {
      "id": "ns.c3.c3g.phase_space_edge",
      "latex": "(q_p,\\sigma_p,x_p,t_p)\\to(q_c,\\sigma_c,x_c,t_c)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "相空間因果邊",
      "label_en": "Phase-space causal edge",
      "definition_zh": "第 17 節將 shell-level 因果邊升級為相空間邊，結合 C3-F 的空間局域性（$\\eta_q\\ge\\eta_0$ 下 $|x_c-x_p|\\lesssim\\lambda_p^{-1}$）與本輪 first-crossing 時間排序 $t_p<t_c$、尺度界 $|q_c-q_p|\\le C_L$。",
      "definition_en": "Section 17 upgrades the shell-level causal edge to a phase-space edge, combining C3-F's spatial locality ($|x_c-x_p|\\lesssim\\lambda_p^{-1}$ when $\\eta_q\\ge\\eta_0$) with this round's first-crossing temporal order $t_p<t_c$ and scale bound $|q_c-q_p|\\le C_L$.",
      "defining_relation": "|q_c-q_p|\\le C_L,\\quad t_p<t_c,\\quad |x_c-x_p|\\lesssim\\lambda_p^{-1}",
      "notes": "局域效率 $\\eta_q$、半徑 $R_q$ 為 C3-F 既有符號，本節僅重用以構造新的相空間邊，未在此重新定義。"
    },
    {
      "id": "ns.c3.c3g.ancestry_ray",
      "latex": "v_0\\to v_1\\to v_2\\to\\cdots",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "因果祖源鏈節點",
      "label_en": "Ancestry ray node",
      "definition_zh": "第 18–20 節：由 Critical Activation DAG 中依 first-crossing 排序串連的節點鏈 $v_n$；第 19 節證明沿無限鏈頻率必 $\\sup_nq_n=\\infty$，第 33 節將其座標具體化為 $(x_n,t_n,\\lambda_n,\\sigma_n)$ 供 renormalization 使用。",
      "definition_en": "Sections 18–20: nodes $v_n$ chained by first-crossing order within the Critical Activation DAG; Section 19 proves any infinite such chain forces $\\sup_nq_n=\\infty$; Section 33 instantiates its coordinates as $(x_n,t_n,\\lambda_n,\\sigma_n)$ for the renormalization construction.",
      "defining_relation": "v_0\\to v_1\\to\\cdots\\to v_N,\\quad q_N\\to\\infty",
      "notes": "同一記號 $v_n$ 於第 33 節被重用來標記 rescaled velocity profile $v_n(y,s)$（見該條目），兩種用法透過同一祖源點串接，非同型物件。"
    },
    {
      "id": "ns.c3.c3g.r_minus",
      "latex": "r_->1",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "尺度幾何增長率",
      "label_en": "Geometric scale growth rate",
      "definition_zh": "第 20 節假設沿 ray 的尺度滿足 $\\lambda_{n+1}\\ge r_-\\lambda_n$，用以套用 C3-F 的 ancestry-cone theorem 得出 $x_n\\to x_\\ast$、$t_n\\to T_\\ast$。",
      "definition_en": "Section 20 assumes the scales along the ray satisfy $\\lambda_{n+1}\\ge r_-\\lambda_n$, invoked to apply C3-F's ancestry-cone theorem yielding $x_n\\to x_\\ast$ and $t_n\\to T_\\ast$.",
      "defining_relation": "\\lambda_{n+1}\\ge r_-\\lambda_n,\\quad r_->1"
    },
    {
      "id": "ns.c3.c3g.triad_conservation",
      "latex": "\\dot e_k+\\dot e_p+\\dot e_q=0,\\quad s_kk\\dot e_k+s_pp\\dot e_p+s_qq\\dot e_q=0",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Triad 能量／helicity 守恆恆等式",
      "label_en": "Triad energy/helicity conservation identities",
      "definition_zh": "第 26 節命題 26.1 引用的 triadwise 守恆律：$e_k,e_p,e_q$ 為 triad 三波數之能量，$s_k,s_p,s_q$ 為對應 helicity 符號；兩恆等式僅將轉移向量限制在一維方向 $\\mathbf v_\\tau$。",
      "definition_en": "The triadwise conservation laws invoked in Proposition 26.1 (Section 26): $e_k,e_p,e_q$ are the energies of the three triad wavenumbers, $s_k,s_p,s_q$ their helicity signs; together the identities confine the transfer vector to the one-dimensional direction $\\mathbf v_\\tau$.",
      "defining_relation": "\\dot e_k+\\dot e_p+\\dot e_q=0,\\quad s_kk\\dot e_k+s_pp\\dot e_p+s_qq\\dot e_q=0",
      "notes": "此處 $k,p,q$ 為 triad 波數標籤，承襲自 C3-A 框架，與本輪前段作 dyadic shell 指標的 $p,q$ 用法不同。"
    },
    {
      "id": "ns.c3.c3g.Theta_tau",
      "latex": "\\Theta_\\tau(t)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "帶號轉移係數",
      "label_en": "Signed transfer coefficient",
      "definition_zh": "命題 26.1 核心構造：$\\dot{\\mathbf e}=\\Theta_\\tau(t)\\mathbf v_\\tau$ 中的純量係數；守恆代數未固定其正負號，故 $\\Theta_\\tau$ 可正可負，donor/receiver 角色可反轉，是 monotone depletion no-go 的關鍵。",
      "definition_en": "The core construction of Proposition 26.1 (Section 26): the scalar coefficient in $\\dot{\\mathbf e}=\\Theta_\\tau(t)\\mathbf v_\\tau$; the conservation algebra fixes no sign for it, so $\\Theta_\\tau$ can be positive or negative and donor/receiver roles can reverse — the key to the monotone-depletion no-go.",
      "defining_relation": "\\dot{\\mathbf e}=\\Theta_\\tau(t)\\mathbf v_\\tau",
      "notes": "下標 $\\tau$ 標記 triad，與 $\\tau_{q,\\sigma}$（first-crossing 時刻）字母相同但意義無關。"
    },
    {
      "id": "ns.c3.c3g.v_tau",
      "latex": "\\mathbf v_\\tau",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Triad 轉移方向向量",
      "label_en": "Triad transfer direction vector",
      "definition_zh": "第 26–27 節固定的一維轉移方向向量，是 $\\dot{\\mathbf e}=\\Theta_\\tau(t)\\mathbf v_\\tau$ 中唯一由守恆律限定的部分；第 27 節以 $\\Theta(t)=\\sin t$ 構造出符合守恆恆等式但能量前後交換的反例。",
      "definition_en": "The fixed one-dimensional transfer-direction vector of Sections 26–27, the only part of $\\dot{\\mathbf e}=\\Theta_\\tau(t)\\mathbf v_\\tau$ pinned down by the conservation laws; Section 27 builds an explicit counterexample with $\\Theta(t)=\\sin t$ satisfying the conservation identities while energy oscillates back and forth."
    },
    {
      "id": "ns.c3.c3g.use_p",
      "latex": "\\operatorname{Use}(p)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "天真 parent 使用量",
      "label_en": "Naive parent-use functional",
      "definition_zh": "第 25、29 節提出並否證的功能量：試圖以 $\\operatorname{Use}(p)\\le$ parent 初始能量、或 $\\operatorname{Use}(p)=$ children 總輸出 來界定 parent 可用資源，因守恆代數僅給 signed exchange 而不成立。",
      "definition_en": "The functional proposed and refuted in Sections 25 and 29: attempts to bound a parent's usable resource by $\\operatorname{Use}(p)\\le$ its initial energy, or define it as total children output; fails because conservation algebra gives only signed exchange, not monotone depletion.",
      "defining_relation": "\\operatorname{Use}(p)\\le\\text{initial energy of }p"
    },
    {
      "id": "ns.c3.c3g.ledger_p",
      "latex": "\\operatorname{Ledger}(p)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "Parent 資源帳本",
      "label_en": "Parent resource ledger",
      "definition_zh": "第 29 節提出取代 $\\operatorname{Use}(p)$ 的正確結構：由儲存振幅、進帳 recharge、出帳 transfer、黏性損耗、相位反轉、重用次數組成，平衡式為「終存量＝初存量＋recharge－帶號轉出－黏性不可逆損耗」。",
      "definition_en": "The correct multi-component structure proposed in Section 29 to replace $\\operatorname{Use}(p)$: a ledger of stored amplitude, incoming recharge, outgoing transfer, viscous loss, phase reversals, and reuse times, with balance \"ending stock = initial stock + recharge − outgoing signed transfer − viscous irreversible loss.\"",
      "defining_relation": "\\operatorname{Ledger}(p)=\\left\\langle\\text{stored amplitude},\\text{incoming recharge},\\text{outgoing transfer},\\text{viscous loss},\\text{phase reversals},\\text{reuse times}\\right\\rangle"
    },
    {
      "id": "ns.c3.c3g.a_m_seq",
      "latex": "a_m\\le\\rho a_{m-1}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "視窗索引振幅序列",
      "label_en": "Window-indexed amplitude sequence",
      "definition_zh": "第 24 節考慮某固定 shell 振幅在連續黏性視窗間若無非線性補注僅滿足線性遞迴 $a_m\\le\\rho a_{m-1}$，則 $M$ 個視窗後仍達 $\\beta$ 需初始儲量 $a_0\\ge\\beta\\rho^{-M}$，呈指數增長要求。",
      "definition_en": "Section 24 considers a fixed shell amplitude across successive viscous windows: absent nonlinear recharge it obeys the linear recurrence $a_m\\le\\rho a_{m-1}$, so remaining $\\ge\\beta$ after $M$ windows requires an initial reserve $a_0\\ge\\beta\\rho^{-M}$, growing exponentially in $M$.",
      "defining_relation": "a_m\\le\\rho^ma_0,\\qquad a_0\\ge\\beta\\rho^{-M}"
    },
    {
      "id": "ns.c3.c3g.S_j",
      "latex": "S_j",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "第 j 視窗正規化補注",
      "label_en": "Normalized recharge in window j",
      "definition_zh": "第 25 節 recharge recurrence 引入的量：第 $j$ 個黏性視窗中非線性來源對振幅的正規化補注貢獻，顯示 reusable parent 的資源實為「儲存振幅＋折現 recharge 歷史」而非單一初始 token。",
      "definition_en": "Introduced in the Section 25 recharge recurrence: the normalized nonlinear-source contribution to the amplitude within the $j$-th viscous window; shows a reusable parent's resource is really \"stored amplitude plus discounted recharge history,\" not a single initial token.",
      "defining_relation": "a_M\\le\\rho^Ma_0+\\sum_{j=1}^M\\rho^{M-j}S_j"
    },
    {
      "id": "ns.c3.c3g.rescaled_profile",
      "latex": "v_n(y,s)=\\lambda_n^{-1}u\\left(x_n+\\lambda_n^{-1}y,t_n+\\lambda_n^{-2}s\\right)",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "重標度速度剖面",
      "label_en": "Rescaled velocity profile",
      "definition_zh": "第 33 節（C3-H 展望）對 causal ray 上第 $n$ 個祖源點 $(x_n,t_n,\\lambda_n,\\sigma_n)$ 作 N–S 臨界重標度所得之場，將尺度 $\\lambda_n$ 送到 unit scale，是後續 compactness/rigidity 分析（H1–H6）的出發點。",
      "definition_en": "Section 33's (C3-H preview) critical N–S rescaling of $u$ about the $n$-th ancestry point $(x_n,t_n,\\lambda_n,\\sigma_n)$ on the causal ray, sending the scale $\\lambda_n$ to unit scale; the starting point for the subsequent compactness/rigidity analysis (H1–H6).",
      "defining_relation": "v_n(y,s)=\\lambda_n^{-1}u(x_n+\\lambda_n^{-1}y,\\ t_n+\\lambda_n^{-2}s)",
      "notes": "與第 18–20 節作 DAG 節點用的 $v_n$ 共用記號；此處具體指重標度後的函數。"
    },
    {
      "id": "ns.c3.c3g.v_infty",
      "latex": "v_n\\to v_\\infty",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "重標度極限剖面",
      "label_en": "Rescaled limit profile",
      "definition_zh": "第 34 節 H1（Compactness class）中 $v_n$ 抽取子列後的極限；H2（Nontriviality）要求 first-crossing threshold 在極限中保留使 $v_\\infty\\not\\equiv0$，是 C3-H 待解的核心對象。",
      "definition_en": "The limit of a subsequence of $v_n$ under H1 (Compactness class, Section 34); H2 (Nontriviality) requires the first-crossing threshold to survive in the limit so that $v_\\infty\\not\\equiv0$ — the central open object for C3-H.",
      "defining_relation": "v_n\\to v_\\infty,\\qquad v_\\infty\\not\\equiv0"
    },
    {
      "id": "ns.c3.c3g.c3h_target",
      "latex": "\\textbf{C3-H — Ancestry Renormalization and Rigidity Interface}",
      "series": "NS",
      "first_appearance": "C3-G",
      "label_zh": "下一輪目標 C3-H",
      "label_en": "Next-round target C3-H",
      "definition_zh": "第 33 節本輪結尾正式命名的下一輪主題：對 causal ancestry ray 做臨界重標度，檢驗能否產生比既有 critical-element/profile-decomposition 更強、可被排除的 rigidity 結構；本輪僅定義目標與 proof obligations（H1–H6），未解決。",
      "definition_en": "The next round's theme, formally named at the end of this round (Section 33): apply critical rescaling to the causal ancestry ray and test whether it yields a rigidity structure stronger than, and excludable via, existing critical-element/profile-decomposition theory; this round only states the target and proof obligations (H1–H6), leaving it open.",
      "notes": "屬本輪向前指涉的命名，非本輪已解決之內容。"
    },
    {
      "id": "ns.c3.c3h.ancestry_node",
      "latex": "\\mathfrak a_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "血緣鏈節點",
      "label_en": "Ancestry ray node",
      "definition_zh": "第1節定義 $\\mathfrak a_n=(q_n,\\sigma_n,x_n,t_n)$，為血緣鏈上第 $n$ 個 first-crossing 事件的座標包。",
      "definition_en": "Sec. 1 defines $\\mathfrak a_n=(q_n,\\sigma_n,x_n,t_n)$ as the coordinate tuple of the n-th first-crossing event on the ancestry ray."
    },
    {
      "id": "ns.c3.c3h.lambda_n",
      "latex": "\\lambda_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "血緣節點 dyadic 尺度",
      "label_en": "Ancestry-node dyadic scale",
      "definition_zh": "$\\lambda_n=2^{q_n}\\to\\infty$ 是第 $n$ 個血緣節點的 dyadic 尺度，為 rescaling 的核心縮放因子。",
      "definition_en": "$\\lambda_n=2^{q_n}\\to\\infty$ is the dyadic scale of ancestry node n, the core scaling factor of the rescaling.",
      "defining_relation": "\\lambda_n=2^{q_n}"
    },
    {
      "id": "ns.c3.c3h.beta_star",
      "latex": "\\beta_\\ast",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "臨界 first-crossing 閾值",
      "label_en": "Critical first-crossing threshold",
      "definition_zh": "固定臨界閾值，由第1節 $a_{q_n}^{\\sigma_n}(t_n)=\\beta_\\ast$ 定義，本輪成為極限 profile $w_\\ast$ 的精確 $L^\\infty$ 錨定值。",
      "definition_en": "The fixed threshold defined by $a_{q_n}^{\\sigma_n}(t_n)=\\beta_\\ast$ in Sec. 1, which this round also fixes as the exact $L^\\infty$ anchor value of the limit profile $w_\\ast$.",
      "defining_relation": "a_{q_n}^{\\sigma_n}(t_n)=\\frac{\\|u_{q_n}^{\\sigma_n}(t_n)\\|_\\infty}{\\nu\\lambda_n}=\\beta_\\ast"
    },
    {
      "id": "ns.c3.c3h.a_q_sigma",
      "latex": "a_q^{\\sigma}(t)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "正規化 dyadic-helical 振幅比",
      "label_en": "Normalized dyadic-helical amplitude ratio",
      "definition_zh": "承接先前輪次的正規化振幅比 $a_q^\\sigma(t)=\\|u_q^\\sigma(t)\\|_\\infty/(\\nu2^q)$，第4節給出其 rescaling 恆等式。",
      "definition_en": "The normalized amplitude ratio $a_q^\\sigma(t)=\\|u_q^\\sigma(t)\\|_\\infty/(\\nu2^q)$ inherited from earlier rounds, with its rescaling identity given in Sec. 4."
    },
    {
      "id": "ns.c3.c3h.v_n",
      "latex": "v_n(y,s)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "血緣中心化黏性正規化速度場",
      "label_en": "Ancestry-centered viscosity-normalized rescaled velocity",
      "definition_zh": "第2節核心定義 $v_n(y,s)=\\frac1{\\nu\\lambda_n}u(x_n+y/\\lambda_n,t_n+s/(\\nu\\lambda_n^2))$，即以血緣節點為中心的 N–S critical rescaling。",
      "definition_en": "The central Sec. 2 definition $v_n(y,s)=\\frac1{\\nu\\lambda_n}u(x_n+y/\\lambda_n,t_n+s/(\\nu\\lambda_n^2))$, the ancestry-centered N–S critical rescaling.",
      "defining_relation": "v_n(y,s)=\\frac1{\\nu\\lambda_n}u\\left(x_n+\\frac{y}{\\lambda_n},t_n+\\frac{s}{\\nu\\lambda_n^2}\\right)",
      "notes": "第0節曾以「$v_0\\to v_1\\to\\cdots$」表示血緣鏈節點，與此處速度場 $v_n$ 是不同物件，源文本有此符號重疊。"
    },
    {
      "id": "ns.c3.c3h.pi_n",
      "latex": "\\pi_n(y,s)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "血緣中心化重整化壓力場",
      "label_en": "Ancestry-centered rescaled pressure",
      "definition_zh": "第2節定義 $\\pi_n(y,s)=\\frac1{\\nu^2\\lambda_n^2}p(x_n+y/\\lambda_n,t_n+s/(\\nu\\lambda_n^2))$，是與 $v_n$ 配對的重整化壓力場。",
      "definition_en": "Sec. 2 defines $\\pi_n(y,s)=\\frac1{\\nu^2\\lambda_n^2}p(x_n+y/\\lambda_n,t_n+s/(\\nu\\lambda_n^2))$, the rescaled pressure paired with $v_n$.",
      "defining_relation": "\\pi_n(y,s)=\\frac1{\\nu^2\\lambda_n^2}p\\left(x_n+\\frac{y}{\\lambda_n},t_n+\\frac{s}{\\nu\\lambda_n^2}\\right)"
    },
    {
      "id": "ns.c3.c3h.rescaled_lifespan",
      "latex": "-\\nu\\lambda_n^2t_n<s<\\nu\\lambda_n^2(T_\\ast-t_n)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "重整化生存時間區間",
      "label_en": "Rescaled lifespan interval",
      "definition_zh": "第3節：$v_n$ 的時間定義域，因 $t_n\\uparrow T_\\ast$ 使 backward lifespan 趨於 $(-\\infty,0]$。",
      "definition_en": "Sec. 3: the time domain of $v_n$, whose backward lifespan tends to $(-\\infty,0]$ as $t_n\\uparrow T_\\ast$."
    },
    {
      "id": "ns.c3.c3h.dyadic_scaling_identity",
      "latex": "\\Delta_jP^\\sigma v_n(y,s)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "Dyadic 尺度縮放恆等式",
      "label_en": "Dyadic scaling identity",
      "definition_zh": "第4節恆等式，把 rescaled 場的 dyadic-helical 分量 $\\Delta_jP^\\sigma v_n$ 精確對應回原場在 $q_n+j$ 層的分量。",
      "definition_en": "The Sec. 4 identity exactly relating the rescaled field's dyadic-helical piece $\\Delta_jP^\\sigma v_n$ back to the original field's piece at level $q_n+j$.",
      "defining_relation": "\\Delta_jP^\\sigma v_n(y,s)=\\frac1{\\nu\\lambda_n}\\left[\\Delta_{q_n+j}P^\\sigma u\\right]\\left(x_n+\\frac{y}{\\lambda_n},t_n+\\frac{s}{\\nu\\lambda_n^2}\\right)"
    },
    {
      "id": "ns.c3.c3h.sigma_star",
      "latex": "\\sigma_\\ast",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "固定 helicity 子序列符號",
      "label_en": "Fixed helicity subsequence sign",
      "definition_zh": "第6節：固定 helicity 子序列符號，取子序列使 $\\sigma_n=\\sigma_\\ast$ 對所有 $n$ 成立。",
      "definition_en": "Sec. 6: the fixed helicity sign of a subsequence with $\\sigma_n=\\sigma_\\ast$ for all n."
    },
    {
      "id": "ns.c3.c3h.w_star",
      "latex": "w_\\ast(y)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "Unit-shell 極限 profile",
      "label_en": "Unit-shell snapshot limit profile",
      "definition_zh": "定理9.1：$\\Delta_0P^{\\sigma_\\ast}v_n(0)\\to w_\\ast$ 於 $C^\\infty_{\\rm loc}$，滿足 $|w_\\ast(0)|\\ge\\frac12\\beta_\\ast$ 與 helical 關係 $\\nabla\\times w_\\ast=\\sigma_\\ast Dw_\\ast$。",
      "definition_en": "Thm 9.1: $\\Delta_0P^{\\sigma_\\ast}v_n(0)\\to w_\\ast$ in $C^\\infty_{\\rm loc}$, satisfying $|w_\\ast(0)|\\ge\\frac12\\beta_\\ast$ and the helical relation $\\nabla\\times w_\\ast=\\sigma_\\ast Dw_\\ast$.",
      "defining_relation": "\\Delta_0P^{\\sigma_\\ast}v_n(0)\\to w_\\ast,\\qquad \\nabla\\times w_\\ast=\\sigma_\\ast Dw_\\ast",
      "notes": "算子 $D$ 未在本輪重新定義，可能承襲自更早輪次（如 C3-A）的 helical/Beltrami 分解，信心中等。"
    },
    {
      "id": "ns.c3.c3h.local_mass_constants",
      "latex": "r_0,\\ c_0",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "局部臨界質量下界常數",
      "label_en": "Local critical-mass lower-bound constants",
      "definition_zh": "第10節：固定常數 $r_0>0,c_0>0$ 使 $\\|w_n\\|_{L^3(B_{r_0})}\\ge c_0\\beta_\\ast$，給出局部臨界質量下界。",
      "definition_en": "Sec. 10: fixed constants $r_0>0,c_0>0$ giving the local critical-mass lower bound $\\|w_n\\|_{L^3(B_{r_0})}\\ge c_0\\beta_\\ast$."
    },
    {
      "id": "ns.c3.c3h.w_n",
      "latex": "w_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "錨定 unit shell",
      "label_en": "Anchored unit shell",
      "definition_zh": "第21節定義 $w_n=\\Delta_0P^{\\sigma_\\ast}v_n$，即錨定 unit shell，滿足 $\\|w_n(0)\\|_\\infty=\\beta_\\ast$。",
      "definition_en": "Sec. 21 defines $w_n=\\Delta_0P^{\\sigma_\\ast}v_n$, the anchored unit shell satisfying $\\|w_n(0)\\|_\\infty=\\beta_\\ast$.",
      "defining_relation": "w_n=\\Delta_0P^{\\sigma_\\ast}v_n"
    },
    {
      "id": "ns.c3.c3h.vn_critical_norms",
      "latex": "\\|v_n(0)\\|_3,\\ \\|v_n(0)\\|_{\\dot H^{1/2}}",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "$v_n$ 的發散臨界範數",
      "label_en": "Diverging critical norms of $v_n$",
      "definition_zh": "定理13.1與第14節：$\\|v_n(0)\\|_3\\to\\infty$ 且 $\\|v_n(0)\\|_{\\dot H^{1/2}}\\to\\infty$，由 scaling 加 Seregin 必要 blow-up 定理導出。",
      "definition_en": "Thm 13.1 & Sec. 14: $\\|v_n(0)\\|_3\\to\\infty$ and $\\|v_n(0)\\|_{\\dot H^{1/2}}\\to\\infty$, derived from scaling combined with Seregin's necessary blow-up theorems.",
      "defining_relation": "\\|v_n(0)\\|_3=\\frac1\\nu\\|u(t_n)\\|_3\\to\\infty,\\qquad \\|v_n(0)\\|_{\\dot H^{1/2}}=\\frac1\\nu\\|u(t_n)\\|_{\\dot H^{1/2}}\\to\\infty"
    },
    {
      "id": "ns.c3.c3h.v_n_mu",
      "latex": "v_{n,\\mu}(y)=\\mu v_n(\\mu y)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "次級 N–S 尺度變換",
      "label_en": "Secondary N–S scaling map",
      "definition_zh": "第18節：次級尺度變換 $v_{n,\\mu}(y)=\\mu v_n(\\mu y)$，證明 $\\|v_{n,\\mu}\\|_3=\\|v_n\\|_3$ 恆成立。",
      "definition_en": "Sec. 18: the secondary scaling map $v_{n,\\mu}(y)=\\mu v_n(\\mu y)$, showing $\\|v_{n,\\mu}\\|_3=\\|v_n\\|_3$ always holds."
    },
    {
      "id": "ns.c3.c3h.z_n",
      "latex": "z_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "振幅正規化場（非法重整化嘗試）",
      "label_en": "Amplitude-normalized field (illegitimate attempt)",
      "definition_zh": "第19節：$z_n=v_n/\\|v_n(0)\\|_3$ 使 $\\|z_n(0)\\|_3=1$，但方程混入發散係數 $M_n$，故非法。",
      "definition_en": "Sec. 19: $z_n=v_n/\\|v_n(0)\\|_3$ gives $\\|z_n(0)\\|_3=1$, but its equation carries a diverging coefficient $M_n$, making it illegitimate."
    },
    {
      "id": "ns.c3.c3h.m_n",
      "latex": "M_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "發散振幅係數",
      "label_en": "Diverging amplitude coefficient",
      "definition_zh": "第19節：$M_n=\\|v_n(0)\\|_3\\to\\infty$，是 $z_n$ 方程非線性項前的發散係數。",
      "definition_en": "Sec. 19: $M_n=\\|v_n(0)\\|_3\\to\\infty$, the diverging coefficient multiplying the nonlinear term in the $z_n$ equation."
    },
    {
      "id": "ns.c3.c3h.r_n",
      "latex": "r_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "臨界背景缺陷",
      "label_en": "Critical background defect",
      "definition_zh": "第21節定義 $r_n=v_n-w_n$，是扣除錨定 shell 後承載發散臨界質量的背景缺陷。",
      "definition_en": "Sec. 21 defines $r_n=v_n-w_n$, the background defect carrying the diverging critical mass after subtracting the anchored shell.",
      "defining_relation": "r_n=v_n-w_n",
      "notes": "依方向分為 D-IR/D-UV/D-SP/D-CORE，對應分量 $r_n^{IR},r_n^{UV},r_n^{SP},r_n^{CORE}$。"
    },
    {
      "id": "ns.c3.c3h.d_ir",
      "latex": "D-IR",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "紅外缺陷方向",
      "label_en": "Infrared defect direction (D-IR)",
      "definition_zh": "第22節：紅外缺陷方向 $|\\xi|\\ll1$，對應 $r_n^{IR}$ 分量。",
      "definition_en": "Sec. 22: the infrared escape direction $|\\xi|\\ll1$, corresponding to component $r_n^{IR}$."
    },
    {
      "id": "ns.c3.c3h.d_uv",
      "latex": "D-UV",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "紫外多重性缺陷方向",
      "label_en": "Ultraviolet multiplicity defect direction (D-UV)",
      "definition_zh": "第22節：紫外缺陷方向 $|\\xi|\\gg1$，對應 $r_n^{UV}$ 分量。",
      "definition_en": "Sec. 22: the ultraviolet escape direction $|\\xi|\\gg1$, corresponding to component $r_n^{UV}$."
    },
    {
      "id": "ns.c3.c3h.d_sp",
      "latex": "D-SP",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "空間缺陷方向",
      "label_en": "Spatial defect direction (D-SP)",
      "definition_zh": "第22節：空間缺陷方向 $|y|\\to\\infty$，對應 $r_n^{SP}$ 分量。",
      "definition_en": "Sec. 22: the spatial escape direction $|y|\\to\\infty$, corresponding to component $r_n^{SP}$."
    },
    {
      "id": "ns.c3.c3h.d_core",
      "latex": "D-CORE",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "核心多尺度壅塞",
      "label_en": "Core multiscale congestion (D-CORE)",
      "definition_zh": "第22節：核心多尺度壅塞，指臨界質量在 anchored cone 內跨尺度累積而不逃離，對應 $r_n^{CORE}$。",
      "definition_en": "Sec. 22: core multiscale congestion, where critical mass accumulates across scales inside the anchored cone rather than escaping; corresponds to $r_n^{CORE}$."
    },
    {
      "id": "ns.c3.c3h.xrendefect",
      "latex": "\\operatorname{XRenDefect}_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "X-缺陷證書",
      "label_en": "X-Defect Certificate",
      "definition_zh": "第23節證書，打包 anchor 與四類方向性缺陷分量連同 provenance 供分析。",
      "definition_en": "The Sec. 23 certificate bundling the anchor with the four directional defect components and provenance for analysis.",
      "defining_relation": "\\operatorname{XRenDefect}_n=\\left\\langle w_n,r_n^{IR},r_n^{UV},r_n^{SP},r_n^{CORE},\\operatorname{Prov}_n\\right\\rangle"
    },
    {
      "id": "ns.c3.c3h.prov_n",
      "latex": "\\operatorname{Prov}_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "來源證明記錄",
      "label_en": "Provenance record",
      "definition_zh": "出現於 XRenDefect/XRenCert 中，記錄第 $n$ 個 rescaled 物件的來源軌跡，記號承襲自 X-Integration 框架。",
      "definition_en": "Appears in XRenDefect/XRenCert, recording the provenance trail of the n-th rescaled object; notation inherited from the X-Integration framework."
    },
    {
      "id": "ns.c3.c3h.ancestry_edge",
      "latex": "(p_n,\\sigma_n^p,x_n^p,t_n^p)\\to(q_n,\\sigma_n^c,x_n^c,t_n^c)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "因果血緣邊（親代-子代對）",
      "label_en": "Causal ancestry edge (parent-child pair)",
      "definition_zh": "第24節：親代-子代因果邊，是 $d_n,y_n^p,\\delta_n$ 等 rescaled 邊量的共同基礎。",
      "definition_en": "Sec. 24: the parent-child causal edge, the common basis for the rescaled edge quantities $d_n,y_n^p,\\delta_n$."
    },
    {
      "id": "ns.c3.c3h.d_n",
      "latex": "d_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "尺度偏移",
      "label_en": "Scale offset",
      "definition_zh": "第24節定義 $d_n=p_n-q_n\\in\\{-C_L,\\ldots,C_L\\}$，衡量親代與子代 dyadic 尺度的有限偏移。",
      "definition_en": "Sec. 24 defines $d_n=p_n-q_n\\in\\{-C_L,\\ldots,C_L\\}$, the finite dyadic-scale offset between parent and child."
    },
    {
      "id": "ns.c3.c3h.y_n_p",
      "latex": "y_n^p",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "重整化親代-子代空間位移",
      "label_en": "Rescaled parent-child spatial displacement",
      "definition_zh": "第25節定義 $y_n^p=\\lambda_n(x_n^p-x_n^c)$，滿足 $|y_n^p|\\le C$，是重整化後的親代-子代空間位移。",
      "definition_en": "Sec. 25 defines $y_n^p=\\lambda_n(x_n^p-x_n^c)$, satisfying $|y_n^p|\\le C$, the rescaled parent-child spatial displacement."
    },
    {
      "id": "ns.c3.c3h.delta_n",
      "latex": "\\delta_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "重整化時間間隔",
      "label_en": "Rescaled (normalized) time lag",
      "definition_zh": "第26節核心定義 $\\delta_n=\\nu\\lambda_n^2(t_n^c-t_n^p)$，滿足 $0<\\delta_n\\le\\theta$ 但缺乏一致正下界，是因果坍縮 No-Go 的關鍵量。",
      "definition_en": "The central Sec. 26 definition $\\delta_n=\\nu\\lambda_n^2(t_n^c-t_n^p)$, satisfying $0<\\delta_n\\le\\theta$ with no uniform positive lower bound, key to the causal-collapse No-Go.",
      "defining_relation": "\\delta_n=\\nu\\lambda_n^2(t_n^c-t_n^p)"
    },
    {
      "id": "ns.c3.c3h.theta",
      "latex": "\\theta",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "黏性視窗上界參數",
      "label_en": "Viscous-window bound parameter",
      "definition_zh": "第26節中作為 $\\delta_n$ 上界出現的黏性視窗常數，應承襲自更早輪次（如 C3-E）。",
      "definition_en": "The viscous-window constant bounding $\\delta_n$ in Sec. 26, likely inherited from an earlier round (e.g. C3-E)."
    },
    {
      "id": "ns.c3.c3h.g_timegap",
      "latex": "G_{\\rm time-gap}",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "時間間隔極限合法性防護",
      "label_en": "Time-gap limit-legality guard",
      "definition_zh": "第28節防護條件 $G_{\\rm time-gap}:\\delta_\\ast>0$，是 No-Go 27.1 的正式化，缺此則因果邊在極限中坍縮為同時發生。",
      "definition_en": "The Sec. 28 guard $G_{\\rm time-gap}:\\delta_\\ast>0$ formalizing No-Go 27.1; without it the causal edge collapses to simultaneity in the limit.",
      "defining_relation": "G_{\\rm time-gap}:\\ \\delta_\\ast>0"
    },
    {
      "id": "ns.c3.c3h.motif",
      "latex": "\\mathfrak m_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "重整化血緣母題",
      "label_en": "Renormalized ancestry motif",
      "definition_zh": "第30節定義 $\\mathfrak m_n=\\langle d_n,\\sigma_n^p,\\sigma_n^c,y_n^p,\\delta_n,\\eta_n,\\mathcal C_n\\rangle$，是 coherent 子序列下收斂的血緣邊 metadata 母題。",
      "definition_en": "Sec. 30 defines $\\mathfrak m_n=\\langle d_n,\\sigma_n^p,\\sigma_n^c,y_n^p,\\delta_n,\\eta_n,\\mathcal C_n\\rangle$, the ancestry-edge metadata motif convergent on a coherent subsequence.",
      "defining_relation": "\\mathfrak m_n=\\left\\langle d_n,\\sigma_n^p,\\sigma_n^c,y_n^p,\\delta_n,\\eta_n,\\mathcal C_n\\right\\rangle"
    },
    {
      "id": "ns.c3.c3h.eta_n",
      "latex": "\\eta_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "母題相干係數",
      "label_en": "Motif coherence parameter",
      "definition_zh": "第30節：母題 $\\mathfrak m_n$ 的相干係數 $\\eta_n\\in[0,1]$，coherent 子序列要求 $\\eta_n\\ge\\eta_0>0$。",
      "definition_en": "Sec. 30: the coherence parameter $\\eta_n\\in[0,1]$ within $\\mathfrak m_n$, required to satisfy $\\eta_n\\ge\\eta_0>0$ on a coherent subsequence."
    },
    {
      "id": "ns.c3.c3h.triad_class",
      "latex": "\\mathcal C_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "三元交互類別標籤",
      "label_en": "Triad interaction class label",
      "definition_zh": "第30節：$\\mathfrak m_n$ 中的有限三元交互類別標籤，描述該血緣邊涉及的 helical triad 類型。",
      "definition_en": "Sec. 30: the finite triad-interaction class label within $\\mathfrak m_n$, describing the helical triad type of that ancestry edge."
    },
    {
      "id": "ns.c3.c3h.v_infty",
      "latex": "v_\\infty",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "候選 ancient 解",
      "label_en": "Candidate ancient-solution limit",
      "definition_zh": "第32節：假設性 ancient solution 極限 $v_n\\to v_\\infty$，本輪僅建立 backward lifespan 條件而非 full-field compactness。",
      "definition_en": "Sec. 32: the hypothetical ancient-solution limit $v_n\\to v_\\infty$; this round establishes only the backward-lifespan condition, not full-field compactness."
    },
    {
      "id": "ns.c3.c3h.trichotomy",
      "latex": "\\text{Branch A / B / C}",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "重整化三分法",
      "label_en": "Renormalization trichotomy",
      "definition_zh": "第37節將 rescaling 結果分三支：A 全域緊性、B unit-shell 緊但 defect 發散、C metadata 收斂但 $\\delta_n\\to0$。",
      "definition_en": "Sec. 37 splits the rescaling outcome into three branches: A (global compactness), B (unit shell compact, defect diverges), C (metadata converges but $\\delta_n\\to0$)."
    },
    {
      "id": "ns.c3.c3h.xrencert",
      "latex": "\\operatorname{XRenCert}_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "X-整合極限證書",
      "label_en": "X-Integration limit certificate",
      "definition_zh": "第41節頂層證書，彙整 $v_n,w_n,r_n,\\mathfrak m_n,\\beta_\\ast$ 與 provenance，供極限稽核使用。",
      "definition_en": "The Sec. 41 top-level certificate collecting $v_n,w_n,r_n,\\mathfrak m_n,\\beta_\\ast$ and provenance for the limit audit.",
      "defining_relation": "\\operatorname{XRenCert}_n=\\left\\langle v_n,w_n,r_n,\\mathfrak m_n,\\beta_\\ast,\\operatorname{Prov}_n\\right\\rangle"
    },
    {
      "id": "ns.c3.c3h.audit_guards",
      "latex": "\\text{G-ANCHOR, G-DEFECT, G-ANCIENT, G-COMPACT, G-NONLINEAR, G-HEL}",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "XRenCert 稽核防護項",
      "label_en": "XRenCert audit guards",
      "definition_zh": "第41節列出的六項稽核，檢查 anchor、defect、ancient limit、compactness topology、nonlinear passage、helicity 是否於極限中保存。",
      "definition_en": "The six Sec. 41 checklist items verifying whether the anchor, defect, ancient limit, compactness topology, nonlinear passage, and helicity survive the limit."
    },
    {
      "id": "ns.c3.c3h.theta_hat_n",
      "latex": "\\widehat\\Theta_n",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "重整化 ETN 狀態",
      "label_en": "Renormalized ETN state",
      "definition_zh": "第42節定義 $\\widehat\\Theta_n=\\mathcal R_{\\lambda_n,x_n,t_n}\\Theta$，是 True ETN 狀態經 zoom operator 作用後的重整化血緣深度層。",
      "definition_en": "Sec. 42 defines $\\widehat\\Theta_n=\\mathcal R_{\\lambda_n,x_n,t_n}\\Theta$, the renormalized-ancestry-depth layer of the True ETN state under the zoom operator.",
      "defining_relation": "\\widehat\\Theta_n=\\mathcal R_{\\lambda_n,x_n,t_n}\\Theta"
    },
    {
      "id": "ns.c3.c3h.zoom_operator",
      "latex": "\\mathcal R_{\\lambda,x,t}",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "N–S 臨界縮放算子",
      "label_en": "N–S critical zoom operator",
      "definition_zh": "第42節：作用於 True ETN 狀態 $\\Theta$ 的抽象 N–S 臨界縮放算子。",
      "definition_en": "Sec. 42: the abstract N–S critical zoom operator acting on the True ETN state $\\Theta$."
    },
    {
      "id": "ns.c3.c3h.core_functional",
      "latex": "\\mathfrak C_{R,M}(v_n)",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "相空間核心泛函",
      "label_en": "Phase-space core functional",
      "definition_zh": "第44節（C3-I 義務 I1）：定義於 $B_R\\times\\{2^{-M}\\lesssim|\\xi|\\lesssim2^M\\}$ 上，衡量 $v_n$ 的 critical mass 以建立 anchor 下界。",
      "definition_en": "Sec. 44 (C3-I obligation I1): defined on $B_R\\times\\{2^{-M}\\lesssim|\\xi|\\lesssim2^M\\}$, measuring $v_n$'s critical mass to establish the anchor lower bound."
    },
    {
      "id": "ns.c3.c3h.two_threshold",
      "latex": "\\beta_0<\\beta_1",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "雙閾值 first-crossing 參數",
      "label_en": "Two-threshold first-crossing parameters",
      "definition_zh": "第44節（I6）：提議的雙閾值 first-crossing 參數，用於嘗試建立一致時間間隔下界。",
      "definition_en": "Sec. 44 (I6): the proposed two-threshold first-crossing parameters, aimed at establishing a uniform time-gap lower bound."
    },
    {
      "id": "ns.c3.c3h.delta_0",
      "latex": "\\delta_0",
      "series": "NS",
      "first_appearance": "C3-H",
      "label_zh": "目標一致時間間隔下界",
      "label_en": "Target uniform time-gap lower bound",
      "definition_zh": "第44節：$\\delta_n\\ge\\delta_0>0$ 中欲證得的目標一致時間間隔下界，目前為未解問題。",
      "definition_en": "Sec. 44: the target uniform time-gap lower bound $\\delta_0$ in $\\delta_n\\ge\\delta_0>0$, currently an open problem."
    },
    {
      "id": "ns.c3.c3i.critical_shell_amplitude",
      "latex": "a_q^\\sigma(t)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "臨界殼層振幅",
      "label_en": "critical shell amplitude",
      "definition_zh": "沿用 C3-G 的第 $q$ 殼、helicity $\\sigma$ 正規化振幅（第1節），定義為 $a_q^\\sigma(t)=\\|u_q^\\sigma(t)\\|_\\infty/(\\nu\\lambda_q)$，是本輪 frontier crossing 判準的基本量。",
      "definition_en": "The normalized shell-$q$, helicity-$\\sigma$ amplitude carried over from C3-G (Section 1), $a_q^\\sigma(t)=\\|u_q^\\sigma(t)\\|_\\infty/(\\nu\\lambda_q)$, underlying this round's frontier-crossing criterion.",
      "defining_relation": "a_q^\\sigma(t)=\\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}"
    },
    {
      "id": "ns.c3.c3i.beta_star",
      "latex": "\\beta_\\ast",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "固定臨界門檻",
      "label_en": "fixed critical threshold",
      "definition_zh": "第1節選定的固定門檻常數，滿足 $0<\\beta_\\ast<c_\\dagger$，用以定義 first frontier crossing 與 UV shellwise cap。",
      "definition_en": "The fixed threshold constant chosen in Section 1, satisfying $0<\\beta_\\ast<c_\\dagger$, defining the first frontier crossing and the UV shellwise cap.",
      "defining_relation": "0<\\beta_\\ast<c_\\dagger"
    },
    {
      "id": "ns.c3.c3i.c_dagger",
      "latex": "c_\\dagger",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "耗散尺度上界常數",
      "label_en": "dissipation-scale bound constant",
      "definition_zh": "第1節中由 dissipation-wavenumber unboundedness 保證，在 hypothetical blow-up 下會被任意高殼層超過的固定常數，界定 $\\beta_\\ast$ 的可選範圍上界。",
      "definition_en": "The fixed constant in Section 1, guaranteed by dissipation-wavenumber unboundedness to be exceeded by arbitrarily high shells under hypothetical blow-up, bounding $\\beta_\\ast$ from above."
    },
    {
      "id": "ns.c3.c3i.frontier_level_q",
      "latex": "Q",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "frontier 層級",
      "label_en": "frontier level",
      "definition_zh": "第2節引入的整數參數，標記 frequency frontier 位置，是本輪 first frontier crossing 分析與 rescaling 的中心索引。",
      "definition_en": "The integer parameter introduced in Section 2 marking the frequency-frontier position; the central index for this round's first-crossing analysis and rescaling."
    },
    {
      "id": "ns.c3.c3i.first_crossing_time_tq",
      "latex": "T_Q",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "第一次 frontier 跨越時刻",
      "label_en": "first frontier crossing time",
      "definition_zh": "第2節定義為某殼 $q\\ge Q$ 首次達到 $a_q^\\sigma\\ge\\beta_\\ast$ 的最早時刻；定理3.1證明 $T_Q\\uparrow T_\\ast$。",
      "definition_en": "Defined in Section 2 as the earliest time some shell $q\\ge Q$ first reaches $a_q^\\sigma\\ge\\beta_\\ast$; Theorem 3.1 proves $T_Q\\uparrow T_\\ast$.",
      "defining_relation": "T_Q=\\inf\\left\\{t\\in(0,T_\\ast):\\exists q\\ge Q,\\ \\sigma\\in\\{+,-\\},\\quad a_q^\\sigma(t)\\ge\\beta_\\ast\\right\\}"
    },
    {
      "id": "ns.c3.c3i.crossing_shell_qq_sigmaq",
      "latex": "(q_Q,\\sigma_Q)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "跨越殼層指標與符號",
      "label_en": "crossing shell index and sign",
      "definition_zh": "第4節由連續性論證取得，在 $T_Q$ 恰好達到門檻的殼層指標與 helicity 符號。",
      "definition_en": "The shell index and helicity sign obtained in Section 4 via continuity, at which the threshold is exactly attained at $T_Q$.",
      "defining_relation": "a_{q_Q}^{\\sigma_Q}(T_Q)=\\beta_\\ast"
    },
    {
      "id": "ns.c3.c3i.locality_constant_cl",
      "latex": "C_L",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "局部性常數",
      "label_en": "locality constant",
      "definition_zh": "第5節起沿用的固定常數，界定 crossing shell 與其 causal parent 間的殼層距離上界，如 $Q\\le q_Q\\le Q+C_L$。",
      "definition_en": "A fixed constant used from Section 5 onward, bounding the shell distance between a crossing shell and its causal parent, e.g. $Q\\le q_Q\\le Q+C_L$."
    },
    {
      "id": "ns.c3.c3i.parent_shell_pq_sigmap",
      "latex": "(p_Q,\\sigma_P)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "因果 parent 殼層與符號",
      "label_en": "causal parent shell and sign",
      "definition_zh": "第5節取得的 earlier parent 殼層指標與符號，滿足 $Q-C_L\\le p_Q<Q$ 且早於 $T_Q$ 已跨越門檻。",
      "definition_en": "The earlier parent shell index and sign obtained in Section 5, satisfying $Q-C_L\\le p_Q<Q$ and crossing the threshold before $T_Q$."
    },
    {
      "id": "ns.c3.c3i.parent_crossing_time_tau",
      "latex": "\\tau_{p_Q,\\sigma_P}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "parent 殼層跨越時刻",
      "label_en": "parent shell crossing time",
      "definition_zh": "第5節中 parent 殼本身跨越門檻的時刻，滿足 $\\tau_{p_Q,\\sigma_P}<T_Q$，建立 first frontier crossing 的因果來源。",
      "definition_en": "The time at which the parent shell itself crosses the threshold, satisfying $\\tau_{p_Q,\\sigma_P}<T_Q$, establishing the causal source of the first frontier crossing."
    },
    {
      "id": "ns.c3.c3i.rescaling_center_xq",
      "latex": "x_Q",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "rescaling 空間中心",
      "label_en": "rescaling spatial center",
      "definition_zh": "第6節選取、位於 child shell near-max 區域的空間中心點，作為 frontier-centered parabolic rescaling 的基準點。",
      "definition_en": "The spatial center chosen in Section 6, located in the child shell's near-max region, serving as the base point for the frontier-centered parabolic rescaling."
    },
    {
      "id": "ns.c3.c3i.rescaled_field_vq",
      "latex": "V_Q(y,s)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "frontier 中心化重新縮放場",
      "label_en": "frontier-centered rescaled field",
      "definition_zh": "第6節以 $x_Q,T_Q,\\lambda_Q$ 對 $u$ 作 parabolic rescaling 所得的場，從第7節起大量使用其 $s=0$ snapshot。",
      "definition_en": "The field obtained in Section 6 by parabolically rescaling $u$ using $x_Q,T_Q,\\lambda_Q$; its $s=0$ snapshot is used extensively from Section 7 onward.",
      "defining_relation": "V_Q(y,s)=\\frac1{\\nu\\lambda_Q}u\\left(x_Q+\\frac y{\\lambda_Q},T_Q+\\frac{s}{\\nu\\lambda_Q^2}\\right)",
      "notes": "全文中 \"$V_Q(0)$\" 指時刻切片 $V_Q(\\cdot,0)$，非空間點 $y=0$ 之取值。"
    },
    {
      "id": "ns.c3.c3i.relative_crossing_index_jq",
      "latex": "j_Q",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "相對 frontier 之跨越指標",
      "label_en": "frontier-relative crossing index",
      "definition_zh": "第8節定義 $j_Q=q_Q-Q$，即 crossing shell 相對 frontier $Q$ 的位移，滿足 $0\\le j_Q\\le C_L$。",
      "definition_en": "Defined in Section 8 as $j_Q=q_Q-Q$, the crossing shell's offset relative to the frontier $Q$, satisfying $0\\le j_Q\\le C_L$."
    },
    {
      "id": "ns.c3.c3i.one_sided_besov_cap",
      "latex": "\\dot B^{-1}_{\\infty,\\infty}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "單側臨界 Besov 型 cap",
      "label_en": "one-sided critical Besov-type cap",
      "definition_zh": "第9節指出定理8.1給出僅對 $j\\ge0$ 成立的 $\\dot B^{-1}_{\\infty,\\infty}$ 型上界，不可等同於含 $j<0$ 的完整 Besov norm 控制。",
      "definition_en": "Section 9 identifies Theorem 8.1's bound as a $\\dot B^{-1}_{\\infty,\\infty}$-type cap holding only for $j\\ge0$, not upgradable to control of the full norm including $j<0$.",
      "defining_relation": "\\sup_{j\\ge0,\\ \\sigma}2^{-j}\\|\\Delta_jP^\\sigma V_Q(0)\\|_\\infty\\le\\beta_\\ast"
    },
    {
      "id": "ns.c3.c3i.freq_cutoff_level_m",
      "latex": "M",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "高頻截止層級",
      "label_en": "high-frequency cutoff level",
      "definition_zh": "第12節固定的自然數，界定 mid/high-side finite frequency band $P_{[0,M]}$ 的上端。",
      "definition_en": "The fixed natural number in Section 12 bounding the upper end of the finite frequency band $P_{[0,M]}$."
    },
    {
      "id": "ns.c3.c3i.spatial_cutoff_radius_r",
      "latex": "R",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "空間截止半徑",
      "label_en": "spatial cutoff radius",
      "definition_zh": "第12節固定的正數，決定 spatial cutoff $\\chi_R$ 的支撐尺度。",
      "definition_en": "The fixed positive number in Section 12 determining the support scale of the spatial cutoff $\\chi_R$."
    },
    {
      "id": "ns.c3.c3i.freq_band_projector_p0m",
      "latex": "P_{[0,M]}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "有限頻帶投影",
      "label_en": "finite frequency-band projector",
      "definition_zh": "第12節定義 $P_{[0,M]}=\\sum_{j=0}^M\\Delta_j$；第14節與 $P_{<0},P_{>M}$ 構成頻率三分解 $I=P_{<0}+P_{[0,M]}+P_{>M}$。",
      "definition_en": "Defined in Section 12 as $P_{[0,M]}=\\sum_{j=0}^M\\Delta_j$; with $P_{<0},P_{>M}$ it forms the Section 14 frequency trichotomy $I=P_{<0}+P_{[0,M]}+P_{>M}$.",
      "defining_relation": "P_{[0,M]}=\\sum_{j=0}^{M}\\Delta_j"
    },
    {
      "id": "ns.c3.c3i.spatial_cutoff_chir",
      "latex": "\\chi_R",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "空間光滑截斷",
      "label_en": "smooth spatial cutoff",
      "definition_zh": "第12節定義、在 $B_{2R}$ 內支撐且在 $B_R$ 上為1的光滑函數，用以分離 spatial core 與 far-space。",
      "definition_en": "The smooth cutoff function from Section 12, supported in $B_{2R}$ and equal to $1$ on $B_R$, separating the spatial core from far-space."
    },
    {
      "id": "ns.c3.c3i.finite_core_cqrm",
      "latex": "C_{Q;R,M}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "高側有限相空間核",
      "label_en": "finite high-side phase-space core",
      "definition_zh": "第12節定義的 $\\chi_RP_{[0,M]}V_Q$；定理13.1證明其 $L^3$ norm 對所有 $Q$ 一致地被 $C(R,M)\\beta_\\ast$ 界住。",
      "definition_en": "Defined in Section 12 as $\\chi_RP_{[0,M]}V_Q$; Theorem 13.1 shows its $L^3$ norm is uniformly bounded by $C(R,M)\\beta_\\ast$ over all $Q$.",
      "defining_relation": "C_{Q;R,M}=\\chi_RP_{[0,M]}V_Q"
    },
    {
      "id": "ns.c3.c3i.complementary_projectors_ir_uv",
      "latex": "P_{<0},\\ P_{>M}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "IR／UV 互補頻率投影",
      "label_en": "complementary IR/UV frequency projectors",
      "definition_zh": "第14節頻率三分解中，$P_{<0}$ 對應 $q<Q$ 的相對低頻投影，$P_{>M}$ 對應 $j>M$ 的高頻投影。",
      "definition_en": "In Section 14's frequency trichotomy, $P_{<0}$ projects onto the relative-IR range $q<Q$ and $P_{>M}$ onto the UV range $j>M$."
    },
    {
      "id": "ns.c3.c3i.vq_ir_component",
      "latex": "V_Q^{IR}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "IR 分量（D-IR）",
      "label_en": "IR component (D-IR)",
      "definition_zh": "第14、16節定義 $V_Q^{IR}=P_{<0}V_Q$，即 relative infrared reservoir，是 first frontier child 因果祖先所在側。",
      "definition_en": "Defined in Sections 14 and 16 as $V_Q^{IR}=P_{<0}V_Q$, the relative infrared reservoir carrying the first frontier child's causal ancestry.",
      "defining_relation": "V_Q^{IR}=P_{<0}V_Q"
    },
    {
      "id": "ns.c3.c3i.vq_uv_component",
      "latex": "V_Q^{UV}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "UV 分量（D-UV）",
      "label_en": "UV component (D-UV)",
      "definition_zh": "第14、16節定義 $V_Q^{UV}=P_{>M}V_Q$，其發散只能源於 shell 或 spatial multiplicity，而非單殼振幅爆炸。",
      "definition_en": "Defined in Sections 14 and 16 as $V_Q^{UV}=P_{>M}V_Q$; its divergence can only arise from shell or spatial multiplicity, not single-shell amplitude blow-up.",
      "defining_relation": "V_Q^{UV}=P_{>M}V_Q"
    },
    {
      "id": "ns.c3.c3i.vq_sp_component",
      "latex": "V_Q^{SP}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "空間逃逸分量（D-SP）",
      "label_en": "spatial escape component (D-SP)",
      "definition_zh": "第14、16節定義 $V_Q^{SP}=(1-\\chi_R)P_{[0,M]}V_Q$，即有限頻窗內遠離 ancestry center 的 spatial multiplicity/escape defect。",
      "definition_en": "Defined in Sections 14 and 16 as $V_Q^{SP}=(1-\\chi_R)P_{[0,M]}V_Q$, the spatial multiplicity/escape defect within the finite frequency window, away from the ancestry center.",
      "defining_relation": "V_Q^{SP}=(1-\\chi_R)P_{[0,M]}V_Q"
    },
    {
      "id": "ns.c3.c3i.d_core_defect",
      "latex": "\\text{D-CORE}_{[0,M],R}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "核心壅塞缺陷",
      "label_en": "core-congestion defect",
      "definition_zh": "第17節命名、對應有限相空間核 $C_{Q;R,M}$ 的 defect 類型，已被定理13.1真正排除為 global $L^3$ divergence 來源。",
      "definition_en": "The defect type named in Section 17, corresponding to the finite phase-space core $C_{Q;R,M}$, ruled out by Theorem 13.1 as a source of global $L^3$ divergence."
    },
    {
      "id": "ns.c3.c3i.nonlocal_remainder_remq",
      "latex": "\\operatorname{Rem}_Q",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "非局部餘項",
      "label_en": "nonlocal remainder",
      "definition_zh": "第19節在 eventual local-source dominance 假設下，child window 的 nonlocal remainder，滿足 $\\operatorname{Rem}_Q\\le\\varepsilon\\beta_\\ast$。",
      "definition_en": "Under the eventual local-source dominance hypothesis in Section 19, the nonlocal remainder of the child window, satisfying $\\operatorname{Rem}_Q\\le\\varepsilon\\beta_\\ast$.",
      "defining_relation": "\\operatorname{Rem}_Q\\le\\varepsilon\\beta_\\ast"
    },
    {
      "id": "ns.c3.c3i.relative_parent_index_jparent",
      "latex": "j_{\\rm parent}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "相對 parent 頻率指標",
      "label_en": "relative parent frequency index",
      "definition_zh": "第19節將 child first crossing 的 immediate causal source 限制在 $-C_L\\le j_{\\rm parent}\\le C_L$，經 frontier minimality 進一步壓到 $-C_L\\le j_{\\rm parent}<0$。",
      "definition_en": "Section 19 confines the immediate causal source of the child's first crossing to $-C_L\\le j_{\\rm parent}\\le C_L$, tightened by frontier minimality to $-C_L\\le j_{\\rm parent}<0$.",
      "defining_relation": "-C_L\\le j_{\\rm parent}<0"
    },
    {
      "id": "ns.c3.c3i.phase_efficiency_etaq",
      "latex": "\\eta_q",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "相位/局部性效率",
      "label_en": "phase/locality efficiency",
      "definition_zh": "第20節引入的 local production phase efficiency 下界量，假設 $\\eta_q\\ge\\eta_0>0$，支持 D-SP 不主導 immediate source 的論證。",
      "definition_en": "The local-production phase-efficiency quantity from Section 20, assumed to satisfy $\\eta_q\\ge\\eta_0>0$, supporting the argument that D-SP cannot dominate the immediate source."
    },
    {
      "id": "ns.c3.c3i.coherent_source_radius_rstar",
      "latex": "R_\\ast",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "一致相干來源半徑",
      "label_en": "coherent-source radius",
      "definition_zh": "第20、21節選定的固定半徑，使固定比例的 nonlinear source 來自 $O(\\lambda_q^{-1})$ 鄰域內，即 $|y-y_Q|\\le R_\\ast$。",
      "definition_en": "The fixed radius from Sections 20-21 ensuring a fixed fraction of the nonlinear source lies within an $O(\\lambda_q^{-1})$ neighborhood, i.e. $|y-y_Q|\\le R_\\ast$."
    },
    {
      "id": "ns.c3.c3i.parent_original_coords",
      "latex": "q_{\\rm parent},\\ x_{\\rm parent}",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "原座標下的 parent 位置",
      "label_en": "parent location in original coordinates",
      "definition_zh": "定理21.1將 rescaled parent 換回原 coordinates 所得的頻率指標與空間位置，量化 one-generation decoupling 的尺度。",
      "definition_en": "The parent's frequency index and spatial location in original coordinates from Theorem 21.1, quantifying the scale of the one-generation decoupling.",
      "defining_relation": "q_{\\rm parent}\\in[Q-C_L,Q-1],\\quad|x_{\\rm parent}-x_Q|\\lesssim\\lambda_Q^{-1}"
    },
    {
      "id": "ns.c3.c3i.generation_core_cn",
      "latex": "\\mathcal C_n",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "第 n 代 frontier 核",
      "label_en": "generation-$n$ frontier core",
      "definition_zh": "第33節定義的每代 moving phase-space core，由 $|j|\\le C_L$、$|y-y_n|\\le R_\\ast$ 與 $I_n$ 界定，是 Defect Re-entry Ledger 的分析單元。",
      "definition_en": "The per-generation moving phase-space core defined in Section 33, delimited by $|j|\\le C_L$, $|y-y_n|\\le R_\\ast$ and $I_n$; the analytical unit of the Defect Re-entry Ledger.",
      "defining_relation": "\\mathcal C_n=\\left\\{|j|\\le C_L,\\quad|y-y_n|\\le R_\\ast,\\quad I_n\\right\\}",
      "notes": "$I_n$ 在源文中未獨立展開定義，僅作為 $\\mathcal C_n$ 定義式的第三個限制條件出現。"
    },
    {
      "id": "ns.c3.c3i.entry_functional_entryn",
      "latex": "\\operatorname{Entry}_n(D)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "第 n 代缺陷進入量",
      "label_en": "generation-$n$ defect entry contribution",
      "definition_zh": "第33節定義 defect 分量 $D_n$ 對 $\\mathcal C_n$ 的 ancestry-relevant nonlinear source contribution；級數 $\\sum_n\\operatorname{Entry}_n(D)$ 收斂與否是下一輪 C3-J 的核心問題。",
      "definition_en": "Defined in Section 33 as a defect component $D_n$'s ancestry-relevant nonlinear source contribution to $\\mathcal C_n$; whether $\\sum_n\\operatorname{Entry}_n(D)$ converges is the central question posed for the next round, C3-J."
    },
    {
      "id": "ns.c3.c3i.spatial_multiplicity_fn",
      "latex": "f_N(x)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "空間多重性玩具構造",
      "label_en": "spatial-multiplicity toy construction",
      "definition_zh": "第26節由 $N$ 個遠距平移波包疊加而成，$f_N=\\sum_{m=1}^N\\phi(x-x_m)$，證明 bounded shell $L^\\infty$ 不蘊含 bounded global shell $L^3$（$\\|f_N\\|_3\\sim N^{1/3}\\|\\phi\\|_3$）。",
      "definition_en": "The Section 26 sum of $N$ widely-separated translated wave packets, $f_N=\\sum_{m=1}^N\\phi(x-x_m)$, showing bounded shell $L^\\infty$ does not imply bounded global shell $L^3$ (since $\\|f_N\\|_3\\sim N^{1/3}\\|\\phi\\|_3$).",
      "defining_relation": "f_N(x)=\\sum_{m=1}^{N}\\phi(x-x_m)"
    },
    {
      "id": "ns.c3.c3i.uv_packet_family_phij",
      "latex": "\\phi_j(x)",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "UV 多尺度臨界波包族",
      "label_en": "UV multiscale critical packet family",
      "definition_zh": "第28節定義的 divergence-free critical packet family $\\phi_j(x)=2^j\\phi(2^j(x-x_j))$，用於構造 UV multiscale scalar no-go model。",
      "definition_en": "The divergence-free critical packet family defined in Section 28 as $\\phi_j(x)=2^j\\phi(2^j(x-x_j))$, used to build the UV multiscale scalar no-go model.",
      "defining_relation": "\\phi_j(x)=2^j\\phi(2^j(x-x_j))"
    },
    {
      "id": "ns.c3.c3i.uv_multiscale_sum_fm",
      "latex": "F_M",
      "series": "NS",
      "first_appearance": "C3-I",
      "label_zh": "多尺度反例疊加場",
      "label_en": "multiscale counter-example sum",
      "definition_zh": "第28節定義 $F_M=\\beta\\sum_{j=0}^M\\phi_j$，在 disjoint-packet bookkeeping 下顯示 $\\|F_M\\|_3\\sim\\beta M^{1/3}$ 發散，即 finite energy 加 uniform shell cap 仍不足控制 global $L^3$。",
      "definition_en": "Defined in Section 28 as $F_M=\\beta\\sum_{j=0}^M\\phi_j$; under ideal disjoint-packet bookkeeping $\\|F_M\\|_3\\sim\\beta M^{1/3}$ diverges, showing finite energy plus a uniform shell cap still cannot control global $L^3$.",
      "defining_relation": "F_M=\\beta\\sum_{j=0}^{M}\\phi_j"
    },
    {
      "id": "ns.c3.c3j.high_pass_profile",
      "latex": "h",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "高通輪廓函數",
      "label_en": "high-pass profile function",
      "definition_zh": "第2節引入的固定光滑函數 $h\\in C^\\infty([0,\\infty))$，滿足 $0\\le h\\le1$、$h(r)=0$（$r\\le1$）、$h(r)=1$（$r\\ge2$）且 $h'\\ge0$，用以構造移動頻率前緣的 Fourier 乘子 $A_\\Lambda$。",
      "definition_en": "The fixed smooth cutoff profile $h\\in C^\\infty([0,\\infty))$ introduced in Sec. 2, with $0\\le h\\le1$, $h(r)=0$ for $r\\le1$, $h(r)=1$ for $r\\ge2$, $h'\\ge0$; used to build the moving-frontier Fourier multiplier $A_\\Lambda$.",
      "defining_relation": "h(r)=0\\ (r\\le1),\\quad h(r)=1\\ (r\\ge2),\\quad h'\\ge0",
      "notes": "第5節取 $m(r)=h(r)^2$ 作為 $M_\\Lambda$ 的 symbol，用於證明 gauge sweep 的符號。"
    },
    {
      "id": "ns.c3.c3j.moving_frontier",
      "latex": "\\Lambda(t)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動頻率前緣",
      "label_en": "moving frequency frontier",
      "definition_zh": "第2節引入的隨時間變化正值頻率前緣 $\\Lambda(t)>0$，取代先前各輪的固定整數 frontier $Q$，是本輪頻譜 gauge sweep 分析的核心自由度。",
      "definition_en": "The time-dependent positive frequency frontier $\\Lambda(t)>0$ introduced in Sec. 2, replacing the fixed integer frontier $Q$ of earlier rounds; the core moving degree of freedom behind this round's spectral gauge-sweep analysis.",
      "notes": "與 C3-G/H/I 使用的離散整數 frontier $Q$ 相關但不同：$\\Lambda(t)$ 連續可微，使 $\\dot\\Lambda(t)$ 有意義。"
    },
    {
      "id": "ns.c3.c3j.moving_multiplier",
      "latex": "A_{\\Lambda(t)}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動頻率 Fourier 乘子",
      "label_en": "moving-frontier Fourier multiplier",
      "definition_zh": "第2節定義的 self-adjoint Fourier multiplier，symbol 為 $a_\\Lambda(\\xi)=h(|\\xi|/\\Lambda(t))$，把 $h$ 依 $\\Lambda(t)$ 縮放到頻率空間，構成移動 high-pass filter。",
      "definition_en": "The self-adjoint Fourier multiplier defined in Sec. 2 with symbol $a_\\Lambda(\\xi)=h(|\\xi|/\\Lambda(t))$, rescaling $h$ in frequency space by the moving frontier $\\Lambda(t)$ to act as a moving high-pass filter.",
      "defining_relation": "a_\\Lambda(\\xi)=h\\left(\\frac{|\\xi|}{\\Lambda(t)}\\right)",
      "notes": "常簡寫 $A_\\Lambda$；因 $\\Lambda(t)$ 隨 $t$ 變化，$A_\\Lambda$ 不與 $\\partial_t$ 交換，這正是 gauge sweep $\\mathcal G_\\Lambda$ 的來源。"
    },
    {
      "id": "ns.c3.c3j.squared_multiplier",
      "latex": "M_\\Lambda",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "平方頻率乘子",
      "label_en": "squared frequency multiplier",
      "definition_zh": "第2節定義 $M_\\Lambda=A_\\Lambda^2$，symbol 為 $m(|\\xi|/\\Lambda)$（$m(r)=h(r)^2$，第5節），用於構造二次型能量 $E_\\Lambda=\\frac12\\langle u,M_\\Lambda u\\rangle$。",
      "definition_en": "Defined in Sec. 2 as $M_\\Lambda=A_\\Lambda^2$, symbol $m(|\\xi|/\\Lambda)$ with $m(r)=h(r)^2$ (Sec. 5); used to form the quadratic energy $E_\\Lambda=\\frac12\\langle u,M_\\Lambda u\\rangle$.",
      "defining_relation": "M_\\Lambda=A_\\Lambda^2",
      "notes": "第5節證明 $\\dot\\Lambda\\ge0\\Rightarrow\\partial_tm(|\\xi|/\\Lambda)\\le0$（因 $m'\\ge0$），此即 $\\mathcal G_\\Lambda\\le0$ 的關鍵引理。"
    },
    {
      "id": "ns.c3.c3j.moving_spectral_energy",
      "latex": "E_\\Lambda(t)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動頻譜能量",
      "label_en": "moving spectral energy",
      "definition_zh": "第3節定義 $E_\\Lambda(t)=\\frac12\\|A_{\\Lambda(t)}u(t)\\|_2^2=\\frac12\\langle u,M_\\Lambda u\\rangle$，即被移動 high-pass filter 篩出的能量，是定理4.1（C3-J.1）平衡律的主角。",
      "definition_en": "Defined in Sec. 3 as $E_\\Lambda(t)=\\frac12\\|A_{\\Lambda(t)}u(t)\\|_2^2=\\frac12\\langle u,M_\\Lambda u\\rangle$, the energy captured by the moving high-pass filter; the central quantity of Theorem 4.1 (C3-J.1).",
      "defining_relation": "E_\\Lambda(t)=\\frac12\\|A_{\\Lambda(t)}u(t)\\|_2^2=\\frac12\\langle u,M_\\Lambda u\\rangle",
      "notes": "當 $\\Lambda$ 固定時退化為第25節的 gauge-corrected spectral balance。"
    },
    {
      "id": "ns.c3.c3j.spectral_gauge_sweep",
      "latex": "\\mathcal G_\\Lambda",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "頻譜 gauge sweep",
      "label_en": "spectral gauge sweep",
      "definition_zh": "定理4.1（C3-J.1）中 $\\dot E_\\Lambda$ 分解出的第一項，$\\mathcal G_\\Lambda=\\frac12\\langle u,\\dot M_\\Lambda u\\rangle$，代表純因前緣移動造成的能量重新標記而非真實非線性轉移；第5節證明 $\\dot\\Lambda\\ge0\\Rightarrow\\mathcal G_\\Lambda\\le0$。",
      "definition_en": "The first term in the Theorem 4.1 decomposition of $\\dot E_\\Lambda$: $\\mathcal G_\\Lambda=\\frac12\\langle u,\\dot M_\\Lambda u\\rangle$, pure relabeling from the moving frontier, not genuine nonlinear transfer; Sec. 5 shows $\\dot\\Lambda\\ge0\\Rightarrow\\mathcal G_\\Lambda\\le0$.",
      "defining_relation": "\\mathcal G_\\Lambda=\\frac12\\langle u,\\dot M_\\Lambda u\\rangle",
      "notes": "Guard code G-SWEEP-F（第32節）保存此項；時間積分記為 $G_\\Lambda=\\int\\mathcal G_\\Lambda$（第29節，見 spectral ledger）。"
    },
    {
      "id": "ns.c3.c3j.spectral_nonlinear_transfer",
      "latex": "\\Phi_\\Lambda",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "頻譜非線性轉移通量",
      "label_en": "genuine spectral nonlinear transfer",
      "definition_zh": "定理4.1分解的第二項，$\\Phi_\\Lambda=-\\langle B(u,u),M_\\Lambda u\\rangle$，代表真正因非線性項進入 filtered high side 的頻譜轉移，扣除 $\\mathcal G_\\Lambda$ 後才是 genuine dynamics。",
      "definition_en": "The second term in the Theorem 4.1 decomposition: $\\Phi_\\Lambda=-\\langle B(u,u),M_\\Lambda u\\rangle$, the genuine nonlinear spectral transfer into the filtered high side, distinct from the gauge sweep $\\mathcal G_\\Lambda$.",
      "defining_relation": "\\Phi_\\Lambda=-\\langle B(u,u),M_\\Lambda u\\rangle",
      "notes": "積分版 $F_\\Lambda=\\int\\Phi_\\Lambda$（第29節）；其正變差 $\\operatorname{Var}_+(\\Phi_\\Lambda)$（第26節）是 C3-J.5 no-go 的主角。"
    },
    {
      "id": "ns.c3.c3j.abs_freq",
      "latex": "\\operatorname{AbsFreq}=q",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "絕對頻率殼層標記",
      "label_en": "absolute-shell frequency label",
      "definition_zh": "第7節引入的 X-Integration 保存量，記錄 Fourier content 所在的絕對（非相對）殼層 $q$；第6節以通用符號 $r$ 表示同一概念（如定理15.1的固定殼層）。與 $\\operatorname{RelFreq}$ 不得互相取代。",
      "definition_en": "The X-Integration bookkeeping quantity from Sec. 7, recording the absolute (not frontier-relative) shell index $q$; the generic variable $r$ (Sec. 6, Thm 15.1) denotes the same notion. Must not be conflated with $\\operatorname{RelFreq}$.",
      "notes": "對應 guard code G-ABS（第32節）。"
    },
    {
      "id": "ns.c3.c3j.rel_freq",
      "latex": "\\operatorname{RelFreq}=q-Q",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "相對頻率殼層指標",
      "label_en": "relative-shell (frontier-relative) index",
      "definition_zh": "第7節定義 $\\operatorname{RelFreq}=q-Q$（第6節記作 $j_Q=r-Q$，或簡寫 $j=q-Q$），是殼層相對於移動前緣 $Q$ 的位置；No-Go 6.1 證明此標記改變（如 UV→IR）不蘊含真正的向下頻譜轉移。",
      "definition_en": "Defined in Sec. 7 as $\\operatorname{RelFreq}=q-Q$ (written $j_Q=r-Q$ in Sec. 6, or $j=q-Q$), the shell's position relative to moving frontier $Q$; No-Go 6.1 shows a change in this label does not imply genuine downscale spectral transfer.",
      "defining_relation": "\\operatorname{RelFreq}=q-Q",
      "notes": "對應 guard code G-REL（第32節）。與 AbsFreq 的區分是本輪 no-go 論證核心。"
    },
    {
      "id": "ns.c3.c3j.moving_spatial_core",
      "latex": "\\chi(t,x)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動空間核心截止函數",
      "label_en": "moving spatial-core cutoff",
      "definition_zh": "第8節定義 $\\chi(t,x)=\\chi_0\\left(\\frac{x-X(t)}{R(t)}\\right)$（$0\\le\\chi_0\\le1$），以移動中心 $X(t)$ 與可能收縮的半徑 $R(t)$ 描述局部核心，$z=(x-X(t))/R(t)$（第11節）；是 $\\Lambda(t)$ 的空間對應物。",
      "definition_en": "Defined in Sec. 8 as $\\chi(t,x)=\\chi_0\\left(\\frac{x-X(t)}{R(t)}\\right)$ ($0\\le\\chi_0\\le1$), with moving center $X(t)$ and shrinking/expanding radius $R(t)$, $z=(x-X(t))/R(t)$ (Sec. 11); the spatial counterpart of $\\Lambda(t)$.",
      "defining_relation": "\\chi(t,x)=\\chi_0\\left(\\frac{x-X(t)}{R(t)}\\right)",
      "notes": "第11節給出 $\\partial_t\\chi=-\\left[\\dot X(t)+\\frac{\\dot R(t)}{R(t)}(x-X(t))\\right]\\cdot\\nabla\\chi$，是 $\\mathcal G_\\chi$ 的來源。"
    },
    {
      "id": "ns.c3.c3j.local_kinetic_energy",
      "latex": "E_\\chi(t)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "局部核心動能",
      "label_en": "local core-weighted kinetic energy",
      "definition_zh": "第8節定義 $E_\\chi(t)=\\int\\chi(t,x)\\frac{|u(x,t)|^2}{2}\\,dx$，是定理10.1（C3-J.2）空間版能量平衡律的主角，對應頻譜版的 $E_\\Lambda$。",
      "definition_en": "Defined in Sec. 8 as $E_\\chi(t)=\\int\\chi(t,x)\\frac{|u(x,t)|^2}{2}\\,dx$, the central quantity of the spatial-core balance Theorem 10.1 (C3-J.2), the spatial analogue of $E_\\Lambda$.",
      "defining_relation": "E_\\chi(t)=\\int\\chi(t,x)\\frac{|u(x,t)|^2}{2}\\,dx"
    },
    {
      "id": "ns.c3.c3j.spatial_gauge_sweep",
      "latex": "\\mathcal G_\\chi",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "空間 gauge sweep",
      "label_en": "spatial-core gauge sweep",
      "definition_zh": "定理10.1（C3-J.2）中 $\\dot E_\\chi$ 分解的第一項，$\\mathcal G_\\chi=\\int\\frac{|u|^2}{2}\\partial_t\\chi\\,dx$，代表純因核心中心/半徑移動造成的能量重新標記（第11-12節），非真實 packet transport。",
      "definition_en": "The first term in the Theorem 10.1 decomposition of $\\dot E_\\chi$: $\\mathcal G_\\chi=\\int\\frac{|u|^2}{2}\\partial_t\\chi\\,dx$, pure relabeling from the moving center/radius of the core (Sec. 11-12), not genuine packet transport.",
      "defining_relation": "\\mathcal G_\\chi=\\int\\frac{|u|^2}{2}\\partial_t\\chi\\,dx",
      "notes": "對應 guard G-SWEEP-X；積分版記為 $G_\\chi=\\int\\mathcal G_\\chi$（第30節）。No-Go 12.1 的核心。"
    },
    {
      "id": "ns.c3.c3j.advective_boundary_flux",
      "latex": "\\Phi_\\chi^{\\rm adv}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "對流／壓力邊界通量",
      "label_en": "advective/pressure boundary flux",
      "definition_zh": "定理10.1分解的第二項，$\\Phi_\\chi^{\\rm adv}=\\int\\left(\\frac{|u|^2}{2}+p\\right)u\\cdot\\nabla\\chi\\,dx$，是真正流體/壓力通過局部邊界的通量。",
      "definition_en": "The second term of Theorem 10.1: $\\Phi_\\chi^{\\rm adv}=\\int\\left(\\frac{|u|^2}{2}+p\\right)u\\cdot\\nabla\\chi\\,dx$, the genuine advective/pressure flux across the localized boundary.",
      "defining_relation": "\\Phi_\\chi^{\\rm adv}=\\int\\left(\\frac{|u|^2}{2}+p\\right)u\\cdot\\nabla\\chi\\,dx",
      "notes": "積分版 $F_\\chi^{adv}$（第30節）。"
    },
    {
      "id": "ns.c3.c3j.diffusive_boundary_flux",
      "latex": "\\Phi_\\chi^{\\rm diff}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "黏性邊界通量",
      "label_en": "viscous diffusive boundary flux",
      "definition_zh": "定理10.1分解的第三項，$\\Phi_\\chi^{\\rm diff}=\\nu\\int\\frac{|u|^2}{2}\\Delta\\chi\\,dx$，是黏性擴散穿越局部化邊界的貢獻。",
      "definition_en": "The third term of Theorem 10.1: $\\Phi_\\chi^{\\rm diff}=\\nu\\int\\frac{|u|^2}{2}\\Delta\\chi\\,dx$, the viscous-diffusion contribution across the localized boundary.",
      "defining_relation": "\\Phi_\\chi^{\\rm diff}=\\nu\\int\\frac{|u|^2}{2}\\Delta\\chi\\,dx",
      "notes": "積分版 $F_\\chi^{diff}$（第30節）。"
    },
    {
      "id": "ns.c3.c3j.reentry_certificate",
      "latex": "\\operatorname{ReEntryCert}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "真正 re-entry 證書",
      "label_en": "genuine re-entry certificate",
      "definition_zh": "第13節定義六元組 $\\operatorname{ReEntryCert}=\\langle$絕對來源身分, moving gauge, 真實邊界通量, 非線性頻譜通量, 黏性擴散, commutators$\\rangle$，第31節精煉為加法分解 $\\operatorname{Entry}=\\operatorname{GaugeSweep}+\\operatorname{PhysicalFlux}+\\operatorname{SpectralTransfer}+\\operatorname{Diffusion}+\\operatorname{Commutator}$；只有扣除 GaugeSweep 後的部分才算 genuine re-entry。",
      "definition_en": "The 6-tuple from Sec. 13, $\\operatorname{ReEntryCert}=\\langle$absolute source identity, moving gauge, true boundary flux, nonlinear spectral flux, viscous diffusion, commutators$\\rangle$; refined in Sec. 31 into $\\operatorname{Entry}=\\operatorname{GaugeSweep}+\\operatorname{PhysicalFlux}+\\operatorname{SpectralTransfer}+\\operatorname{Diffusion}+\\operatorname{Commutator}$, with only the non-gauge remainder counting as genuine re-entry.",
      "notes": "本輪最核心的方法論定義，統合 $\\mathcal G_\\Lambda,\\Phi_\\Lambda,\\mathcal G_\\chi,\\Phi_\\chi^{adv},\\Phi_\\chi^{diff},[\\chi,A_\\Lambda]$ 等所有先前定義的項。"
    },
    {
      "id": "ns.c3.c3j.phase_space_commutator",
      "latex": "[\\chi,A_\\Lambda]",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "相空間局部化交換子",
      "label_en": "phase-space localization commutator",
      "definition_zh": "第14節指出同時做空間截止 $\\chi$ 與頻率截止 $A_\\Lambda$ 時 $\\chi A_\\Lambda\\ne A_\\Lambda\\chi$，其交換子一般不可忽略（尤其在 $R\\sim\\Lambda^{-1}$ 的 ancestry core 中為 order-one）；X-Integration 以 $G_{\\rm COMM}$ 保存其來源項。",
      "definition_en": "Sec. 14 notes that simultaneous spatial cutoff $\\chi$ and frequency cutoff $A_\\Lambda$ give $\\chi A_\\Lambda\\ne A_\\Lambda\\chi$; the commutator is generally non-negligible (order-one when $R\\sim\\Lambda^{-1}$), and X-Integration records its source as $G_{\\rm COMM}$.",
      "defining_relation": "\\chi A_\\Lambda\\ne A_\\Lambda\\chi",
      "notes": "$G_{\\rm COMM}$ 等同 guard code G-COMM（第32節），無獨立顯式公式。"
    },
    {
      "id": "ns.c3.c3j.direct_reuse_bound",
      "latex": "C_L",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "絕對殼層直接複用上界",
      "label_en": "absolute-shell direct-reuse bound",
      "definition_zh": "常數 $C_L$ 源自 C3-G 的 parent window 條件 $Q-C_L\\le p<Q$；本輪定理15.1（第15節）新證明固定絕對殼層 $r$ 在所有整數前緣 $Q$ 中最多只能作 $C_L$ 個 frontier levels 的 direct local parent。",
      "definition_en": "The constant $C_L$ originates from C3-G's parent-window condition $Q-C_L\\le p<Q$; this round's new Theorem 15.1 (Sec. 15) proves a fixed absolute shell $r$ can be direct local parent for at most $C_L$ frontier levels among all integer $Q$.",
      "defining_relation": "r\\in[Q-C_L,\\,Q-1]\\iff Q\\in[r+1,\\,r+C_L]",
      "notes": "$C_L$ 符號承自 C3-G；C3-J 的新貢獻是此複用次數上界定理本身（gauge-invariant no-double-counting statement，第16節）。"
    },
    {
      "id": "ns.c3.c3j.shell_helicity_component",
      "latex": "u_q^\\sigma",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "絕對殼層螺旋分量",
      "label_en": "shell helicity-sign component",
      "definition_zh": "第17節設 $u_q^\\sigma=\\Delta_qP^\\sigma u$，滿足 $\\partial_tu_q^\\sigma=\\nu\\Delta u_q^\\sigma-\\Delta_qP^\\sigma\\mathbb P\\nabla\\cdot(u\\otimes u)$，是第18-24節固定殼層時間正則性分析的對象。",
      "definition_en": "Sec. 17 sets $u_q^\\sigma=\\Delta_qP^\\sigma u$, obeying $\\partial_tu_q^\\sigma=\\nu\\Delta u_q^\\sigma-\\Delta_qP^\\sigma\\mathbb P\\nabla\\cdot(u\\otimes u)$; the object of the fixed-shell time-regularity analysis in Sec. 18-24.",
      "notes": "定理18.1證明 $\\|\\partial_tu_q^\\sigma(t)\\|_\\infty\\le C[\\nu\\lambda_q^{7/2}E_0^{1/2}+\\lambda_q^4E_0]$，是 $a_q^\\sigma$ Lipschitz 界的基礎。$\\Delta_q,P^\\sigma$ 為系列既有算子。"
    },
    {
      "id": "ns.c3.c3j.normalized_shell_amplitude",
      "latex": "a_q^\\sigma(t)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "正規化殼層振幅",
      "label_en": "normalized shell amplitude",
      "definition_zh": "第19節精確定義 $a_q^\\sigma(t)=\\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}$（第0節回顧 C3-I 時已用同記號但未給出此精確正規化），是 hysteresis 理論（定理21.1）的判準量。",
      "definition_en": "Sec. 19 gives the precise definition $a_q^\\sigma(t)=\\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}$ (used informally without this exact normalization when recalling C3-I in Sec. 0); the threshold quantity driving the hysteresis theorem (Thm 21.1).",
      "defining_relation": "a_q^\\sigma(t)=\\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}",
      "notes": "第0節的 $T_Q=\\inf\\{t:\\exists q\\ge Q,\\sigma,\\ a_q^\\sigma(t)\\ge\\beta_*\\}$ 承自 C3-I，用同一記號 $a_q^\\sigma$ 但單一門檻 $\\beta_*$，不同於本輪雙門檻 $\\beta_0,\\beta_1$。"
    },
    {
      "id": "ns.c3.c3j.lipschitz_constant",
      "latex": "L_q",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "殼層 Lipschitz 常數",
      "label_en": "shell Lipschitz constant",
      "definition_zh": "第19節證明 $|a_q^\\sigma(t)-a_q^\\sigma(s)|\\le L_q|t-s|$，其中 $L_q\\le C\\left[\\lambda_q^{5/2}E_0^{1/2}+\\frac{\\lambda_q^3}{\\nu}E_0\\right]$，是定理21.1上界與第23節正規化時間間隙下界的關鍵輸入。",
      "definition_en": "Sec. 19 proves $|a_q^\\sigma(t)-a_q^\\sigma(s)|\\le L_q|t-s|$ with $L_q\\le C\\left[\\lambda_q^{5/2}E_0^{1/2}+\\frac{\\lambda_q^3}{\\nu}E_0\\right]$; the key input to the Theorem 21.1 upcrossing bound and Sec. 23's time-gap lower bound.",
      "defining_relation": "L_q\\le C\\left[\\lambda_q^{5/2}E_0^{1/2}+\\frac{\\lambda_q^3}{\\nu}E_0\\right]"
    },
    {
      "id": "ns.c3.c3j.hysteresis_thresholds",
      "latex": "\\beta_0,\\beta_1",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "雙門檻 hysteresis 參數",
      "label_en": "two-threshold hysteresis pair",
      "definition_zh": "第20節取 $0<\\beta_0<\\beta_1$，定義一次 complete upcrossing 為不相交區間 $[s_m,t_m]$ 滿足 $a_q^\\sigma(s_m)\\le\\beta_0$、$a_q^\\sigma(t_m)\\ge\\beta_1$，排除門檻附近 infinitesimal jitter 被誤記為 re-entry。",
      "definition_en": "Sec. 20 fixes $0<\\beta_0<\\beta_1$, defining a complete upcrossing as a disjoint interval $[s_m,t_m]$ with $a_q^\\sigma(s_m)\\le\\beta_0$ and $a_q^\\sigma(t_m)\\ge\\beta_1$, excluding infinitesimal threshold jitter from counting as re-entry.",
      "notes": "對應 guard G-HYST（第32節）。與第0節回顧 C3-I 的單一門檻 $\\beta_*$ 不同，是本輪新引入的雙門檻機制。"
    },
    {
      "id": "ns.c3.c3j.upcrossing_count",
      "latex": "N_q^{\\rm up}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "完整 upcrossing 次數",
      "label_en": "complete-upcrossing count",
      "definition_zh": "定理21.1（C3-J.3）證明固定 $(q,\\sigma)$ 在 $[0,T_*)$ 內的 complete upcrossing 次數 $N_q^{\\rm up}\\le1+\\frac{L_qT_*}{\\beta_1-\\beta_0}<\\infty$，即同一絕對殼層的 separated hysteretic reactivation 次數有限。",
      "definition_en": "Theorem 21.1 (C3-J.3) proves the complete-upcrossing count for fixed $(q,\\sigma)$ on $[0,T_*)$ satisfies $N_q^{\\rm up}\\le1+\\frac{L_qT_*}{\\beta_1-\\beta_0}<\\infty$: finitely many separated hysteretic reactivations per fixed absolute shell.",
      "defining_relation": "N_q^{\\rm up}\\le1+\\frac{L_qT_*}{\\beta_1-\\beta_0}",
      "notes": "第39節 K2 提議研究加權和 $\\sum_qw_qN_q^{up}$ 是否有 scale-critical 上界，留待 C3-K；本輪認為 ordinary energy 權重仍太弱。"
    },
    {
      "id": "ns.c3.c3j.normalized_time_gap",
      "latex": "\\delta_q",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "黏性正規化時間間隙",
      "label_en": "viscous-normalized hysteresis time gap",
      "definition_zh": "第23節定義 $\\delta_q=\\nu\\lambda_q^2\\Delta t_q$（$\\Delta t_q\\ge\\frac{\\beta_1-\\beta_0}{L_q}$ 為每次 upcrossing 最短物理時間），只得到弱下界 $\\delta_q\\gtrsim C\\lambda_q^{-1}$，故 $\\delta_q\\to0$ 未被排除；C3-J.4 證明僅用 energy inequality 無法修復此崩塌。",
      "definition_en": "Sec. 23 defines $\\delta_q=\\nu\\lambda_q^2\\Delta t_q$ (with $\\Delta t_q\\ge\\frac{\\beta_1-\\beta_0}{L_q}$ the minimum time per upcrossing), obtaining only $\\delta_q\\gtrsim C\\lambda_q^{-1}$, so $\\delta_q\\to0$ is not excluded; C3-J.4 shows energy inequality alone cannot repair this.",
      "defining_relation": "\\delta_q=\\nu\\lambda_q^2\\Delta t_q",
      "notes": "呼應 C3-H 的 $\\delta_n\\to0$ causal-limit collapse（同類量，符號隨輪次改變）。K7（第39節）將其重述為待證 $\\nu\\lambda_q^2\\Delta t_q\\ge\\delta_0>0$。"
    },
    {
      "id": "ns.c3.c3j.positive_flux_variation",
      "latex": "\\operatorname{Var}_+(\\Phi_\\Lambda)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "正頻譜通量變差",
      "label_en": "positive spectral flux variation",
      "definition_zh": "第26節定義 $\\operatorname{Var}_+(\\Phi_\\Lambda)=\\int_0^T[\\Phi_\\Lambda(t)]_+\\,dt$，計數真正「進入 high side」的累積量；C3-J.5（命題27.1）證明 signed energy identity 只控制 $\\int\\Phi_\\Lambda$，不控制此正變差或 $\\int|\\Phi_\\Lambda|$。",
      "definition_en": "Sec. 26 defines $\\operatorname{Var}_+(\\Phi_\\Lambda)=\\int_0^T[\\Phi_\\Lambda(t)]_+\\,dt$, counting cumulative genuine entry into the high side; C3-J.5 (Prop. 27.1) proves the signed energy identity controls only $\\int\\Phi_\\Lambda$, not this positive variation or $\\int|\\Phi_\\Lambda|$.",
      "defining_relation": "\\operatorname{Var}_+(\\Phi_\\Lambda)=\\int_0^T[\\Phi_\\Lambda(t)]_+\\,dt",
      "notes": "Guard G-NET/VAR（第32節）要求區分 signed net flux 與此正變差。K3（第39節）提議逐殼層版本 $\\int[\\Phi_q]_+dt$。"
    },
    {
      "id": "ns.c3.c3j.sign_changing_counterexample",
      "latex": "\\Phi_N(t)=N\\sin(Nt)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "符號反轉反例函數",
      "label_en": "sign-changing counterexample function",
      "definition_zh": "命題27.1（C3-J.5）證明中的具體構造：$\\Phi_N(t)=N\\sin(Nt)$，其 signed integral 可保持 $O(1)$（甚至沿特定週期為零），但 $\\int[\\Phi_N]_+\\,dt\\sim cN$，展示 signed balance identity 本身不控制 total positive flux variation。",
      "definition_en": "The explicit construction in the proof of Prop. 27.1 (C3-J.5): $\\Phi_N(t)=N\\sin(Nt)$, whose signed integral stays $O(1)$ (even vanishing along special periods) while $\\int[\\Phi_N]_+\\,dt\\sim cN$, showing a signed balance identity alone cannot control total positive flux variation.",
      "defining_relation": "\\Phi_N(t)=N\\sin(Nt),\\quad\\int[\\Phi_N]_+\\,dt\\sim cN",
      "notes": "純代數反例（algebraic counter-ledger），非直接來自 N–S 方程本身。"
    },
    {
      "id": "ns.c3.c3j.spectral_ledger",
      "latex": "\\Delta E_\\Lambda+D_\\Lambda=G_\\Lambda+F_\\Lambda",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動頻譜 re-entry 帳本",
      "label_en": "moving spectral re-entry ledger",
      "definition_zh": "第29節給出的時間積分版平衡式，其中 $G_\\Lambda=\\int\\mathcal G_\\Lambda$、$F_\\Lambda=\\int\\Phi_\\Lambda$、$D_\\Lambda=\\nu\\int_0^T\\|\\nabla A_\\Lambda u\\|_2^2dt$；說明觀察到的 $\\Delta E_\\Lambda$ 不能直接稱為 spectral transfer，須先扣除 $G_\\Lambda$。",
      "definition_en": "The time-integrated ledger identity from Sec. 29, with $G_\\Lambda=\\int\\mathcal G_\\Lambda$, $F_\\Lambda=\\int\\Phi_\\Lambda$, dissipation $D_\\Lambda=\\nu\\int_0^T\\|\\nabla A_\\Lambda u\\|_2^2dt$; an observed $\\Delta E_\\Lambda$ cannot be called spectral transfer until $G_\\Lambda$ is subtracted.",
      "notes": "大寫 $G_\\Lambda,F_\\Lambda$（時間積分）與第4節逐點 $\\mathcal G_\\Lambda,\\Phi_\\Lambda$ 需區分。$\\Lambda$ 固定時 $G_\\Lambda=0$，退化為第25節的 gauge-corrected balance。"
    },
    {
      "id": "ns.c3.c3j.spatial_ledger",
      "latex": "\\Delta E_\\chi+D_\\chi=G_\\chi+F_\\chi^{adv}+F_\\chi^{diff}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動空間 re-entry 帳本",
      "label_en": "moving spatial re-entry ledger",
      "definition_zh": "第30節的空間版帳本，$G_\\chi=\\int\\mathcal G_\\chi$、$F_\\chi^{adv}=\\int\\Phi_\\chi^{adv}$、$F_\\chi^{diff}=\\int\\Phi_\\chi^{diff}$；說明 $\\Delta E_\\chi>0$ 不是 genuine packet inflow 的證明，可能純由邊界掃過造成。",
      "definition_en": "The spatial analogue ledger from Sec. 30, with $G_\\chi=\\int\\mathcal G_\\chi$, $F_\\chi^{adv}=\\int\\Phi_\\chi^{adv}$, $F_\\chi^{diff}=\\int\\Phi_\\chi^{diff}$; $\\Delta E_\\chi>0$ is not by itself a certificate of genuine packet inflow, since it may come purely from boundary sweep.",
      "notes": "對應頻譜版的 spectral ledger 條目。"
    },
    {
      "id": "ns.c3.c3j.ancestry_core",
      "latex": "\\mathcal C_n",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "移動相空間 ancestry core",
      "label_en": "moving phase-space ancestry core",
      "definition_zh": "第31節提到「真正的 moving ancestry core：$\\mathcal C_n$」，同時具有 center、spatial radius、frequency frontier、time window 四個可變自由度；合法 re-entry certificate 須針對此物件分離 GaugeSweep 與其餘四類貢獻。",
      "definition_en": "Sec. 31 refers to the true moving ancestry core $\\mathcal C_n$, simultaneously varying in center, spatial radius, frequency frontier, and time window; a legal re-entry certificate must separate GaugeSweep from the other four contributions for this object.",
      "notes": "記號可能承自 C3-F（Phase-Space Ancestry Cone）；本檔案未重新給出精確構造式，與 C3-F cone 的確切關係有不確定性。"
    },
    {
      "id": "ns.c3.c3j.entry_type_taxonomy",
      "latex": "\\text{Type }0,\\ldots,\\text{Type }5",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "Defect re-entry 六型分類",
      "label_en": "six-way defect re-entry taxonomy",
      "definition_zh": "第33節提出新分類：Type 0 gauge pseudo-entry（只有 GaugeSweep≠0）、Type 1 direct local entry、Type 2 spectral transport entry、Type 3 spatial transport entry、Type 4 diffusive entry、Type 5 mixed phase-space entry；取代第7節較粗的 R-GAUGE/R-DYN 二分法。",
      "definition_en": "Sec. 33 proposes: Type 0 gauge pseudo-entry (only GaugeSweep≠0), Type 1 direct local entry, Type 2 spectral transport entry, Type 3 spatial transport entry, Type 4 diffusive entry, Type 5 mixed phase-space entry; refining the coarser R-GAUGE/R-DYN split of Sec. 7.",
      "notes": "R-GAUGE（$q$ fixed, $Q$ moved）大致對應 Type 0；R-DYN 對應 Type 1-5 之並集。"
    },
    {
      "id": "ns.c3.c3j.transport_congestion_branches",
      "latex": "\\text{Branch A},\\ \\text{Branch B}",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "傳輸 vs 擁塞二分支",
      "label_en": "transport-vs-congestion dichotomy",
      "definition_zh": "第37節提出 Branch A（genuine repeated transport：扣除 gauge 後 positive flux variation 持續很大）與 Branch B（frontier sweeps through pre-existing structure：re-entry 主要是既有 multiscale field 被移動 gauge 重新標記）；直接導出下一輪 C3-K 的 K5/K6 proof obligations。",
      "definition_en": "Sec. 37 introduces Branch A (genuine repeated transport: gauge-corrected positive flux variation stays persistently large) versus Branch B (frontier sweeps through pre-existing structure: re-entry is mostly relabeling by the moving gauge); this directly motivates the K5/K6 obligations of the next round, C3-K.",
      "notes": "本輪收尾、開啟 C3-K 的核心概念分野（\"transport problem\" vs \"occupancy problem\"，第37節末）。"
    },
    {
      "id": "ns.c3.c3j.occupancy_functional",
      "latex": "\\mathfrak O(q,t)",
      "series": "NS",
      "first_appearance": "C3-J",
      "label_zh": "絕對殼層佔用泛函",
      "label_en": "absolute-shell occupancy functional",
      "definition_zh": "第39節 K1 提議定義的泛函 $\\mathfrak O(q,t)$，記錄絕對殼層 $q$ 的 critical amplitude、spatial packet multiplicity、helicity、phase efficiency，避免 relative-gauge 混淆；本輪僅提出構想，未給出精確公式，留待 C3-K 建構。",
      "definition_en": "The functional $\\mathfrak O(q,t)$ proposed in K1 (Sec. 39), meant to record an absolute shell $q$'s critical amplitude, spatial packet multiplicity, helicity, and phase efficiency, avoiding relative-gauge confusion; only proposed here, with construction left to C3-K.",
      "notes": "屬 forward-looking proof obligation，非本輪已證定義；與 Branch B 的 \"growing absolute multiscale occupancy\"（K5）呼應。"
    },
    {
      "id": "ns.c3.c3k.fixed_threshold",
      "latex": "\\beta",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "固定臨界閾值",
      "label_en": "Fixed critical threshold",
      "definition_zh": "本輪捨棄 C3-J 的 moving-frontier 判準，改用固定常數 β>0（§1）作為每個 absolute shell 的臨界啟動閾值，用以定義 absolute active set A_{q,σ}(β)。",
      "definition_en": "A fixed constant β>0 (§1) replacing C3-J's moving-frontier criterion as the critical activation threshold for each absolute shell; used to define the absolute active set A_{q,σ}(β).",
      "notes": "與 C3-J 的 moving frontier Q(t) 對照，是本輪 gauge-invariance 的關鍵設計。"
    },
    {
      "id": "ns.c3.c3k.absolute_shell_amplitude",
      "latex": "a_q^\\sigma(t)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "絕對臨界殼層振幅",
      "label_en": "Absolute critical shell amplitude",
      "definition_zh": "第 q 個 dyadic shell、helicity σ 分量的臨界振幅，以 L^∞ 範數對 νλ_q 正規化（§1）；是 gauge-invariant 的 absolute 版本，取代 C3-J 中依賴 moving frontier 的振幅。",
      "definition_en": "The critical amplitude of the σ-helicity component at dyadic shell q, normalized by νλ_q (§1); the gauge-invariant 'absolute' version replacing the moving-frontier-dependent amplitude used in C3-J.",
      "defining_relation": "a_q^\\sigma(t)=\\frac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}",
      "notes": "λ_q=2^q 為標準 dyadic 尺度；σ∈{+,-} 為 helicity sign，兩者皆屬本系列既有記號。"
    },
    {
      "id": "ns.c3.c3k.helical_shell_velocity",
      "latex": "u_q^\\sigma",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "螺旋殼層速度分量",
      "label_en": "Helical shell-projected velocity",
      "definition_zh": "速度場先做 helicity 投影 P^σ、再做 Littlewood–Paley 第 q 殼層投影 Δ_q 後的分量，定義為 u_q^σ=Δ_qP^σu（§1），是 a_q^σ 的構成要素。",
      "definition_en": "The velocity field after helicity projection P^σ followed by Littlewood–Paley shell projection Δ_q, defined as u_q^σ=Δ_qP^σu (§1); the building block of a_q^σ.",
      "notes": "區別於 §16 A_q^{loc} 所用未做 helicity 分解的一般 shell 分量 u_m。"
    },
    {
      "id": "ns.c3.c3k.absolute_active_set",
      "latex": "A_{q,\\sigma}(\\beta)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "絕對活躍集合",
      "label_en": "Absolute active set",
      "definition_zh": "殼層 (q,σ) 上振幅 a_q^σ(t) 超過固定閾值 β 的時間集合（§1）；此定義完全不依賴 C3-J 的 moving frontier Q(t)，因此是 gauge-invariant 的核心對象。",
      "definition_en": "The set of times at which the shell (q,σ) amplitude a_q^σ(t) exceeds the fixed threshold β (§1); independent of C3-J's moving frontier Q(t), making it the round's core gauge-invariant object.",
      "defining_relation": "A_{q,\\sigma}(\\beta)=\\left\\{t\\in(0,T_\\ast):a_q^\\sigma(t)\\ge\\beta\\right\\}",
      "notes": "是本輪 occupancy measure μ_β（§5）與 occupation moments M_1(β)、M_2(β)（§27）的基礎；hypothetical blow-up 要求存在 β_*>0 使任意高頻殼層滿足 A_{q,σ}(β_*)≠∅（§8）。"
    },
    {
      "id": "ns.c3.c3k.occupancy_measure",
      "latex": "\\mu_\\beta",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "占用測度",
      "label_en": "Occupancy measure",
      "definition_zh": "在 shell-time 空間上定義的 weighted measure，以 λ_q 加權絕對活躍集合（§5）；定理證明其總質量 μ_β(ℤ×{+,-}×(0,T_*)) 有限，可視為 absolute shell activation tension measure（§9）。",
      "definition_en": "A weighted measure on shell-time space, weighting the absolute active set by λ_q (§5); its total mass μ_β(ℤ×{+,-}×(0,T_*)) is proven finite, and can be read as an absolute shell activation tension measure (§9).",
      "defining_relation": "d\\mu_\\beta(q,\\sigma,t)=\\lambda_q1_{A_{q,\\sigma}(\\beta)}(t)\\,dt",
      "notes": "blow-up 要求 supp μ_β 在 frequency 方向不有界但 ‖μ_β‖<∞（§9），即 'finite mass escaping to infinite frequency'。§35 中 μ_q 為其 q-marginal，供 M_s(μ) 使用。"
    },
    {
      "id": "ns.c3.c3k.high_freq_active_count",
      "latex": "N_{\\ge Q}(t;\\beta)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "高頻活躍殼層計數",
      "label_en": "High-frequency active-shell count",
      "definition_zh": "時刻 t 時，頻率不低於 Q 且處於活躍狀態的 (q,σ) 殼層個數（§6）；推論 6.1 證明 ∫₀^{T*}N_{≥Q}(t;β)dt ≤ CE_0/(ν³β²λ_Q)，即以 O(λ_Q^{-1}) 衰減。",
      "definition_en": "The number of shells (q,σ) with frequency at least Q that are active at time t (§6); Corollary 6.1 shows ∫₀^{T*}N_{≥Q}(t;β)dt ≤ CE_0/(ν³β²λ_Q), i.e. O(λ_Q^{-1}) decay.",
      "defining_relation": "N_{\\ge Q}(t;\\beta)=\\#\\left\\{(q,\\sigma):q\\ge Q,\\ a_q^\\sigma(t)\\ge\\beta\\right\\}",
      "notes": "此處 Q 僅為 dyadic 頻率索引閾值，與 §0 中 C3-J moving-frontier 記號 Q(t) 無關，需注意區分。"
    },
    {
      "id": "ns.c3.c3k.hysteresis_thresholds",
      "latex": "\\beta_0,\\ \\beta_1,\\ \\beta_m",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "雙閾值遲滯門檻",
      "label_en": "Two-threshold hysteresis levels",
      "definition_zh": "承襲 C3-J 的下閾值 β_0、上閾值 β_1（0<β_0<β_1）與中點 β_m=(β_0+β_1)/2（§10），用以定義 separated complete upcrossing 事件，為 C3-K.2 加權遲滯計數定理之輸入。",
      "definition_en": "The lower threshold β_0, upper threshold β_1 (0<β_0<β_1), and midpoint β_m=(β_0+β_1)/2 carried over from C3-J (§10), used to define separated complete upcrossing events; inputs to the C3-K.2 weighted hysteretic count theorem.",
      "notes": "源自 C3-J 的 two-threshold hysteresis 架構；本輪新意在於將其與 absolute shell (q,σ) 及 λ_q 權重結合。"
    },
    {
      "id": "ns.c3.c3k.upcrossing_count",
      "latex": "N_{q,\\sigma}^{up}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "分離完整上穿次數",
      "label_en": "Separated complete upcrossing count",
      "definition_zh": "殼層 (q,σ) 上振幅 a_q^σ 從 β_0 完整上穿至 β_1 的分離事件次數（§10）；C3-J 僅有逐殼層有限性 N_q^{up}<∞，本輪細分至 helicity σ 並代入全域加權求和（C3-K.2）。",
      "definition_en": "The number of separated events in which a_q^σ completely up-crosses from β_0 to β_1 on shell (q,σ) (§10); C3-J only established per-shell finiteness N_q^{up}<∞, while this round refines it by helicity σ and feeds it into a globally weighted sum (C3-K.2).",
      "notes": "下界（§11）：|A_{q,σ}(β_m)| ≥ N_{q,σ}^{up}·(β_1-β_0)/(2L_q)，因每次上穿至少需要 Δt≥(β_1-β_0)/(2L_q) 的活躍時間。"
    },
    {
      "id": "ns.c3.c3k.lipschitz_const",
      "latex": "L_q",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "殼層 Lipschitz 常數",
      "label_en": "Fixed-shell Lipschitz constant",
      "definition_zh": "滿足 |a_q^σ(t)-a_q^σ(s)|≤L_q|t-s| 的殼層 Lipschitz 常數，源自 C3-J（§10）；本輪關鍵新用法是比值 λ_q/L_q 作為遲滯計數的權重（§12-13），高頻漸近有 λ_q/L_q≳cλ_q^{-2}。",
      "definition_en": "The Lipschitz constant satisfying |a_q^σ(t)-a_q^σ(s)|≤L_q|t-s|, originating in C3-J (§10); this round's key new use is the ratio λ_q/L_q as a weight for hysteretic counts (§12-13), with high-frequency asymptotic λ_q/L_q≳cλ_q^{-2}.",
      "notes": "§13 明確警告：精確定理應保留 λ_q/L_q，λ_q^{-2} 只是 high-frequency asymptotic interpretation，非嚴格等式。"
    },
    {
      "id": "ns.c3.c3k.weighted_hysteretic_count",
      "latex": "\\sum_{q,\\sigma}\\frac{\\lambda_q}{L_q}N_{q,\\sigma}^{up}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "加權遲滯活化計數（定理 C3-K.2）",
      "label_en": "Weighted hysteretic activation count (Thm C3-K.2)",
      "definition_zh": "定理 C3-K.2（§12）：將所有 absolute shells 的 upcrossing 次數以 λ_q/L_q 加權後全域求和，其和被初始能量 E_0 控制；比 C3-J 逐殼層有限性更強，因為是單一 global 界，但即使如此仍不排除 one-new-shell-per-scale 的無限 genealogy（§14 no-go）。",
      "definition_en": "Theorem C3-K.2 (§12): the λ_q/L_q-weighted sum of upcrossing counts over all absolute shells is globally bounded by the initial energy E_0; strictly stronger than C3-J's per-shell finiteness since it is one global bound, yet still does not rule out an infinite one-new-shell-per-scale genealogy (§14 no-go).",
      "defining_relation": "\\sum_{q,\\sigma}\\frac{\\lambda_q}{L_q}N_{q,\\sigma}^{up}\\le\\frac{CE_0}{\\nu^3\\beta_m^2(\\beta_1-\\beta_0)}",
      "notes": "稱作 gauge-invariant 版本的 Zeno no-go：即使每個 q 只取 N_{q,σ}^{up}=1，∑λ_q^{-2}<∞ 仍容許無限多尺度各發生一次上穿。"
    },
    {
      "id": "ns.c3.c3k.local_energy_transfer",
      "latex": "T_q^{loc}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "局域非線性能量轉移",
      "label_en": "Local nonlinear energy transfer",
      "definition_zh": "限制在頻率鄰域 |p-q|,|r-q|≤C_L 內三重交互作用的 bounded-ratio 局域能量轉移（§15），可再依 helicity 細分；滿足上界 |T_q^{loc}(t)|≤CνA_q^{loc}(t)λ_q^2E_q^*(t)（§16）。",
      "definition_en": "The bounded-ratio local nonlinear energy transfer restricted to triadic interactions within frequency neighborhood |p-q|,|r-q|≤C_L (§15), further splittable by helicity class; obeys the bound |T_q^{loc}(t)|≤CνA_q^{loc}(t)λ_q^2E_q^*(t) (§16).",
      "defining_relation": "T_q^{loc}=-\\sum_{\\substack{|p-q|\\le C_L\\\\|r-q|\\le C_L}}\\left\\langle\\Delta_q\\mathbb P(u_p\\cdot\\nabla u_r),u_q\\right\\rangle",
      "notes": "C_L 為固定 locality 半徑常數，與 §16 的 C_* 各自獨立。定理 17.1 給出 subthreshold 積分界 ∑_q∫_{S_q(β)}|T_q^{loc}|dt≤CβE_0。"
    },
    {
      "id": "ns.c3.c3k.local_energy_packet",
      "latex": "\\mathcal E_q^\\ast",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "局域能量包",
      "label_en": "Local energy packet",
      "definition_zh": "殼層 q 鄰域（半徑 C_*）內所有 shell 的 L^2 能量總和（§15），用於界定 T_q^{loc} 的上界；透過 ∑_qλ_q^2E_q^*≤C‖∇u‖_2^2（有限重疊）於定理 17.1 證明中被控制。",
      "definition_en": "The total L^2 energy summed over shells within radius C_* of shell q (§15), used to bound T_q^{loc}; controlled via ∑_qλ_q^2E_q^*≤C‖∇u‖_2^2 (finite overlap) in the proof of Theorem 17.1.",
      "defining_relation": "\\mathcal E_q^\\ast=\\sum_{|m-q|\\le C_\\ast}\\|u_m\\|_2^2",
      "notes": "半徑常數 C_* 與 T_q^{loc} 的 C_L 是不同的固定局域化參數。"
    },
    {
      "id": "ns.c3.c3k.local_active_amplitude",
      "latex": "A_q^{loc}(t)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "局域活躍振幅",
      "label_en": "Local active amplitude",
      "definition_zh": "殼層 q 鄰域內（半徑 C_*）各 shell 振幅的最大值（§16），不做 helicity 分解，用於定義 subthreshold region S_q(β)。",
      "definition_en": "The maximum shell amplitude over shells within radius C_* of q (§16), without helicity decomposition; used to define the subthreshold region S_q(β).",
      "defining_relation": "A_q^{loc}(t)=\\max_{|m-q|\\le C_\\ast}\\frac{\\|u_m(t)\\|_\\infty}{\\nu\\lambda_m}",
      "notes": "與 a_q^σ 不同：a_q^σ 是單一殼層、單一 helicity 的絕對振幅；A_q^{loc} 是鄰域內、未做 helicity 區分的局域最大振幅。"
    },
    {
      "id": "ns.c3.c3k.subthreshold_region",
      "latex": "S_q(\\beta)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "次閾值區域",
      "label_en": "Subthreshold region",
      "definition_zh": "局域活躍振幅低於 β 的時間集合（§17，定理 C3-K.3）；定理 17.1 證明此區域上局域能量轉移的時間積分被 CβE_0 一致控制，即 finite subthreshold local turnover。",
      "definition_en": "The set of times where the local active amplitude stays below β (§17, Thm C3-K.3); Theorem 17.1 shows the time-integrated local energy transfer on this region is uniformly bounded by CβE_0 — finite subthreshold local turnover.",
      "defining_relation": "S_q(\\beta)=\\left\\{t:A_q^{loc}(t)<\\beta\\right\\}",
      "notes": "與 A_{q,σ}(β)（振幅≥β 的絕對活躍集合）互補：S_q(β) 用 <β 且基於 local-max 版本 A_q^{loc} 而非單一 helicity 的 a_q^σ。"
    },
    {
      "id": "ns.c3.c3k.local_variation",
      "latex": "\\mathcal V_{\\rm loc}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "局域變異總量",
      "label_en": "Local (turnover) variation",
      "definition_zh": "所有殼層局域能量轉移絕對值的時空總積分（§19）；若其發散，由 §4 worldvolume 界與 §17 subthreshold 界聯合可知，發散必須集中在 finite weighted active worldvolume 上。",
      "definition_en": "The total spacetime integral of the absolute local energy transfer summed over all shells (§19); if divergent, the §4 worldvolume bound and §17 subthreshold bound jointly force the divergence to concentrate on a finite weighted active worldvolume.",
      "defining_relation": "\\mathcal V_{\\rm loc}=\\sum_q\\int|T_q^{loc}|\\,dt",
      "notes": "§18 以文字先描述此 'absolute variation'，§19 正式賦予符號 V_loc。"
    },
    {
      "id": "ns.c3.c3k.congestion_variation_principle",
      "latex": "\\textbf{Congestion--Variation Principle}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "擁塞—變異原理",
      "label_en": "Congestion–Variation Principle",
      "definition_zh": "§19 命名的 gauge-invariant 原理：若 V_loc 發散，其發散必然集中於（定理 4.1 給出的）有限加權 active worldvolume 上，而非分散於整個 spacetime。",
      "definition_en": "The gauge-invariant principle named in §19: if V_loc diverges, the divergence must concentrate on the finite weighted active worldvolume established by Theorem 4.1, rather than spreading across all spacetime.",
      "notes": "是本輪 occupancy（M_1(β)<∞，§4/27）與 subthreshold turnover（定理 17.1，§17）兩結果的聯合推論。"
    },
    {
      "id": "ns.c3.c3k.heterochiral_triad",
      "latex": "\\tau=(k,p,q),\\quad e_{\\rm uniq}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "異手性三元組與唯一符能量",
      "label_en": "Heterochiral triad and unique-sign modal energy",
      "definition_zh": "三元組 τ=(k,p,q) 標記一組 heterochiral 交互作用（§21，沿用 C3-A/B/D 的三元組架構）；e_uniq 為該三元組中唯一符 helicity mode 攜帶的能量，滿足 R_τ=r_τ·ė_uniq。",
      "definition_en": "The triad τ=(k,p,q) labels a heterochiral interaction (§21, reusing the triad framework from C3-A/B/D); e_uniq is the energy carried by the triad's unique-sign helicity mode, satisfying R_τ=r_τ·ė_uniq.",
      "notes": "§22/39 交替以 ė_τ 或 ΔE_τ 表示同一 per-triad modal energy（速率 vs 增量形式），源文本身有此記號漂移，指涉同一物理量。"
    },
    {
      "id": "ns.c3.c3k.critical_pair_production_rate",
      "latex": "\\mathcal R_\\tau",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "臨界螺旋對產生率",
      "label_en": "Critical helical pair-production rate",
      "definition_zh": "三元組 τ 的臨界對產生率，精確等於唯一符 mode 波數 r_τ 乘其能量變化率（§21）；對 local triad（k~p~q~λ_τ）漸近等價於比 ordinary energy transfer |ė| 多一個頻率因子的 λ_τ|ė_uniq|，是 One-Frequency-Moment Gap 的定量核心。",
      "definition_en": "The critical pair-production rate for triad τ, exactly equal to the unique-sign mode's wavenumber r_τ times its energy rate (§21); for local triads (k~p~q~λ_τ) it is asymptotically λ_τ|ė_uniq| — one frequency factor above ordinary energy transfer |ė| — the quantitative core of the One-Frequency-Moment Gap.",
      "defining_relation": "\\mathcal R_\\tau=r_\\tau\\dot e_{\\rm uniq}\\qquad\\text{(local triad: }|\\mathcal R_\\tau|\\asymp\\lambda_\\tau|\\dot e_{\\rm uniq}|\\text{)}",
      "notes": "聚合版本 R（無 τ 下標，§20）由 C3-A/B 證明 hypothetical blow-up 要求 ∫[R]_+dt=∞，非本輪首次定義；本輪新增的是逐三元組 λ-權重關係。"
    },
    {
      "id": "ns.c3.c3k.one_frequency_moment_gap",
      "latex": "\\textbf{One-Frequency-Moment Gap}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "單一頻率矩缺口",
      "label_en": "One-Frequency-Moment Gap",
      "definition_zh": "本輪標題性結果（定理 C3-K.4，§22）：ordinary energy transfer 變異的有限性（∑_τ∫|ė_τ|dt<∞）並不能控制多一個 λ 權重的 critical pair-production 變異（∑_τ∫λ_τ|ė_τ|dt），兩者在 scaling 上完全相容。",
      "definition_en": "The round's headline result (Thm C3-K.4, §22): finiteness of the ordinary energy-transfer variation (∑_τ∫|ė_τ|dt<∞) does not control the one-λ-weighted critical pair-production variation (∑_τ∫λ_τ|ė_τ|dt), the two being fully scaling-compatible.",
      "defining_relation": "\\sum_\\tau\\int|\\dot e_\\tau|\\,dt<\\infty\\ \\not\\Longrightarrow\\ \\sum_\\tau\\int\\lambda_\\tau|\\dot e_\\tau|\\,dt<\\infty",
      "notes": "由 §23 的抽象 X_n/Y_n 例子具體示範；§39 結論以 ∑|ΔE_τ|<∞ 與 ∑λ_τ|ΔE_τ|=∞ 的形式重述同一缺口。下一輪 C3-L 的核心問題即由此展開。"
    },
    {
      "id": "ns.c3.c3k.toy_geometric_ledger",
      "latex": "X_n,\\ Y_n",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "抽象幾何轉移帳本（玩具例）",
      "label_en": "Abstract geometric transfer ledger (toy example)",
      "definition_zh": "§23 具體示範 One-Frequency-Moment Gap 的抽象數列：X_n=λ_n^{-1} 模擬每代 ordinary energy transfer（∑X_n<∞），Y_n=λ_nX_n=1 模擬 critical weighted transfer（∑Y_n=∞）；作者明言此非 N–S 解的實際建構，僅為 scaling 相容性示範。",
      "definition_en": "The abstract sequence pair from §23 concretely illustrating the One-Frequency-Moment Gap: X_n=λ_n^{-1} models per-generation ordinary energy transfer (∑X_n<∞), while Y_n=λ_nX_n=1 models the critical weighted transfer (∑Y_n=∞); explicitly not a Navier–Stokes solution construction, only a scaling-compatibility demonstration.",
      "notes": "λ_n=2^n，與 λ_q 同型但索引 n 為抽象世代，非實際 shell。"
    },
    {
      "id": "ns.c3.c3k.critical_l2_stock",
      "latex": "C_q",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "臨界 L^2 儲量",
      "label_en": "Critical L^2 stock",
      "definition_zh": "殼層 q 的正規化臨界能量儲量，定義為 C_q=λ_q‖u_q‖_2^2/ν^2（§25）；若 a_q≥β，由 Bernstein 下界得 C_q≥cβ^2，即每個臨界活躍殼層攜帶 O(1) 正規化儲量，儘管其 ordinary 能量代價沿幾何殼層可求和。",
      "definition_en": "The normalized critical energy stock at shell q, defined as C_q=λ_q‖u_q‖_2^2/ν^2 (§25); if a_q≥β, the Bernstein lower bound gives C_q≥cβ^2, so every critical-active shell carries O(1) normalized stock even though its ordinary energy cost is summable along geometric shells.",
      "notes": "展示 '無限多個 O(1) critical tokens 仍可具有 finite ordinary energy' 的另一版本 moment gap，與 §26 的 E_q 互為對照。"
    },
    {
      "id": "ns.c3.c3k.counter_ledger_energy",
      "latex": "E_q",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "抽象計數帳本能量",
      "label_en": "Abstract counter-ledger energy",
      "definition_zh": "§26 抽象取用的逐殼層能量下界值 E_q=cν^2β^2λ_q^{-1}，滿足 ∑_qE_q<∞，但正規化臨界儲量 λ_qE_q/ν^2=cβ^2 對每個尺度都保持固定正值；明言為 scaling counter-ledger，非 N–S field construction。",
      "definition_en": "The per-shell energy lower-bound value abstractly taken in §26, E_q=cν^2β^2λ_q^{-1}, satisfying ∑_qE_q<∞ while the normalized critical stock λ_qE_q/ν^2=cβ^2 stays fixed and positive at every scale; explicitly flagged as a scaling counter-ledger, not an N–S field construction.",
      "notes": "與 C_q（§25）呈現同一 scaling 現象的第二個示範版本。"
    },
    {
      "id": "ns.c3.c3k.occupation_moment_one",
      "latex": "M_1(\\beta)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "第一占用矩",
      "label_en": "First occupation moment",
      "definition_zh": "即定理 4.1/C3-K.1 所證的 absolute active-worldvolume budget，於 §27 正式命名為 M_1(β)；由 energy inequality 自動控制，M_1(β)≤CE_0/(ν^3β^2)。",
      "definition_en": "The absolute active-worldvolume budget proved in Thm 4.1/C3-K.1, formally named M_1(β) in §27; automatically controlled by the energy inequality, M_1(β)≤CE_0/(ν^3β^2).",
      "defining_relation": "M_1(\\beta)=\\sum_{q,\\sigma}\\lambda_q\\left|A_{q,\\sigma}(\\beta)\\right|<\\infty",
      "notes": "對應論文標題 'Absolute Occupancy Worldvolume'；§29 稱其為 C2 dissipation wavenumber Λ∈L^1\\setminus L^{5/2} 的 shell-resolved 精細化版本。"
    },
    {
      "id": "ns.c3.c3k.occupation_moment_two",
      "latex": "M_2(\\beta)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "第二占用矩",
      "label_en": "Second occupation moment",
      "definition_zh": "§27/定理 C3-K.5 引入的高階占用矩，對應 local critical viscous window 的自然速率 νλ_q^2；不像 M_1(β) 被 energy inequality 自動控制，其是否必然發散（'parabolic Zeno cascade' 特徵）是下一輪 C3-L 的 L1 開放問題。",
      "definition_en": "The higher occupation moment introduced in §27/Thm C3-K.5, matching the natural rate νλ_q^2 of the local critical viscous window; unlike M_1(β) it is not automatically controlled by the energy inequality — whether it must diverge (the 'parabolic Zeno cascade' signature) is left as open problem L1 for the next round, C3-L.",
      "defining_relation": "M_2(\\beta)=\\sum_{q,\\sigma}\\lambda_q^2\\left|A_{q,\\sigma}(\\beta)\\right|",
      "notes": "例示：若 |A_q|~λ_q^{-2}，則 M_1~∑λ_q^{-1}<∞ 但 M_2~∑1=∞，此即 §28 的 One-Moment Occupancy Barrier。"
    },
    {
      "id": "ns.c3.c3k.general_occupation_moment",
      "latex": "M_s(\\mu)",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "一般化 s 階占用矩",
      "label_en": "General s-th occupation moment",
      "definition_zh": "§35 True ETN 更新中將 M_1、M_2 推廣為占用測度 μ（即 μ_β，μ_q 為其 shell-q marginal）的一般 s 階頻率矩 M_s(μ)=∑_qλ_q^sμ_q；本輪顯示 M_0<∞（適當正規化）並不控制 M_1，凸顯 True ETN non-collapse 需處理 finite mass escaping to infinity while higher moments diverge。",
      "definition_en": "In the §35 True ETN update, M_1 and M_2 are generalized to an arbitrary s-th frequency moment M_s(μ)=∑_qλ_q^sμ_q of the occupancy measure μ (i.e. μ_β, with μ_q its shell-q marginal); this round shows M_0<∞ (under suitable normalization) does not control M_1, highlighting that True ETN non-collapse must handle finite mass escaping to infinity while higher moments diverge.",
      "notes": "s=1,2 時分別與 M_1(β)、M_2(β) 一致（取 μ=μ_β）。"
    },
    {
      "id": "ns.c3.c3k.x_integration_variation_certificates",
      "latex": "\\operatorname{EnergyVariation},\\ \\operatorname{CriticalWeightedVariation}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "X-Integration 能量變異／臨界加權變異證書",
      "label_en": "X-Integration EnergyVariation / CriticalWeightedVariation certificates",
      "definition_zh": "§34 為 X-Integration Unified Program 新增的強制規範：每份 transfer certificate 必須分開記錄 ordinary EnergyVariation（如 ∑E）與 CriticalWeightedVariation（如 ∑λE），不得因前者有限就推論後者有限。",
      "definition_en": "A new hard requirement in §34 for the X-Integration Unified Program: every transfer certificate must separately record ordinary EnergyVariation (e.g. ∑E) and CriticalWeightedVariation (e.g. ∑λE), and must not infer finiteness of the latter from finiteness of the former.",
      "notes": "直接對應 One-Frequency-Moment Gap 的 bookkeeping 落實；與新增守衛 G_MOMENT 搭配使用。"
    },
    {
      "id": "ns.c3.c3k.moment_guard",
      "latex": "G_{\\rm MOMENT}",
      "series": "NS",
      "first_appearance": "C3-K",
      "label_zh": "頻率矩守衛",
      "label_en": "Moment guard",
      "definition_zh": "§34 新增的 X-Integration 檢查項，用以偵測證明是否『偷偷』把一個 λ^1 的量提升為 λ^2（或更高）的量而未經合法論證，直接防範 One-Frequency-Moment Gap 相關的推理錯誤。",
      "definition_en": "A new X-Integration check introduced in §34 to detect whether a proof silently raises a λ^1 quantity to λ^2 (or higher) without legitimate justification — a direct safeguard against reasoning errors around the One-Frequency-Moment Gap.",
      "notes": "與 EnergyVariation/CriticalWeightedVariation 證書配合使用。"
    },
    {
      "id": "ns.c3.c3l.m1_beta",
      "latex": "M_1(\\beta)",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "一階絕對佔用矩",
      "label_en": "First absolute occupancy moment",
      "definition_zh": "第0、6節重述自 C3-K 的一階頻率佔用矩：對每個 dyadic shell 在振幅超閾值集合 $A_q(\\beta)$ 上的測度以 $\\lambda_q$ 加權求和，C3-K 已證其恆有限。C3-L 第14節將它與新導出的 $M_{5/2}(\\beta)=\\infty$ 並列，構成「低階有限＋高階發散」的 moment-escape signature。",
      "definition_en": "The first-order occupancy moment recapped from C3-K (Secs. 0, 6): the λ_q-weighted measure of the amplitude threshold-occupancy sets A_q(β), summed over shells, proved finite in C3-K. C3-L pairs its finiteness with the newly derived divergence M_{5/2}(β)=∞ (Sec. 14) to form the round's signature escape dichotomy.",
      "defining_relation": "M_1(\\beta) = \\sum_q \\lambda_q |A_q(\\beta)| < \\infty",
      "notes": "Section 0 recaps a helicity-refined version M_1(\\beta)=\\sum_{q,\\sigma}\\lambda_q|A_{q,\\sigma}(\\beta)| with an extra chirality index σ; Secs. 6/14 onward drop σ. Origin is C3-K (NS_C3K_AbsoluteOccupancy_OneMomentGap); listed here because C3-L's central new result is its pairing against M_{5/2}(β)."
    },
    {
      "id": "ns.c3.c3l.a_q",
      "latex": "a_q(t)",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "正規化 shell 振幅",
      "label_en": "Normalized shell amplitude",
      "definition_zh": "第1節定義的正規化 shell 振幅：$L^\\infty$ shell 速度大小除以黏性與該 shell 頻率之積，是本輪所有矩（$J_q$、$\\mathfrak M_2^{amp}$、$M_1$、$M_{5/2}$）的共同建構元。Bernstein 界給出 $a_q(t)\\le C\\|u_0\\|_2\\lambda_q^{1/2}/\\nu$（第12節），為定理13.1提供關鍵上界。",
      "definition_en": "The normalized shell amplitude defined in Sec. 1: the shell velocity's L^∞ norm divided by viscosity times the shell's dyadic frequency, the common building block of every moment in this round (J_q, 𝔐_2^amp, M_1, M_{5/2}). The Bernstein bound a_q(t)≤C‖u_0‖_2λ_q^{1/2}/ν (Sec. 12) drives Theorem 13.1.",
      "defining_relation": "a_q(t) = \\dfrac{\\|u_q(t)\\|_\\infty}{\\nu\\lambda_q}",
      "notes": "A helicity-refined variant a_q^\\sigma(t)=\\|u_q^\\sigma(t)\\|_\\infty/(\\nu\\lambda_q) is also given in Sec. 1 \"if needed,\" but is not used in this round's actual theorems (4.1, 9.1, 13.1)."
    },
    {
      "id": "ns.c3.c3l.dissipation_wavenumber",
      "latex": "\\Lambda(t),\\ Q(t)",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "耗散波數與頻率指標",
      "label_en": "Dissipation wavenumber and frequency index",
      "definition_zh": "第2節重述的耗散波數 $\\Lambda(t)=\\lambda_{Q(t)}$：$Q(t)$ 是高於此即落入黏性主導小量門檻的 shell 頻率指標。C2 已證 $\\Lambda\\in L^1$；本輪新增判準是 $\\Lambda$ 是否落在 $L^2$，藉此把發散 carrier 分成 Branch A（$\\Lambda\\notin L^2$）與 Branch B（$\\Lambda\\in L^2$）。",
      "definition_en": "The dissipation wavenumber Λ(t)=λ_{Q(t)} recapped in Sec. 2, where Q(t) is the shell-frequency index above which nonlinear amplitude is viscosity-dominated. C2 proved Λ∈L^1; this round's new criterion is whether Λ additionally lies in L^2, splitting the divergence carrier into Branch A (Λ∉L^2) versus Branch B (Λ∈L^2).",
      "defining_relation": "\\Lambda(t) = \\lambda_{Q(t)}",
      "notes": "Under NS scaling Λ_λ(t)=λΛ(λ^2t), so ∫Λ^2dt is scale-invariant, making Λ∉L^2 a genuine critical frontier-spike mechanism, one scaling level above C2's Λ∈L^1 bookkeeping."
    },
    {
      "id": "ns.c3.c3l.j_q",
      "latex": "J_q",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "頻率局部化臨界通行費積分",
      "label_en": "Frequency-localized critical toll integral",
      "definition_zh": "第3節由 Cheskidov–Dai 判準逆否推出的量：在活躍窗口 $q\\le Q(t)$ 內對 $\\lambda_q\\|u_q(t)\\|_\\infty$ 於 $(T_*/2,T_*)$ 取時間積分。定理4.1證明 $\\limsup_qJ_q>c_*$（外部判準給出）蘊含 $\\sum_qJ_q=\\infty$，等價於 $\\nu\\mathfrak M_2^{amp}=\\infty$。",
      "definition_en": "The frequency-localized critical toll integral (Sec. 3), obtained from the contrapositive of the Cheskidov–Dai criterion: the time integral of λ_q‖u_q(t)‖_∞ over the active window q≤Q(t) on (T_*/2,T_*). Theorem 4.1 shows limsup_q J_q>c_* (from the external criterion) forces ∑_q J_q=∞, equivalent to ν𝔐_2^amp=∞.",
      "defining_relation": "J_q = \\int_{T_\\ast/2}^{T_\\ast} 1_{\\{q\\le Q(t)\\}}\\,\\lambda_q\\|u_q(t)\\|_\\infty\\,dt",
      "notes": "c_* is the small universal/viscosity-normalized threshold constant of the external Cheskidov–Dai criterion (Sec. 3); not otherwise used."
    },
    {
      "id": "ns.c3.c3l.omega_q",
      "latex": "\\omega_q",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "shell 局部化渦度",
      "label_en": "Shell-localized vorticity",
      "definition_zh": "第5節：在頻率環 $|\\xi|\\sim\\lambda_q$ 上 curl 與 Biot–Savart 皆為光滑環狀乘子，給出 $\\|\\omega_q\\|_\\infty\\asymp\\lambda_q\\|u_q\\|_\\infty$。藉此定理4.1可改寫成本文所稱「Critical Vorticity-Moment Escape」：$\\int\\sum_{q\\le Q(t)}\\|\\omega_q(t)\\|_\\infty dt=\\infty$。",
      "definition_en": "Shell-localized vorticity (Sec. 5): since curl and Biot–Savart are smooth annular multipliers on |ξ|∼λ_q, ‖ω_q‖_∞≍λ_q‖u_q‖_∞. This recasts Theorem 4.1 as the paper's named result \"Critical Vorticity-Moment Escape\": ∫∑_{q≤Q(t)}‖ω_q(t)‖_∞dt=∞.",
      "defining_relation": "\\|\\omega_q\\|_\\infty \\asymp \\lambda_q\\|u_q\\|_\\infty",
      "notes": "Equivalence holds up to universal annular constants; ω_q is distinct from the generic full vorticity field ω=∇×u."
    },
    {
      "id": "ns.c3.c3l.m2_amp",
      "latex": "\\mathfrak M_2^{amp}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "振幅加權二階矩",
      "label_en": "Amplitude-weighted second moment",
      "definition_zh": "第6節命名的振幅加權二階矩，把定理4.1的發散量正式量化為 $\\mathfrak M_2^{amp}=\\int_{T_*/2}^{T_*}\\sum_{q\\le Q(t)}\\lambda_q^2a_q(t)dt$。Blow-up 蘊含此矩必為無限，是第7–9節 Branch A/B 拆分的出發量。",
      "definition_en": "The amplitude-weighted second moment named in Sec. 6, formalizing Theorem 4.1's divergent quantity as 𝔐_2^amp=∫_{T_*/2}^{T_*}∑_{q≤Q(t)}λ_q^2a_q(t)dt. Blow-up forces 𝔐_2^amp=∞, the starting quantity for the Branch A/B split of Secs. 7–9.",
      "defining_relation": "\\mathfrak M_2^{amp} = \\int_{T_\\ast/2}^{T_\\ast}\\sum_{q\\le Q(t)}\\lambda_q^2a_q(t)\\,dt",
      "notes": "Equal to ν^{-1} times the vorticity-moment sum of Sec. 5, and to the J_q-sum via Theorem 4.1's equivalence; relabeled 𝔗_spec in Sec. 27."
    },
    {
      "id": "ns.c3.c3l.a_q_beta",
      "latex": "A_q(\\beta)",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "振幅超閾值時間集合",
      "label_en": "Amplitude threshold-occupancy time set",
      "definition_zh": "第6節定義的振幅超閾值時間集合 $A_q(\\beta)=\\{t:a_q(t)\\ge\\beta\\}$，是 $M_1(\\beta)$ 與新定義 $M_{5/2}(\\beta)$ 共用的測度基礎；第13節另用限制版本 $A_q(\\beta)\\cap(T_*/2,T_*)$。",
      "definition_en": "The amplitude threshold-occupancy time set A_q(β)={t:a_q(t)≥β} (Sec. 6), the shared measure underlying both M_1(β) and the newly introduced M_{5/2}(β); Sec. 13 uses the restricted form A_q(β)∩(T_*/2,T_*).",
      "defining_relation": "A_q(\\beta) = \\{t : a_q(t)\\ge\\beta\\}",
      "notes": "Sec. 16 gives an abstract example |A_q|∼λ_q^{-2} showing M_1<∞ and M_{5/2}=∞ can coexist with finite total occupied time (a \"Zeno-pack\") — bookkeeping-consistent, not an NS construction."
    },
    {
      "id": "ns.c3.c3l.m2_threshold_split",
      "latex": "\\mathfrak M_{2,<\\beta},\\ \\mathfrak M_{2,\\ge\\beta}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "次閾值／閾上二階矩分解",
      "label_en": "Subthreshold / at-or-above-threshold second-moment split",
      "definition_zh": "第7–9節：固定 $\\beta>0$ 把 $a_q$ 拆成 $a_q<\\beta$ 與 $a_q\\ge\\beta$ 兩部分，使 $\\mathfrak M_2^{amp}=\\mathfrak M_{2,<\\beta}+\\mathfrak M_{2,\\ge\\beta}$。第8節證次閾值部分被 $C\\beta\\int\\Lambda^2dt$ 控制，故定理9.1（C3-L.2）給出：若 $\\Lambda\\in L^2$ 則對每個 $\\beta>0$ 都有 $\\mathfrak M_{2,\\ge\\beta}=\\infty$。",
      "definition_en": "For fixed β>0, Secs. 7–9 split a_q into a_q<β and a_q≥β so that 𝔐_2^amp=𝔐_{2,<β}+𝔐_{2,≥β}; Sec. 8 bounds the subthreshold part by Cβ∫Λ^2dt. Theorem 9.1 (C3-L.2) then concludes that if Λ∈L^2, the threshold-and-above part 𝔐_{2,≥β} must diverge for every fixed β>0.",
      "defining_relation": "\\mathfrak M_2^{amp}=\\mathfrak M_{2,<\\beta}+\\mathfrak M_{2,\\ge\\beta},\\qquad \\mathfrak M_{2,<\\beta}\\le C\\beta\\int_{T_\\ast/2}^{T_\\ast}\\Lambda(t)^2\\,dt",
      "notes": "This is Theorem 9.1, named \"Critical-Moment Carrier Dichotomy\" (C3-L.2)."
    },
    {
      "id": "ns.c3.c3l.m_5_2_beta",
      "latex": "M_{5/2}(\\beta)",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "5/2 階活躍佔用矩",
      "label_en": "Active 5/2-occupancy moment",
      "definition_zh": "第13–14節新定義的高階活躍佔用矩，由定理13.1（C3-L.3）導出：若 $\\Lambda\\in L^2$，對每個 $\\beta>0$ 都有 $M_{5/2}(\\beta)=\\infty$。與 $M_1(\\beta)<\\infty$（第14節）並列，是本輪標誌性 moment-escape signature。",
      "definition_en": "The new higher occupancy moment of Secs. 13–14, derived in Theorem 13.1 (C3-L.3): if Λ∈L^2, then for every β>0, M_{5/2}(β)=∞. Paired with M_1(β)<∞ (Sec. 14), this is C3-L's signature \"finite low moment + divergent high moment\" result.",
      "defining_relation": "M_{5/2}(\\beta) = \\sum_q \\lambda_q^{5/2}\\left|A_q(\\beta)\\cap(T_\\ast/2,T_\\ast)\\right| = \\infty \\quad (\\Lambda\\in L^2)",
      "notes": "Derived via the Bernstein-based bound λ_q^2a_q≤C‖u_0‖_2λ_q^{5/2}/ν (Secs. 12–13). Recurs as the Branch B characterization in Secs. 15, 30, 36–37."
    },
    {
      "id": "ns.c3.c3l.lambda2_plus",
      "latex": "\\lambda_2^+",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "應變張量中間特徵值正部",
      "label_en": "Positive part of the middle strain eigenvalue",
      "definition_zh": "第23–24節：應變特徵值排序 $\\lambda_1\\le\\lambda_2\\le\\lambda_3$（因 $\\operatorname{tr}S=0$）中，取中間特徵值正部 $\\lambda_2^+=\\max\\{\\lambda_2,0\\}$。引用 Miller 定理（$\\lambda_2^+\\in L_t^rL_x^p,\\ 2/r+3/p=2$）於 $p=3,r=2$ 特化，得 C3-L.5：hypothetical blow-up 蘊含 $\\int_0^{T_*}\\|\\lambda_2^+(t)\\|_3^2dt=\\infty$。",
      "definition_en": "The positive part of the middle strain eigenvalue, λ_2^+=max{λ_2,0}, from the ordering λ_1≤λ_2≤λ_3 (tr S=0) in Secs. 23–24. Specializing Miller's criterion (λ_2^+∈L_t^rL_x^p, 2/r+3/p=2, 3/2<p≤∞) to p=3, r=2 gives C3-L.5: hypothetical blow-up forces ∫_0^{T_*}‖λ_2^+(t)‖_3^2dt=∞.",
      "defining_relation": "\\lambda_2^+ = \\max\\{\\lambda_2,0\\}; \\qquad \\int_0^{T_\\ast}\\|\\lambda_2^+(t)\\|_3^2\\,dt=\\infty",
      "notes": "This is the paper's second (geometric) escape channel, parallel to but not proved implied by/implying the spectral moment escape (NG-L4, open). Reused as 𝔗_strain in Sec. 27; Sec. 32 explicitly disclaims that this is an alignment theorem (ω need not align with the middle eigenvector)."
    },
    {
      "id": "ns.c3.c3l.lambda2_plus_besov_endpoint",
      "latex": "\\lambda_2^+ \\in L_t^2\\dot B^{-1}_{\\infty,\\infty}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "中間應變特徵值臨界端點 Besov 判準",
      "label_en": "Endpoint Besov criterion for the middle strain eigenvalue",
      "definition_zh": "第25節引入 Guo–O（2025）延伸判準：若 $\\lambda_2^+\\in L^2(0,T;\\dot B^{-1}_{\\infty,\\infty})$ 則局部強解可光滑延拓，是 Miller 判準的端點加強版。故 hypothetical singularity 必須另外滿足逆否 $\\lambda_2^+\\notin L_t^2\\dot B^{-1}_{\\infty,\\infty}$。",
      "definition_en": "Sec. 25 imports the Guo–O (2025) extension criterion, an endpoint sharpening of Miller's: λ_2^+∈L^2(0,T;Ḃ^{-1}_{∞,∞}) implies smooth extension of the local strong solution. Hypothetical singularity must therefore also satisfy the contrapositive λ_2^+∉L_t^2Ḃ^{-1}_{∞,∞}.",
      "defining_relation": "\\lambda_2^+\\in L^2\\!\\left(0,T;\\dot B^{-1}_{\\infty,\\infty}\\right) \\ \\Rightarrow\\ \\text{smooth extension}",
      "notes": "First use of this reference (Z. Guo, C.-J. O, Applied Mathematics Letters 160 (2025)) in the series; combined with C3-L.5 into Theorem 26.1's two parallel necessary conditions."
    },
    {
      "id": "ns.c3.c3l.vortex_stretching_integral",
      "latex": "\\mathcal V_S",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "渦線伸展積分",
      "label_en": "Vortex-stretching integral",
      "definition_zh": "第18–20節由渦度方程與恆等式 $\\frac12\\frac d{dt}\\|\\omega\\|_2^2+\\nu\\|\\nabla\\omega\\|_2^2=\\int\\omega\\cdot S\\omega\\,dx$ 導出的渦線伸展積分。命題20.1（C3-L.4）證明任何想把 energy dissipation 免費升級成 enstrophy dissipation 的論證，都必須另外控制 $\\mathcal V_S=\\int\\omega\\cdot S\\omega$，此即「moment-raising geometry debt」。",
      "definition_en": "The vortex-stretching integral (Secs. 18–20), arising from the vorticity equation's exact enstrophy identity (1/2)d/dt‖ω‖_2^2+ν‖∇ω‖_2^2=∫ω·Sω dx. Proposition 20.1 (C3-L.4) shows any argument upgrading energy dissipation to enstrophy dissipation must separately control 𝒱_S=∫ω·Sω — the \"vortex-stretching geometry debt\" of raising one moment.",
      "defining_relation": "\\mathcal V_S = \\int \\omega\\cdot S\\omega",
      "notes": "Scaling audit (Sec. 21) shows ∫‖∇ω_λ‖_2^2dt and the stretching integral both scale by exactly λ under u_λ=λu(λx,λ^2t), so the enstrophy identity carries no hidden scale advantage over the energy identity. Feeds directly into guard G_RAISE (Sec. 28)."
    },
    {
      "id": "ns.c3.c3l.t_spec",
      "latex": "\\mathfrak T_{\\rm spec}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "ETN 頻譜張力通道",
      "label_en": "ETN spectral tension channel",
      "definition_zh": "第27節 True ETN 框架下把定理4.1的發散量標記為「頻譜張力通道」：$\\mathfrak T_{\\rm spec}=\\int\\sum_{q\\le Q(t)}\\lambda_q^2a_q(t)dt=\\infty$，數值上等同 $\\mathfrak M_2^{amp}$。",
      "definition_en": "The \"spectral tension channel\" of the True ETN framework (Sec. 27), labeling Theorem 4.1's divergent quantity: 𝔗_spec=∫∑_{q≤Q(t)}λ_q^2a_q(t)dt=∞, numerically identical to 𝔐_2^amp.",
      "defining_relation": "\\mathfrak T_{\\rm spec} = \\int \\sum_{q\\le Q(t)}\\lambda_q^2a_q(t)\\,dt = \\infty",
      "notes": "Paired with 𝔗_strain; X-Integration requires the two channels' provenance never be collapsed into one scalar (Sec. 27)."
    },
    {
      "id": "ns.c3.c3l.t_strain",
      "latex": "\\mathfrak T_{\\rm strain}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "ETN 應變張力通道",
      "label_en": "ETN strain (geometric) tension channel",
      "definition_zh": "第27節與 $\\mathfrak T_{\\rm spec}$ 並列的第二必發散通道：$\\mathfrak T_{\\rm strain}=\\int\\|\\lambda_2^+(t)\\|_3^2dt=\\infty$。X-Integration 規定兩者 provenance 不可壓成同一 scalar。",
      "definition_en": "The second must-diverge channel paired with 𝔗_spec (Sec. 27): 𝔗_strain=∫‖λ_2^+(t)‖_3^2dt=∞, numerically identical to the C3-L.5 quantity.",
      "defining_relation": "\\mathfrak T_{\\rm strain} = \\int \\|\\lambda_2^+(t)\\|_3^2\\,dt = \\infty",
      "notes": "Whether 𝔗_spec=∞ implies or is implied by 𝔗_strain=∞ is explicitly left open (NG-L4); C3-M is defined to investigate exactly this coupling."
    },
    {
      "id": "ns.c3.c3l.g_raise",
      "latex": "G_{\\rm RAISE}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "X-Guard：矩提升護則",
      "label_en": "X-Guard: Moment-Raising guard",
      "definition_zh": "第28節新增 X-Guard：任何從低階矩 $M_s$ 推出 $M_{s+1}$ 有限的證明，須指出提高導數的確切方程、新增 source term、其獨立界，以及是否只是把缺失矩藏進非線性幾何。對 enstrophy 路線，此護則輸出的 debt 即 $\\omega\\cdot S\\omega$。",
      "definition_en": "A new X-Guard (Sec. 28): any proof deriving finite M_{s+1} from a lower moment M_s must name the exact derivative-raising equation, the new source term, its independent bound, and whether the missing moment is merely hidden in nonlinear geometry. For the enstrophy route, this guard's output debt is ω·Sω.",
      "defining_relation": "G_{\\rm RAISE}\\text{輸出 debt} = \\omega\\cdot S\\omega",
      "notes": "Operationalizes Proposition 20.1 / 𝒱_S as a reusable checklist; \"M_s\", \"M_{s+1}\" here are schematic generic-order moment labels, not literally M_1/M_{5/2}."
    },
    {
      "id": "ns.c3.c3l.g_geom",
      "latex": "G_{\\rm GEOM}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "X-Guard：幾何不可塌縮護則",
      "label_en": "X-Guard: Geometry Non-Collapse guard",
      "definition_zh": "第29節新增 X-Guard：禁止直接從純量梯度範數 $\\|\\nabla u\\|$ 偏大推出 $\\lambda_2^+$ 偏大，因特徵值正負號、渦度方向、shell 間抵消、空間局部化等資訊皆會被純量範數抹除，中間特徵值資訊須獨立保存。",
      "definition_en": "A new X-Guard (Sec. 29) forbidding the inference from a large scalar gradient norm ‖∇u‖ to a pointwise-large λ_2^+, since eigenvalue signs, vorticity orientation, inter-shell cancellation, and spatial localization are all erased by a scalar norm; middle-eigenvalue information must be kept independently.",
      "defining_relation": "\\|\\nabla u\\|\\text{ large} \\ \\not\\Rightarrow\\ \\lambda_2^+\\text{ large}",
      "notes": "Formalizes the prohibition underlying NG-L3 as a standing methodological guard for future rounds."
    },
    {
      "id": "ns.c3.c3l.ng_l1",
      "latex": "\\text{NG-L1}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "路線 NG-L1",
      "label_en": "Route NG-L1",
      "definition_zh": "第33節裁定為 FALSE：一階矩有限（$M_1<\\infty$）不蘊含下一階矩有限；Branch B（$M_1(\\beta)<\\infty$ 但 $M_{5/2}(\\beta)=\\infty$）正是具體反例。",
      "definition_en": "Ruled FALSE in Sec. 33: finiteness of the first moment (M_1<∞) does not imply the next moment is finite; Branch B (M_1(β)<∞ yet M_{5/2}(β)=∞) is the concrete witness.",
      "defining_relation": "M_1<\\infty \\ \\Rightarrow\\ M_2<\\infty \\quad (\\text{FALSE})",
      "notes": "\"M_2\" is used loosely/schematically here for \"the next critical moment\"; the round's precise counterexample instead uses M_{5/2}(β), not a literally-defined M_2(β)."
    },
    {
      "id": "ns.c3.c3l.ng_l2",
      "latex": "\\text{NG-L2}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "路線 NG-L2",
      "label_en": "Route NG-L2",
      "definition_zh": "第33節裁定為「無 stretching 控制則 FALSE」：$\\int\\|u\\|_{\\dot H^1}^2dt<\\infty$ 不能免費升級成 $\\int\\|u\\|_{\\dot H^2}^2dt<\\infty$，是命題20.1的直接結論。",
      "definition_en": "Ruled FALSE without stretching control (Sec. 33): ∫‖u‖_{Ḣ^1}^2dt<∞ cannot be upgraded for free to ∫‖u‖_{Ḣ^2}^2dt<∞ — the direct content of Proposition 20.1.",
      "defining_relation": "\\int\\|u\\|_{\\dot H^1}^2dt<\\infty \\ \\Rightarrow\\ \\int\\|u\\|_{\\dot H^2}^2dt<\\infty \\quad (\\text{FALSE without stretching control})",
      "notes": "Restates Proposition 20.1 (C3-L.4, Sec. 20) as a formal no-go catalog entry."
    },
    {
      "id": "ns.c3.c3l.ng_l3",
      "latex": "\\text{NG-L3}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "路線 NG-L3",
      "label_en": "Route NG-L3",
      "definition_zh": "第33節裁定為 NOT ESTABLISHED：不得由渦度大直接斷言 $\\lambda_2^+$ 逐點大，對應 X-Guard $G_{\\rm GEOM}$（第29節）所防範的推論漏洞。",
      "definition_en": "Ruled NOT ESTABLISHED in Sec. 33: large vorticity cannot be used to directly assert pointwise-large λ_2^+, the exact inferential gap that guard G_GEOM (Sec. 29) blocks.",
      "notes": "No exact defining formula given in the source (qualitative non-implication); see G_GEOM (Sec. 29) for the guard this motivates."
    },
    {
      "id": "ns.c3.c3l.ng_l4",
      "latex": "\\text{NG-L4}",
      "series": "NS",
      "first_appearance": "C3-L",
      "label_zh": "路線 NG-L4",
      "label_en": "Route NG-L4",
      "definition_zh": "第26、33節：目前只證明頻譜通道與應變通道皆為 blow-up 必要條件，尚未證明其一蘊含另一，是留給 C3-M（Critical Vorticity–Strain Coupling Rigidity）的核心開放問題。",
      "definition_en": "Secs. 26 & 33: only that both the spectral and strain channels are separately necessary for blow-up has been shown; neither implication is proved — the open problem handed to C3-M (Critical Vorticity–Strain Coupling Rigidity).",
      "defining_relation": "\\mathfrak T_{\\rm spec}=\\infty \\ \\Rightarrow\\ \\mathfrak T_{\\rm strain}=\\infty \\quad (\\text{OPEN, not proved either direction})",
      "notes": "The paper's headline open problem; explicitly not to be \"偷寫成因果等價\" (silently treated as causal equivalence) per Sec. 26."
    },
    {
      "id": "ns.c3.c3m.stretching_density",
      "latex": "\\mathcal S_\\omega(x,t)",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "伸展密度",
      "label_en": "stretching density",
      "definition_zh": "第1節定義的逐點量 S_ω = ω·Sω，即渦量與應變耦合產生的局部伸展率密度，是 enstrophy identity 的來源項。",
      "definition_en": "The pointwise quantity defined in Sec. 1 as S_ω = ω·Sω, the local vortex-stretching rate density forming the source term of the enstrophy identity.",
      "defining_relation": "\\mathcal S_\\omega(x,t)=\\omega\\cdot S\\omega"
    },
    {
      "id": "ns.c3.c3m.vorticity_direction",
      "latex": "\\xi",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "渦量方向",
      "label_en": "vorticity direction",
      "definition_zh": "第2節在 ω≠0 處定義的單位渦量向量 ξ=ω/|ω|，用以將伸展密度分解為 |ω|²α。",
      "definition_en": "The unit vorticity vector ξ=ω/|ω| defined in Sec. 2 where ω≠0, used to factor the stretching density as |ω|²α."
    },
    {
      "id": "ns.c3.c3m.alpha_stretching_rate",
      "latex": "\\alpha",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "瞬時伸展率",
      "label_en": "instantaneous stretching rate",
      "definition_zh": "第2節定義的沿渦量方向瞬時伸展率 α=ξ·Sξ，是全篇 eigenframe 分解的核心純量。",
      "definition_en": "The instantaneous stretching rate along the vorticity direction, α=ξ·Sξ, defined in Sec. 2 as the central scalar of the whole eigenframe decomposition.",
      "defining_relation": "\\alpha=\\xi\\cdot S\\xi"
    },
    {
      "id": "ns.c3.c3m.strain_eigenvalues",
      "latex": "\\lambda_1\\le\\lambda_2\\le\\lambda_3",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "排序應變特徵值",
      "label_en": "ordered strain eigenvalues",
      "definition_zh": "第3節定義應變張量 S 依大小排序的三個特徵值 λ1≤λ2≤λ3，由不可壓縮性給出無跡條件 λ1+λ2+λ3=0。",
      "definition_en": "The three eigenvalues of the strain tensor S, ordered as λ1≤λ2≤λ3 and defined in Sec. 3, obeying the trace-free condition λ1+λ2+λ3=0 from incompressibility."
    },
    {
      "id": "ns.c3.c3m.eigenvectors",
      "latex": "e_1,e_2,e_3",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "正交特徵向量",
      "label_en": "orthonormal eigenframe vectors",
      "definition_zh": "第3節定義對應 λ1,λ2,λ3 的正交單位特徵向量 e1,e2,e3，構成本輪全部分解所依賴的應變 eigenframe。",
      "definition_en": "The orthonormal unit eigenvectors e1,e2,e3 corresponding to λ1,λ2,λ3, defined in Sec. 3, forming the strain eigenframe underlying every decomposition in this round."
    },
    {
      "id": "ns.c3.c3m.orientation_weights",
      "latex": "c_i",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "方向權重",
      "label_en": "orientation weights",
      "definition_zh": "第3節定義的 c_i=|ξ·e_i|²，即渦量方向在第 i 個應變特徵向量上的投影平方，滿足 c1+c2+c3=1。",
      "definition_en": "Defined in Sec. 3 as c_i=|ξ·e_i|², the squared projection of the vorticity direction onto the i-th strain eigenvector, satisfying c1+c2+c3=1."
    },
    {
      "id": "ns.c3.c3m.stretching_orientation_identity",
      "latex": "\\alpha=\\lambda_2+(\\lambda_3-\\lambda_2)c_3-(\\lambda_2-\\lambda_1)c_1",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "精確伸展-方向恆等式",
      "label_en": "exact stretching-orientation identity",
      "definition_zh": "定理4.1（第4節）給出的核心恆等式，將 α 精確分解為 middle baseline、principal stretching surplus 與 compressive alignment depletion 三個 typed 分量之和。",
      "definition_en": "The central identity of Theorem 4.1 (Sec. 4), decomposing α exactly into a middle baseline plus a principal-stretching surplus minus a compressive-alignment depletion term.",
      "defining_relation": "\\alpha=\\lambda_2+(\\lambda_3-\\lambda_2)c_3-(\\lambda_2-\\lambda_1)c_1"
    },
    {
      "id": "ns.c3.c3m.alpha_plus",
      "latex": "\\alpha_+",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "正部伸展率",
      "label_en": "positive stretching part",
      "definition_zh": "第6節用 [x]_+=max{x,0} 定義的 α 正部 α_+，滿足上界 α_+≤λ2^++√2|S|c3。",
      "definition_en": "The positive part of α, α_+, via [x]_+=max{x,0}, defined in Sec. 6 and bounded by α_+≤λ2^++√2|S|c3.",
      "defining_relation": "\\alpha_+\\le\\lambda_2^++\\sqrt2|S|c_3"
    },
    {
      "id": "ns.c3.c3m.lambda2_plus_carrier",
      "latex": "\\lambda_2^+",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "中間特徵值正部（承載角色）",
      "label_en": "positive middle eigenvalue (carrier role)",
      "definition_zh": "承接自 C3-L 的 strain channel 量 λ2^+=[λ2]_+，本輪（第6–8節）將其確立為 pointwise stretching 的 Carrier M 候選承擔者。",
      "definition_en": "The strain-channel quantity λ2^+=[λ2]_+ inherited from C3-L; this round (Sec. 6–8) establishes it as the Carrier M candidate bearer of pointwise stretching.",
      "notes": "與 C3-L 的 λ2^+∉L_t^2L_x^3 channel 為同一符號，本輪賦予其 pointwise carrier 角色。"
    },
    {
      "id": "ns.c3.c3m.carrier_m",
      "latex": "\\lambda_2^+\\gtrsim\\alpha_+",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "中間應變承載者",
      "label_en": "Carrier M (middle-strain carrier)",
      "definition_zh": "第8節定義的第一種 pointwise carrier 選項，指正伸展主要由中間特徵值正部 λ2^+ 承擔。",
      "definition_en": "Defined in Sec. 8 as the first pointwise carrier option, where positive stretching is borne mainly by the positive middle eigenvalue λ2^+.",
      "defining_relation": "\\lambda_2^+\\gtrsim\\alpha_+"
    },
    {
      "id": "ns.c3.c3m.carrier_p",
      "latex": "c_3\\gtrsim\\frac{\\alpha_+}{|S|}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "主軸對齊承載者",
      "label_en": "Carrier P (principal-alignment carrier)",
      "definition_zh": "第8節定義的第二種 pointwise carrier 選項，指正伸展需由渦量對最伸長特徵向量 e3 的對齊程度 c3 承擔。",
      "definition_en": "Defined in Sec. 8 as the second pointwise carrier option, where positive stretching must instead be borne by the alignment weight c3 toward the most-stretching eigenvector e3.",
      "defining_relation": "c_3\\gtrsim\\frac{\\alpha_+}{|S|}"
    },
    {
      "id": "ns.c3.c3m.carrier_dichotomy",
      "latex": "\\textbf{Middle-Strain / Principal-Alignment Carrier Dichotomy}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "承載者二分法",
      "label_en": "Middle-Strain / Principal-Alignment Carrier Dichotomy",
      "definition_zh": "定理C3-M.3（第10節）：hypothetical blow-up 必使 λ2^+|ω|² 與 |S||ξ·e3|²|ω|² 兩個時空積分至少一個發散，是 Carrier M/P 的積分版本。",
      "definition_en": "Theorem C3-M.3 (Sec. 10): any hypothetical blow-up forces at least one of the spacetime integrals of λ2^+|ω|² or |S||ξ·e3|²|ω|² to diverge — the integral-level version of the Carrier M/P dichotomy.",
      "defining_relation": "\\int_0^{T_*}\\int\\lambda_2^+|\\omega|^2\\,dxdt=\\infty\\quad\\text{or}\\quad\\int_0^{T_*}\\int|S||\\xi\\cdot e_3|^2|\\omega|^2\\,dxdt=\\infty"
    },
    {
      "id": "ns.c3.c3m.betchov_identity",
      "latex": "\\int_{\\mathbb R^3}\\omega\\cdot S\\omega\\,dx=-4\\int_{\\mathbb R^3}\\det S\\,dx",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "Betchov恆等式",
      "label_en": "Betchov identity",
      "definition_zh": "第12節引入的全域恆等式，將全空間 enstrophy 生成積分 ∫ω·Sω 與應變行列式積分 -4∫detS 等同，是本輪 orientation collapse 論證的關鍵外部輸入。",
      "definition_en": "The global identity introduced in Sec. 12 equating the whole-space enstrophy-production integral ∫ω·Sω to -4∫detS, the key external input driving this round's orientation-collapse argument.",
      "defining_relation": "\\int_{\\mathbb R^3}\\omega\\cdot S\\omega\\,dx=-4\\int_{\\mathbb R^3}\\det S\\,dx"
    },
    {
      "id": "ns.c3.c3m.enstrophy_strain_norm_identity",
      "latex": "\\int|\\omega|^2dx=2\\int|S|^2dx",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "渦量-應變範數恆等式",
      "label_en": "enstrophy-strain norm identity",
      "definition_zh": "第12節伴隨 Betchov 恆等式一併給出的全域恆等式 ∫|ω|²dx=2∫|S|²dx，將渦量與應變張量的 L² 範數聯繫起來。",
      "definition_en": "A companion global identity stated alongside the Betchov identity in Sec. 12, ∫|ω|²dx=2∫|S|²dx, linking the L² norms of vorticity and the strain tensor."
    },
    {
      "id": "ns.c3.c3m.global_orientation_collapse",
      "latex": "\\text{local orientation}\\overset{\\int dx}{\\longrightarrow}\\text{global strain determinant}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "全域方向坍縮",
      "label_en": "Global Orientation Collapse",
      "definition_zh": "第13節的 No-Go 13.1：全域 enstrophy-production 恆等式無法單獨還原局部 vorticity–strain 特徵向量對齊資訊，方向資訊在積分中真正坍縮為應變行列式。",
      "definition_en": "No-Go 13.1 (Sec. 13): the global enstrophy-production identity cannot by itself recover local vorticity–strain eigenvector alignment information — orientation data genuinely collapses into the strain determinant under integration."
    },
    {
      "id": "ns.c3.c3m.g_betchov",
      "latex": "G_{\\rm BETCHOV}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "Betchov稽核旗標",
      "label_en": "Betchov non-collapse audit guard",
      "definition_zh": "第14節引入的 X-Integration 稽核符號：任何試圖由全域 stretching 積分反推局部方向資訊的論證，必須先通過此 Betchov non-collapse audit。",
      "definition_en": "The X-Integration audit symbol introduced in Sec. 14: any argument inferring local orientation from a global stretching integral must first pass this Betchov non-collapse audit."
    },
    {
      "id": "ns.c3.c3m.det_s_bound",
      "latex": "-\\det S\\le\\frac12\\lambda_2^+|S|^2",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "應變行列式上界",
      "label_en": "strain-determinant upper bound",
      "definition_zh": "第16節證明的代數引理：當 λ2>0 時 -detS≤(1/2)λ2^+|S|²，是連結 Betchov 恆等式與中間特徵值承載角色的直接代數橋樑。",
      "definition_en": "The algebraic lemma proved in Sec. 16: for λ2>0, -detS≤(1/2)λ2^+|S|², the direct algebraic bridge linking the Betchov identity to the middle-eigenvalue carrier role.",
      "defining_relation": "-\\det S\\le\\frac12\\lambda_2^+|S|^2"
    },
    {
      "id": "ns.c3.c3m.two_positive_eigenvalue_geometry",
      "latex": "\\lambda_2>0",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "雙正特徵值幾何",
      "label_en": "two-positive-eigenvalue geometry",
      "definition_zh": "定理C3-M.5（第17節）：全域正 enstrophy 生成只能由 λ2>0（兩個正應變特徵值、一個負特徵值）的幾何區域承擔，並非任意 analytic artefact。",
      "definition_en": "Theorem C3-M.5 (Sec. 17): positive global enstrophy production can only be carried by regions with λ2>0 (two positive strain eigenvalues, one negative), not an arbitrary analytic artefact."
    },
    {
      "id": "ns.c3.c3m.local_eigenframe_alignment",
      "latex": "\\xi(x,t)\\cdot e_i(x,t)",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "局部特徵框對齊",
      "label_en": "local eigenframe alignment",
      "definition_zh": "第20節定義的第一類幾何：同一點上渦量方向相對應變特徵向量的對齊程度 ξ·e_i，由 G-ORI guard 保存。",
      "definition_en": "The first geometric type defined in Sec. 20: the alignment ξ·e_i of the vorticity direction with the strain eigenvectors at the same point, preserved by the G-ORI guard."
    },
    {
      "id": "ns.c3.c3m.spatial_direction_coherence",
      "latex": "\\xi(x,t)-\\xi(y,t)",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "空間方向相干性",
      "label_en": "spatial direction coherence",
      "definition_zh": "第20節定義的第二類幾何：不同空間點間渦量方向的差異 ξ(x)-ξ(y)，是 Constantin–Fefferman 型 geometric depletion 實際作用的對象，由 G-DIR guard 保存並與 G-ORI 區分。",
      "definition_en": "The second geometric type defined in Sec. 20: the variation ξ(x)-ξ(y) of vorticity direction between distinct spatial points, the actual target of Constantin–Fefferman-type geometric depletion, preserved by the G-DIR guard and kept distinct from G-ORI."
    },
    {
      "id": "ns.c3.c3m.bmo_log_weighted",
      "latex": "\\mathrm{bmo}_{1/|\\log r|}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "對數加權bmo空間",
      "label_en": "logarithmically weighted bmo space",
      "definition_zh": "第22節引入、來自 Grujić 2026 preprint 的方向正則性空間，在 L^{3/2,∞} critical point-concentration 情境下，渦量方向落入此空間即可產生 logarithmic depletion。",
      "definition_en": "The direction-regularity space introduced in Sec. 22 from Grujić's 2026 preprint; under L^{3/2,∞} critical point-concentration, the vorticity direction lying in this space suffices to produce logarithmic depletion."
    },
    {
      "id": "ns.c3.c3m.directional_roughness_debt",
      "latex": "\\xi\\notin\\mathrm{bmo}_{1/|\\log r|}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "方向粗糙度債",
      "label_en": "Directional Roughness Debt",
      "definition_zh": "C3-M.6（第23節，條件式結果）：在 Grujić 2026 的 critical point-concentration 情境下，hypothetical blow-up 必須支付渦量方向不落入 bmo_{1/|log r|} 的債務才能存活。",
      "definition_en": "C3-M.6 (Sec. 23, conditional): under Grujić's 2026 critical point-concentration scenario, any hypothetical blow-up must pay the debt of the vorticity direction failing to lie in bmo_{1/|log r|} to survive.",
      "defining_relation": "\\xi\\notin\\mathrm{bmo}_{1/|\\log r|}"
    },
    {
      "id": "ns.c3.c3m.miller_orthogonality",
      "latex": "\\left\\langle-\\Delta S,\\omega\\otimes\\omega\\right\\rangle=0",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "Miller逆耦合正交性",
      "label_en": "Miller reverse-coupling orthogonality",
      "definition_zh": "第25節引用 Evan Miller 2026 結果：對 divergence-free 場，⟨-ΔS,ω⊗ω⟩=0 恆成立，是 C3-M.7 no-go 的外部定理基礎。",
      "definition_en": "The Evan Miller 2026 result cited in Sec. 25: for divergence-free fields, ⟨-ΔS,ω⊗ω⟩=0 identically, the external theorem underlying the C3-M.7 no-go.",
      "defining_relation": "\\left\\langle-\\Delta S,\\omega\\otimes\\omega\\right\\rangle=0"
    },
    {
      "id": "ns.c3.c3m.p_st_projection",
      "latex": "P_{st}(\\omega\\otimes\\omega)",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "渦量至應變投影耦合項",
      "label_en": "vorticity-to-strain coupling projection",
      "definition_zh": "第26節指出應變方程中含有的 P_st(ω⊗ω) 型耦合項，是 Miller orthogonality 所針對、被排除為單獨 blow-up driver 的對象。",
      "definition_en": "The coupling term P_st(ω⊗ω) noted in Sec. 26 as appearing in the strain equation — the object targeted by the Miller orthogonality result and ruled out as a sole blow-up driver."
    },
    {
      "id": "ns.c3.c3m.frak_t_necessary_conditions",
      "latex": "\\mathfrak T_{\\rm spec},\\ \\mathfrak T_{\\lambda_2}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "必要條件量（頻譜／中間應變）",
      "label_en": "spectral / middle-strain necessary-condition quantities",
      "definition_zh": "第28節並列的兩個 Fraktur 記號，分別代表承接自 C3-L 的頻譜矩逃逸量與中間應變逃逸量，均在 blow-up 下必須發散至無窮。",
      "definition_en": "The two Fraktur symbols paired in Sec. 28, denoting the spectral-moment-escape and middle-strain-escape necessary quantities inherited from C3-L, each required to diverge to infinity under blow-up."
    },
    {
      "id": "ns.c3.c3m.chi_n_core",
      "latex": "\\chi_n(x),\\ \\chi_{x_0,R}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "祖源核心截斷函數",
      "label_en": "ancestry-core cutoff function",
      "definition_zh": "第29節與第38節（N1）引入的光滑截斷函數：χn 用於 ancestry core，χ_{x0,R} 為以 x0 為中心、半徑 R 的一般版本，用來 localize Betchov 恆等式。",
      "definition_en": "The smooth cutoff functions introduced in Sec. 29 and Sec. 38 (N1): χn for the ancestry core, and the general centered version χ_{x0,R}, used to localize the Betchov identity."
    },
    {
      "id": "ns.c3.c3m.frak_b_chi",
      "latex": "\\mathfrak B_\\chi",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "局部化Betchov缺項",
      "label_en": "localized Betchov defect (integral)",
      "definition_zh": "第29節定義的局部化 Betchov 積分（以 χn 截斷 ω·Sω+4detS 所得），記錄被全域 Betchov 積分抹去、在截斷區域內殘留的局部 orientation/cancellation 資訊。",
      "definition_en": "The localized Betchov integral defined in Sec. 29 (χn cutting off ω·Sω+4detS), recording the local orientation/cancellation information that survives inside the cutoff region despite vanishing under global Betchov integration.",
      "defining_relation": "\\mathfrak B_{\\chi_n}=\\int\\chi_n\\left(\\omega\\cdot S\\omega+4\\det S\\right)dx"
    },
    {
      "id": "ns.c3.c3m.b_betchov_density",
      "latex": "b(x,t)",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "局部Betchov缺項密度",
      "label_en": "localized Betchov defect density",
      "definition_zh": "第30節定義的逐點密度 b(x,t)=ω·Sω+4detS，全空間積分為零但局部一般非零，是第33節 Θ_stretch tuple 中對應分量的來源。",
      "definition_en": "The pointwise density defined in Sec. 30 as b(x,t)=ω·Sω+4detS, globally integrating to zero yet generically nonzero locally; the source of the corresponding component in the Sec. 33 Θ_stretch tuple.",
      "defining_relation": "b(x,t)=\\omega\\cdot S\\omega+4\\det S"
    },
    {
      "id": "ns.c3.c3m.spatial_compensation_identity",
      "latex": "\\int\\chi b=-\\int(1-\\chi)b",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "空間補償恆等式",
      "label_en": "spatial compensation identity",
      "definition_zh": "第30–31節導出的精確恆等式：任何 core 內的正 Betchov 缺項，必須由 core 外的等量負貢獻精確補償。",
      "definition_en": "The exact identity derived in Sec. 30–31: any positive Betchov defect inside a core region must be exactly compensated by an equal negative contribution outside it.",
      "defining_relation": "\\int\\chi b=-\\int(1-\\chi)b"
    },
    {
      "id": "ns.c3.c3m.spatial_betchov_compensation_debt",
      "latex": "\\textbf{Spatial Betchov Compensation Debt}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "空間Betchov補償債",
      "label_en": "Spatial Betchov Compensation Debt",
      "definition_zh": "C3-M.8（第31節）：ancestry core 內若出現顯著的 local orientation surplus，不能作為孤立純量源存在，必伴隨此補償債務，其能否轉為 boundary flux 或 nonlocal transport 尚未知。",
      "definition_en": "C3-M.8 (Sec. 31): a significant local orientation surplus inside the ancestry core cannot exist as an isolated scalar source — it necessarily carries this compensation debt, whose convertibility into boundary flux or nonlocal transport remains unknown."
    },
    {
      "id": "ns.c3.c3m.theta_stretch_etn",
      "latex": "\\Theta_{\\rm stretch}=\\left\\langle|\\omega|,\\lambda_2,\\lambda_3-\\lambda_2,c_1,c_3,\\xi\\text{-coherence},b_{\\rm Betchov}\\right\\rangle",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "伸展張力（True ETN更新）",
      "label_en": "stretching tension (updated True ETN)",
      "definition_zh": "第33節將原本單純寫成 ω·Sω 的伸展張力 Θ_stretch 重新定義為七分量 typed geometry state tuple，取代單一純量表示。",
      "definition_en": "Sec. 33 redefines the stretching tension Θ_stretch — previously written simply as ω·Sω — as a seven-component typed geometry-state tuple, replacing the single-scalar representation.",
      "defining_relation": "\\Theta_{\\rm stretch}=\\left\\langle|\\omega|,\\lambda_2,\\lambda_3-\\lambda_2,c_1,c_3,\\xi\\text{-coherence},b_{\\rm Betchov}\\right\\rangle"
    },
    {
      "id": "ns.c3.c3m.x_integration_guards",
      "latex": "\\text{G-EIG, G-ORI, G-DIR, G-REV, G-COMP}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "X-Integration更新守衛組",
      "label_en": "updated X-Integration guard suite",
      "definition_zh": "第32節新增的守衛：G-EIG 保存 (λ1,λ2,λ3)、G-ORI 保存 (c1,c2,c3)、G-DIR 保存空間方向相干性且不得與 G-ORI 混同、G-REV 保存 Miller 逆耦合正交性、G-COMP 要求局部 Betchov surplus 必須保存其全域補償來源，與 G-BETCHOV 共同構成本輪稽核集合。",
      "definition_en": "The guards added in Sec. 32: G-EIG preserves (λ1,λ2,λ3), G-ORI preserves (c1,c2,c3), G-DIR preserves spatial direction coherence distinct from G-ORI, G-REV preserves the Miller reverse-coupling orthogonality, and G-COMP requires any local Betchov surplus to retain its global compensation source, together with G-BETCHOV forming this round's audit set."
    },
    {
      "id": "ns.c3.c3m.c3n_frontier",
      "latex": "\\textbf{C3-N — Localized Betchov Compensation and Strain Self-Amplification Rigidity}",
      "series": "NS",
      "first_appearance": "C3-M",
      "label_zh": "下一輪前沿：C3-N",
      "label_en": "next-round frontier: C3-N",
      "definition_zh": "第37–38節命名的下一輪主題「C3-N — Localized Betchov Compensation and Strain Self-Amplification Rigidity」，由本輪的 orientation collapse 結論直接引出，並列出 N1–N7 共七項待證目標。",
      "definition_en": "The next round, named in Sec. 37–38 as \"C3-N — Localized Betchov Compensation and Strain Self-Amplification Rigidity,\" follows directly from this round's orientation-collapse conclusion and lists seven proof obligations N1–N7."
    },
    {
      "id": "ns.c3.c3n.velocity_gradient_a",
      "latex": "A",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "速度梯度張量",
      "label_en": "Velocity gradient tensor",
      "definition_zh": "第1節定義速度梯度 $A_{ij}=\\partial_j u_i$，分解為對稱應變 $S$ 與反對稱部分 $\\Omega$，是本輪 $\\operatorname{tr}(A^3)$ 與 $F_B$ 全部代數的基礎物件。",
      "definition_en": "Defined in Section 1 as the velocity gradient $A_{ij}=\\partial_j u_i$, split into symmetric strain $S$ and antisymmetric part $\\Omega$; the base object for all of this round's $\\operatorname{tr}(A^3)$ and $F_B$ algebra.",
      "defining_relation": "A_{ij}=\\partial_j u_i,\\quad A=S+\\Omega"
    },
    {
      "id": "ns.c3.c3n.rotation_tensor_omega",
      "latex": "\\Omega",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "反對稱旋轉張量",
      "label_en": "Antisymmetric rotation tensor",
      "definition_zh": "第1節中 $A=S+\\Omega$ 分解的反對稱部分，第2節用 $\\Omega^2=\\frac14(\\omega\\otimes\\omega-|\\omega|^2I)$ 推出 Betchov density 公式。",
      "definition_en": "The antisymmetric part in the decomposition $A=S+\\Omega$ from Section 1; Section 2 uses $\\Omega^2=\\frac14(\\omega\\otimes\\omega-|\\omega|^2I)$ to derive the Betchov density formula.",
      "defining_relation": "\\Omega=\\frac12(A-A^\\top),\\quad \\Omega^2=\\frac14\\left(\\omega\\otimes\\omega-|\\omega|^2I\\right)",
      "notes": "From Section 5 onward the same letter is reused for a smooth bounded domain $\\Omega\\subset\\mathbb R^3$, unrelated to this antisymmetric tensor; disambiguate by context."
    },
    {
      "id": "ns.c3.c3n.betchov_density_bb",
      "latex": "b_B",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "局部 Betchov 密度",
      "label_en": "Localized Betchov density",
      "definition_zh": "第2節定義的渦拉伸與應變自放大逐點差量 $b_B=\\omega\\cdot S\\omega+4\\det S=\\frac43\\operatorname{tr}(A^3)$，是全篇 localized Betchov boundary theorem 的核心密度。",
      "definition_en": "The pointwise vortex-stretching/strain-self-amplification mismatch $b_B=\\omega\\cdot S\\omega+4\\det S=\\frac43\\operatorname{tr}(A^3)$ defined in Section 2, the core density of the round's localized Betchov boundary theorem.",
      "defining_relation": "b_B=\\omega\\cdot S\\omega+4\\det S=\\frac43\\operatorname{tr}(A^3)"
    },
    {
      "id": "ns.c3.c3n.betchov_current_fb",
      "latex": "F_B",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "Betchov 空間散度流",
      "label_en": "Betchov spatial divergence current",
      "definition_zh": "第3節定義使 $\\operatorname{tr}(A^3)=\\nabla\\cdot F_B$ 成立的向量場，定理4.1與9.1把 $b_B$ 的局部積分精確化為 $F_B$ 的邊界通量。",
      "definition_en": "The vector field defined in Section 3 satisfying $\\operatorname{tr}(A^3)=\\nabla\\cdot F_B$; Theorems 4.1 and 9.1 turn the localized integral of $b_B$ exactly into a boundary flux of $F_B$.",
      "defining_relation": "F_B=\\left(A^2-\\frac12\\operatorname{tr}(A^2)I\\right)u"
    },
    {
      "id": "ns.c3.c3n.transition_annulus_ar",
      "latex": "\\mathcal A_R",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "過渡環域",
      "label_en": "Transition annulus",
      "definition_zh": "第8節定義的環形區域 $\\mathcal A_R=B_{2R}(x_0)\\setminus B_R(x_0)$，是 $\\nabla\\chi_R$ 的支撐處，後續所有 ball-scale 估計都以此環域上的量表示。",
      "definition_en": "The annular region $\\mathcal A_R=B_{2R}(x_0)\\setminus B_R(x_0)$ defined in Section 8, where $\\nabla\\chi_R$ is supported; all subsequent ball-scale estimates are expressed over this annulus."
    },
    {
      "id": "ns.c3.c3n.scaled_cutoff_chir",
      "latex": "\\chi_R",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "尺度化 cutoff 函數",
      "label_en": "Scaled cutoff family",
      "definition_zh": "第8節定義 $\\chi_R(x)=\\chi_0((x-x_0)/R)$，由基底 bump function $\\chi_0$（於 $B_1$ 為1，支撐於 $B_2$）依尺度 $R$ 縮放而得，用於全部 ball-scale 與 dimensionless compensation 估計。",
      "definition_en": "Defined in Section 8 as $\\chi_R(x)=\\chi_0((x-x_0)/R)$, rescaling a base bump function $\\chi_0$ (equal to 1 on $B_1$, supported in $B_2$) by radius $R$; used throughout the ball-scale and dimensionless compensation estimates."
    },
    {
      "id": "ns.c3.c3n.annular_velocity_amplitude",
      "latex": "a_R",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "環域臨界速度振幅",
      "label_en": "Annular critical velocity amplitude",
      "definition_zh": "第9節定義的無量綱量，以黏性 $\\nu$ 正規化環域 $\\mathcal A_R$ 上速度的 $L^\\infty$ 振幅，是定理9.1中控制 Betchov defect 的兩因子之一。",
      "definition_en": "The dimensionless quantity from Section 9 normalizing the $L^\\infty$ velocity amplitude on $\\mathcal A_R$ by viscosity $\\nu$; one of the two factors bounding the Betchov defect in Theorem 9.1.",
      "defining_relation": "a_R=\\frac{R\\|u\\|_{L^\\infty(\\mathcal A_R)}}{\\nu}"
    },
    {
      "id": "ns.c3.c3n.annular_gradient_stock",
      "latex": "d_R",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "正規化環域梯度庫存",
      "label_en": "Normalized annular gradient stock",
      "definition_zh": "第9節定義的無量綱量，正規化環域 $\\mathcal A_R$ 上速度梯度平方的積分，是定理9.1 $\\widehat{\\mathfrak B}_R\\le Ca_Rd_R$ 中的另一因子。",
      "definition_en": "The dimensionless normalized integral of squared velocity gradient over $\\mathcal A_R$, defined in Section 9; the other factor in Theorem 9.1's bound $\\widehat{\\mathfrak B}_R\\le Ca_Rd_R$.",
      "defining_relation": "d_R=\\frac{R}{\\nu^2}\\int_{\\mathcal A_R}|\\nabla u|^2dx"
    },
    {
      "id": "ns.c3.c3n.normalized_betchov_defect",
      "latex": "\\widehat{\\mathfrak B}_R",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "正規化局部 Betchov 缺量",
      "label_en": "Normalized localized Betchov defect",
      "definition_zh": "第9節定義的無量綱局部 Betchov 缺量，定理9.1證明其被 $Ca_Rd_R$ 上界控制，是 dimensionless boundary compensation 的核心度量。",
      "definition_en": "The dimensionless localized Betchov defect defined in Section 9; Theorem 9.1 bounds it by $Ca_Rd_R$, making it the central measure of dimensionless boundary compensation.",
      "defining_relation": "\\widehat{\\mathfrak B}_R=\\frac{R^3}{\\nu^3}\\left|\\int\\chi_Rb_Bdx\\right|\\le Ca_Rd_R"
    },
    {
      "id": "ns.c3.c3n.magnitude_current_au",
      "latex": "Au",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "應變/渦量強度流",
      "label_en": "Strain/enstrophy magnitude current",
      "definition_zh": "第10節指出 $\\operatorname{tr}(A^2)=\\nabla\\cdot(Au)$（其中 $(Au)_i=u_j\\partial_ju_i$），第42節 G-B2 guard 稱其為 $\\operatorname{tr}(A^2)$ 的 magnitude current。",
      "definition_en": "Section 10 shows $\\operatorname{tr}(A^2)=\\nabla\\cdot(Au)$ (with $(Au)_i=u_j\\partial_ju_i$); Section 42's G-B2 guard names it the magnitude current for $\\operatorname{tr}(A^2)$.",
      "defining_relation": "\\operatorname{tr}(A^2)=\\nabla\\cdot(Au),\\quad (Au)_i=u_j\\partial_ju_i"
    },
    {
      "id": "ns.c3.c3n.pressure_current_fp",
      "latex": "F_p",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "壓力 Hessian 散度流",
      "label_en": "Pressure-Hessian divergence current",
      "definition_zh": "第15節定義的向量場，引理15.1證明 $\\nabla\\cdot F_p=S:\\nabla^2p$，使壓力項也化為邊界散度流；第24節強調此為 provenance-preserving 變換而非 locality 定理。",
      "definition_en": "The vector field defined in Section 15; Lemma 15.1 shows $\\nabla\\cdot F_p=S:\\nabla^2p$, converting the pressure term into a boundary divergence current too, though Section 24 stresses this is a provenance-preserving transform, not a locality theorem.",
      "defining_relation": "F_p=\\left(\\nabla^2p-\\Delta p\\,I\\right)u,\\quad \\nabla\\cdot F_p=S:\\nabla^2p"
    },
    {
      "id": "ns.c3.c3n.local_strain_energy",
      "latex": "E_S^\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "局部應變能量",
      "label_en": "Local strain energy",
      "definition_zh": "第16節定義的 cutoff 加權應變能量，是定理17.1 exact local strain self-amplification balance 的主變數。",
      "definition_en": "The cutoff-weighted strain energy defined in Section 16; the primary variable of Theorem 17.1's exact local strain self-amplification balance.",
      "defining_relation": "E_S^\\chi(t)=\\frac12\\int\\chi(t,x)|S(x,t)|^2dx"
    },
    {
      "id": "ns.c3.c3n.boundary_gauge_package",
      "latex": "\\mathcal C_\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "邊界/規範修正包",
      "label_en": "Boundary/gauge correction package",
      "definition_zh": "定理17.1中把 cutoff motion、advection、viscous localization、Betchov mismatch 與 pressure Hessian 五種貢獻打包成的修正項，是本輪最核心的新物件，將 local strain growth分解為 bulk self-amplification加此項。",
      "definition_en": "The correction term bundling cutoff motion, advection, viscous localization, Betchov mismatch, and pressure-Hessian contributions in Theorem 17.1; this round's most central new object, splitting local strain growth into bulk self-amplification plus this package.",
      "defining_relation": "\\mathcal C_\\chi=\\frac12\\int|S|^2\\left(\\partial_t\\chi+u\\cdot\\nabla\\chi+\\nu\\Delta\\chi\\right)dx+\\frac13\\int\\nabla\\chi\\cdot F_B\\,dx+\\int\\nabla\\chi\\cdot F_p\\,dx"
    },
    {
      "id": "ns.c3.c3n.miller_projection_pst",
      "latex": "P_{st}",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "Miller 無散無跡投影算子",
      "label_en": "Miller's solenoidal-traceless projection operator",
      "definition_zh": "第20節引用 Miller strain decomposition 的投影算子，用於把 full N–S strain equation 寫成 $\\partial_tS-\\nu\\Delta S+\\frac23P_{st}(S^2)+P_{st}(\\cdots)=0$，本輪僅援引以比較 self-amplification model，未重新定義。",
      "definition_en": "The projection operator from Miller's strain decomposition, quoted in Section 20 to write the full N-S strain equation as $\\partial_tS-\\nu\\Delta S+\\frac23P_{st}(S^2)+P_{st}(\\cdots)=0$; invoked only to compare against the self-amplification model, not redefined here.",
      "notes": "External notation from E. Miller's papers (refs 3, 6), first invoked in this round's Section 20 rather than defined in it; its precise operator definition is not restated in this document."
    },
    {
      "id": "ns.c3.c3n.bulk_ssa_achi",
      "latex": "\\mathcal A_\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "局部整體應變自放大量",
      "label_en": "Local bulk strain self-amplification",
      "definition_zh": "第21節定義為定理17.1中唯一保留在 bulk 體積內的三次生成項，滿足 $\\frac d{dt}E_S^\\chi+\\nu\\int\\chi|\\nabla S|^2=\\mathcal A_\\chi+\\mathcal C_\\chi$，是第21節 dichotomy 與第41節 rigidity ratio $\\mathfrak D_n$ 的分母。",
      "definition_en": "Defined in Section 21 as the sole cubic production term retained in the bulk volume in Theorem 17.1, satisfying $\\frac d{dt}E_S^\\chi+\\nu\\int\\chi|\\nabla S|^2=\\mathcal A_\\chi+\\mathcal C_\\chi$; anchors the Section 21 dichotomy and is the denominator of the rigidity ratio $\\mathfrak D_n$ in Section 41.",
      "defining_relation": "\\mathcal A_\\chi=-2\\int\\chi\\det S\\,dx",
      "notes": "Section 34 redefines the same symbol as $\\mathcal A_\\chi=-4\\int\\chi\\det S$ (coefficient 4, not 2) when pairing with $\\mathcal V_\\chi$ for the Betchov VS/SSA sign convention; the two boxed definitions are numerically inconsistent in the source. This entry follows the Section 21/Theorem 17.1 usage, which is the one carried through Sections 26 and 39-41."
    },
    {
      "id": "ns.c3.c3n.term_c_gauge",
      "latex": "\\frac12\\int|S|^2\\partial_t\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "C-GAUGE：移動核心再分類項",
      "label_en": "C-GAUGE: moving-core reclassification term",
      "definition_zh": "第22節將 $\\mathcal C_\\chi$ 拆解出的第一型分量，對應 cutoff 隨時間移動造成的 core reclassification。",
      "definition_en": "The first labeled component of $\\mathcal C_\\chi$'s decomposition in Section 22, corresponding to core reclassification caused by a time-moving cutoff."
    },
    {
      "id": "ns.c3.c3n.term_c_adv",
      "latex": "\\frac12\\int|S|^2u\\cdot\\nabla\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "C-ADV：邊界物理平流項",
      "label_en": "C-ADV: physical advection-through-boundary term",
      "definition_zh": "第22節 $\\mathcal C_\\chi$ 分解中對應速度場把應變能量平流穿過 cutoff 邊界的分量。",
      "definition_en": "The component of $\\mathcal C_\\chi$'s Section 22 decomposition corresponding to the velocity field advecting strain energy through the cutoff boundary."
    },
    {
      "id": "ns.c3.c3n.term_c_diff",
      "latex": "\\frac\\nu2\\int|S|^2\\Delta\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "C-DIFF：黏性邊界修正項",
      "label_en": "C-DIFF: viscous boundary correction term",
      "definition_zh": "第22節 $\\mathcal C_\\chi$ 分解中由黏性擴散作用於 cutoff 邊界產生的修正分量。",
      "definition_en": "The viscous-diffusion correction component of $\\mathcal C_\\chi$'s Section 22 decomposition, arising at the cutoff boundary."
    },
    {
      "id": "ns.c3.c3n.term_c_betchov",
      "latex": "\\frac13\\int\\nabla\\chi\\cdot F_B",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "C-BETCHOV：渦拉伸/應變自放大失配流",
      "label_en": "C-BETCHOV: vortex-stretching/SSA mismatch current",
      "definition_zh": "第22節 $\\mathcal C_\\chi$ 分解中直接來自 $F_B$ 的分量，承載 vortex-stretching 與 strain-self-amplification 的局部失配。",
      "definition_en": "The component of $\\mathcal C_\\chi$'s Section 22 decomposition coming directly from $F_B$, carrying the local mismatch between vortex stretching and strain self-amplification."
    },
    {
      "id": "ns.c3.c3n.term_c_press",
      "latex": "\\int\\nabla\\chi\\cdot F_p",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "C-PRESS：壓力 Hessian 邊界流",
      "label_en": "C-PRESS: pressure-Hessian boundary current",
      "definition_zh": "第22節 $\\mathcal C_\\chi$ 分解中由壓力 Hessian 流 $F_p$ 貢獻的邊界分量。",
      "definition_en": "The boundary component of $\\mathcal C_\\chi$'s Section 22 decomposition contributed by the pressure-Hessian current $F_p$."
    },
    {
      "id": "ns.c3.c3n.generation_betchov_current",
      "latex": "\\mathfrak J_{B,n}",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "第n代 Betchov 邊界流",
      "label_en": "Generation-n Betchov boundary current",
      "definition_zh": "第33節對 ancestry core半徑 $R_n$ 定義，恰等於該球內 $\\int(\\omega\\cdot S\\omega+4\\det S)dx$，是 $\\operatorname{XBetchov}_n$ 證書元組的第一分量。",
      "definition_en": "Defined in Section 33 over the ancestry core radius $R_n$, exactly equal to $\\int_{B_{R_n}}(\\omega\\cdot S\\omega+4\\det S)dx$; the first component of the $\\operatorname{XBetchov}_n$ certificate tuple.",
      "defining_relation": "\\mathfrak J_{B,n}=\\frac43\\int_{\\partial B_{R_n}(x_n)}F_B\\cdot n\\,dS=\\int_{B_{R_n}(x_n)}\\left(\\omega\\cdot S\\omega+4\\det S\\right)dx"
    },
    {
      "id": "ns.c3.c3n.xbetchov_certificate",
      "latex": "\\operatorname{XBetchov}_n",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "XBetchov 證書元組",
      "label_en": "XBetchov certificate tuple",
      "definition_zh": "第33節定義的 X-Integration 型元組 $\\langle\\mathfrak J_{B,n},\\mathcal V_n,\\mathcal A_n,\\operatorname{ProvBoundary}_n\\rangle$，收錄第n代 core 的 Betchov 邊界流、渦拉伸、應變自放大表示 $\\mathcal A_n=-4\\int\\det S$ 與邊界來源標記。",
      "definition_en": "The X-Integration-style tuple $\\langle\\mathfrak J_{B,n},\\mathcal V_n,\\mathcal A_n,\\operatorname{ProvBoundary}_n\\rangle$ defined in Section 33, recording generation-n's Betchov boundary current, vortex stretching, strain-self-amplification representation $\\mathcal A_n=-4\\int\\det S$, and boundary-provenance tag."
    },
    {
      "id": "ns.c3.c3n.local_vortex_stretching",
      "latex": "\\mathcal V_\\chi",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "局部渦拉伸量",
      "label_en": "Local vortex-stretching functional",
      "definition_zh": "第34節定義 $\\mathcal V_\\chi=\\int\\chi\\,\\omega\\cdot S\\omega$，與同節的 $\\mathcal A_\\chi=-4\\int\\chi\\det S$ 配對給出 $\\mathcal V_\\chi-\\mathcal A_\\chi=-\\frac43\\int\\nabla\\chi\\cdot F_B$，是定理35.1 local VS/SSA imbalance 的主角。",
      "definition_en": "Defined in Section 34 as $\\mathcal V_\\chi=\\int\\chi\\,\\omega\\cdot S\\omega$, paired there with $\\mathcal A_\\chi=-4\\int\\chi\\det S$ to give $\\mathcal V_\\chi-\\mathcal A_\\chi=-\\frac43\\int\\nabla\\chi\\cdot F_B$; the central quantity of Theorem 35.1's local VS/SSA imbalance result.",
      "notes": "Pairs specifically with the Section 34 variant of $\\mathcal A_\\chi$ (coefficient $-4$), not the Section 21/Theorem 17.1 variant (coefficient $-2$) used elsewhere in this table."
    },
    {
      "id": "ns.c3.c3n.bulk_boundary_ratio",
      "latex": "\\mathfrak D_n",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "bulk/boundary 漸近比率",
      "label_en": "Bulk/boundary asymptotic ratio",
      "definition_zh": "第41節定義的新 rigidity target，其極限行為（$\\to0$、$\\to-1$、或振盪/無界）分類出下一輪 C3-O 要研究的三種可能分支。",
      "definition_en": "The new rigidity target defined in Section 41 closing this round; its limiting behavior ($\\to0$, $\\to-1$, or oscillatory/unbounded) classifies the three branches to be studied in the next round, C3-O.",
      "defining_relation": "\\mathfrak D_n=\\frac{\\mathcal C_{\\chi_n}}{-2\\int\\chi_n\\det S}"
    },
    {
      "id": "ns.c3.c3n.true_etn_strain_state",
      "latex": "\\Theta_n^{strain}",
      "series": "NS",
      "first_appearance": "C3-N",
      "label_zh": "True ETN 應變張力態",
      "label_en": "True ETN strain-tension state",
      "definition_zh": "第43節將 True ETN 框架具體化為應變版本的八分量元組，滿足 exact balance $\\dot E_{S,n}+D_{S,n}=A_{SSA,n}+\\sum J_n$，把本輪所有邊界/bulk修正正式收進一個 typed local balance。",
      "definition_en": "Section 43 instantiates the True ETN framework as an eight-component strain-specific tuple satisfying the exact balance $\\dot E_{S,n}+D_{S,n}=A_{SSA,n}+\\sum J_n$, formally folding all of this round's boundary/bulk corrections into one typed local balance.",
      "defining_relation": "\\Theta_n^{strain}=\\left\\langle E_{S,n},D_{S,n},A_{SSA,n},J_{B,n},J_{adv,n},J_{press,n},J_{diff,n},J_{gauge,n}\\right\\rangle,\\quad \\dot E_{S,n}+D_{S,n}=A_{SSA,n}+\\sum J_{n}"
    },
    {
      "id": "ns.c3.c3o.chi_adjoint",
      "latex": "\\chi",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "伴隨截斷函數",
      "label_en": "Adjoint cutoff",
      "definition_zh": "第2節定義：在 ancestry window $I=[t_0,t_1]$ 上解 backward adjoint 方程 $\\partial_t\\chi+u\\cdot\\nabla\\chi+\\nu\\Delta\\chi=0$、終端條件 $\\chi(t_1,x)=\\chi_1(x)$ 的截斷函數；其軌跡稱 Adjoint Ancestry Tube，backward 跟隨速度場並 parabolic 擴散，自動吸收原本 cutoff 移動造成的 gauge/advection/diffusion 誤差。",
      "definition_en": "Defined in Sec. 2 as the cutoff solving the backward adjoint equation $\\partial_t\\chi+u\\cdot\\nabla\\chi+\\nu\\Delta\\chi=0$ on $I=[t_0,t_1]$ with terminal data $\\chi(t_1,x)=\\chi_1(x)$; its trajectory (the \"Adjoint Ancestry Tube\") backward-follows the velocity drift and diffuses parabolically, automatically absorbing gauge/advection/diffusion cutoff-error terms.",
      "defining_relation": "\\partial_t\\chi+u\\cdot\\nabla\\chi+\\nu\\Delta\\chi=0,\\qquad \\chi(t_1,x)=\\chi_1(x)",
      "notes": "與 C3-N 中泛用的 $\\chi$（產生一般 $\\mathcal C_\\chi$ 修正項）不同，本輪把 $\\chi$ 特化為此 adjoint PDE 的解，使 $\\mathcal C_\\chi$ 的 gauge/advection/diffusion 部分 exact 消失，只剩 $B_\\chi$。"
    },
    {
      "id": "ns.c3.c3o.chi_1",
      "latex": "\\chi_1",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "終端截斷",
      "label_en": "Terminal cutoff",
      "definition_zh": "第2節：終端時刻 $t_1$ 的給定截斷 $0\\le\\chi_1\\le1$，localized near child ancestry core，作為 adjoint cutoff 方程的終端條件。",
      "definition_en": "Sec. 2: a prescribed cutoff $0\\le\\chi_1\\le1$ at terminal time $t_1$, localized near the child ancestry core, serving as terminal datum for the adjoint cutoff equation.",
      "defining_relation": "0\\le\\chi_1\\le1,\\qquad \\chi(t_1,x)=\\chi_1(x)"
    },
    {
      "id": "ns.c3.c3o.j_corr",
      "latex": "J_{\\rm corr}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "修正電流",
      "label_en": "Correction current",
      "definition_zh": "定理4.1（C3-O.1）定義的合併修正項，$J_{\\rm corr}=\\frac13F_B+F_p$，把 Betchov force $F_B$ 與 pressure current $F_p$ 合併成 adjoint balance 右端唯一的邊界型 correction。",
      "definition_en": "Defined in Thm. 4.1 (C3-O.1) as the combined correction $J_{\\rm corr}=\\frac13F_B+F_p$, merging the Betchov force $F_B$ and pressure current $F_p$ into the single boundary-type correction on the RHS of the adjoint balance.",
      "defining_relation": "J_{\\rm corr}=\\frac13F_B+F_p"
    },
    {
      "id": "ns.c3.c3o.e_chi",
      "latex": "E_\\chi(t)",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "局部應變能量",
      "label_en": "Localized strain energy",
      "definition_zh": "第5節定義，$E_\\chi(t)=\\frac12\\int\\chi|S|^2dx$，adjoint-cutoff 加權的局部應變能量，為 gauge-clean balance $E_\\chi'+D_\\chi=A_\\chi+B_\\chi$ 的能量項。",
      "definition_en": "Defined in Sec. 5 as $E_\\chi(t)=\\frac12\\int\\chi|S|^2dx$, the adjoint-cutoff-weighted local strain energy in the gauge-clean balance $E_\\chi'+D_\\chi=A_\\chi+B_\\chi$.",
      "defining_relation": "E_\\chi(t)=\\frac12\\int\\chi|S|^2dx"
    },
    {
      "id": "ns.c3.c3o.d_chi",
      "latex": "D_\\chi(t)",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "局部耗散",
      "label_en": "Localized dissipation",
      "definition_zh": "第5節定義，$D_\\chi(t)=\\nu\\int\\chi|\\nabla S|^2dx\\ge0$，adjoint-cutoff 加權的局部黏性耗散項。",
      "definition_en": "Defined in Sec. 5 as $D_\\chi(t)=\\nu\\int\\chi|\\nabla S|^2dx\\ge0$, the adjoint-cutoff-weighted local viscous dissipation.",
      "defining_relation": "D_\\chi(t)=\\nu\\int\\chi|\\nabla S|^2dx"
    },
    {
      "id": "ns.c3.c3o.a_chi",
      "latex": "A_\\chi(t)",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "局部自放大項",
      "label_en": "Localized self-amplification term",
      "definition_zh": "第5節定義，$A_\\chi(t)=-2\\int\\chi\\det S\\,dx$，局部 strain self-amplification（SSA）產生項，是 balance 方程唯一的 bulk（非邊界）來源。",
      "definition_en": "Defined in Sec. 5 as $A_\\chi(t)=-2\\int\\chi\\det S\\,dx$, the local strain self-amplification (SSA) production term — the sole bulk (non-boundary) source in the balance equation.",
      "defining_relation": "A_\\chi(t)=-2\\int\\chi\\det S\\,dx",
      "notes": "勿與 C3-N recap（第0節）中速度梯度張量 $A=\\nabla u$ 混淆；兩者皆記作 \"A\" 但意義完全不同。"
    },
    {
      "id": "ns.c3.c3o.b_chi",
      "latex": "B_\\chi(t)",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "局部修正/邊界電流",
      "label_en": "Localized correction/boundary current",
      "definition_zh": "第5節定義，$B_\\chi(t)=\\int\\nabla\\chi\\cdot J_{\\rm corr}\\,dx$；在 adjoint cutoff 下，這是 C3-N 之 $\\mathcal C_\\chi$ 僅存的部分——gauge/advection/diffusion 項因 $\\chi$ 滿足 adjoint 方程而消失。",
      "definition_en": "Defined in Sec. 5 as $B_\\chi(t)=\\int\\nabla\\chi\\cdot J_{\\rm corr}\\,dx$; under the adjoint cutoff this is all that remains of C3-N's $\\mathcal C_\\chi$, since the gauge/advection/diffusion piece vanishes once $\\chi$ solves the adjoint equation.",
      "defining_relation": "B_\\chi(t)=\\int\\nabla\\chi\\cdot J_{\\rm corr}\\,dx"
    },
    {
      "id": "ns.c3.c3o.delta_e_i",
      "latex": "\\Delta E_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "窗口能量變化",
      "label_en": "Window energy change",
      "definition_zh": "第6節定義，$\\Delta E_I=E_\\chi(t_1)-E_\\chi(t_0)$，窗口 $I=[t_0,t_1]$ 上局部應變能量淨變化；$\\Delta E_I>0$ 即第7節的 positive local strain-growth window。",
      "definition_en": "Defined in Sec. 6 as $\\Delta E_I=E_\\chi(t_1)-E_\\chi(t_0)$, the net change of local strain energy over $I=[t_0,t_1]$; $\\Delta E_I>0$ defines a positive local strain-growth window (Sec. 7).",
      "defining_relation": "\\Delta E_I=E_\\chi(t_1)-E_\\chi(t_0)"
    },
    {
      "id": "ns.c3.c3o.d_i",
      "latex": "D_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "窗口耗散",
      "label_en": "Window-integrated dissipation",
      "definition_zh": "第6節定義，$D_I=\\int_ID_\\chi(t)\\,dt\\ge0$，窗口 $I$ 上累積的黏性耗散。",
      "definition_en": "Defined in Sec. 6 as $D_I=\\int_ID_\\chi(t)\\,dt\\ge0$, the cumulative viscous dissipation over window $I$.",
      "defining_relation": "D_I=\\int_ID_\\chi(t)\\,dt"
    },
    {
      "id": "ns.c3.c3o.a_i",
      "latex": "A_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "窗口自放大量",
      "label_en": "Window-integrated self-amplification",
      "definition_zh": "第6節定義，$A_I=\\int_IA_\\chi(t)\\,dt$，窗口累積 bulk SSA production；其正負決定第8節 Growth-Carrier Dichotomy 的分支。",
      "definition_en": "Defined in Sec. 6 as $A_I=\\int_IA_\\chi(t)\\,dt$, cumulative bulk SSA production over $I$; its sign determines which branch of the Growth-Carrier Dichotomy (Sec. 8) applies.",
      "defining_relation": "A_I=\\int_IA_\\chi(t)\\,dt"
    },
    {
      "id": "ns.c3.c3o.b_i",
      "latex": "B_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "窗口修正/邊界電流",
      "label_en": "Window-integrated correction/boundary current",
      "definition_zh": "第6節定義，$B_I=\\int_IB_\\chi(t)\\,dt$，滿足 window balance $\\Delta E_I+D_I=A_I+B_I$；第27節再分解為 $B_I=B_I^B+B_I^p$。",
      "definition_en": "Defined in Sec. 6 as $B_I=\\int_IB_\\chi(t)\\,dt$, satisfying $\\Delta E_I+D_I=A_I+B_I$; further split in Sec. 27 as $B_I=B_I^B+B_I^p$.",
      "defining_relation": "B_I=\\int_IB_\\chi(t)\\,dt"
    },
    {
      "id": "ns.c3.c3o.rho_i",
      "latex": "\\rho_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "邊界/自放大比",
      "label_en": "Boundary-to-bulk ratio",
      "definition_zh": "第9節定義（$A_I>0$ 時），$\\rho_I=B_I/A_I$；growth 要求 $\\rho_I>-1$，$\\rho_I\\le-1$ 為 C3-O.3 Hard Depletion Barrier（第10節）排除區。本輪核心 growth-carrier 分類指標，餵入第14節 O-A/O-B/O-C 與第23節 BD-1至BD-4。",
      "definition_en": "Defined in Sec. 9 (for $A_I>0$) as $\\rho_I=B_I/A_I$; growth forces $\\rho_I>-1$, and $\\rho_I\\le-1$ is excluded by the Hard Depletion Barrier (C3-O.3, Sec. 10). The round's central growth-carrier classifier, feeding the O-A/O-B/O-C regimes (Sec. 14) and BD-1…BD-4 plane (Sec. 23).",
      "defining_relation": "\\rho_I=\\frac{B_I}{A_I}",
      "notes": "全文最終結論（第29、36節）：$\\rho$ 只能作 local strain-energy growth-carrier classifier，不能單獨作 regularity parameter。"
    },
    {
      "id": "ns.c3.c3o.kappa_i",
      "latex": "\\kappa_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "抵消走廊參數",
      "label_en": "Cancellation-corridor parameter",
      "definition_zh": "第11節定義（$A_I>0$ 時），$\\kappa_I=1+\\rho_I=(\\Delta E_I+D_I)/A_I$，growth window 恆正。第12節 C3-O.4：$\\rho_I\\to-1^+$（即 $\\kappa_I\\to0^+$）而 $\\Delta E_I+D_I$ 未同比例趨零時，需 $A_I\\to\\infty$ 且 $|B_I|\\sim A_I$（cancellation-precision debt）。",
      "definition_en": "Defined in Sec. 11 (for $A_I>0$) as $\\kappa_I=1+\\rho_I=(\\Delta E_I+D_I)/A_I$, positive on growth windows. Sec. 12 (C3-O.4): as $\\rho_I\\to-1^+$ ($\\kappa_I\\to0^+$) with $\\Delta E_I+D_I$ not vanishing proportionally, $A_I\\to\\infty$ and $|B_I|\\sim A_I$ — the cancellation-precision debt.",
      "defining_relation": "\\kappa_I=1+\\rho_I=\\frac{\\Delta E_I+D_I}{A_I}"
    },
    {
      "id": "ns.c3.c3o.p_st",
      "latex": "P_{st}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "(Miller) 應變空間投影算子",
      "label_en": "Miller's strain-space projection operator",
      "definition_zh": "第15節引入（源自 Miller 的分解，本文未重新給出其明確定義），用以把 full strain equation 寫成 $\\partial_tS-\\nu\\Delta S+\\frac23P_{st}(S^2)+\\mathcal P_{NS}=0$，並建構 $\\mathcal N_{SSA}$ 與 $\\mathcal P_{NS}$。",
      "definition_en": "Introduced in Sec. 15 (from Miller's decomposition; explicit definition not restated here) to write the full strain equation as $\\partial_tS-\\nu\\Delta S+\\frac23P_{st}(S^2)+\\mathcal P_{NS}=0$, and to build both $\\mathcal N_{SSA}$ and $\\mathcal P_{NS}$.",
      "notes": "與 $\\mathcal P_{NS}$（calligraphic，算子缺陷場）及 $\\mathfrak P_I$（fraktur，純量 diagnostic）為三個不同的 \"P\" 家族記號，勿混淆。"
    },
    {
      "id": "ns.c3.c3o.p_ns",
      "latex": "\\mathcal P_{NS}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "NS算子缺陷（被丟棄項）",
      "label_en": "NS operator defect (omitted perturbation)",
      "definition_zh": "第15節定義，$\\mathcal P_{NS}=P_{st}\\left((u\\cdot\\nabla)S+\\frac13S^2+\\frac14\\omega\\otimes\\omega\\right)$，即 full N–S strain equation 相對 SSA model 被丟棄的算子級項。第16節證 $\\langle\\mathcal P_{NS},S\\rangle=0$（正交但不代表小，命題17.1/C3-O.5），是本輪 no-go 論證與第31節 $\\Theta^{op}$ 的核心對象。",
      "definition_en": "Defined in Sec. 15 as $\\mathcal P_{NS}=P_{st}\\left((u\\cdot\\nabla)S+\\frac13S^2+\\frac14\\omega\\otimes\\omega\\right)$, the operator-level perturbation omitted from full N–S relative to the SSA model. Sec. 16 shows $\\langle\\mathcal P_{NS},S\\rangle=0$ (orthogonal but not small — Prop. 17.1/C3-O.5), the central object of this round's no-go argument and of $\\Theta^{op}$.",
      "defining_relation": "\\mathcal P_{NS}=P_{st}\\left((u\\cdot\\nabla)S+\\frac13S^2+\\frac14\\omega\\otimes\\omega\\right)",
      "notes": "G-OP guard：$B/A\\to0$ 不得推出 $\\mathcal P_{NS}\\to0$。G-PROJ guard：全域正交 $\\langle\\mathcal P_{NS},S\\rangle=0$ 只是 orthogonality，不是 smallness。"
    },
    {
      "id": "ns.c3.c3o.n_ssa",
      "latex": "\\mathcal N_{SSA}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "SSA非線性項",
      "label_en": "SSA nonlinearity",
      "definition_zh": "第31節（True ETN更新）定義，$\\mathcal N_{SSA}=\\frac23P_{st}(S^2)$，strain self-amplification model 自身的非線性項，作為 $\\Theta^{op}$ 第一分量。",
      "definition_en": "Defined in Sec. 31 (True ETN update) as $\\mathcal N_{SSA}=\\frac23P_{st}(S^2)$, the nonlinearity of the SSA model itself, the first component of $\\Theta^{op}$.",
      "defining_relation": "\\mathcal N_{SSA}=\\frac23P_{st}(S^2)"
    },
    {
      "id": "ns.c3.c3o.mathfrak_p_i",
      "latex": "\\mathfrak P_I",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "尺度不變算子缺陷比",
      "label_en": "Scale-invariant operator-defect ratio",
      "definition_zh": "第20節定義的 candidate diagnostic，$\\mathfrak P_I=\\frac{\\int_I\\|\\mathcal P_{NS}(t)\\|_{\\dot H^{-1}}^2dt}{\\nu^2\\int_I\\|S(t)\\|_{\\dot H^1}^2dt}$（分母非零時）；第21節證其在 N–S parabolic scaling 下不變。第22節強調僅是 candidate：未證 $\\mathfrak P_I<\\varepsilon\\Rightarrow$SSA近似，亦未證 $\\mathfrak P_I\\gg1\\Rightarrow$regularity，作用是防止「zero energy pairing」被偷換成「small operator」。",
      "definition_en": "The candidate diagnostic of Sec. 20, $\\mathfrak P_I=\\frac{\\int_I\\|\\mathcal P_{NS}(t)\\|_{\\dot H^{-1}}^2dt}{\\nu^2\\int_I\\|S(t)\\|_{\\dot H^1}^2dt}$ (denominator nonzero); shown scale-invariant under NS parabolic scaling in Sec. 21. Sec. 22: only a candidate — neither $\\mathfrak P_I<\\varepsilon\\Rightarrow$SSA-approximation nor $\\mathfrak P_I\\gg1\\Rightarrow$regularity is proved; it exists to block \"zero energy pairing\" from being conflated with \"small operator.\"",
      "defining_relation": "\\mathfrak P_I=\\frac{\\int_I\\|\\mathcal P_{NS}(t)\\|_{\\dot H^{-1}}^2dt}{\\nu^2\\int_I\\|S(t)\\|_{\\dot H^1}^2dt}",
      "notes": "與 $\\rho_I$ 共組第23節 Balance–Dynamics plane 之 BD-1（both small）、BD-2（balance-SSA但operator-large）、BD-3（cancellation corridor）、BD-4（boundary driven）四區。"
    },
    {
      "id": "ns.c3.c3o.theta_bal",
      "latex": "\\Theta^{bal}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "平衡層狀態",
      "label_en": "Balance-layer state",
      "definition_zh": "第31節 True ETN 更新定義，$\\Theta^{bal}=(E,D,A,B,\\rho,\\kappa)$，局部應變態的「平衡層」，由本輪能量/耗散/自放大/邊界電流/比值/走廊參數家族組成，對比 $\\Theta^{op}$。",
      "definition_en": "Defined in Sec. 31's True ETN update as $\\Theta^{bal}=(E,D,A,B,\\rho,\\kappa)$, the \"balance layer\" of the local strain state, contrasted with $\\Theta^{op}$.",
      "defining_relation": "\\Theta^{bal}=(E,D,A,B,\\rho,\\kappa)",
      "notes": "分量沿用 $E_\\chi,D_\\chi,A_\\chi,B_\\chi,\\rho_I,\\kappa_I$（或其窗口版）的通用寫法，未加下標。"
    },
    {
      "id": "ns.c3.c3o.theta_op",
      "latex": "\\Theta^{op}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "算子層狀態",
      "label_en": "Operator-layer state",
      "definition_zh": "第31節定義，$\\Theta^{op}=(\\mathcal N_{SSA},\\mathcal P_{NS},\\mathfrak P,\\operatorname{Prov})$，局部應變態的「算子層」。核心結論：$\\Theta^{bal}$ convergence 不推出 $\\Theta^{op}$ convergence——即第30節 Balance Fixed Point / Dynamics Fixed Point Separation 在狀態層級的正式化。",
      "definition_en": "Defined in Sec. 31 as $\\Theta^{op}=(\\mathcal N_{SSA},\\mathcal P_{NS},\\mathfrak P,\\operatorname{Prov})$, the \"operator layer.\" Central conclusion: $\\Theta^{bal}$ convergence does not imply $\\Theta^{op}$ convergence — formalizing the Balance/Dynamics Fixed Point Separation (Sec. 30).",
      "defining_relation": "\\Theta^{op}=\\left(\\mathcal N_{SSA},\\mathcal P_{NS},\\mathfrak P,\\operatorname{Prov}\\right)"
    },
    {
      "id": "ns.c3.c3o.prov",
      "latex": "\\operatorname{Prov}",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "溯源分量",
      "label_en": "Provenance component",
      "definition_zh": "僅在第31節 $\\Theta^{op}$ 元組末位出現一次，文中未再展開定義。",
      "definition_en": "Appears only once, as the fourth component of $\\Theta^{op}$ in Sec. 31; not elaborated elsewhere in the text.",
      "notes": "信心低：本檔案未給出 $\\operatorname{Prov}$ 的明確定義式；字面推測與 ancestry/adjoint tube 的來源追蹤有關，屬未經文本證實的推測，需後續 round 或原始定義確認。"
    },
    {
      "id": "ns.c3.c3o.b_i_betchov",
      "latex": "B_I^B",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "窗口Betchov電流分量",
      "label_en": "Window Betchov-current component",
      "definition_zh": "第27節定義，$B_I^B=\\frac13\\int_I\\int\\nabla\\chi\\cdot F_B$，$B_I=B_I^B+B_I^p$ 分解中源自 $F_B$ 的部分；第28節示 $\\rho_I\\to-1^+$ 時，$B_I^B,B_I^p$ 至少一者須達 $O(A_I)$ 量級。",
      "definition_en": "Defined in Sec. 27 as $B_I^B=\\frac13\\int_I\\int\\nabla\\chi\\cdot F_B$, the Betchov-driven part of $B_I=B_I^B+B_I^p$; Sec. 28 shows as $\\rho_I\\to-1^+$ at least one of $B_I^B,B_I^p$ must reach $O(A_I)$ magnitude.",
      "defining_relation": "B_I^B=\\frac13\\int_I\\int\\nabla\\chi\\cdot F_B"
    },
    {
      "id": "ns.c3.c3o.b_i_pressure",
      "latex": "B_I^p",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "窗口壓力電流分量",
      "label_en": "Window pressure-current component",
      "definition_zh": "第27節定義，$B_I^p=\\int_I\\int\\nabla\\chi\\cdot F_p$，源自 $F_p$ 的部分；與 $B_I^B$ 合稱把 boundary-dominated branch 再分為 Betchov-current dominated / pressure-current dominated。",
      "definition_en": "Defined in Sec. 27 as $B_I^p=\\int_I\\int\\nabla\\chi\\cdot F_p$; together with $B_I^B$ refines the boundary-dominated branch into Betchov- vs. pressure-current-dominated sub-branches.",
      "defining_relation": "B_I^p=\\int_I\\int\\nabla\\chi\\cdot F_p"
    },
    {
      "id": "ns.c3.c3o.f_b",
      "latex": "F_B",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "Betchov型力場",
      "label_en": "Betchov-type force field",
      "definition_zh": "源自 C3-N（第0節 recap），$F_B=\\left(A^2-\\frac12\\operatorname{tr}(A^2)I\\right)u$，$A=\\nabla u$；本輪用於構成 $J_{\\rm corr}=\\frac13F_B+F_p$ 及窗口分量 $B_I^B$。",
      "definition_en": "Inherited from C3-N (recapped Sec. 0), $F_B=\\left(A^2-\\frac12\\operatorname{tr}(A^2)I\\right)u$ with $A=\\nabla u$; used this round in $J_{\\rm corr}=\\frac13F_B+F_p$ and its window component $B_I^B$.",
      "defining_relation": "F_B=\\left(A^2-\\frac12\\operatorname{tr}(A^2)I\\right)u,\\qquad A=\\nabla u",
      "notes": "此處 $A=\\nabla u$（速度梯度張量）與本輪的 $A_\\chi,A_I$（self-amplification 項）為不同記號，僅字母相同。"
    },
    {
      "id": "ns.c3.c3o.f_p",
      "latex": "F_p",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "壓力電流",
      "label_en": "Pressure current",
      "definition_zh": "源自 C3-N（第0節 recap），$F_p=(\\nabla^2p-\\Delta p\\,I)u$，含 nonlocal pressure Hessian；第26節強調 gauge-clean 不等於 boundary-small，主因即 $F_p$ 可能很大。",
      "definition_en": "Inherited from C3-N (recapped Sec. 0), $F_p=(\\nabla^2p-\\Delta p\\,I)u$, involving the nonlocal pressure Hessian; Sec. 26 stresses gauge-clean does not imply boundary-small, precisely because $F_p$ can be large.",
      "defining_relation": "F_p=(\\nabla^2p-\\Delta p\\,I)u"
    },
    {
      "id": "ns.c3.c3o.g_adj",
      "latex": "G\\text{-}ADJ",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：伴隨截斷",
      "label_en": "X-Integration guard: adjoint cutoff",
      "definition_zh": "第32節 X-Integration hard guards之一：計算 ratio 時應優先使用 adjoint cutoff，或明確做完整的 gauge subtraction。",
      "definition_en": "One of the Sec. 32 X-Integration hard guards: ratio computations must use the adjoint cutoff, or perform a complete explicit gauge subtraction."
    },
    {
      "id": "ns.c3.c3o.g_grow",
      "latex": "G\\text{-}GROW",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：僅限成長窗口",
      "label_en": "X-Integration guard: growth windows only",
      "definition_zh": "第32節守則：ratio 只能在 $\\Delta E>0$ 的 growth windows 中作 growth-carrier 判斷。",
      "definition_en": "Sec. 32 guard: the ratio may only be used for growth-carrier judgments within $\\Delta E>0$ growth windows."
    },
    {
      "id": "ns.c3.c3o.g_ratio",
      "latex": "G\\text{-}RATIO",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：比值下界",
      "label_en": "X-Integration guard: ratio lower bound",
      "definition_zh": "第32節守則：若 $A>0$，positive growth 要求 $\\rho>-1$。",
      "definition_en": "Sec. 32 guard: if $A>0$, positive growth requires $\\rho>-1$."
    },
    {
      "id": "ns.c3.c3o.g_cancel",
      "latex": "G\\text{-}CANCEL",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：保存總量",
      "label_en": "X-Integration guard: preserve gross quantities",
      "definition_zh": "第32節守則：若 $\\rho\\to-1$，必須保存 gross 的 $A,B$，不能只保存 residual $A+B$。",
      "definition_en": "Sec. 32 guard: if $\\rho\\to-1$, the gross quantities $A,B$ must be preserved, not merely the residual $A+B$."
    },
    {
      "id": "ns.c3.c3o.g_op",
      "latex": "G\\text{-}OP",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：比值趨零不推小算子",
      "label_en": "X-Integration guard: ratio-to-zero does not imply small operator",
      "definition_zh": "第32節守則：$B/A\\to0$ 不得推出 $\\mathcal P_{NS}\\to0$。",
      "definition_en": "Sec. 32 guard: $B/A\\to0$ must not be taken to imply $\\mathcal P_{NS}\\to0$."
    },
    {
      "id": "ns.c3.c3o.g_proj",
      "latex": "G\\text{-}PROJ",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：正交非微小",
      "label_en": "X-Integration guard: orthogonality is not smallness",
      "definition_zh": "第32節守則：全域 $\\langle\\mathcal P_{NS},S\\rangle=0$ 只是 orthogonality，不是 smallness。",
      "definition_en": "Sec. 32 guard: the global identity $\\langle\\mathcal P_{NS},S\\rangle=0$ is only orthogonality, not smallness."
    },
    {
      "id": "ns.c3.c3o.g_press",
      "latex": "G\\text{-}PRESS",
      "series": "NS",
      "first_appearance": "C3-O",
      "label_zh": "X-Integration守則：壓力/Betchov分開保存",
      "label_en": "X-Integration guard: keep pressure/Betchov separate",
      "definition_zh": "第32節守則：pressure 與 Betchov correction 必須分開保存，不可合併後只看總量。",
      "definition_en": "Sec. 32 guard: the pressure and Betchov corrections must be kept separate, not merged and tracked only as a total."
    },
    {
      "id": "ns.c3.c3p.p_st",
      "latex": "P_{st}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "對稱無跡投影算子",
      "label_en": "symmetric trace-free projection",
      "definition_zh": "貫穿全篇、承接自先前各輪、將張量投影到對稱無跡子空間、用以建構 $\\mathcal N_{SSA}$、$\\mathcal N_{SV}$、$\\mathcal P_{SSA}$、$\\mathcal Q_{SV}$、$\\mathcal G$ 等本輪所有 operator-level 量的投影算子（§2–§6）。",
      "definition_en": "The projection onto symmetric trace-free tensors, inherited from earlier rounds and used throughout Sec. 2-6 to build every operator-level quantity of this round ($\\mathcal N_{SSA}$, $\\mathcal N_{SV}$, $\\mathcal P_{SSA}$, $\\mathcal Q_{SV}$, $\\mathcal G$).",
      "notes": "本輪未重新給出其正式定義式，僅延續使用。"
    },
    {
      "id": "ns.c3.c3p.n_ssa",
      "latex": "\\mathcal N_{SSA}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "應變自增強非線性項",
      "label_en": "strain self-amplification nonlinearity",
      "definition_zh": "§3 定義為 $\\mathcal N_{SSA}=\\frac23P_{st}(S^2)$，構成 SSA model 方程 $\\partial_tS-\\Delta S+\\mathcal N_{SSA}=0$ 的非線性核心。",
      "definition_en": "Defined in Sec. 3 as $\\mathcal N_{SSA}=\\frac23P_{st}(S^2)$, the nonlinear core of the SSA model equation $\\partial_tS-\\Delta S+\\mathcal N_{SSA}=0$.",
      "defining_relation": "\\mathcal N_{SSA}=\\frac23P_{st}(S^2)"
    },
    {
      "id": "ns.c3.c3p.p_ssa",
      "latex": "\\mathcal P_{SSA}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "SSA 模型殘餘算子",
      "label_en": "SSA-model residual operator",
      "definition_zh": "§3 定義為 $\\mathcal P_{SSA}=P_{st}\\left((u\\cdot\\nabla)S+\\frac13S^2+\\frac14\\omega\\otimes\\omega\\right)$、衡量 full N–S 相對 SSA model 的偏差、其 smallness 於 §11 被證明不能作 regularity 判準。",
      "definition_en": "Defined in Sec. 3 as $\\mathcal P_{SSA}=P_{st}\\left((u\\cdot\\nabla)S+\\frac13S^2+\\frac14\\omega\\otimes\\omega\\right)$, measuring full N-S's deviation from the SSA model, whose smallness Sec. 11 proves cannot serve as a regularity criterion."
    },
    {
      "id": "ns.c3.c3p.n_sv",
      "latex": "\\mathcal N_{SV}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "應變–渦度交互非線性項",
      "label_en": "strain-vorticity interaction nonlinearity",
      "definition_zh": "§4 引 Miller 2026 定義為 $\\mathcal N_{SV}=-\\frac12P_{st}(\\omega\\otimes\\omega)$，構成對 $S^0\\in L^2_{st}$ 有整體光滑解之 SV interaction model 的非線性核心。",
      "definition_en": "Defined in Sec. 4 via Miller (2026) as $\\mathcal N_{SV}=-\\frac12P_{st}(\\omega\\otimes\\omega)$, the nonlinear core of the SV interaction model, which has global smooth solutions for $S^0\\in L^2_{st}$.",
      "defining_relation": "\\mathcal N_{SV}=-\\frac12P_{st}(\\omega\\otimes\\omega)"
    },
    {
      "id": "ns.c3.c3p.q_sv",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "SV 模型殘餘算子",
      "label_en": "SV-model residual operator",
      "definition_zh": "§4 定義為 $\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)$，是本輪最核心的 operator-level 量，貫穿 Miller 定理、$d_{SV}$、$\\mathfrak Q_{SV}$ 及兩張 X-certificate。",
      "definition_en": "Defined in Sec. 4 as $\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)$, this round's single most central operator-level quantity, running through Miller's theorems, $d_{SV}$, $\\mathfrak Q_{SV}$, and both X-certificates.",
      "defining_relation": "\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)"
    },
    {
      "id": "ns.c3.c3p.model_gap_g",
      "latex": "\\mathcal G",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "兩模型差距算子",
      "label_en": "two-model gap operator",
      "definition_zh": "定理5.1／§6 定義為 $\\mathcal G=\\mathcal Q_{SV}-\\mathcal P_{SSA}=P_{st}\\left(\\frac23S^2+\\frac12\\omega\\otimes\\omega\\right)$，由推論6.1給出 $\\|\\mathcal Q_{SV}\\|+\\|\\mathcal P_{SSA}\\|\\ge\\|\\mathcal G\\|$，除非其本身很小否則 full dynamics 不能同時接近兩個 model。",
      "definition_en": "Defined via Thm 5.1/Sec. 6 as $\\mathcal G=\\mathcal Q_{SV}-\\mathcal P_{SSA}=P_{st}\\left(\\frac23S^2+\\frac12\\omega\\otimes\\omega\\right)$, satisfying $\\|\\mathcal Q_{SV}\\|+\\|\\mathcal P_{SSA}\\|\\ge\\|\\mathcal G\\|$ (Cor. 6.1), so unless it is itself small, full dynamics cannot stay close to both models simultaneously.",
      "defining_relation": "\\mathcal G=\\mathcal Q_{SV}-\\mathcal P_{SSA}=P_{st}\\left(\\frac23S^2+\\frac12\\omega\\otimes\\omega\\right)"
    },
    {
      "id": "ns.c3.c3p.d_sv",
      "latex": "d_{SV}(t)",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "到正則 SV 模型的算子距離",
      "label_en": "operator distance to the regular SV model",
      "definition_zh": "§12 定義為 $d_{SV}(t)=\\|\\mathcal Q_{SV}(t)\\|_2/\\|-\\Delta S(t)\\|_2$，由 Miller Theorem 1.9 導出 blow-up 必要條件 $\\limsup d_{SV}\\ge1$（§9, 推論9.1），是 C3-P.2 operator escape 的核心量。",
      "definition_en": "Defined in Sec. 12 as $d_{SV}(t)=\\|\\mathcal Q_{SV}(t)\\|_2/\\|-\\Delta S(t)\\|_2$, forced by Miller's Theorem 1.9 to satisfy $\\limsup d_{SV}\\ge1$ at blow-up (Sec. 9, Cor. 9.1), the central quantity of the C3-P.2 operator-escape result.",
      "defining_relation": "d_{SV}(t)=\\frac{\\|\\mathcal Q_{SV}(t)\\|_2}{\\|-\\Delta S(t)\\|_2}"
    },
    {
      "id": "ns.c3.c3p.d_ssa",
      "latex": "d_{SSA}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "到 SSA 模型的算子距離",
      "label_en": "operator distance to the SSA model",
      "definition_zh": "§12 引入、與 $d_{SV}$ 對照、以 $\\mathcal P_{SSA}$ 為核心但未給出顯式公式、其 smallness 不是 regularity 保證甚至可與 blow-up 相容的距離概念。",
      "definition_en": "Introduced in Sec. 12 as the counterpart to $d_{SV}$, centered on $\\mathcal P_{SSA}$ but never given an explicit formula in this round, whose smallness is not a regularity guarantee and may even be blow-up-compatible.",
      "notes": "僅於 §12 標題與 §36 的 $\\operatorname{XOp}_n$ tuple 中以 $d_{SSA,n}$ 出現，本文未給出正式定義式，需與 $d_{SV}$ 的顯式公式區分。"
    },
    {
      "id": "ns.c3.c3p.rho_ba",
      "latex": "\\rho=B/A",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "局部應變平衡比（承自 C3-O）",
      "label_en": "local strain-balance ratio (from C3-O)",
      "definition_zh": "承自 C3-O 的 $\\rho=B/A$（即 $B_\\chi/A_\\chi$），§13 新賦予其與 $d_{SV}$ 對照角色，證明 $\\rho\\to0$ 可與 $d_{SV}\\gtrsim1$ 同時成立（\"Balance-SSA / Operator-large\" regime），並於 §34 作 Interface P2 門檻 $\\rho>-1$。",
      "definition_en": "Inherited from C3-O as $\\rho=B/A$ (i.e. $B_\\chi/A_\\chi$), newly given in Sec. 13 a contrastive role against $d_{SV}$ — showing $\\rho\\to0$ can coexist with $d_{SV}\\gtrsim1$ (the \"Balance-SSA / Operator-large\" regime) — and used as the Interface P2 threshold $\\rho>-1$ in Sec. 34.",
      "notes": "此處 $A,B$ 為 C3-O 的 $A_\\chi,B_\\chi$ 簡寫，與 §14 起另行引入、代表速度梯度矩陣的 $A=\\nabla u$ 是同符號複用但無關的兩個記號。"
    },
    {
      "id": "ns.c3.c3p.velocity_grad_a",
      "latex": "A=\\nabla u",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "速度梯度矩陣",
      "label_en": "velocity gradient matrix",
      "definition_zh": "§14 為 pressure Poisson 分析引入 $A=\\nabla u$，用以寫出 $-\\Delta p=\\operatorname{tr}(A^2)$。",
      "definition_en": "The notation $A=\\nabla u$ introduced in Sec. 14 for the pressure-Poisson analysis, used to write $-\\Delta p=\\operatorname{tr}(A^2)$."
    },
    {
      "id": "ns.c3.c3p.pressure_source_f",
      "latex": "f",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "壓力 Poisson 方程源項",
      "label_en": "pressure Poisson source term",
      "definition_zh": "§14 定義為 $f=\\operatorname{tr}(A^2)=\\partial_iu_j\\partial_ju_i$，滿足 $-\\Delta p=f$ 及 §15 的 $\\|f(t)\\|_1\\le\\|\\nabla u(t)\\|_2^2$，是後續 near/far 分解的源頭。",
      "definition_en": "Defined in Sec. 14 as $f=\\operatorname{tr}(A^2)=\\partial_iu_j\\partial_ju_i$, satisfying $-\\Delta p=f$ and the Sec. 15 bound $\\|f(t)\\|_1\\le\\|\\nabla u(t)\\|_2^2$, the source for the subsequent near/far decomposition.",
      "defining_relation": "f=\\operatorname{tr}(A^2)=-\\Delta p"
    },
    {
      "id": "ns.c3.c3p.kappa",
      "latex": "\\kappa\\ge4",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "核心/遠場尺度比參數",
      "label_en": "core-to-far scale ratio parameter",
      "definition_zh": "§16 引入、滿足 $\\kappa\\ge4$、控制 cutoff 半徑 $\\kappa R$ 相對核心半徑 $R$ 比例、貫穿全篇 far-pressure 衰減估計與 decoupling 條件的自由參數。",
      "definition_en": "The free parameter of Sec. 16, satisfying $\\kappa\\ge4$, controlling the cutoff radius $\\kappa R$ relative to the core radius $R$, driving every far-pressure decay estimate and decoupling condition in the round."
    },
    {
      "id": "ns.c3.c3p.cutoff_eta",
      "latex": "\\eta_{\\kappa R}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "光滑截斷函數",
      "label_en": "smooth cutoff function",
      "definition_zh": "§16 定義、滿足 $\\eta_{\\kappa R}=1$ on $B_{\\kappa R}(x_0)$ 且支撐於 $B_{2\\kappa R}(x_0)$、用以將源項 $f$ 分割為 $f_{\\rm near},f_{\\rm far}$ 的 smooth cutoff。",
      "definition_en": "The smooth cutoff of Sec. 16, equal to 1 on $B_{\\kappa R}(x_0)$ and supported in $B_{2\\kappa R}(x_0)$, used to split the source $f$ into $f_{\\rm near}, f_{\\rm far}$."
    },
    {
      "id": "ns.c3.c3p.near_far_source",
      "latex": "f_{\\rm near},\\ f_{\\rm far}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "近場／遠場壓力源分解",
      "label_en": "near/far pressure-source split",
      "definition_zh": "§16 依 cutoff 將壓力源 $f$ 分解為 $f_{\\rm near}=\\eta_{\\kappa R}f$ 與 $f_{\\rm far}=(1-\\eta_{\\kappa R})f$，其中 $f_{\\rm far}=0$ on $B_{\\kappa R}(x_0)$，是 §17 far pressure harmonic 性質的起點。",
      "definition_en": "Sec. 16 splits the pressure source $f$ via the cutoff into $f_{\\rm near}=\\eta_{\\kappa R}f$ and $f_{\\rm far}=(1-\\eta_{\\kappa R})f$, with $f_{\\rm far}=0$ on $B_{\\kappa R}(x_0)$, the starting point for far pressure being harmonic in Sec. 17."
    },
    {
      "id": "ns.c3.c3p.near_far_pressure",
      "latex": "p_{\\rm near},\\ p_{\\rm far}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "近場／遠場壓力分解",
      "label_en": "near/far pressure split",
      "definition_zh": "§16 由對應近/遠場源反解出 $p_{\\rm near}=(-\\Delta)^{-1}f_{\\rm near}$ 與 $p_{\\rm far}=(-\\Delta)^{-1}f_{\\rm far}$，其中 $p_{\\rm far}$ 在核心 $B_{\\kappa R}(x_0)$ 內滿足 §17 的 $\\Delta p_{\\rm far}=0$。",
      "definition_en": "Sec. 16 solves $p_{\\rm near}=(-\\Delta)^{-1}f_{\\rm near}$ and $p_{\\rm far}=(-\\Delta)^{-1}f_{\\rm far}$ from the respective sources, with $p_{\\rm far}$ satisfying $\\Delta p_{\\rm far}=0$ inside the core $B_{\\kappa R}(x_0)$ per Sec. 17."
    },
    {
      "id": "ns.c3.c3p.h_far",
      "latex": "H_{\\rm far}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "遠場壓力 Hessian",
      "label_en": "far-pressure Hessian",
      "definition_zh": "§17 定義為 $H_{\\rm far}=\\nabla^2p_{\\rm far}$，在 ancestry core 中是 trace-free harmonic 對稱張量場，其導數衰減由定理18.1給出 $\\|\\nabla^mH_{\\rm far}\\|_{L^\\infty(B_R)}\\le C_m(\\kappa R)^{-3-m}\\|f\\|_1$。",
      "definition_en": "Defined in Sec. 17 as $H_{\\rm far}=\\nabla^2p_{\\rm far}$, a trace-free harmonic symmetric tensor field in the ancestry core, whose derivative decay Theorem 18.1 bounds as $\\|\\nabla^mH_{\\rm far}\\|_{L^\\infty(B_R)}\\le C_m(\\kappa R)^{-3-m}\\|f\\|_1$.",
      "defining_relation": "H_{\\rm far}=\\nabla^2p_{\\rm far},\\qquad \\operatorname{tr}H_{\\rm far}=0"
    },
    {
      "id": "ns.c3.c3p.riesz_kernel_k",
      "latex": "K_{ab}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "Riesz 對偶核",
      "label_en": "Riesz-pair kernel",
      "definition_zh": "§18 中滿足 $|\\nabla^mK_{ab}(z)|\\le C_m|z|^{-3-m}$、用以推導定理18.1遠場 Hessian 衰減估計的 Calderón–Zygmund 型核。",
      "definition_en": "The Calderon-Zygmund-type kernel of Sec. 18 satisfying $|\\nabla^mK_{ab}(z)|\\le C_m|z|^{-3-m}$, used to derive Theorem 18.1's far-Hessian decay estimate."
    },
    {
      "id": "ns.c3.c3p.h0",
      "latex": "H_0(t)",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "核心點遠場壓力 Hessian（五維矩陣）",
      "label_en": "core-point far-pressure Hessian (5D matrix)",
      "definition_zh": "C3-P.3／§19 定義為 $H_0(t)=H_{\\rm far}(x_0,t)$，屬 5 維對稱無跡常數矩陣，滿足 $|H_0|\\le C\\kappa^{-3}R^{-3}\\|\\nabla u\\|_2^2$，依 §28 不能被壓力 gauge 吸收，代表真正的 far-field dynamical channel。",
      "definition_en": "Defined by C3-P.3/Sec. 19 as $H_0(t)=H_{\\rm far}(x_0,t)$, a constant matrix in the 5-dimensional space of symmetric trace-free $3\\times3$ matrices obeying $|H_0|\\le C\\kappa^{-3}R^{-3}\\|\\nabla u\\|_2^2$, which Sec. 28 shows cannot be absorbed by the pressure gauge and so is a genuine dynamical far-field channel.",
      "defining_relation": "H_0(t)=H_{\\rm far}(x_0,t)"
    },
    {
      "id": "ns.c3.c3p.e_far",
      "latex": "E_{\\rm far}(x)",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "遠場壓力 Hessian 餘項",
      "label_en": "far-pressure Hessian remainder",
      "definition_zh": "C3-P.3／§19 由 $H_{\\rm far}(x)=H_0+E_{\\rm far}(x)$ 定義的空間變化餘項，滿足 $\\|E_{\\rm far}\\|_{L^\\infty(B_R)}\\le C\\kappa^{-4}R^{-3}\\|\\nabla u\\|_2^2$，比 $H_0$ 多一階 $\\kappa^{-1}$ 衰減。",
      "definition_en": "The spatially varying remainder defined by C3-P.3/Sec. 19 via $H_{\\rm far}(x)=H_0+E_{\\rm far}(x)$, obeying $\\|E_{\\rm far}\\|_{L^\\infty(B_R)}\\le C\\kappa^{-4}R^{-3}\\|\\nabla u\\|_2^2$, one extra order of $\\kappa^{-1}$ decay beyond $H_0$.",
      "defining_relation": "H_{\\rm far}(x)=H_0+E_{\\rm far}(x)"
    },
    {
      "id": "ns.c3.c3p.enstrophy_r",
      "latex": "\\mathfrak E_R(t)",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "尺度不變重整化 enstrophy 數",
      "label_en": "scale-invariant rescaled enstrophy number",
      "definition_zh": "§21 定義為 $\\mathfrak E_R(t)=R\\|\\nabla u(t)\\|_2^2/\\nu^2$，在 N–S rescaling 下不變，由 C3-P.4（§23）與 §38 顯示 far-pressure decoupling 真正門檻為 $\\kappa^{-3}\\mathfrak E_R\\to0$ 而非單純 $\\kappa\\to\\infty$。",
      "definition_en": "Defined in Sec. 21 as $\\mathfrak E_R(t)=R\\|\\nabla u(t)\\|_2^2/\\nu^2$, invariant under N-S rescaling; C3-P.4 (Sec. 23) and Sec. 38 show the true far-pressure decoupling threshold is $\\kappa^{-3}\\mathfrak E_R\\to0$, not merely $\\kappa\\to\\infty$.",
      "defining_relation": "\\mathfrak E_R(t)=\\frac{R\\|\\nabla u(t)\\|_2^2}{\\nu^2}"
    },
    {
      "id": "ns.c3.c3p.h_far_hat",
      "latex": "\\widehat H_{\\rm far}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "重整化遠場壓力 Hessian",
      "label_en": "normalized far-pressure Hessian",
      "definition_zh": "§22 定義為 $\\widehat H_{\\rm far}=(R^4/\\nu^2)H_{\\rm far}$，將其上界改寫成 $\\mathfrak E_R$ 的函數 $\\sup_{B_R}|\\widehat H_{\\rm far}-\\widehat H_0|\\le C\\kappa^{-4}\\mathfrak E_R$。",
      "definition_en": "Defined in Sec. 22 as $\\widehat H_{\\rm far}=(R^4/\\nu^2)H_{\\rm far}$, recasting the earlier bound as $\\sup_{B_R}|\\widehat H_{\\rm far}-\\widehat H_0|\\le C\\kappa^{-4}\\mathfrak E_R$ in terms of $\\mathfrak E_R$."
    },
    {
      "id": "ns.c3.c3p.h0_hat",
      "latex": "\\widehat H_0",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "重整化核心遠場矩陣",
      "label_en": "normalized core far-pressure matrix",
      "definition_zh": "§22 中 $H_0$ 的 rescaled 版本，滿足 $|\\widehat H_0|\\le C\\kappa^{-3}\\mathfrak E_R$，是 C3-P.4 decoupling 定理（§23）與 §38 threshold 分析的核心量。",
      "definition_en": "The rescaled version of $H_0$ in Sec. 22, obeying $|\\widehat H_0|\\le C\\kappa^{-3}\\mathfrak E_R$, the central quantity in the C3-P.4 decoupling theorem (Sec. 23) and the Sec. 38 threshold analysis."
    },
    {
      "id": "ns.c3.c3p.f_p_far",
      "latex": "F_{p,{\\rm far}}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "遠場壓力流",
      "label_en": "far-pressure current",
      "definition_zh": "§26 將 C3-N 的 pressure current $F_p=(\\nabla^2p-\\Delta pI)u$ 在核心 harmonic 條件下化簡為 $F_{p,{\\rm far}}=H_{\\rm far}u=H_0u+E_{\\rm far}u$，連接壓力 Hessian 與 boundary current。",
      "definition_en": "Sec. 26 simplifies C3-N's pressure current $F_p=(\\nabla^2p-\\Delta pI)u$ under the core-harmonic condition to $F_{p,{\\rm far}}=H_{\\rm far}u=H_0u+E_{\\rm far}u$, bridging the pressure Hessian and the boundary current.",
      "defining_relation": "F_{p,{\\rm far}}=H_{\\rm far}u=H_0u+E_{\\rm far}u"
    },
    {
      "id": "ns.c3.c3p.bp_h0_rem",
      "latex": "B_p^{H_0},\\ B_p^{rem}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "主導矩陣流／餘項邊界流",
      "label_en": "leading-matrix / remainder boundary currents",
      "definition_zh": "§27, §29 將遠場邊界流 $B_p^{far}=\\int\\nabla\\chi\\cdot H_{\\rm far}u$ 拆為常數矩陣主導項 $B_p^{H_0}=\\int\\nabla\\chi\\cdot H_0u$（可分部積分寫成 $-\\int\\chi H_0:S\\,dx$）與多帶一個 $\\kappa^{-1}$ 衰減的空間變化餘項 $B_p^{rem}=\\int\\nabla\\chi\\cdot E_{\\rm far}u$。",
      "definition_en": "Sec. 27 and 29 split the far boundary current $B_p^{far}=\\int\\nabla\\chi\\cdot H_{\\rm far}u$ into the constant-matrix leading term $B_p^{H_0}=\\int\\nabla\\chi\\cdot H_0u$ (rewritable by parts as $-\\int\\chi H_0:S\\,dx$) and the remainder $B_p^{rem}=\\int\\nabla\\chi\\cdot E_{\\rm far}u$, which carries one extra order of $\\kappa^{-1}$ decay."
    },
    {
      "id": "ns.c3.c3p.bchi_split",
      "latex": "B_\\chi^B,\\ B_\\chi^p",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "Betchov／壓力邊界流分解",
      "label_en": "Betchov/pressure boundary-current split",
      "definition_zh": "§32 將 adjoint balance 的 $B_\\chi$ 分解為 local algebraic 的 Betchov current $B_\\chi^B=\\frac13\\int\\nabla\\chi\\cdot F_B$ 與 nonlocal 的 pressure current $B_\\chi^p=\\int\\nabla\\chi\\cdot F_p$，強調二者 provenance 不同不可混同。",
      "definition_en": "Sec. 32 splits the adjoint balance's $B_\\chi$ into the local-algebraic Betchov current $B_\\chi^B=\\frac13\\int\\nabla\\chi\\cdot F_B$ and the nonlocal pressure current $B_\\chi^p=\\int\\nabla\\chi\\cdot F_p$, stressing that their provenance must not be conflated.",
      "defining_relation": "B_\\chi=B_\\chi^B+B_\\chi^p",
      "notes": "$F_B=u(\\nabla u)^2$ 為承自 Betchov identity（C3-N）的 local algebraic current，與含 $\\nabla^2p$ 的 nonlocal $F_p$ 形成本節對比。"
    },
    {
      "id": "ns.c3.c3p.xop_n",
      "latex": "\\operatorname{XOp}_n",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "雙算子 X-certificate",
      "label_en": "two-operator X-certificate",
      "definition_zh": "§36 定義為 $\\operatorname{XOp}_n=\\langle\\mathcal P_{SSA,n},\\mathcal Q_{SV,n},\\mathcal G_n,d_{SSA,n},d_{SV,n},\\operatorname{Prov}_n\\rangle$，由五條守衛（G-MODEL, G-REGMODEL, G-BLOWMODEL, G-GAP, G-ORTH）防止兩模型距離被混為單一量的六元組驗證憑證。",
      "definition_en": "Defined in Sec. 36 as $\\operatorname{XOp}_n=\\langle\\mathcal P_{SSA,n},\\mathcal Q_{SV,n},\\mathcal G_n,d_{SSA,n},d_{SV,n},\\operatorname{Prov}_n\\rangle$, a six-tuple verification certificate guarded by five rules (G-MODEL, G-REGMODEL, G-BLOWMODEL, G-GAP, G-ORTH) that prevent the two model-distances from being conflated into one quantity."
    },
    {
      "id": "ns.c3.c3p.xpressure_n",
      "latex": "\\operatorname{XPressure}_n",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "壓力 X-certificate",
      "label_en": "pressure X-certificate",
      "definition_zh": "§37 定義為 $\\operatorname{XPressure}_n=\\langle p_{\\rm near},H_{0,n},E_{{\\rm far},n},\\kappa,\\mathfrak E_{R_n},\\operatorname{ProvFar}\\rangle$，由五條守衛（G-PNEAR, G-PFAR, G-H0, G-PREM, G-PENST）防止單憑 $\\kappa^{-3}$ 衰減就宣布 far pressure 可忽略的六元組驗證憑證。",
      "definition_en": "Defined in Sec. 37 as $\\operatorname{XPressure}_n=\\langle p_{\\rm near},H_{0,n},E_{{\\rm far},n},\\kappa,\\mathfrak E_{R_n},\\operatorname{ProvFar}\\rangle$, a six-tuple verification certificate guarded by five rules (G-PNEAR, G-PFAR, G-H0, G-PREM, G-PENST) that block declaring far pressure negligible on $\\kappa^{-3}$ decay alone."
    },
    {
      "id": "ns.c3.c3p.q_sv_frak",
      "latex": "\\mathfrak Q_{SV}",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "Miller 支持的整合算子債",
      "label_en": "Miller-backed integrated operator debt",
      "definition_zh": "§35 提出、用以取代 C3-O 的 $\\mathfrak P$、直接對應 Miller theorem-backed 積分恆等式、要求 hypothetical blow-up 滿足 $\\mathfrak Q_{SV}=\\infty$ 的新診斷量。",
      "definition_en": "The new diagnostic proposed in Sec. 35 to supersede C3-O's $\\mathfrak P$, tied directly to Miller's theorem-backed integral identity, forcing hypothetical blow-up to satisfy $\\mathfrak Q_{SV}=\\infty$.",
      "defining_relation": "\\mathfrak Q_{SV}=\\int\\frac{\\|\\mathcal Q_{SV}\\|_2^2}{\\|S\\|_{\\dot H^1}^2}dt"
    },
    {
      "id": "ns.c3.c3p.h_n_frak",
      "latex": "\\mathsf H_n",
      "series": "NS",
      "first_appearance": "C3-P",
      "label_zh": "C3-Q 用重整化調和矩陣序列",
      "label_en": "rescaled harmonic-matrix sequence for C3-Q",
      "definition_zh": "§41（Q1）為下一輪 C3-Q 定義的 ancestry sequence 版本 $\\mathsf H_n=(R_n^4/\\nu^2)H_{0,n}$，準備分析其抽子序列後趨於 0、趨於非零常數 $\\mathsf H_\\ast$、或發散三種情形。",
      "definition_en": "The ancestry-sequence version $\\mathsf H_n=(R_n^4/\\nu^2)H_{0,n}$ defined in Sec. 41 (Q1) to set up the next round C3-Q, distinguishing the subsequential outcomes $\\mathsf H_n\\to0$, $\\mathsf H_n\\to\\mathsf H_\\ast\\ne0$, or $|\\mathsf H_n|\\to\\infty$."
    },
    {
      "id": "ns.c3.c3q.n_raw",
      "latex": "\\mathcal N_{\\rm raw}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "原始應變非線性算子",
      "label_en": "Raw strain nonlinearity",
      "definition_zh": "未投影的對稱非線性矩陣（§2），滿足完整應變方程 ∂_tS − νΔS + 𝒩_raw + ∇²p = 0；是本輪 Pressure–Projection 分解的出發點。",
      "definition_en": "The unprojected symmetric nonlinear matrix (§2), appearing in the full strain equation ∂_tS − νΔS + 𝒩_raw + ∇²p = 0; the starting object for this round's pressure-projection decomposition.",
      "defining_relation": "\\mathcal N_{\\rm raw} = (u\\cdot\\nabla)S + S^2 + \\frac14\\omega\\otimes\\omega - \\frac14|\\omega|^2I"
    },
    {
      "id": "ns.c3.c3q.n_proj",
      "latex": "\\mathcal N_{\\rm proj}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "投影應變非線性算子",
      "label_en": "Projected strain nonlinearity",
      "definition_zh": "定理3.1（§3）定義為 𝒩_proj = P_st 𝒩_raw；與 ∇²p 構成正交分解 𝒩_raw = 𝒩_proj − ∇²p，且 ‖𝒩_raw‖² = ‖𝒩_proj‖² + ‖∇²p‖²（定理4.1，§4）。",
      "definition_en": "Defined in Theorem 3.1 (§3) as 𝒩_proj = P_st 𝒩_raw; together with ∇²p it gives the orthogonal decomposition 𝒩_raw = 𝒩_proj − ∇²p and the Pythagoras identity ‖𝒩_raw‖² = ‖𝒩_proj‖² + ‖∇²p‖² (Thm 4.1, §4).",
      "defining_relation": "\\mathcal N_{\\rm proj} = P_{st}\\mathcal N_{\\rm raw}",
      "notes": "§7 rewrites it as P_st((u·∇)S+S²+¼ω⊗ω) after using P_st(|ω|²I)=0; combined with 𝒩_SV's −½ coefficient this reproduces 𝒬_SV's ¾ω⊗ω coefficient from §0 exactly, so there is no inconsistency between the two stated forms."
    },
    {
      "id": "ns.c3.c3q.n_sv",
      "latex": "\\mathcal N_{SV}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "SV模型非線性項",
      "label_en": "SV-model nonlinearity",
      "definition_zh": "§7定義為 𝒩_SV = −(1/2)P_st(ω⊗ω)，用以將 Miller 算子改寫為 𝒬_SV = 𝒩_proj − 𝒩_SV，區分「投影完整非線性」與「應變–渦度自放大模型非線性」。",
      "definition_en": "Defined in §7 as 𝒩_SV = −(1/2)P_st(ω⊗ω), used to rewrite the Miller operator as 𝒬_SV = 𝒩_proj − 𝒩_SV, separating the projected full nonlinearity from the strain-vorticity self-amplification model nonlinearity.",
      "defining_relation": "\\mathcal N_{SV} = -\\frac12 P_{st}(\\omega\\otimes\\omega)"
    },
    {
      "id": "ns.c3.c3q.q_sv",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "Miller算子",
      "label_en": "Miller operator",
      "definition_zh": "承自C3-P（§0），定義為 P_st((u·∇)S+S²+(3/4)ω⊗ω)；本輪新證明 𝒬_SV = 𝒩_proj − 𝒩_SV（§7），並明確警告它不是 ∇²p 在同一Pythagoras分解中的互補分量（§8）。",
      "definition_en": "Inherited from C3-P (§0), defined as P_st((u·∇)S+S²+(3/4)ω⊗ω); this round newly proves 𝒬_SV = 𝒩_proj − 𝒩_SV (§7) and explicitly warns it is NOT the complementary component to ∇²p in the same Pythagorean decomposition as 𝒩_proj (§8).",
      "defining_relation": "\\mathcal Q_{SV} = P_{st}\\left((u\\cdot\\nabla)S + S^2 + \\frac34\\omega\\otimes\\omega\\right)",
      "notes": "§8 forbids writing ‖𝒩_raw‖² = ‖𝒬_SV‖² + ‖∇²p‖² — this does NOT hold in general, unlike the true Pythagoras pair (𝒩_proj, ∇²p)."
    },
    {
      "id": "ns.c3.c3q.p_st",
      "latex": "P_{st}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "應變約束正交投影",
      "label_en": "Strain-constraint orthogonal projection",
      "definition_zh": "§1定義為 L²_st 的 L²正交投影算子；關鍵性質 P_st(∇²p)=0，即壓力Hessian完全落在約束補空間中，是本輪一切正交性論證的基礎。",
      "definition_en": "Defined in §1 as the L²-orthogonal projection onto L²_st; the key property P_st(∇²p)=0 (pressure Hessian lies entirely in the complement space) underlies every orthogonality argument in this round.",
      "defining_relation": "P_{st}(\\nabla^2p)=0"
    },
    {
      "id": "ns.c3.c3q.l2_st",
      "latex": "L^2_{st}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "應變約束子空間",
      "label_en": "Strain constraint subspace",
      "definition_zh": "§1定義為全空間應變約束子空間，S(t)、∂_tS、ΔS（光滑regime下）皆位於其中，並與Hessian matrix fields等constraint-complement方向正交。",
      "definition_en": "Defined in §1 as the whole-space strain constraint subspace containing S(t), ∂_tS, and ΔS (in the smooth regime), orthogonal to Hessian-matrix-field complement directions."
    },
    {
      "id": "ns.c3.c3q.pressure_source_f",
      "latex": "f",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "壓力Poisson源項",
      "label_en": "Pressure Poisson source",
      "definition_zh": "§5定義為 −Δp=f:=tr((∇u)²)（§16寫作tr(A²)）；分解為近/遠場 f=f_near+f_far（§16），是遠場調和壓力分析的出發點，且 ‖∇²p‖₂=‖f‖₂（定理5.1）。",
      "definition_en": "Defined in §5 via −Δp=f:=tr((∇u)²) (written tr(A²) in §16); split into near/far parts f=f_near+f_far (§16), the starting point of the far-pressure harmonic analysis, with ‖∇²p‖₂=‖f‖₂ (Thm 5.1).",
      "defining_relation": "-\\Delta p = f := \\operatorname{tr}((\\nabla u)^2)"
    },
    {
      "id": "ns.c3.c3q.h_p0",
      "latex": "H_p^0",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "各向異性壓力Hessian",
      "label_en": "Anisotropic (trace-free) pressure Hessian",
      "definition_zh": "§6定義為壓力Hessian的無跡部分 H_p^0=∇²p−(1/3)(Δp)I；恆等式 ‖H_p^0‖²=(2/3)‖f‖² 顯示它佔壓力源範數的固定比例，非任意小殘量。",
      "definition_en": "Defined in §6 as the trace-free part of the pressure Hessian, H_p^0=∇²p−(1/3)(Δp)I; the identity ‖H_p^0‖²=(2/3)‖f‖² shows it is a fixed fraction of the source norm, not an arbitrarily small residual.",
      "defining_relation": "H_p^0 = \\nabla^2p - \\frac13(\\Delta p)I,\\qquad \\|H_p^0\\|_2^2=\\frac23\\|f\\|_2^2"
    },
    {
      "id": "ns.c3.c3q.g_projcomp",
      "latex": "G_{\\rm PROJCOMP}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "投影來源守則",
      "label_en": "Projection-provenance guard",
      "definition_zh": "§10新增的X-Integration guard：任何同時使用壓力與投影應變算子的論證，須分別標記「range channel」P_st𝒩_raw與「complement channel」−(I−P_st)𝒩_raw=∇²p，不得將兩者當同一純量力相減。",
      "definition_en": "X-Integration guard introduced in §10: any argument mixing pressure and the projected strain operator must separately tag the range channel P_st𝒩_raw and the complement channel −(I−P_st)𝒩_raw=∇²p, and may not subtract them as if they were the same scalar force.",
      "notes": "§38 lists a further set of seven named guards (G-PORTH…G-REDIST); the document does not explicitly state whether these supersede or merely refine G_PROJCOMP."
    },
    {
      "id": "ns.c3.c3q.g_porth",
      "latex": "G-PORTH",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：壓力—投影正交",
      "label_en": "Guard: pressure-projection orthogonality",
      "definition_zh": "§38新增守則，保存核心事實 𝒩_proj⊥∇²p，供後續論證引用而不必重證。",
      "definition_en": "Guard added in §38 preserving the core fact 𝒩_proj⊥∇²p for later arguments to cite without re-proving.",
      "defining_relation": "\\mathcal N_{\\rm proj}\\perp\\nabla^2p"
    },
    {
      "id": "ns.c3.c3q.g_qshift",
      "latex": "G-QSHIFT",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：算子非投影非線性",
      "label_en": "Guard: operator is not the projected nonlinearity",
      "definition_zh": "§38守則，強調 𝒬_SV 不等於 𝒩_proj，禁止誤套Pythagoras恆等式（即不可寫 ‖𝒩_raw‖²=‖𝒬_SV‖²+‖∇²p‖²）。",
      "definition_en": "Guard from §38 emphasizing 𝒬_SV ≠ 𝒩_proj, forbidding misapplication of the Pythagoras identity with 𝒬_SV in place of 𝒩_proj."
    },
    {
      "id": "ns.c3.c3q.g_oploc",
      "latex": "G-OPLOC",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：算子逃逸定位",
      "label_en": "Guard: operator-escape localization",
      "definition_zh": "§38守則，要求任何 global operator escape 的論證必須標明其 core/exterior carrier（見§13-15的Q_c/Q_e二分法）。",
      "definition_en": "Guard from §38 requiring any global-operator-escape argument to specify its core/exterior carrier (cf. the Q_c/Q_e dichotomy of §13-15)."
    },
    {
      "id": "ns.c3.c3q.g_harm",
      "latex": "G-HARM",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：調和壓力主項",
      "label_en": "Guard: harmonic pressure leading object",
      "definition_zh": "§38守則，標記遠場調和壓力的主要對象為 H_0∈Sym_0(3)，供後續章節（§16以下）引用。",
      "definition_en": "Guard from §38 tagging the far-harmonic-pressure leading object as H_0∈Sym_0(3), referenced by later sections (§16 onward)."
    },
    {
      "id": "ns.c3.c3q.g_halign",
      "latex": "G-HALIGN",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：壓力對齊係數",
      "label_en": "Guard: pressure alignment coefficient",
      "definition_zh": "§38守則，保存對齊係數定義 ζ_R=−H_0:M_R/(|H_0||M_R|)，供後續耦合論證引用。",
      "definition_en": "Guard from §38 preserving the alignment-coefficient definition ζ_R=−H_0:M_R/(|H_0||M_R|) for later coupling arguments.",
      "defining_relation": "\\zeta_R=-\\frac{H_0:M_R}{|H_0||M_R|}"
    },
    {
      "id": "ns.c3.c3q.g_phorizon",
      "latex": "G-PHORIZON",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：壓力視界",
      "label_en": "Guard: pressure horizon",
      "definition_zh": "§38守則，要求檢查遠場壓力解耦時必須核對 κ^{-3}𝔈_R（或pressure-work版本）的大小，對應§27-28的兩個視界。",
      "definition_en": "Guard from §38 requiring any far-pressure decoupling claim to check the size of κ^{-3}𝔈_R (or its pressure-work analogue), corresponding to the two horizons of §27-28."
    },
    {
      "id": "ns.c3.c3q.g_redist",
      "latex": "G-REDIST",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "守則：無跡重分配",
      "label_en": "Guard: trace-free redistribution",
      "definition_zh": "§38守則，提醒trace-free遠場壓力並非sign-definite的耗散/depletion機制（見§17,33-34）。",
      "definition_en": "Guard from §38 reminding that trace-free far pressure is not a sign-definite dissipation/depletion mechanism (cf. §17, §33-34)."
    },
    {
      "id": "ns.c3.c3q.d_sv",
      "latex": "d_{SV}(t)",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "Miller逃逸比值",
      "label_en": "Miller escape ratio",
      "definition_zh": "§11正式定義 d_SV(t)=‖𝒬_SV(t)‖₂/‖−ΔS(t)‖₂；Miller定理要求若 T*<∞ 則 limsup_{t↑T*} d_SV(t)≥1。",
      "definition_en": "Formally defined in §11 as d_SV(t)=‖𝒬_SV(t)‖₂/‖−ΔS(t)‖₂; Miller's theorem requires limsup_{t↑T*} d_SV(t)≥1 whenever T*<∞.",
      "defining_relation": "d_{SV}(t) = \\frac{\\|\\mathcal Q_{SV}(t)\\|_2}{\\|-\\Delta S(t)\\|_2}"
    },
    {
      "id": "ns.c3.c3q.chi_cutoff",
      "latex": "\\chi",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "核心/外部截斷函數",
      "label_en": "Core/exterior cutoff function",
      "definition_zh": "§12引入的截斷函數 0≤χ≤1，用以將全域算子範數與消散範數分裂為核心（χ）與外部（1−χ）兩部分，是§13定理13.1二分法的工具。",
      "definition_en": "Cutoff function 0≤χ≤1 introduced in §12, splitting global operator and dissipation norms into core (χ) and exterior (1−χ) parts; the tool behind the Theorem 13.1 dichotomy of §13."
    },
    {
      "id": "ns.c3.c3q.q_core_ext",
      "latex": "Q_{\\rm c},\\ Q_{\\rm e}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "核心/外部算子範數",
      "label_en": "Core/exterior operator norms",
      "definition_zh": "§12定義 Q_c²=∫χ|𝒬_SV|²dx、Q_e²=∫(1−χ)|𝒬_SV|²dx，滿足 Q_c²+Q_e²=‖𝒬_SV‖₂²，是定理13.1二分法的基礎量。",
      "definition_en": "Defined in §12 as Q_c²=∫χ|𝒬_SV|²dx and Q_e²=∫(1−χ)|𝒬_SV|²dx, satisfying Q_c²+Q_e²=‖𝒬_SV‖₂²; the basic quantities behind the Theorem 13.1 dichotomy.",
      "defining_relation": "Q_{\\rm c}^2=\\int\\chi|\\mathcal Q_{SV}|^2dx,\\quad Q_{\\rm e}^2=\\int(1-\\chi)|\\mathcal Q_{SV}|^2dx"
    },
    {
      "id": "ns.c3.c3q.d_core_ext",
      "latex": "D_{\\rm c},\\ D_{\\rm e}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "核心/外部消散範數",
      "label_en": "Core/exterior dissipation norms",
      "definition_zh": "§12定義 D_c²=∫χ|ΔS|²dx、D_e²=∫(1−χ)|ΔS|²dx，滿足 D_c²+D_e²=‖ΔS‖₂²，與Q_c,Q_e配對用於算子逃逸定位二分法（定理13.1）。",
      "definition_en": "Defined in §12 as D_c²=∫χ|ΔS|²dx and D_e²=∫(1−χ)|ΔS|²dx, satisfying D_c²+D_e²=‖ΔS‖₂²; paired with Q_c,Q_e in the operator-escape localization dichotomy (Theorem 13.1).",
      "defining_relation": "D_{\\rm c}^2=\\int\\chi|\\Delta S|^2dx,\\quad D_{\\rm e}^2=\\int(1-\\chi)|\\Delta S|^2dx"
    },
    {
      "id": "ns.c3.c3q.q_op_core_defect",
      "latex": "\\text{Q-OP-CORE},\\ \\text{Q-OP-DEFECT}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "算子逃逸核心/缺陷分支",
      "label_en": "Operator-escape core/defect branches",
      "definition_zh": "§14由 limsup d_SV≥1 沿子序列 t_n 推出：固定核心截斷 χ_n 後，至少一分支無窮多次成立——Q-OP-CORE 或 Q-OP-DEFECT。",
      "definition_en": "Derived in §14 from limsup d_SV≥1 along a subsequence t_n with fixed core cutoff χ_n: at least one of the two branches, Q-OP-CORE or Q-OP-DEFECT, holds infinitely often.",
      "defining_relation": "Q_{{\\rm c},n}\\ge(1-\\varepsilon)D_{{\\rm c},n}\\ \\text{(Q-OP-CORE)},\\qquad Q_{{\\rm e},n}\\ge(1-\\varepsilon)D_{{\\rm e},n}\\ \\text{(Q-OP-DEFECT)}",
      "notes": "§15 cautions this is observational localization only — \"operator field is large inside core\", not proof that its provenance is purely local (nonlocal projection P_st can still mix in exterior sources)."
    },
    {
      "id": "ns.c3.c3q.kappa",
      "latex": "\\kappa",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "遠場分離因子",
      "label_en": "Far-field separation factor",
      "definition_zh": "承自C3-P並在本輪系統化使用（§16起，κ≥4），為壓力源與ancestry核心 B_R(x_0) 的距離倍率（κR）；本輪的遠場補償界（§24）、enstrophy debt 𝔈_R≳κ²（§25-26）與兩個壓力視界（§27-28）均以κ為尺度參數。",
      "definition_en": "Inherited from C3-P and used systematically from §16 onward (κ≥4); the multiple of R by which the pressure source is separated from the ancestry ball B_R(x_0) (distance κR). This round's far-pressure compensation bound (§24), enstrophy debt 𝔈_R≳κ² (§25-26), and the two pressure horizons (§27-28) are all scaled by κ."
    },
    {
      "id": "ns.c3.c3q.ancestry_ball",
      "latex": "B_R(x_0)",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "Ancestry核心球",
      "label_en": "Ancestry core ball",
      "definition_zh": "§16半徑R、心x_0的空間球，作為壓力源近/遠場分解（f_near/f_far）與遠場壓力調和性（p_far在B_R(x_0)內harmonic）的參考域。",
      "definition_en": "The radius-R ball centered at x_0 (§16), the reference domain for the near/far pressure-source split and for the harmonicity of p_far inside B_R(x_0)."
    },
    {
      "id": "ns.c3.c3q.h0",
      "latex": "H_0",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "遠場調和壓力Hessian",
      "label_en": "Far-harmonic pressure Hessian",
      "definition_zh": "§16定義為 H_0=∇²p_far(x_0)∈Sym_0(3)，滿足 |H_0|≤Cκ^{-3}R^{-3}‖∇u‖₂²；因無跡，特徵值滿足h_1<0<h_3（§17），故非sign-definite depletion。",
      "definition_en": "Defined in §16 as H_0=∇²p_far(x_0)∈Sym_0(3), bounded by |H_0|≤Cκ^{-3}R^{-3}‖∇u‖₂²; being trace-free its eigenvalues satisfy h_1<0<h_3 (§17), so it cannot act as sign-definite depletion.",
      "defining_relation": "H_0=\\nabla^2p_{\\rm far}(x_0),\\qquad |H_0|\\le C\\kappa^{-3}R^{-3}\\|\\nabla u\\|_2^2"
    },
    {
      "id": "ns.c3.c3q.p_far",
      "latex": "p_{\\rm far}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "遠場壓力",
      "label_en": "Far pressure",
      "definition_zh": "§16由 f=f_near+f_far 誘導的壓力分量；在ancestry球B_R(x_0)內為調和函數，其Hessian在x_0處即為H_0。",
      "definition_en": "The pressure component induced by f_far (§16); harmonic inside the ancestry ball B_R(x_0), with Hessian equal to H_0 at x_0."
    },
    {
      "id": "ns.c3.c3q.e_far",
      "latex": "E_{\\rm far}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "遠場壓力空間變異項",
      "label_en": "Far-pressure spatial-variation remainder",
      "definition_zh": "§0、§16中 ∇²p_far=H_0+E_far 的剩餘項，相對常數項H_0多帶一個κ^{-1}的空間變異抑制因子；是Θ_p張量的分量之一（§39）。",
      "definition_en": "The remainder term in ∇²p_far=H_0+E_far (§0, §16), carrying an extra κ^{-1} spatial-variation suppression factor relative to the constant leading term H_0; a component of the Θ_p tuple (§39).",
      "defining_relation": "\\nabla^2p_{\\rm far}=H_0+E_{\\rm far}"
    },
    {
      "id": "ns.c3.c3q.h0_eigenvalues",
      "latex": "h_1\\le h_2\\le h_3",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "H_0的特徵值",
      "label_en": "Eigenvalues of H_0",
      "definition_zh": "§17，H_0的三個排序特徵值；因無跡滿足h_1+h_2+h_3=0，故必有h_1<0<h_3（定理17.1：無均勻調和壓力耗散）。",
      "definition_en": "The three ordered eigenvalues of H_0 (§17); trace-freeness forces h_1+h_2+h_3=0, hence h_1<0<h_3 (Theorem 17.1: no uniform harmonic-pressure depletion).",
      "defining_relation": "h_1+h_2+h_3=0,\\qquad h_1<0<h_3"
    },
    {
      "id": "ns.c3.c3q.dt_lambda_p_far",
      "latex": "(D_t\\lambda_i)_{p,{\\rm far}}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "遠場壓力對應變特徵值的物質導數貢獻",
      "label_en": "Far-pressure material-derivative contribution to a strain eigenvalue",
      "definition_zh": "§18定義：在局部應變特徵向量e_i簡單的點，遠場壓力對特徵值λ_i物質導數的貢獻為 −e_i^⊤H_0e_i。",
      "definition_en": "Defined in §18: at a point where the local strain eigenvector e_i is simple, the far-pressure contribution to the material derivative of eigenvalue λ_i equals −e_i^⊤H_0e_i.",
      "defining_relation": "(D_t\\lambda_i)_{p,{\\rm far}} = -e_i^\\top H_0e_i"
    },
    {
      "id": "ns.c3.c3q.h_i_s",
      "latex": "h_i^{(S)}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "應變特徵框架下的壓力分量",
      "label_en": "Pressure component in the local strain eigenframe",
      "definition_zh": "§18/§33定義為 h_i^{(S)}=e_i^⊤H_0e_i，滿足Σh_i^{(S)}=0；促進中間應變λ_2^+成長需h_2^{(S)}<0，但無跡性迫使另一方向h_j^{(S)}>0，構成§33的Trace-Free Redistribution Debt。",
      "definition_en": "Defined in §18/§33 as h_i^{(S)}=e_i^⊤H_0e_i, satisfying Σh_i^{(S)}=0; promoting middle-strain growth λ_2^+ requires h_2^{(S)}<0, but trace-freeness forces some other h_j^{(S)}>0 — the §33 Trace-Free Redistribution Debt.",
      "defining_relation": "h_i^{(S)}=e_i^\\top H_0e_i,\\qquad h_1^{(S)}+h_2^{(S)}+h_3^{(S)}=0"
    },
    {
      "id": "ns.c3.c3q.sym0_3",
      "latex": "\\operatorname{Sym}_0(3)",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "三維無跡對稱矩陣空間",
      "label_en": "Space of trace-free symmetric 3×3 matrices",
      "definition_zh": "H_0與M_R所棲身的5維向量空間（§16,19）；本輪多處利用其有限維(5D)特性論證遠場壓力motif的緊緻性（§35）與多核matrix packing的可能性（§42 R4）。",
      "definition_en": "The 5-dimensional space in which H_0 and M_R live (§16, §19); its finite-dimensionality is repeatedly used to argue compactness of the far-pressure motif (§35) and the possibility of multi-core matrix packing (§42, R4)."
    },
    {
      "id": "ns.c3.c3q.chi_r",
      "latex": "\\chi_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "尺度R局部化截斷",
      "label_en": "Scale-R localization cutoff",
      "definition_zh": "§19引入之尺度R局部化函數，用於定義局部應變均值矩陣M_R（§19）與局部應變存量𝔖_R（§22）。",
      "definition_en": "The scale-R localization function introduced in §19, used to define the local strain-mean matrix M_R (§19) and the local strain stock 𝔖_R (§22)."
    },
    {
      "id": "ns.c3.c3q.m_r",
      "latex": "M_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "局部應變均值矩陣",
      "label_en": "Local strain mean matrix",
      "definition_zh": "§19定義 M_R=∫χ_R S dx；與H_0的Frobenius內積構成遠場壓力對局部應變能的貢獻 B_{H_0}=−H_0:M_R。",
      "definition_en": "Defined in §19 as M_R=∫χ_R S dx; its Frobenius inner product with H_0 gives the far-pressure contribution to local strain energy, B_{H_0}=−H_0:M_R.",
      "defining_relation": "M_R=\\int\\chi_R S\\,dx"
    },
    {
      "id": "ns.c3.c3q.b_h0",
      "latex": "B_{H_0}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "常數遠場矩陣的壓力流貢獻",
      "label_en": "Constant far-matrix pressure-current contribution",
      "definition_zh": "§19-20定義 B_{H_0}=−H_0:M_R=ζ_R|H_0||M_R|，衡量遠場調和壓力對核心局部應變能瞬時成長的貢獻，是C3-N/O pressure current架構的延伸。",
      "definition_en": "Defined in §19-20 as B_{H_0}=−H_0:M_R=ζ_R|H_0||M_R|, measuring the instantaneous contribution of the far-harmonic pressure to local strain-energy growth in the core, extending the C3-N/O pressure-current framework.",
      "defining_relation": "B_{H_0}=-H_0:M_R"
    },
    {
      "id": "ns.c3.c3q.zeta_r",
      "latex": "\\zeta_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "壓力對齊係數",
      "label_en": "Pressure alignment coefficient",
      "definition_zh": "§20定義 ζ_R=−H_0:M_R/(|H_0||M_R|)∈[-1,1]；ζ_R>0是遠場壓力能支援局部正應變能成長所需的matrix anti-alignment條件。",
      "definition_en": "Defined in §20 as ζ_R=−H_0:M_R/(|H_0||M_R|)∈[-1,1]; ζ_R>0 is the matrix-anti-alignment condition needed for far pressure to support positive local strain-energy growth.",
      "defining_relation": "\\zeta_R=-\\frac{H_0:M_R}{|H_0||M_R|}\\in[-1,1]"
    },
    {
      "id": "ns.c3.c3q.h_hat_r",
      "latex": "\\widehat H_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "標準化遠場壓力矩陣",
      "label_en": "Normalized far-pressure matrix",
      "definition_zh": "§21定義 Ĥ_R=(R⁴/ν²)H_0，是§23-29無因次化壓力視界與enstrophy debt估計的核心量，滿足 |Ĥ_R|≤Cκ^{-3}𝔈_R。",
      "definition_en": "Defined in §21 as Ĥ_R=(R⁴/ν²)H_0, the central quantity in the dimensionless pressure-horizon and enstrophy-debt estimates of §23-29, satisfying |Ĥ_R|≤Cκ^{-3}𝔈_R.",
      "defining_relation": "\\widehat H_R=\\frac{R^4}{\\nu^2}H_0"
    },
    {
      "id": "ns.c3.c3q.m_hat_r",
      "latex": "\\widehat M_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "標準化局部應變均值",
      "label_en": "Normalized local strain mean",
      "definition_zh": "§21定義 M̂_R=(1/(νR))M_R；由Cauchy–Schwarz得界 |M̂_R|≤C𝔖_R^{1/2}（§22）。",
      "definition_en": "Defined in §21 as M̂_R=(1/(νR))M_R; bounded via Cauchy–Schwarz by |M̂_R|≤C𝔖_R^{1/2} (§22).",
      "defining_relation": "\\widehat M_R=\\frac{1}{\\nu R}M_R"
    },
    {
      "id": "ns.c3.c3q.b_hat_h0",
      "latex": "\\widehat B_{H_0}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "標準化壓力流",
      "label_en": "Normalized pressure-work / current",
      "definition_zh": "§21定義 B̂_{H_0}=(R³/ν³)B_{H_0}=−Ĥ_R:M̂_R；定理24.1界為 Cκ^{-3}𝔈_R^{3/2}，是§25-28 enstrophy debt與pressure-work horizon的核心量。",
      "definition_en": "Defined in §21 as B̂_{H_0}=(R³/ν³)B_{H_0}=−Ĥ_R:M̂_R; bounded by Cκ^{-3}𝔈_R^{3/2} in Theorem 24.1, the central quantity behind the §25-28 enstrophy debt and pressure-work horizon.",
      "defining_relation": "\\widehat B_{H_0}=\\frac{R^3}{\\nu^3}B_{H_0}=-\\widehat H_R:\\widehat M_R,\\qquad |\\widehat B_{H_0}|\\le C\\kappa^{-3}\\mathfrak E_R^{3/2}"
    },
    {
      "id": "ns.c3.c3q.strain_stock",
      "latex": "\\mathfrak S_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "局部應變存量",
      "label_en": "Local strain stock",
      "definition_zh": "§22定義 𝔖_R=(R/ν²)∫χ_R|S|²dx，滿足𝔖_R≤𝔈_R（§23），用以界定標準化局部應變均值M̂_R的大小。",
      "definition_en": "Defined in §22 as 𝔖_R=(R/ν²)∫χ_R|S|²dx, satisfying 𝔖_R≤𝔈_R (§23); used to bound the size of the normalized mean M̂_R.",
      "defining_relation": "\\mathfrak S_R=\\frac{R}{\\nu^2}\\int\\chi_R|S|^2dx"
    },
    {
      "id": "ns.c3.c3q.rescaled_enstrophy",
      "latex": "\\mathfrak E_R",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "重標度全域enstrophy",
      "label_en": "Rescaled global enstrophy",
      "definition_zh": "沿用C3-P定義 𝔈_R=(R/ν²)‖∇u‖₂²（§23）；本輪核心新結果為：固定大小的遠場壓力補償迫使 𝔈_R≳κ²（§25，C3-Q.6「Far-Pressure Enstrophy Debt」），驅動§27-28兩個壓力視界的尺度。",
      "definition_en": "Carried over from C3-P as 𝔈_R=(R/ν²)‖∇u‖₂² (§23); this round's key new result is that any fixed-size far-pressure compensation forces 𝔈_R≳κ² (§25, C3-Q.6 \"Far-Pressure Enstrophy Debt\"), which sets the scale of the two pressure horizons of §27-28.",
      "defining_relation": "\\mathfrak E_R=\\frac{R}{\\nu^2}\\|\\nabla u\\|_2^2,\\qquad \\mathfrak E_R\\gtrsim\\kappa^2"
    },
    {
      "id": "ns.c3.c3q.hessian_pressure_horizon",
      "latex": "\\textbf{Hessian Pressure Horizon}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "Hessian壓力視界",
      "label_en": "Hessian pressure horizon",
      "definition_zh": "§27命名的尺度：欲使遠場Hessian |Ĥ_R|≤ε，κ須滿足κ≳(𝔈_R/ε)^{1/3}；其重標度半徑隨𝔈_R^{1/3}擴張。",
      "definition_en": "Named scale defined in §27: forcing the far Hessian |Ĥ_R|≤ε requires κ≳(𝔈_R/ε)^{1/3}; its rescaled radius grows like 𝔈_R^{1/3}.",
      "defining_relation": "\\kappa\\gtrsim\\left(\\frac{\\mathfrak E_R}{\\varepsilon}\\right)^{1/3}"
    },
    {
      "id": "ns.c3.c3q.pressure_work_horizon",
      "latex": "\\textbf{pressure-work horizon}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "壓力功視界",
      "label_en": "Pressure-work horizon",
      "definition_zh": "§28命名的尺度：欲使 |B̂_{H_0}|≤ε，κ須滿足κ≳𝔈_R^{1/2}ε^{-1/3}；可能比Hessian壓力視界更大。",
      "definition_en": "Named scale defined in §28: forcing |B̂_{H_0}|≤ε requires κ≳𝔈_R^{1/2}ε^{-1/3}; may exceed the Hessian-amplitude horizon in size.",
      "defining_relation": "\\kappa\\gtrsim\\mathfrak E_R^{1/2}\\varepsilon^{-1/3}"
    },
    {
      "id": "ns.c3.c3q.trace_free_redistribution_debt",
      "latex": "\\textbf{Trace-Free Redistribution Debt}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "無跡重分配債",
      "label_en": "Trace-free redistribution debt",
      "definition_zh": "§33命名的結構性事實：壓力欲促進中間應變λ_2^+成長（需h_2^{(S)}<0）時，無跡性Σh_i^{(S)}=0迫使必在另一特徵方向同時支付相反符號的重分配（h_j^{(S)}>0）。",
      "definition_en": "Structural fact named in §33: whenever pressure promotes middle-strain growth λ_2^+ (requiring h_2^{(S)}<0), trace-freeness Σh_i^{(S)}=0 forces it to simultaneously pay an opposite-sign redistribution (h_j^{(S)}>0) in another eigendirection."
    },
    {
      "id": "ns.c3.c3q.h_hat_star",
      "latex": "\\widehat H_\\ast",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "遠場壓力緊緻極限",
      "label_en": "Far-pressure compactness limit",
      "definition_zh": "§35，由|Ĥ_n|有界及Sym_0(3)為5維空間，可取子序列使Ĥ_n→Ĥ_*；此僅是leading far-harmonic motif的緊緻性，不代表完整壓力場緊緻。",
      "definition_en": "In §35, boundedness of |Ĥ_n| together with the 5-dimensionality of Sym_0(3) yields a subsequential limit Ĥ_n→Ĥ_*; this is only compactness of the leading far-harmonic motif, not of the full pressure field.",
      "defining_relation": "\\widehat H_n\\to\\widehat H_\\ast",
      "notes": "§36 restates the same limit without the hat, as \"H_*\" — likely the identical object; a minor notational inconsistency in the source."
    },
    {
      "id": "ns.c3.c3q.survivor_qa",
      "latex": "Q-A",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "存活分支Q-A：核心算子／壓力解耦",
      "label_en": "Survivor branch Q-A: core operator / pressure-decoupled",
      "definition_zh": "§37分類之一：Q_c≳D_c 且 Ĥ_far→0；singular debt真正位於core投影動力學本身。",
      "definition_en": "One of the §37 classification branches: Q_c≳D_c and Ĥ_far→0; the singular debt genuinely resides in the core's projected dynamics itself."
    },
    {
      "id": "ns.c3.c3q.survivor_qb",
      "latex": "Q-B",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "存活分支Q-B：核心算子／壓力活躍",
      "label_en": "Survivor branch Q-B: core operator / pressure-active",
      "definition_zh": "§37分類之一：Q_c≳D_c 且 Ĥ_far不趨於0；需要算子逃逸與5D遠場壓力矩陣共同維持singular dynamics。",
      "definition_en": "One of the §37 classification branches: Q_c≳D_c and Ĥ_far does not vanish; operator escape and the 5D far-pressure matrix must jointly sustain the singular dynamics."
    },
    {
      "id": "ns.c3.c3q.survivor_qc",
      "latex": "Q-C",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "存活分支Q-C：外部算子／壓力活躍",
      "label_en": "Survivor branch Q-C: exterior operator / pressure-active",
      "definition_zh": "§37分類之一：global Miller debt主要位於exterior，但透過調和壓力矩陣影響core，稱為「defect-fed pressure ancestry」。",
      "definition_en": "One of the §37 classification branches: the global Miller debt sits mainly in the exterior but still influences the core via the harmonic pressure matrix — termed \"defect-fed pressure ancestry.\""
    },
    {
      "id": "ns.c3.c3q.survivor_qd",
      "latex": "Q-D",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "存活分支Q-D：外部算子／壓力解耦",
      "label_en": "Survivor branch Q-D: exterior operator / pressure-decoupled",
      "definition_zh": "§37分類之一：全域singular算子債與目前ancestry core分離，暗示所選核心可能非完整singular driver，需重新selection或多核genealogy。",
      "definition_en": "One of the §37 classification branches: the global singular operator debt is decoupled from the currently chosen ancestry core, implying the chosen core may not be the complete singular driver and a re-selection or multi-core genealogy is needed."
    },
    {
      "id": "ns.c3.c3q.theta_op",
      "latex": "\\Theta_{\\rm op}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "投影算子張力",
      "label_en": "Projected operator tension",
      "definition_zh": "§39 True ETN更新的三分量之一：Θ_op=(𝒬_SV,d_SV,core/exterior carrier)；與Θ_p構成projection-complement關係，但非同一observable。",
      "definition_en": "One of three components in the §39 True-ETN update: Θ_op=(𝒬_SV, d_SV, core/exterior carrier); stands in a projection-complement relation to Θ_p without being the same observable.",
      "defining_relation": "\\Theta_{\\rm op}=(\\mathcal Q_{SV},d_{SV},\\text{core/exterior carrier})"
    },
    {
      "id": "ns.c3.c3q.theta_p",
      "latex": "\\Theta_{\\rm p}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "約束壓力張力",
      "label_en": "Constraint pressure tension",
      "definition_zh": "§39 True ETN更新之一：Θ_p=(H_0,E_far,ζ,𝔈_R,κ)，彙整本輪遠場壓力的全部量化描述子。",
      "definition_en": "One of the §39 True-ETN components: Θ_p=(H_0,E_far,ζ,𝔈_R,κ), collecting this round's full set of quantitative far-pressure descriptors.",
      "defining_relation": "\\Theta_{\\rm p}=(H_0,E_{\\rm far},\\zeta,\\mathfrak E_R,\\kappa)",
      "notes": "Uses \"ζ\" without the R subscript, unlike §20's ζ_R; almost certainly the same alignment coefficient, generically written."
    },
    {
      "id": "ns.c3.c3q.theta_strain",
      "latex": "\\Theta_{\\rm strain}",
      "series": "NS",
      "first_appearance": "C3-Q",
      "label_zh": "整體應變幾何張力",
      "label_en": "Bulk strain geometry tension",
      "definition_zh": "§39 True ETN更新第三分量：Θ_strain=(λ_1,λ_2,λ_3,ξ,det S)，代表應變特徵值、渦度方向與行列式構成的整體幾何資料。",
      "definition_en": "The third §39 True-ETN component: Θ_strain=(λ_1,λ_2,λ_3,ξ,det S), the bulk geometric data of strain eigenvalues, vorticity direction, and determinant.",
      "defining_relation": "\\Theta_{\\rm strain}=(\\lambda_1,\\lambda_2,\\lambda_3,\\xi,\\det S)",
      "notes": "ξ (vorticity direction) appears here without being defined within this document; inherited from earlier rounds in the series."
    },
    {
      "id": "ns.c3.c3r.beta_star",
      "latex": "\\beta_\\ast",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "飽和門檻常數",
      "label_en": "saturation threshold constant",
      "definition_zh": "第1節固定的正常數，作為 shell 振幅比 $a_q^\\sigma$ 的飽和門檻，貫穿全篇 multi-core 能量與 enstrophy 估計。",
      "definition_en": "A fixed positive constant from Section 1 serving as the saturation threshold for the shell-amplitude ratio $a_q^\\sigma$, used throughout the round's multi-core energy and enstrophy bounds."
    },
    {
      "id": "ns.c3.c3r.r_scale",
      "latex": "R",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "交叉尺度半徑",
      "label_en": "crossing length scale",
      "definition_zh": "第1節由 first-crossing shell 頻率 $q_Q$ 導出的長度尺度 $R=\\lambda_{q_Q}^{-1}$，是本輪所有 core／cluster 估計的基本標度。",
      "definition_en": "The length scale $R=\\lambda_{q_Q}^{-1}$ derived in Section 1 from the first-crossing shell frequency $q_Q$; the basic unit for every core/cluster estimate in this round."
    },
    {
      "id": "ns.c3.c3r.t_q",
      "latex": "T_Q",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "首次前沿穿越時間",
      "label_en": "first-frontier crossing time",
      "definition_zh": "第1節定義，frontier $Q$ 之上任一 shell 比值 $a_q^\\sigma$ 首次達到門檻 $\\beta_\\ast$ 的時刻。",
      "definition_en": "Defined in Section 1 as the first time any shell ratio $a_q^\\sigma$ at frequency $\\ge Q$ reaches the threshold $\\beta_\\ast$."
    },
    {
      "id": "ns.c3.c3r.f_crossing",
      "latex": "f(x)=u_{q_Q}^{\\sigma_Q}(x,T_Q)",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "穿越殼層速度場",
      "label_en": "crossing-shell velocity field",
      "definition_zh": "第2節定義的 first-crossing shell 速度場 $f=u_{q_Q}^{\\sigma_Q}(\\cdot,T_Q)$，其峰值 $M=\\|f\\|_\\infty=\\nu\\beta_\\ast\\lambda$ 是後續所有 core 能量下界的基準振幅。",
      "definition_en": "The velocity field on the first-crossing shell, $f=u_{q_Q}^{\\sigma_Q}(\\cdot,T_Q)$ from Section 2, whose peak amplitude $M=\\|f\\|_\\infty=\\nu\\beta_\\ast\\lambda$ anchors every subsequent core energy bound."
    },
    {
      "id": "ns.c3.c3r.eta",
      "latex": "\\eta",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "近飽和鬆弛參數",
      "label_en": "near-saturation slack parameter",
      "definition_zh": "第3節固定的小參數 $0<\\eta<1/4$，用以定義近飽和集合 $\\Omega_\\eta$ 及核心分離半徑 $r_\\eta=\\eta/(4C_B)$（$C_B$ 為 Bernstein 常數）。",
      "definition_en": "A small fixed parameter $0<\\eta<1/4$ from Section 3 that sets the near-saturation set $\\Omega_\\eta$ and the core-separation radius $r_\\eta=\\eta/(4C_B)$ ($C_B$ the Bernstein constant)."
    },
    {
      "id": "ns.c3.c3r.omega_eta",
      "latex": "\\Omega_\\eta",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "近飽和集合",
      "label_en": "near-saturation set",
      "definition_zh": "第3節定義 $\\Omega_\\eta=\\{x:|f(x)|\\ge(1-\\eta)\\|f\\|_\\infty\\}$，frontier core 中心即取自此集合。",
      "definition_en": "Defined in Section 3 as $\\Omega_\\eta=\\{x:|f(x)|\\ge(1-\\eta)\\|f\\|_\\infty\\}$; frontier-core centers are chosen from this set."
    },
    {
      "id": "ns.c3.c3r.c_eta",
      "latex": "c_\\eta",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "局部振幅持續常數",
      "label_en": "local amplitude persistence constant",
      "definition_zh": "第4節導出的正常數，保證每個 core 球 $B(x_i,r_\\eta R)$ 內振幅至少維持 $c_\\eta M$，是每核心最小能量估計的關鍵常數。",
      "definition_en": "The positive constant from Section 4 guaranteeing amplitude stays above $c_\\eta M$ throughout each core ball $B(x_i,r_\\eta R)$; key to the minimal per-core energy bound."
    },
    {
      "id": "ns.c3.c3r.m_r",
      "latex": "m_R",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "前沿核心多重度",
      "label_en": "frontier core multiplicity",
      "definition_zh": "第6節定理6.1（Frontier Multi-Core Energy Packing）中，同一 first-frontier crossing shell 內 pairwise $O(R)$-分離之近飽和核心個數。",
      "definition_en": "The number of pairwise $O(R)$-separated near-saturation cores on one first-frontier crossing shell, defined and bounded in Theorem 6.1 (Section 6).",
      "defining_relation": "m_R\\le\\frac{C\\|u_0\\|_2^2}{\\nu^2\\beta_\\ast^2R}",
      "notes": "From Section 15 onward the same role is played by the unsubscripted symbol m, the core count of a spatial cluster."
    },
    {
      "id": "ns.c3.c3r.mathfrak_e_r",
      "latex": "\\mathfrak E_R",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "前沿尺度重標化enstrophy",
      "label_en": "rescaled frontier-scale enstrophy",
      "definition_zh": "第8–9節定義的無量綱 enstrophy $\\mathfrak E_R(T_Q)=R\\|\\nabla u(T_Q)\\|_2^2/\\nu^2$，定理9.1證明其隨核心數線性增長。",
      "definition_en": "The dimensionless enstrophy $\\mathfrak E_R(T_Q)=R\\|\\nabla u(T_Q)\\|_2^2/\\nu^2$ from Sections 8-9; Theorem 9.1 shows it grows at least linearly with core count.",
      "defining_relation": "\\mathfrak E_R(T_Q)=\\frac{R\\|\\nabla u(T_Q)\\|_2^2}{\\nu^2},\\qquad \\mathfrak E_R(T_Q)\\ge c\\,m_R\\beta_\\ast^2"
    },
    {
      "id": "ns.c3.c3r.varepsilon_p",
      "latex": "\\varepsilon_p",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "壓力容忍度",
      "label_en": "pressure-decoupling tolerance",
      "definition_zh": "第12節固定的正容忍值，用於定義 certified pressure horizon $\\kappa_{\\rm cert}$ 所保證之遠場 Hessian 上界。",
      "definition_en": "The fixed positive tolerance from Section 12 used to define the far-Hessian bound guaranteed by the certified pressure horizon $\\kappa_{\\rm cert}$."
    },
    {
      "id": "ns.c3.c3r.kappa_cert",
      "latex": "\\kappa_{\\rm cert}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "認證壓力視界",
      "label_en": "certified pressure horizon",
      "definition_zh": "第12節定義之無量綱半徑，由 universal far-pressure estimate保證半徑 $\\kappa_{\\rm cert}R$ 之外遠場 Hessian $\\le\\varepsilon_p$；第13節強調它僅是最壞情況證書半徑而非實際物理耦合長度。",
      "definition_en": "The dimensionless radius from Section 12, guaranteed by the universal far-pressure estimate to make the far Hessian $\\le\\varepsilon_p$ beyond physical radius $\\kappa_{\\rm cert}R$; Section 13 stresses it is a worst-case certificate radius, not the actual physical coupling length.",
      "defining_relation": "\\kappa_{\\rm cert}=\\left(\\frac{C\\mathfrak E_R}{\\varepsilon_p}\\right)^{1/3}"
    },
    {
      "id": "ns.c3.c3r.l_cluster",
      "latex": "L",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "群集半徑尺度",
      "label_en": "cluster radius scale",
      "definition_zh": "第15–16節引入，包住 $m$ 個核心中心的球 $B(x_\\ast,L)$ 之半徑，稠密封裝時 $L\\sim m^{1/3}R$。",
      "definition_en": "The radius of the ball $B(x_\\ast,L)$ enclosing the $m$ core centers, introduced in Sections 15-16; under dense packing $L\\sim m^{1/3}R$."
    },
    {
      "id": "ns.c3.c3r.ell_spread",
      "latex": "\\ell",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "無量綱群集展幅",
      "label_en": "dimensionless cluster spread",
      "definition_zh": "第16節定義 $\\ell=L/R$，衡量群集半徑相對單一核心尺度之展開，並滿足 packing bound $m\\le C\\ell^3$。",
      "definition_en": "The ratio $\\ell=L/R$ defined in Section 16, measuring cluster radius relative to a single core scale, satisfying the packing bound $m\\le C\\ell^3$."
    },
    {
      "id": "ns.c3.c3r.delta_density",
      "latex": "\\delta",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "無量綱核心密度",
      "label_en": "dimensionless core density",
      "definition_zh": "第16–17節定義 $\\delta=m/\\ell^3\\in(0,C]$，用以劃分 dense（$\\delta\\ge\\delta_0$）與 sparse（$\\delta\\to0$）multi-core cluster。",
      "definition_en": "Defined in Sections 16-17 as $\\delta=m/\\ell^3\\in(0,C]$, splitting multi-core clusters into dense ($\\delta\\ge\\delta_0$) and sparse ($\\delta\\to0$) regimes.",
      "defining_relation": "\\delta=\\frac{m}{\\ell^3}"
    },
    {
      "id": "ns.c3.c3r.g_pmerge",
      "latex": "G_{\\rm PMERGE}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "壓力群集合併規則",
      "label_en": "pressure-cluster merge rule",
      "definition_zh": "第20節新增的 X-Integration guard，規定當多核心的 certified pressure neighborhoods顯著重疊時，須於群集尺度合併重做 near/far pressure 劃分，不得重複計入同一遠場來源。",
      "definition_en": "An X-Integration guard added in Section 20 requiring that when multiple cores' certified pressure neighborhoods significantly overlap, the near/far pressure split must be redone at cluster scale, never double-counting one far source as independent."
    },
    {
      "id": "ns.c3.c3r.mathfrak_e_l",
      "latex": "\\mathfrak E_L",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "群集尺度重標化enstrophy",
      "label_en": "cluster-scale rescaled enstrophy",
      "definition_zh": "第21節定義之群集尺度無量綱 enstrophy $\\mathfrak E_L=L\\|\\nabla u\\|_2^2/\\nu^2$，稠密封裝下滿足 $\\mathfrak E_L\\gtrsim\\beta_\\ast^2m^{4/3}$。",
      "definition_en": "The cluster-scale dimensionless enstrophy $\\mathfrak E_L=L\\|\\nabla u\\|_2^2/\\nu^2$ from Section 21, satisfying $\\mathfrak E_L\\gtrsim\\beta_\\ast^2m^{4/3}$ under dense packing.",
      "defining_relation": "\\mathfrak E_L=\\frac{L\\|\\nabla u\\|_2^2}{\\nu^2},\\qquad \\mathfrak E_L\\gtrsim\\beta_\\ast^2m^{4/3}"
    },
    {
      "id": "ns.c3.c3r.kappa_l_cert",
      "latex": "\\kappa_L^{cert}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "群集尺度認證壓力視界",
      "label_en": "cluster-scale certified pressure horizon",
      "definition_zh": "第22節定義之群集版無量綱認證視界 $\\kappa_L^{cert}\\sim\\mathfrak E_L^{1/3}$，稠密情形下滿足 $\\kappa_L^{cert}\\gtrsim m^{4/9}$。",
      "definition_en": "The cluster-level dimensionless certified horizon $\\kappa_L^{cert}\\sim\\mathfrak E_L^{1/3}$ from Section 22, satisfying $\\kappa_L^{cert}\\gtrsim m^{4/9}$ in the dense case.",
      "defining_relation": "\\kappa_L^{cert}\\sim\\mathfrak E_L^{1/3}\\ \\gtrsim\\ m^{4/9}"
    },
    {
      "id": "ns.c3.c3r.r_pl_cert",
      "latex": "R_{p,L}^{cert}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "群集物理壓力視界半徑",
      "label_en": "physical cluster pressure horizon radius",
      "definition_zh": "第22節定義之有量綱群集壓力視界 $R_{p,L}^{cert}=L\\kappa_L^{cert}$，滿足 $R_{p,L}^{cert}\\gtrsim m^{7/9}R$，即 Pressure-Horizon Inflation under Dense Core Merging。",
      "definition_en": "The dimensional cluster pressure horizon $R_{p,L}^{cert}=L\\kappa_L^{cert}$ from Section 22, satisfying $R_{p,L}^{cert}\\gtrsim m^{7/9}R$ — the Pressure-Horizon Inflation under Dense Core Merging.",
      "defining_relation": "R_{p,L}^{cert}=L\\kappa_L^{cert}\\ \\gtrsim\\ m^{7/9}R"
    },
    {
      "id": "ns.c3.c3r.h_star",
      "latex": "H_\\ast",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "群集共同遠場壓力Hessian矩陣",
      "label_en": "cluster common far-pressure Hessian matrix",
      "definition_zh": "第26節定義，整個群集 $B(x_\\ast,L)$ 之遠場調和壓力在中心的 Hessian $H_\\ast=\\nabla^2p_{\\rm far}^{cluster}(x_\\ast)\\in\\operatorname{Sym}_0(3)$，球內展開誤差 $E_\\ast(x)$ 滿足 $\\|E_\\ast\\|_{L^\\infty(B_L)}\\le C\\kappa^{-4}L^{-3}\\|\\nabla u\\|_2^2$。",
      "definition_en": "Defined in Section 26 as the Hessian $H_\\ast=\\nabla^2p_{\\rm far}^{cluster}(x_\\ast)\\in\\operatorname{Sym}_0(3)$ of the cluster's far harmonic pressure at its center; its expansion remainder $E_\\ast(x)$ satisfies $\\|E_\\ast\\|_{L^\\infty(B_L)}\\le C\\kappa^{-4}L^{-3}\\|\\nabla u\\|_2^2$.",
      "defining_relation": "H_\\ast=\\nabla^2p_{\\rm far}^{cluster}(x_\\ast)\\in\\operatorname{Sym}_0(3),\\quad \\nabla^2p_{\\rm far}^{cluster}(x)=H_\\ast+E_\\ast(x)",
      "notes": "Cluster-level common far matrix; plays the role that the single-core H_0 (recalled from C3-Q in Section 0) played before multi-core clustering."
    },
    {
      "id": "ns.c3.c3r.m_i",
      "latex": "M_i",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "局部平均應變矩陣",
      "label_en": "local mean strain matrix",
      "definition_zh": "第27節定義，以局部截止函數 $\\chi_i$ 對核心 $i$ 加權平均之無跡對稱應變 $M_i=\\int\\chi_iS\\,dx\\in\\operatorname{Sym}_0(3)$，是判斷共同遠場壓力能否 positive-support 所有核心之幾何對象。",
      "definition_en": "Defined in Section 27 as the trace-free symmetric mean strain $M_i=\\int\\chi_iS\\,dx\\in\\operatorname{Sym}_0(3)$ of core $i$ weighted by local cutoff $\\chi_i$; the geometric object testing whether a common far matrix can positive-support all cores.",
      "defining_relation": "M_i=\\int\\chi_iS\\,dx\\in\\operatorname{Sym}_0(3)"
    },
    {
      "id": "ns.c3.c3r.b_i_h",
      "latex": "B_i^{H}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "共同矩陣壓力功",
      "label_en": "common-matrix pressure work",
      "definition_zh": "第27節定義 $B_i^H=-H_\\ast:M_i$，為理想化共同遠場矩陣 $H_\\ast$ 對核心 $i$ 所做之壓力功，其對所有 $i$ 同為正即定理28.1（C3-R.5）之判準對象。",
      "definition_en": "Defined in Section 27 as $B_i^H=-H_\\ast:M_i$, the pressure work the idealized common far matrix $H_\\ast$ performs on core $i$; being positive for every $i$ is exactly what Theorem 28.1 (C3-R.5) tests."
    },
    {
      "id": "ns.c3.c3r.conv_hull_criterion",
      "latex": "0\\notin\\operatorname{conv}\\{M_1,\\ldots,M_m\\}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "共同壓力支持之凸包判準",
      "label_en": "common-support convex-hull criterion",
      "definition_zh": "第28節定理28.1（C3-R.5）證明：存在 $H_\\ast$ 使 $-H_\\ast:M_i>0$ 對所有 $i$ 成立，若且唯若 $0\\notin\\operatorname{conv}\\{M_1,\\ldots,M_m\\}$，即凸包分離定理在 $\\operatorname{Sym}_0(3)\\simeq\\mathbb R^5$ 中之應用。",
      "definition_en": "Theorem 28.1 (C3-R.5, Section 28) proves that some $H_\\ast$ makes $-H_\\ast:M_i>0$ for all $i$ if and only if $0\\notin\\operatorname{conv}\\{M_1,\\ldots,M_m\\}$ — the hyperplane-separation theorem applied in $\\operatorname{Sym}_0(3)\\simeq\\mathbb R^5$.",
      "defining_relation": "\\exists H_\\ast:\\ -H_\\ast:M_i>0\\ (\\forall i)\\iff 0\\notin\\operatorname{conv}\\{M_1,\\ldots,M_m\\}"
    },
    {
      "id": "ns.c3.c3r.strain_cone_debt",
      "latex": "\\textbf{Five-Dimensional Strain-Cone Coherence Debt}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "五維應變錐一致性負債",
      "label_en": "five-dimensional strain-cone coherence debt",
      "definition_zh": "第29節命名，指若要單一共同遠場矩陣 positive-support 所有核心，各核心之局部平均應變 $M_i$ 必須全部落在 $\\operatorname{Sym}_0(3)$ 中同一開半空間 $-H_\\ast:M_i>0$ 內的幾何負債。",
      "definition_en": "Named in Section 29: the geometric debt that a single common far matrix can positive-support all cores only if every local mean strain $M_i$ lies in one common open half-space $-H_\\ast:M_i>0$ of $\\operatorname{Sym}_0(3)$."
    },
    {
      "id": "ns.c3.c3r.six_core_obstruction",
      "latex": "\\textbf{Six-Core Pressure Obstruction}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "六核心壓力障礙",
      "label_en": "six-core pressure obstruction",
      "definition_zh": "第30節推論30.1（C3-R.6），因 $\\dim\\operatorname{Sym}_0(3)=5$，Carathéodory 定理給出：若 $0\\in\\operatorname{conv}\\{M_i\\}$，必有至多6個核心之子集合其凸包已含 $0$，故最多六個核心即足以見證無單一共同遠場 STF 矩陣可 positive-support 全部核心。",
      "definition_en": "Corollary 30.1 (C3-R.6, Section 30): since $\\dim\\operatorname{Sym}_0(3)=5$, Carathéodory's theorem gives that if $0\\in\\operatorname{conv}\\{M_i\\}$ then some subset of at most 6 cores already has $0$ in its convex hull, so at most six cores suffice to witness that no single common far STF matrix can positive-support all cores.",
      "defining_relation": "0\\in\\operatorname{conv}\\{M_1,\\ldots,M_m\\}\\ \\Rightarrow\\ \\exists\\,r\\le6:\\ 0\\in\\operatorname{conv}\\{M_{i_1},\\ldots,M_{i_r}\\}"
    },
    {
      "id": "ns.c3.c3r.h_i",
      "latex": "H_i",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "各核心實際遠場矩陣",
      "label_en": "actual per-core far-pressure matrix",
      "definition_zh": "第32節定義 $H_i=H_\\ast+E_i$，核心 $i$ 處相對共同矩陣 $H_\\ast$ 計入空間餘項 $E_i$ 後之實際遠場 Hessian。",
      "definition_en": "Defined in Section 32 as $H_i=H_\\ast+E_i$, the actual far-pressure Hessian at core $i$ once the spatial remainder $E_i$ relative to the common matrix $H_\\ast$ is included."
    },
    {
      "id": "ns.c3.c3r.b_i_far",
      "latex": "B_i^{far}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "實際遠場壓力功",
      "label_en": "actual far-pressure work",
      "definition_zh": "第32節定義 $B_i^{far}=-H_i:M_i=-H_\\ast:M_i-E_i:M_i$，計入空間餘項後核心 $i$ 實際承受之遠場壓力功，用於定理32.1 Robust Convexity Obstruction。",
      "definition_en": "Defined in Section 32 as $B_i^{far}=-H_i:M_i=-H_\\ast:M_i-E_i:M_i$, the actual far-pressure work on core $i$ once the spatial remainder is included; used in Theorem 32.1's Robust Convexity Obstruction."
    },
    {
      "id": "ns.c3.c3r.h_source",
      "latex": "H^{(a)}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "個別遠場來源矩陣",
      "label_en": "individual far-pressure source matrices",
      "definition_zh": "第35節定義，$N$ 個任意遠場來源區域各自貢獻之矩陣 $H^{(a)}\\in\\operatorname{Sym}_0(3)$，其和構成共同矩陣 $H_\\ast$；第35節證明5維並不能限制來源數 $N$。",
      "definition_en": "Defined in Section 35 as the matrices $H^{(a)}\\in\\operatorname{Sym}_0(3)$ contributed by $N$ arbitrary far-source regions, summing to the common matrix $H_\\ast$; Section 35 shows the 5 dimensions do not force $N\\le5$.",
      "defining_relation": "H_\\ast=\\sum_{a=1}^{N}H^{(a)}"
    },
    {
      "id": "ns.c3.c3r.gamma_h",
      "latex": "\\Gamma_H",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "壓力矩陣一致性指標",
      "label_en": "pressure-matrix coherence index",
      "definition_zh": "第36節定義之無量綱指標 $\\Gamma_H=|\\sum_aH^{(a)}|^2/\\sum_a|H^{(a)}|^2\\in[0,N]$，衡量各遠場來源矩陣係相長疊加（大）或大量抵消（小）。",
      "definition_en": "The dimensionless index $\\Gamma_H=|\\sum_aH^{(a)}|^2/\\sum_a|H^{(a)}|^2\\in[0,N]$ from Section 36, measuring whether far-source matrices reinforce coherently (large) or largely cancel (small).",
      "defining_relation": "\\Gamma_H=\\frac{\\left|\\sum_aH^{(a)}\\right|^2}{\\sum_a|H^{(a)}|^2},\\qquad 0\\le\\Gamma_H\\le N",
      "notes": "A different level of cancellation from the C3-O bulk-SSA-vs-boundary-current corridor (rho to -1, Section 38); the two must not be conflated."
    },
    {
      "id": "ns.c3.c3r.d_n",
      "latex": "d_n",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "雙核心分離距離",
      "label_en": "dual-core separation distance",
      "definition_zh": "第41–42節定義 $d_n=|x_n^{anc}-x_n^{op}|$，為 ancestry core $x_n^{anc}$ 與 operator core $x_n^{op}$ 兩種不同選取準則所得中心點間之距離。",
      "definition_en": "Defined in Sections 41-42 as $d_n=|x_n^{anc}-x_n^{op}|$, the distance between the centers picked out by two different selection principles: the ancestry core $x_n^{anc}$ and the operator core $x_n^{op}$.",
      "notes": "Superscripts anc/op denote the ancestry core (C3-G/I first-crossing/helicity criterion) versus the operator core (Miller ratio Q_SV/Delta S criterion); the two need not coincide."
    },
    {
      "id": "ns.c3.c3r.kappa_n_dual",
      "latex": "\\kappa_n^{dual}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "雙核心分離比",
      "label_en": "dual-core separation ratio",
      "definition_zh": "第42節定義之無量綱比 $\\kappa_n^{dual}=d_n/R_n$，劃分 near branch（$O(1)$）與 far branch（$\\to\\infty$）；第43節證明遠支若壓力仍相關則需 $\\mathfrak E_{R_n}\\gtrsim(\\kappa_n^{dual})^2$。",
      "definition_en": "The dimensionless ratio $\\kappa_n^{dual}=d_n/R_n$ from Section 42, splitting the near branch ($O(1)$) from the far branch ($\\to\\infty$); Section 43 shows the far branch, if still pressure-relevant, forces $\\mathfrak E_{R_n}\\gtrsim(\\kappa_n^{dual})^2$.",
      "defining_relation": "\\kappa_n^{dual}=\\frac{d_n}{R_n},\\qquad \\mathfrak E_{R_n}\\gtrsim b_0^{2/3}(\\kappa_n^{dual})^2"
    },
    {
      "id": "ns.c3.c3r.theta_r_multi",
      "latex": "\\Theta_R^{multi}",
      "series": "NS",
      "first_appearance": "C3-R",
      "label_zh": "多核心相空間張力狀態",
      "label_en": "multi-core phase-space tension state",
      "definition_zh": "第50節 True ETN 更新，將單一 ancestry ray 擴充為多核心元組 $\\Theta_R^{multi}=\\langle\\{x_i\\},m_R,\\delta_R,\\mathfrak E_R,\\kappa_{\\rm cert},\\{M_i\\},H_\\ast,\\Gamma_H,\\operatorname{Prov}\\rangle$，可形成 multi-core ancestry hypergraph。",
      "definition_en": "The True-ETN update of Section 50, extending the single ancestry ray into the multi-core tuple $\\Theta_R^{multi}=\\langle\\{x_i\\},m_R,\\delta_R,\\mathfrak E_R,\\kappa_{\\rm cert},\\{M_i\\},H_\\ast,\\Gamma_H,\\operatorname{Prov}\\rangle$, capable of forming a multi-core ancestry hypergraph.",
      "defining_relation": "\\Theta_R^{multi}=\\left\\langle\\{x_i\\}_{i=1}^{m_R},m_R,\\delta_R,\\mathfrak E_R,\\kappa_{\\rm cert},\\{M_i\\},H_\\ast,\\Gamma_H,\\operatorname{Prov}\\right\\rangle"
    },
    {
      "id": "ns.c3.c3s.trace_free_symmetric_space",
      "latex": "\\mathbb{S}_0=\\operatorname{Sym}_0(3)",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "五維無跡對稱矩陣空間",
      "label_en": "trace-free symmetric matrix space",
      "definition_zh": "第1節將所有跡為零的 3×3 對稱矩陣空間記為 $\\mathbb S_0=\\operatorname{Sym}_0(3)=\\{M=M^\\top,\\operatorname{tr}M=0\\}\\simeq\\mathbb R^5$，作為本輪 strain-cone 幾何的載體空間。",
      "definition_en": "Section 1 defines $\\mathbb S_0=\\operatorname{Sym}_0(3)\\simeq\\mathbb R^5$, the five-dimensional space of trace-free symmetric $3\\times3$ matrices that hosts the round's strain-cone geometry."
    },
    {
      "id": "ns.c3.c3s.frobenius_inner_product",
      "latex": "M:N=\\operatorname{tr}(MN)",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "Frobenius 內積",
      "label_en": "Frobenius inner product",
      "definition_zh": "第1節在 $\\mathbb S_0$ 上配置 Frobenius 內積 $M:N=\\operatorname{tr}(MN)$，是全文所有 cone 與 pressure-work 記號「:」的定義來源。",
      "definition_en": "Section 1 equips $\\mathbb S_0$ with the Frobenius inner product $M:N=\\operatorname{tr}(MN)$, which underlies every colon-pairing used throughout the round for cone and pressure-work quantities."
    },
    {
      "id": "ns.c3.c3s.unit_sphere_s4",
      "latex": "S(\\mathbb{S}_0)=\\{K\\in\\mathbb{S}_0:|K|=1\\}\\simeq S^4",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "單位球面 S⁴",
      "label_en": "unit sphere S^4",
      "definition_zh": "第1節將 $\\mathbb S_0$ 中的單位球面記為 $S(\\mathbb S_0)\\simeq S^4$，是所有 normalized strain direction 與 separator 的活動空間。",
      "definition_en": "Section 1 identifies the unit sphere of $\\mathbb S_0$ with $S^4$, the ambient sphere on which every normalized strain direction and separator lives."
    },
    {
      "id": "ns.c3.c3s.m_i_local_mean_strain",
      "latex": "M_i",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "核心局部平均應變",
      "label_en": "core local mean strain",
      "definition_zh": "第2節對每個 core $i$ 定義局部平均應變 $M_i=\\int\\chi_iS\\,dx\\in\\mathbb S_0$，只保留 $M_i\\ne0$ 的 pressure-visible cores。",
      "definition_en": "Section 2 defines each core's local mean strain $M_i=\\int\\chi_iS\\,dx\\in\\mathbb S_0$, retaining only the pressure-visible cores with $M_i\\ne0$.",
      "defining_relation": "M_i=\\int\\chi_iS\\,dx\\in\\mathbb{S}_0"
    },
    {
      "id": "ns.c3.c3s.v_i_normalized_strain_direction",
      "latex": "v_i",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "正規化應變方向",
      "label_en": "normalized strain direction",
      "definition_zh": "第2節將每個 core 的平均應變正規化為方向 $v_i=M_i/|M_i|\\in S^4$。",
      "definition_en": "Section 2 normalizes each core's mean strain to a direction $v_i=M_i/|M_i|\\in S^4$.",
      "defining_relation": "v_i=\\dfrac{M_i}{|M_i|}\\in S^4"
    },
    {
      "id": "ns.c3.c3s.v_direction_set",
      "latex": "V=\\{v_1,\\ldots,v_m\\}",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "正規化方向集合",
      "label_en": "normalized direction set",
      "definition_zh": "第2節把所有 core 的正規化應變方向收集為集合 $V=\\{v_1,\\ldots,v_m\\}\\subset S^4$，是全篇 margin 與 convex-hull 論證的基本物件。",
      "definition_en": "Section 2 collects the cores' normalized strain directions into $V=\\{v_1,\\ldots,v_m\\}\\subset S^4$, the basic object underlying every margin and convex-hull argument in the round."
    },
    {
      "id": "ns.c3.c3s.h_far_pressure_hessian",
      "latex": "H\\in\\mathbb{S}_0",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "共同遠場壓力 Hessian",
      "label_en": "common far-pressure Hessian",
      "definition_zh": "第3節引入共同遠場調和壓力 Hessian $H\\in\\mathbb S_0$（第0節先以 $H_*$ 預告），用以對所有 cores 施加同一支撐方向。",
      "definition_en": "Section 3 introduces the common far harmonic-pressure Hessian $H\\in\\mathbb S_0$ (previewed as $H_*$ in Section 0), which acts on all cores through a single shared direction.",
      "notes": "Section 0 previews this object with a star as H_*; the unstarred H is the working notation from Section 3 onward, and a starred cross-scale limit resurfaces as proof obligation T8 for the next round C3-T."
    },
    {
      "id": "ns.c3.c3s.b_i_h_pressure_work",
      "latex": "B_i^H",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "核心 i 的壓力作功",
      "label_en": "pressure work on core i",
      "definition_zh": "第3節定義 $H$ 對 core $i$ 的主導壓力作功 $B_i^H=-H:M_i=|H||M_i|(K_H:v_i)$。",
      "definition_en": "Section 3 defines the leading pressure work of $H$ on core $i$ as $B_i^H=-H:M_i=|H||M_i|(K_H:v_i)$.",
      "defining_relation": "B_i^H=-H:M_i=|H||M_i|(K_H:v_i)",
      "notes": "Section 15 shows this quantity is additive under merger, B_merge^H = -H:M_merge = sum_i B_i^H."
    },
    {
      "id": "ns.c3.c3s.k_h_pressure_support_direction",
      "latex": "K_H",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "壓力支撐方向",
      "label_en": "pressure support direction",
      "definition_zh": "第3節將共同遠場壓力方向正規化為 $K_H=-H/|H|$，使 $B_i^H>0\\ \\forall i$ 等價於 $K_H:v_i>0\\ \\forall i$。",
      "definition_en": "Section 3 normalizes the common far-pressure direction as $K_H=-H/|H|$, so that $B_i^H>0$ for all $i$ is equivalent to $K_H:v_i>0$ for all $i$.",
      "defining_relation": "K_H=-\\dfrac{H}{|H|}"
    },
    {
      "id": "ns.c3.c3s.gamma_v_strain_cone_margin",
      "latex": "\\gamma(V)",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "最佳應變錐邊際",
      "label_en": "optimal strain-cone margin",
      "definition_zh": "第4節定義最佳 common strain-cone margin $\\gamma(V)=[\\max_{|K|=1}\\min_iK:v_i]_+\\in[0,1]$，是本輪把 yes/no obstruction 升級成定量指標的核心量。",
      "definition_en": "Section 4 defines the optimal common strain-cone margin $\\gamma(V)=[\\max_{|K|=1}\\min_iK:v_i]_+\\in[0,1]$, the central quantity by which the round upgrades the yes/no obstruction into a quantitative one.",
      "defining_relation": "\\gamma(V)=\\left[\\max_{|K|=1}\\min_{1\\le i\\le m}K:v_i\\right]_+,\\qquad [x]_+=\\max\\{x,0\\}"
    },
    {
      "id": "ns.c3.c3s.thm_c3s1_margin_equals_hull_distance",
      "latex": "\\gamma(V)=\\operatorname{dist}(0,\\operatorname{conv}V)",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.1：邊際等於凸包距離定理",
      "label_en": "C3-S.1: Cone Margin = Convex-Hull Distance",
      "definition_zh": "定理5.1（C3-S.1，第5節）證明 $\\gamma(V)=\\operatorname{dist}(0,\\operatorname{conv}V)$，即最佳 strain-cone margin 恰為原點到 $V$ 凸包的歐氏距離。",
      "definition_en": "Theorem 5.1 (C3-S.1, Section 5) proves $\\gamma(V)=\\operatorname{dist}(0,\\operatorname{conv}V)$, that the optimal strain-cone margin equals the Euclidean distance from the origin to the convex hull of $V$.",
      "defining_relation": "\\gamma(V)=\\operatorname{dist}(0,\\operatorname{conv}V)"
    },
    {
      "id": "ns.c3.c3s.eta_h_actual_pressure_efficiency",
      "latex": "\\eta_H(V)=\\min_iK_H:v_i",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "實際壓力支撐效率",
      "label_en": "actual pressure-support efficiency",
      "definition_zh": "第7節定義實際共同遠場矩陣 $H$ 的支撐效率 $\\eta_H(V)=\\min_i\\frac{-H:M_i}{|H||M_i|}=\\min_iK_H:v_i$，並證明 $\\eta_H(V)\\le\\gamma(V)$ 恆成立。",
      "definition_en": "Section 7 defines the actual support efficiency $\\eta_H(V)=\\min_iK_H:v_i$ of a real far matrix $H$, and shows $\\eta_H(V)\\le\\gamma(V)$ always holds.",
      "notes": "Guard G-PEFF (Section 35) explicitly forbids substituting the optimal margin gamma for the actual efficiency eta_H."
    },
    {
      "id": "ns.c3.c3s.gamma_n_cross_scale_branches",
      "latex": "\\gamma_n=\\gamma(V_n)",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "跨尺度邊際與 S-A/S-B 分支",
      "label_en": "cross-scale margin and branches S-A/S-B",
      "definition_zh": "第8節對每個 ancestry scale $n$ 定義 $V_n\\subset S^4$ 與 $\\gamma_n=\\gamma(V_n)$，並分出 uniform coherence 分支 S-A（$\\liminf_n\\gamma_n>0$）與 cone degeneration 分支 S-B（$\\gamma_n\\to0$）。",
      "definition_en": "Section 8 defines, at each ancestry scale $n$, the set $V_n\\subset S^4$ and margin $\\gamma_n=\\gamma(V_n)$, splitting into branch S-A (uniform coherence, $\\liminf_n\\gamma_n>0$) versus branch S-B (cone degeneration, $\\gamma_n\\to0$)."
    },
    {
      "id": "ns.c3.c3s.thm_c3s2_cross_scale_separator_compactness",
      "latex": "K_\\ast\\in S^4,\\qquad K_\\ast:v_{n_k,i}\\ge\\gamma_0/2",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.2：跨尺度分離子緊緻性定理",
      "label_en": "C3-S.2: Cross-Scale Separator Compactness",
      "definition_zh": "定理9.1（C3-S.2，第9節）證明若 $\\gamma_n\\ge\\gamma_0>0$，則由 $S^4$ 緊緻性可抽出子序列與固定分離子 $K_*$，使 $K_*:v_{n_k,i}\\ge\\gamma_0/2$，第10節稱其產生的固定椎狀域為「Cross-Scale Strain-Cone Fixed Motif」（半頂角至多 $\\arccos(\\gamma_0/2)$）。",
      "definition_en": "Theorem 9.1 (C3-S.2, Section 9) shows that $\\gamma_n\\ge\\gamma_0>0$ yields, via compactness of $S^4$, a subsequence and a fixed separator $K_*$ with $K_*:v_{n_k,i}\\ge\\gamma_0/2$, which Section 10 names the \"Cross-Scale Strain-Cone Fixed Motif\" (cone half-angle at most $\\arccos(\\gamma_0/2)$).",
      "notes": "Section 11 reruns the same compactness argument on actual far-pressure directions K_n^H = -H_n/|H_n| -> K_*^H under uniform efficiency eta_{H_n} >= eta_0 > 0, producing what Section 11 calls the renormalized 5D matrix motif."
    },
    {
      "id": "ns.c3.c3s.m_merge_merged_mean_strain",
      "latex": "M_{\\rm merge}=\\sum_{i=1}^{m}M_i",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "併合平均應變",
      "label_en": "merged mean strain",
      "definition_zh": "第12節在固定共同 separator $K$（滿足 $K:M_i\\ge\\gamma|M_i|\\ \\forall i$）之下，定義併合後的平均應變 $M_{\\rm merge}=\\sum_iM_i$。",
      "definition_en": "Section 12 defines the merged mean strain $M_{\\rm merge}=\\sum_iM_i$, under a fixed common separator $K$ satisfying $K:M_i\\ge\\gamma|M_i|$ for all $i$.",
      "defining_relation": "M_{\\rm merge}=\\sum_{i=1}^{m}M_i"
    },
    {
      "id": "ns.c3.c3s.thm_c3s3_cone_inheritance_merger",
      "latex": "|M_{\\rm merge}|\\ge\\gamma\\sum_i|M_i|",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.3：併合下的錐繼承定理",
      "label_en": "C3-S.3: Cone Inheritance under Merger",
      "definition_zh": "定理13.1（C3-S.3，第13節）證明 $|M_{\\rm merge}|\\ge\\gamma\\sum_i|M_i|$ 且 $K:M_{\\rm merge}/|M_{\\rm merge}|\\ge\\gamma$，即 dense cluster 併合不會因向量抵消而摧毀 coherent mean strain，第14節稱此結果為「Merger Rigidity」。",
      "definition_en": "Theorem 13.1 (C3-S.3, Section 13) proves $|M_{\\rm merge}|\\ge\\gamma\\sum_i|M_i|$ and $K:M_{\\rm merge}/|M_{\\rm merge}|\\ge\\gamma$, showing dense-cluster merger cannot cancel coherent mean strain by vector cancellation; Section 14 names this result \"Merger Rigidity\".",
      "defining_relation": "|M_{\\rm merge}|\\ge\\gamma\\sum_i|M_i|,\\qquad K:\\dfrac{M_{\\rm merge}}{|M_{\\rm merge}|}\\ge\\gamma"
    },
    {
      "id": "ns.c3.c3s.mu_i_normalized_mean_magnitude",
      "latex": "\\mu_i=\\dfrac{|M_i|}{\\nu R}",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "正規化核心平均應變大小",
      "label_en": "normalized core mean-strain magnitude",
      "definition_zh": "第16節定義尺度不變量 $\\mu_i=|M_i|/(\\nu R)$，第17節對 pressure-relevant cores 要求 $\\mu_i\\ge\\mu_0>0$。",
      "definition_en": "Section 16 defines the scale-invariant quantity $\\mu_i=|M_i|/(\\nu R)$, required to satisfy $\\mu_i\\ge\\mu_0>0$ for pressure-relevant cores in Section 17.",
      "defining_relation": "\\mu_i=\\dfrac{|M_i|}{\\nu R}"
    },
    {
      "id": "ns.c3.c3s.thm_c3s4_merger_strain_stock_lower_bound",
      "latex": "\\dfrac{R}{\\nu^2}\\int_U|S|^2dx\\ge c\\,\\gamma^2\\mu_0^2m",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.4：併合應變存量下界定理",
      "label_en": "C3-S.4: Merger Strain-Stock Lower Bound",
      "definition_zh": "定理17.1（C3-S.4，第17節）在有界重疊、體積上界 $|U|\\le CmR^3$、cone margin 與 $\\mu_i\\ge\\mu_0$ 假設下，證明 $\\frac{R}{\\nu^2}\\int_U|S|^2dx\\ge c\\gamma^2\\mu_0^2m$。",
      "definition_en": "Theorem 17.1 (C3-S.4, Section 17) proves, under bounded overlap, the volume bound $|U|\\le CmR^3$, cone margin, and $\\mu_i\\ge\\mu_0$, that $\\frac{R}{\\nu^2}\\int_U|S|^2dx\\ge c\\gamma^2\\mu_0^2m$.",
      "defining_relation": "\\dfrac{R}{\\nu^2}\\int_U|S|^2dx\\ge c\\,\\gamma^2\\mu_0^2m"
    },
    {
      "id": "ns.c3.c3s.mathfrak_s_l_cluster_strain_stock",
      "latex": "\\mathfrak{S}_L=\\dfrac{L}{\\nu^2}\\int_U|S|^2",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "叢集尺度正規化應變存量",
      "label_en": "cluster-scale normalized strain stock",
      "definition_zh": "第18節在 $L\\sim m^{1/3}R$ 下定義叢集尺度存量 $\\mathfrak S_L=\\frac{L}{\\nu^2}\\int_U|S|^2$，並由定理17.1導出 $\\mathfrak S_L\\gtrsim\\gamma^2\\mu_0^2m^{4/3}$。",
      "definition_en": "Section 18 defines the cluster-scale stock $\\mathfrak S_L=\\frac{L}{\\nu^2}\\int_U|S|^2$ at $L\\sim m^{1/3}R$, deriving $\\mathfrak S_L\\gtrsim\\gamma^2\\mu_0^2m^{4/3}$ from Theorem 17.1."
    },
    {
      "id": "ns.c3.c3s.thm_c3s5_six_core_near_balance_witness",
      "latex": "\\left|\\sum_{j=1}^r\\alpha_jv_{i_j}\\right|=\\gamma(V),\\qquad r\\le6",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.5：六核近平衡見證定理",
      "label_en": "C3-S.5: Six-Core Near-Balance Witness",
      "definition_zh": "定理20.1（C3-S.5，第20節）用 Carathéodory theorem 證明存在至多 $r\\le6$ 個 core 方向與凸權重 $\\alpha_j$，使 $|\\sum_j\\alpha_jv_{i_j}|=\\gamma(V)$。",
      "definition_en": "Theorem 20.1 (C3-S.5, Section 20) uses the Carathéodory theorem to show there exist at most $r\\le6$ core directions and convex weights $\\alpha_j$ with $|\\sum_j\\alpha_jv_{i_j}|=\\gamma(V)$.",
      "defining_relation": "\\left|\\sum_{j=1}^r\\alpha_jv_{i_j}\\right|=\\gamma(V),\\quad r\\le6,\\ \\alpha_j\\ge0,\\ \\sum_j\\alpha_j=1",
      "notes": "When gamma(V)=0 this collapses to the exact obstruction of Section 21, which the text identifies explicitly as recovering C3-R's original six-core pressure obstruction."
    },
    {
      "id": "ns.c3.c3s.thm_c3s6_robust_six_core_obstruction",
      "latex": "\\exists j:\\ K:v_{i_j}\\le\\gamma(V)",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.6：穩健六核邊際障礙定理",
      "label_en": "C3-S.6: Robust Six-Core Margin Obstruction",
      "definition_zh": "定理22.1（C3-S.6，第22節）證明對任意單位方向 $K\\in S^4$，定理20.1見證中至少有一個核心滿足 $K:v_{i_j}\\le\\gamma(V)$。",
      "definition_en": "Theorem 22.1 (C3-S.6, Section 22) proves that for any unit direction $K\\in S^4$, at least one core in the Theorem 20.1 witness satisfies $K:v_{i_j}\\le\\gamma(V)$.",
      "defining_relation": "\\forall K\\in S^4\\ \\exists j\\in\\{1,\\ldots,r\\}:\\ K:v_{i_j}\\le\\gamma(V)"
    },
    {
      "id": "ns.c3.c3s.normalized_far_pressure_work",
      "latex": "\\widehat H,\\ \\widehat M_i,\\ \\widehat B_i^H",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "正規化遠場壓力作功",
      "label_en": "normalized far-pressure work",
      "definition_zh": "第24節沿用 C3-Q 定義正規化量 $\\widehat H=(R^4/\\nu^2)H$、$\\widehat M_i=M_i/(\\nu R)$ 與正規化壓力作功 $\\widehat B_i^H=-\\widehat H:\\widehat M_i=|\\widehat H||\\widehat M_i|\\eta_i$。",
      "definition_en": "Section 24 defines, following C3-Q, the normalized quantities $\\widehat H=(R^4/\\nu^2)H$, $\\widehat M_i=M_i/(\\nu R)$, and normalized pressure work $\\widehat B_i^H=-\\widehat H:\\widehat M_i=|\\widehat H||\\widehat M_i|\\eta_i$.",
      "defining_relation": "\\widehat H=\\dfrac{R^4}{\\nu^2}H,\\quad \\widehat M_i=\\dfrac{M_i}{\\nu R},\\quad \\widehat B_i^H=|\\widehat H||\\widehat M_i|\\eta_i,\\quad \\eta_i=-\\dfrac{\\widehat H:\\widehat M_i}{|\\widehat H||\\widehat M_i|}",
      "notes": "Notation explicitly inherited from C3-Q (Section 24 opens with \"沿用 C3-Q\")."
    },
    {
      "id": "ns.c3.c3s.b0_fixed_pressure_work_threshold",
      "latex": "b_0>0",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "固定正規化遠場壓力作功門檻",
      "label_en": "fixed normalized far-pressure work threshold",
      "definition_zh": "第0節第7點與第27節將六核見證中每個核心被要求維持的固定正規化遠場壓力作功記為門檻 $b_0>0$，即 $\\widehat B_i^H\\ge b_0$。",
      "definition_en": "Section 0 (point 7) and Section 27 denote by $b_0>0$ the fixed normalized far-pressure work every witness core is required to sustain, i.e. $\\widehat B_i^H\\ge b_0$."
    },
    {
      "id": "ns.c3.c3s.kappa_pressure_source_scale",
      "latex": "\\kappa R",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "壓力源截止尺度比",
      "label_en": "pressure-source cutoff scale ratio",
      "definition_zh": "第25節設共同遠場矩陣 $H$ 來自 $\\kappa R$ 之外，沿用 C3-Q 得到 $|\\widehat H|\\le C\\kappa^{-3}\\mathfrak E_R$。",
      "definition_en": "Section 25 takes the common far matrix $H$ to be sourced outside radius $\\kappa R$, following C3-Q to obtain $|\\widehat H|\\le C\\kappa^{-3}\\mathfrak E_R$."
    },
    {
      "id": "ns.c3.c3s.mathfrak_e_r_enstrophy_quantity",
      "latex": "\\mathfrak{E}_R=\\dfrac{R\\|\\nabla u\\|_2^2}{\\nu^2}",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "R 尺度正規化 enstrophy 量",
      "label_en": "R-scale normalized enstrophy quantity",
      "definition_zh": "第25節給出 $\\mathfrak E_R=R\\|\\nabla u\\|_2^2/\\nu^2$ 的公式，此量沿用自 C3-Q、C3-R 等早期回合，是本輪所有 pressure-support debt 不等式右側的核心量。",
      "definition_en": "Section 25 gives the formula $\\mathfrak E_R=R\\|\\nabla u\\|_2^2/\\nu^2$; carried over from earlier rounds (C3-Q, C3-R), it is the central quantity on the right-hand side of every pressure-support debt inequality in this round.",
      "defining_relation": "\\mathfrak{E}_R=\\dfrac{R\\|\\nabla u\\|_2^2}{\\nu^2}",
      "notes": "Already invoked in Section 0 (point 1, quoting C3-R's E_R >~ m_R beta_*^2) before its formula is restated here in Section 25."
    },
    {
      "id": "ns.c3.c3s.thm_c3s7_cone_degeneration_pressure_debt",
      "latex": "\\mathfrak{E}_R\\ge c\\,b_0^{2/3}\\kappa^2\\gamma^{-2/3}",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "C3-S.7：錐退化壓力債定理",
      "label_en": "C3-S.7: Cone-Degeneration Pressure Debt",
      "definition_zh": "定理27.1（C3-S.7，第27節）結合穩健六核障礙與正規化壓力、應變大小上界，證明固定核心壓力作功 $b_0$ 迫使 $\\mathfrak E_R\\ge c\\,b_0^{2/3}\\kappa^2\\gamma^{-2/3}$，即 cone margin 趨零時 enstrophy debt 發散。",
      "definition_en": "Theorem 27.1 (C3-S.7, Section 27) combines the robust six-core obstruction with bounds on normalized pressure and strain magnitude to prove that fixed per-core pressure work $b_0$ forces $\\mathfrak E_R\\ge c\\,b_0^{2/3}\\kappa^2\\gamma^{-2/3}$, so enstrophy debt diverges as the cone margin vanishes.",
      "defining_relation": "\\mathfrak{E}_R\\ge c\\,b_0^{2/3}\\kappa^2\\gamma^{-2/3}",
      "notes": "Strictly stronger than C3-Q's bound E_R >~ b_0^{2/3} kappa^2 (Section 28), which lacked the gamma^{-2/3} cone-degeneration factor; the round names this the Cone-Degeneration Pressure Debt."
    },
    {
      "id": "ns.c3.c3s.pressure_support_diversification",
      "latex": "\\textbf{Pressure-Support Diversification}",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "壓力支撐多樣化",
      "label_en": "Pressure-Support Diversification",
      "definition_zh": "第29節將定理27.1的 contrapositive 讀法命名為「Pressure-Support Diversification」：若 $\\mathfrak E_R$ 不足以支付 $\\gamma^{-2/3}$ 債務，六核見證中至少一核必須依賴 near pressure、bulk SSA、Betchov current 等其他來源。",
      "definition_en": "Section 29 names the contrapositive reading of Theorem 27.1 \"Pressure-Support Diversification\": if $\\mathfrak E_R$ cannot pay the $\\gamma^{-2/3}$ debt, at least one witness core must draw support from near pressure, bulk SSA, Betchov current, or another source instead of the common far matrix."
    },
    {
      "id": "ns.c3.c3s.lambda2_plus_middle_eigenvalue",
      "latex": "\\lambda_2^+(M)>0",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "中間特徵值正部",
      "label_en": "positive part of the middle eigenvalue",
      "definition_zh": "第32至33節指出 cone coherence 條件 $K_*:M>0$（僅為 $\\mathbb R^5$ 中的 Frobenius 夾角條件）不能推出共同 eigenframe、也不能取代中間特徵值正部條件 $\\lambda_2^+(M)>0$。",
      "definition_en": "Sections 32-33 note that the cone-coherence condition $K_*:M>0$ (a mere Frobenius-angle condition in $\\mathbb R^5$) implies neither a shared eigenframe nor the middle-eigenvalue condition $\\lambda_2^+(M)>0$.",
      "notes": "Explicitly flagged as belonging to the separate C3-L/M middle-eigenvalue channel, which this round states must still be independently maintained (see also no-go NG-S3)."
    },
    {
      "id": "ns.c3.c3s.theta_n_cone_true_etn_update",
      "latex": "\\Theta_n^{cone}=\\left\\langle V_n,\\gamma_n,K_n,K_\\ast,\\text{six-core witness},\\mathfrak{E}_{R_n},\\kappa_n,\\operatorname{Prov}\\right\\rangle",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "True ETN 之 cone 分量更新",
      "label_en": "True ETN cone-component update",
      "definition_zh": "第36節把本輪 multi-core strain geometry 打包為 True ETN 的新分量 $\\Theta_n^{cone}=\\langle V_n,\\gamma_n,K_n,K_*,\\text{six-core witness},\\mathfrak E_{R_n},\\kappa_n,\\operatorname{Prov}\\rangle$，取代原本二元的 $0\\in\\operatorname{conv}V$ 判準。",
      "definition_en": "Section 36 packages this round's multi-core strain geometry into a new True-ETN component $\\Theta_n^{cone}=\\langle V_n,\\gamma_n,K_n,K_*,\\text{six-core witness},\\mathfrak E_{R_n},\\kappa_n,\\operatorname{Prov}\\rangle$, replacing the earlier binary criterion $0\\in\\operatorname{conv}V$."
    },
    {
      "id": "ns.c3.c3s.x_integration_guards",
      "latex": "\\text{G-CMARGIN, G-SEPARATOR, G-MERGE-CONE, G-6NEAR, G-PEFF, G-CDEBT, G-EIGTYPE}",
      "series": "NS",
      "first_appearance": "C3-S",
      "label_zh": "X-Integration 守則更新",
      "label_en": "X-Integration guard updates",
      "definition_zh": "第35節為 X-Integration 框架新增七條具名守則，分別鎖定 margin 公式、跨尺度分離子、併合下不可刪除的 no-cancellation 下界、六核見證、實際效率不可被最佳值取代、cone-degeneration debt，以及禁止把 matrix-cone coherence 升格為 eigenframe coherence。",
      "definition_en": "Section 35 adds seven named guards to the X-Integration framework, respectively locking in the margin formula, the cross-scale separator, the no-cancellation bound under merger, the six-core witness, the rule that actual efficiency may not be replaced by the optimum, the cone-degeneration debt, and the prohibition on upgrading matrix-cone coherence to eigenframe coherence."
    },
    {
      "id": "ns.c3.c3t.sym0",
      "latex": "\\mathbb{S}_0 = \\operatorname{Sym}_0(3)",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "無跡對稱矩陣空間",
      "label_en": "Trace-free symmetric matrix space",
      "definition_zh": "第1節定義 $\\mathbb S_0=\\operatorname{Sym}_0(3)$ 為三維無跡對稱矩陣所成的空間，作為本輪 strain 矩陣幾何的環境空間。",
      "definition_en": "Section 1 defines $\\mathbb S_0=\\operatorname{Sym}_0(3)$ as the space of trace-free symmetric 3×3 matrices, the ambient space for this round's strain-matrix geometry."
    },
    {
      "id": "ns.c3.c3t.k_kappa_eigenvalues",
      "latex": "K\\in\\mathbb S_0,\\ |K|=1,\\ \\kappa_1\\le\\kappa_2\\le\\kappa_3,\\ \\kappa_1<0<\\kappa_3",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "測試矩陣 K 與其特徵值",
      "label_en": "Test matrix K and its eigenvalues",
      "definition_zh": "第1節取任意單位無跡對稱矩陣 $K$，令其特徵值排序為 $\\kappa_1\\le\\kappa_2\\le\\kappa_3$，並由無跡性證得 $\\kappa_1<0<\\kappa_3$。",
      "definition_en": "Section 1 fixes a unit trace-free symmetric matrix $K$ with ordered eigenvalues $\\kappa_1\\le\\kappa_2\\le\\kappa_3$, and trace-freeness forces $\\kappa_1<0<\\kappa_3$."
    },
    {
      "id": "ns.c3.c3t.lambda2",
      "latex": "\\lambda_2(M):=\\kappa_2(M)",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "中間特徵值函數 λ2",
      "label_en": "Middle-eigenvalue function λ2",
      "definition_zh": "自第2節起，$\\lambda_2(M)$ 表示無跡對稱矩陣 $M$ 由小到大排序後的第二個（中間）特徵值，是 Miller regularity criterion 的核心量。",
      "definition_en": "From Section 2 onward, $\\lambda_2(M)$ denotes the second (middle) eigenvalue of a trace-free symmetric matrix $M$ in ascending order, the core quantity of Miller's regularity criterion.",
      "defining_relation": "\\lambda_2(M):=\\kappa_2(M),\\quad \\kappa_1(M)\\le\\kappa_2(M)\\le\\kappa_3(M)",
      "notes": "此量承接自 Miller 的 middle-eigenvalue regularity criterion（References 1–2），本輪不重新定義，只檢驗它與 mean strain cone 的關係。"
    },
    {
      "id": "ns.c3.c3t.v_plus",
      "latex": "V^+=\\tfrac1{\\sqrt6}\\operatorname{diag}(-2,1,1)",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "雙向伸展測試矩陣 V+",
      "label_en": "Two-stretching test matrix V+",
      "definition_zh": "第2節在 $K$ 的特徵基底中構造單位矩陣 $V^+=\\frac1{\\sqrt6}\\operatorname{diag}(-2,1,1)$，滿足 $K:V^+>0$ 且 $\\lambda_2(V^+)>0$。",
      "definition_en": "Section 2 constructs the unit matrix $V^+=\\frac1{\\sqrt6}\\operatorname{diag}(-2,1,1)$ in the eigenbasis of $K$, satisfying $K:V^+>0$ and $\\lambda_2(V^+)>0$.",
      "defining_relation": "V^+=\\tfrac1{\\sqrt6}\\operatorname{diag}(-2,1,1),\\quad \\lambda_2(V^+)=\\tfrac1{\\sqrt6}>0"
    },
    {
      "id": "ns.c3.c3t.v_minus",
      "latex": "V^-=\\tfrac1{\\sqrt6}\\operatorname{diag}(-1,-1,2)",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "單向伸展測試矩陣 V-",
      "label_en": "One-stretching test matrix V-",
      "definition_zh": "第3節構造單位矩陣 $V^-=\\frac1{\\sqrt6}\\operatorname{diag}(-1,-1,2)$，同樣滿足 $K:V^->0$，但 $\\lambda_2(V^-)<0$，與 $V^+$ 形成對照。",
      "definition_en": "Section 3 constructs the unit matrix $V^-=\\frac1{\\sqrt6}\\operatorname{diag}(-1,-1,2)$, which also satisfies $K:V^->0$ but has $\\lambda_2(V^-)<0$, contrasting with $V^+$.",
      "defining_relation": "V^-=\\tfrac1{\\sqrt6}\\operatorname{diag}(-1,-1,2),\\quad \\lambda_2(V^-)=-\\tfrac1{\\sqrt6}<0"
    },
    {
      "id": "ns.c3.c3t.half_space",
      "latex": "\\mathcal H_K^+=\\{M:K:M>0\\}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "K 的開半空間",
      "label_en": "Open half-space of K",
      "definition_zh": "第4節定義 $\\mathcal H_K^+=\\{M:K:M>0\\}$，並以 $V^+,V^-$ 證明此半空間同時包含正、負中間特徵值的矩陣。",
      "definition_en": "Section 4 defines $\\mathcal H_K^+=\\{M:K:M>0\\}$ and uses $V^+,V^-$ to show this half-space contains matrices with both positive and negative middle eigenvalue."
    },
    {
      "id": "ns.c3.c3t.thm_halfspace_nonrigidity",
      "latex": "\\text{C3-T.1 (Thm 4.1)}:\\ \\text{fixed 5D strain half-space}\\not\\Rightarrow\\lambda_2\\text{ sign}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "半空間符號非剛性定理（C3-T.1）",
      "label_en": "Half-Space Signature Non-Rigidity theorem (C3-T.1)",
      "definition_zh": "第4節定理4.1（C3-T.1）證明固定的5維 strain 半空間並不能決定中間特徵值 $\\lambda_2$ 的正負號。",
      "definition_en": "Theorem 4.1 (C3-T.1) in Section 4 proves that a fixed 5-dimensional strain half-space does not determine the sign of the middle eigenvalue $\\lambda_2$."
    },
    {
      "id": "ns.c3.c3t.gamma_coherence",
      "latex": "\\gamma,\\quad K:V\\ge\\gamma\\ \\Rightarrow\\ |V-K|_F\\le\\sqrt{2(1-\\gamma)}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "窄錐相合參數 γ",
      "label_en": "Narrow-cone coherence parameter γ",
      "definition_zh": "第5節設單位矩陣 $K,V\\in\\mathbb S_0$ 滿足 $K:V\\ge\\gamma$，用以量化兩者方向的相合程度，並推出 Frobenius 距離上界。",
      "definition_en": "Section 5 assumes unit matrices $K,V\\in\\mathbb S_0$ with $K:V\\ge\\gamma$, quantifying directional coherence and yielding a Frobenius-distance bound."
    },
    {
      "id": "ns.c3.c3t.thm_eigenvalue_inheritance",
      "latex": "\\text{Thm 5.1};\\ \\text{T-A1/T-A2}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "窄錐特徵值繼承定理與 T-A1/T-A2 分支",
      "label_en": "Narrow-cone eigenvalue inheritance theorem and T-A1/T-A2 branch split",
      "definition_zh": "第5–6節由 Weyl 不等式證明定理5.1，並據此把 uniform cone 分裂為窄錐 T-A1（$\\lambda_2$ 符號被鎖定）與寬／退化錐 T-A2 兩分支。",
      "definition_en": "Sections 5–6 use Weyl's inequality to prove Theorem 5.1, splitting the uniform-cone branch into narrow-cone T-A1 (where the sign of $\\lambda_2$ is locked) and wide/degenerate-cone T-A2.",
      "defining_relation": "|\\lambda_2(V)-\\lambda_2(K)|\\le\\sqrt{2(1-\\gamma)}",
      "notes": "此分支記號沿用第25節 survivor map 的體系，並在 C3-U 的 U6 obligation 中被重新引用。"
    },
    {
      "id": "ns.c3.c3t.local_mean_strain",
      "latex": "M_i=\\int\\chi_i S\\,dx",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "局部平均應變 M_i",
      "label_en": "Local mean strain M_i",
      "definition_zh": "第7節定義 $M_i=\\int\\chi_i S\\,dx$ 為第 $i$ 個核區域上應變張量的未歸一化局部矩，是 uniform cone 分析所依賴的一階量。",
      "definition_en": "Section 7 defines $M_i=\\int\\chi_i S\\,dx$ as the unnormalized local first moment of the strain tensor over core $i$, the first-order quantity the uniform-cone analysis relies on.",
      "defining_relation": "M_i=\\int\\chi_i S\\,dx"
    },
    {
      "id": "ns.c3.c3t.normalized_mean_strain",
      "latex": "\\overline S_i=M_i/m_i",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "歸一化平均應變 S̄_i",
      "label_en": "Normalized mean strain S̄_i",
      "definition_zh": "第7節在 $m_i=\\int\\chi_i\\,dx>0$ 時定義 $\\overline S_i=M_i/m_i$，其方向與特徵值符號與 $M_i$ 相同，但仍只是空間平均而非逐點量。",
      "definition_en": "Section 7 defines $\\overline S_i=M_i/m_i$ for $m_i=\\int\\chi_i\\,dx>0$; it shares $M_i$'s normalized direction and eigenvalue sign but remains a spatial average, not a pointwise quantity.",
      "defining_relation": "\\overline S_i=\\dfrac{M_i}{m_i},\\qquad m_i=\\int\\chi_i\\,dx"
    },
    {
      "id": "ns.c3.c3t.miller_pointwise_lambda2",
      "latex": "\\lambda_2^+(S(x,t))",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "Miller 逐點中間特徵值判準",
      "label_en": "Miller's pointwise middle-eigenvalue criterion",
      "definition_zh": "第7節指出 Miller 的 regularity criterion 使用逐點／空間範數量 $\\lambda_2^+(S(x,t))$，而非核平均量 $\\lambda_2(\\overline S_i)$，兩者的落差是本輪核心問題。",
      "definition_en": "Section 7 notes that Miller's regularity criterion uses the pointwise/spatial-norm quantity $\\lambda_2^+(S(x,t))$, not the core-averaged $\\lambda_2(\\overline S_i)$, and the gap between the two is this round's central problem.",
      "notes": "引自 Miller 2020/2026（References 1–2）的 regularity criterion，本輪不重新推導，只檢驗 mean strain 能否控制它。"
    },
    {
      "id": "ns.c3.c3t.thm_mean_to_pointwise_nogo",
      "latex": "\\text{C3-T.2}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "均值—逐點中間特徵值不可行定理",
      "label_en": "Mean-to-Pointwise Middle-Eigenvalue No-Go (C3-T.2)",
      "definition_zh": "第8節以 $A=\\operatorname{diag}(2,-1,-1)$、$B=\\operatorname{diag}(-1,2,-1)$ 為反例證明 C3-T.2：均值 $(A+B)/2$ 的 $\\lambda_2$ 可與 $A,B$ 各自的 $\\lambda_2$ 異號，故均值中間特徵值不能推出逐點中間特徵值。",
      "definition_en": "Section 8 uses the counterexample $A=\\operatorname{diag}(2,-1,-1)$, $B=\\operatorname{diag}(-1,2,-1)$ to prove C3-T.2: the $\\lambda_2$ of the average $(A+B)/2$ can have opposite sign to $\\lambda_2(A)=\\lambda_2(B)$, so mean middle-eigenvalue sign does not imply pointwise sign.",
      "defining_relation": "\\lambda_2(A)=\\lambda_2(B)=-1<0,\\quad \\lambda_2\\!\\left(\\tfrac{A+B}2\\right)=\\tfrac12>0"
    },
    {
      "id": "ns.c3.c3t.fluctuation_ratio",
      "latex": "\\mathfrak F_i",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "平均波動比 𝔉_i",
      "label_en": "Mean fluctuation ratio 𝔉_i",
      "definition_zh": "第24節正式定義（第9節先行使用）$\\mathfrak F_i=\\|S-\\overline S_i\\|_{L^\\infty(C_i),op}/|\\overline S_i|$，用以量化核內逐點應變偏離其平均值的程度。",
      "definition_en": "Section 24 formally defines (anticipated in Section 9) $\\mathfrak F_i=\\|S-\\overline S_i\\|_{L^\\infty(C_i),op}/|\\overline S_i|$, quantifying how far the pointwise strain on a core deviates from its mean.",
      "defining_relation": "\\mathfrak F_i=\\dfrac{\\|S-\\overline S_i\\|_{L^\\infty(C_i),op}}{|\\overline S_i|}",
      "notes": "第9、24節證明：若 $\\lambda_2(\\overline S_i/|\\overline S_i|)\\ge\\delta>0$ 且 $\\mathfrak F_i<\\delta$，則核上逐點 $\\lambda_2(S(x))>0$；但目前沒有 uniform theorem 保證 $\\mathfrak F_i<\\delta$，此缺口成為 C3-U 的 U5 obligation。"
    },
    {
      "id": "ns.c3.c3t.gamma_n_witnesses",
      "latex": "\\gamma_n\\to0,\\quad v_{n,i}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "錐退化參數與六核見證方向",
      "label_en": "Cone-degeneration parameter and six-core witness directions",
      "definition_zh": "第0、10節承接 C3-S：退化間隙 $\\gamma_n=\\operatorname{dist}(0,\\operatorname{conv}V_n)\\to0$ 對應每尺度至多六個歸一化見證方向 $v_{n,i_1},\\ldots,v_{n,i_r}$（$r\\le6$）。",
      "definition_en": "Sections 0 and 10 carry over from C3-S: the degeneration gap $\\gamma_n=\\operatorname{dist}(0,\\operatorname{conv}V_n)\\to0$ is matched at each scale by at most six normalized witness directions $v_{n,i_1},\\ldots,v_{n,i_r}$ ($r\\le6$).",
      "defining_relation": "\\gamma_n=\\operatorname{dist}(0,\\operatorname{conv}V_n)\\to0,\\qquad \\Big|\\sum_{j=1}^r\\alpha_jv_{n,i_j}\\Big|=\\gamma_n,\\ r\\le6",
      "notes": "$\\gamma_n$ 與 $v_{n,i}$ 皆定義於前一輪 C3-S（NS_C3S_StrainConeMargin_MergerRigidity_v0.1.md），本輪直接沿用作為 cone degeneration branch 的輸入資料。"
    },
    {
      "id": "ns.c3.c3t.pressure_poor_efficiency",
      "latex": "\\eta_H,\\ \\eta_0,\\quad \\eta_H\\le\\eta_0\\ (\\text{pressure-poor})",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "共同壓力效率與 pressure-poor 定義",
      "label_en": "Common-pressure efficiency and pressure-poor definition",
      "definition_zh": "第11、14節定義共同遠場壓力方向對核的效率 $\\eta_H=K_H:v_{n,i}$，並稱核為 pressure-poor，若其效率不超過固定 threshold $\\eta_0>0$，即 $\\eta_H\\le\\eta_0$。",
      "definition_en": "Sections 11 and 14 define the efficiency $\\eta_H=K_H:v_{n,i}$ of a common far-pressure direction on a core, calling the core pressure-poor when its efficiency stays below a fixed threshold, $\\eta_H\\le\\eta_0$.",
      "defining_relation": "\\exists\\,i_\\ast:\\ K_H:v_{n,i_\\ast}\\le\\gamma_n;\\qquad \\eta_H\\le\\eta_0\\ \\Rightarrow\\ \\text{core is pressure-poor}"
    },
    {
      "id": "ns.c3.c3t.enstrophy_branch_split",
      "latex": "\\mathfrak E_{R_n};\\quad \\text{T-B1/T-B2}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "enstrophy 補償界與 T-B1/T-B2 分支",
      "label_en": "Enstrophy compensation bound and T-B1/T-B2 branch split",
      "definition_zh": "第12節證明若見證核要求固定共同遠場壓力功 $b_0>0$，則 enstrophy 須滿足下界，並據此把 cone degeneration 分裂為 T-B1（enstrophy 補償）與 T-B2（pressure-support diversification）。",
      "definition_en": "Section 12 proves that if witness cores demand a fixed common far-pressure work $b_0>0$, the enstrophy must satisfy a lower bound, splitting cone degeneration into T-B1 (enstrophy compensation) and T-B2 (pressure-support diversification).",
      "defining_relation": "\\mathfrak E_{R_n}\\gtrsim b_0^{2/3}\\kappa_n^2\\gamma_n^{-2/3}",
      "notes": "$\\mathfrak E_{R_n}$（rescaled enstrophy）本身承自 C3-S；本輪新增的是此下界如何驅動 T-B1/T-B2 的二選一。"
    },
    {
      "id": "ns.c3.c3t.g_req_decomposition",
      "latex": "G_i^{req}=P_i^{common}+A_i^{alt}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "所需成長量分解",
      "label_en": "Required growth-demand decomposition",
      "definition_zh": "第13節把某見證核的歸一化所需成長量 $G_i^{req}$ 分解為共同遠場壓力貢獻 $P_i^{common}$ 與其餘替代通道貢獻 $A_i^{alt}$（bulk SSA、near pressure、Betchov 等）。",
      "definition_en": "Section 13 decomposes a witness core's normalized required growth demand $G_i^{req}$ into the common far-pressure contribution $P_i^{common}$ and the remaining alternative-channel contribution $A_i^{alt}$ (bulk SSA, near pressure, Betchov, etc.).",
      "defining_relation": "G_i^{req}=P_i^{common}+A_i^{alt},\\qquad P_i^{common}\\le C\\gamma\\kappa^{-3}\\mathfrak E_R^{3/2}"
    },
    {
      "id": "ns.c3.c3t.thm_diversification_bound",
      "latex": "\\text{Thm 13.1}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "壓力支持多樣化下界定理",
      "label_en": "Pressure-Support Diversification Lower Bound (Thm 13.1)",
      "definition_zh": "第13節定理13.1證明：若 $G_i^{req}\\ge g_0>0$ 且共同壓力上界不超過 $g_0/2$，則替代通道必須滿足 $A_i^{alt}\\ge g_0/2$。",
      "definition_en": "Theorem 13.1 in Section 13 proves that if $G_i^{req}\\ge g_0>0$ and the common-pressure bound is at most $g_0/2$, then the alternative channel must satisfy $A_i^{alt}\\ge g_0/2$.",
      "defining_relation": "A_i^{alt}\\ge g_0-C\\gamma\\kappa^{-3}\\mathfrak E_R^{3/2}\\ \\Rightarrow\\ A_i^{alt}\\ge\\tfrac12g_0"
    },
    {
      "id": "ns.c3.c3t.nogo_witness_not_ray",
      "latex": "\\text{No-Go 15.1}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "逐層見證非因果射線否定結果",
      "label_en": "Per-Level Witness ≠ Causal Ray No-Go",
      "definition_zh": "第15節以無窮二元樹上只標記深度 $n$ 節點 $0^{n-1}1$ 為反例，證明每尺度都有 pressure-poor 見證核並不蘊含存在一條無窮 pressure-poor 因果祖先射線。",
      "definition_en": "Section 15 uses the infinite binary tree marking only the depth-$n$ node $0^{n-1}1$ as a counterexample, proving that a pressure-poor witness core at every scale does not imply an infinite pressure-poor causal ancestry ray exists."
    },
    {
      "id": "ns.c3.c3t.thm_heredity_criterion",
      "latex": "\\text{C3-T.3};\\ \\mathcal T,\\ \\mathcal P",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "遺傳性 pressure-poor 射線判準",
      "label_en": "Hereditary Pressure-Poor Ray Criterion (C3-T.3)",
      "definition_zh": "第16節定理 C3-T.3 證明：若局部有限因果祖先樹 $\\mathcal T$ 中的 pressure-poor 節點集 $\\mathcal P$ 內每個節點都有一個 pressure-poor 子節點，則存在無窮 pressure-poor 因果射線。",
      "definition_en": "Theorem C3-T.3 in Section 16 proves that if every node in the pressure-poor node set $\\mathcal P$ of a locally finite causal ancestry tree $\\mathcal T$ has a pressure-poor child, then an infinite pressure-poor causal ray exists.",
      "notes": "此結果為純組合式（PROVED COMBINATORIAL，見第31節），但條件是否適用於實際 N–S 因果樹仍是 OPEN，見 Pressure-Poor Heredity Lemma 條目。"
    },
    {
      "id": "ns.c3.c3t.heredity_lemma_open",
      "latex": "\\text{Pressure-Poor Heredity Lemma}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "壓力貧乏遺傳引理（未決）",
      "label_en": "Pressure-Poor Heredity Lemma (open)",
      "definition_zh": "第17節提出候選引理：$\\eta_H(parent)\\ll1$ 加上 local source dominance 應蘊含存在子節點使 $\\eta_H(child)\\lesssim\\eta_H(parent)+\\varepsilon$，第31節將其狀態列為 OPEN。",
      "definition_en": "Section 17 proposes the candidate lemma that $\\eta_H(parent)\\ll1$ plus local source dominance should imply a child with $\\eta_H(child)\\lesssim\\eta_H(parent)+\\varepsilon$, and Section 31 lists its status as OPEN.",
      "defining_relation": "\\eta_H(parent)\\ll1+\\text{local source dominance}\\ \\Rightarrow\\ \\exists\\,child:\\eta_H(child)\\lesssim\\eta_H(parent)+\\varepsilon",
      "notes": "此為第30節 C3-U 的 U1 proof obligation 之候選陳述，是本輪標定的「真正缺的 lemma」。"
    },
    {
      "id": "ns.c3.c3t.miller_operator",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "Miller 算子 Q_SV",
      "label_en": "Miller operator Q_SV",
      "definition_zh": "第19節指出 Miller operator $\\mathcal Q_{SV}$ 依賴空間導數、二次波動、渦度、平流與非局部投影，是 mean-strain cone 無法直接控制的對象。",
      "definition_en": "Section 19 notes that the Miller operator $\\mathcal Q_{SV}$ depends on spatial derivatives, quadratic fluctuations, vorticity, advection, and nonlocal projection — an object the mean-strain cone cannot directly control.",
      "notes": "此算子命名承自更早輪次（NS_C3P_OperatorEscape_FarPressureMatrix、NS_C3Q_PressureProjection_OperatorLocalization），本輪只引用不重新定義。"
    },
    {
      "id": "ns.c3.c3t.thm_meanmotif_operator_nogo",
      "latex": "\\text{C3-T.4}:\\ \\text{fixed mean-strain motif}\\not\\Rightarrow\\text{small Miller operator defect}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "均值 motif／算子波動分離不可行定理",
      "label_en": "Mean-Motif / Operator-Fluctuation Separation No-Go (C3-T.4)",
      "definition_zh": "第20節定理 C3-T.4 證明均值映射 $M_\\chi(S)=\\int\\chi S\\,dx$ 只有五個純量輸出且核為無窮維，故保持 mean-strain cone 不變仍可任意改變 $\\mathcal Q_{SV}$。",
      "definition_en": "Theorem C3-T.4 in Section 20 proves that the mean map $M_\\chi(S)=\\int\\chi S\\,dx$ has only five scalar outputs with an infinite-dimensional kernel, so fixing the mean-strain cone motif still allows $\\mathcal Q_{SV}$ to vary arbitrarily."
    },
    {
      "id": "ns.c3.c3t.operator_escape_distance",
      "latex": "d_{SV}(t_n)",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "算子逃逸距離 d_SV",
      "label_en": "Operator-escape distance d_SV",
      "definition_zh": "第21節指出固定 cone coherence $\\gamma_n\\ge\\gamma_0$ 與算子逃逸 $d_{SV}(t_n)\\gtrsim1$ 目前完全相容，逃逸可藏在均值 motif 周圍的零均值高頻波動中。",
      "definition_en": "Section 21 notes that fixed cone coherence $\\gamma_n\\ge\\gamma_0$ and operator escape $d_{SV}(t_n)\\gtrsim1$ are currently fully compatible, with the escape hidden in zero-mean high-frequency fluctuations around the coherent mean motif.",
      "notes": "$d_{SV}$ 承接自更早的 operator-escape 系列輪次，本輪只用它具體化 C3-T.4 no-go 在動力學層面的後果。"
    },
    {
      "id": "ns.c3.c3t.two_motifs_angle",
      "latex": "K_\\ast^{cone},\\ K_\\ast^{pressure},\\ \\vartheta_\\ast=\\arccos(K_\\ast^{pressure}:K_\\ast^{cone})",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "錐與壓力兩固定 motif 及其夾角",
      "label_en": "Cone and pressure fixed motifs and their angle",
      "definition_zh": "第22節由 $S^4$ 緊緻性取極限方向 $K_H^\\ast$，指出 cone motif $K_\\ast^{cone}$ 與 pressure motif $K_\\ast^{pressure}$ 一般不相同，其夾角 $\\vartheta_\\ast$ 目前無定理固定。",
      "definition_en": "Section 22 extracts a limiting direction $K_H^\\ast$ via compactness of $S^4$, noting that the cone motif $K_\\ast^{cone}$ and pressure motif $K_\\ast^{pressure}$ need not coincide, and no current theorem fixes their angle $\\vartheta_\\ast$."
    },
    {
      "id": "ns.c3.c3t.etn_tuples",
      "latex": "\\Theta_n^{motif},\\ \\Theta_n^{div}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "True ETN 狀態元組更新",
      "label_en": "True ETN state-tuple update",
      "definition_zh": "第28節把 uniform branch 的狀態記為 $\\Theta_n^{motif}=\\langle K_\\ast,\\gamma_0,\\lambda_2(K_\\ast),\\{\\mathfrak F_i\\},d_{SV},K_H^\\ast,\\operatorname{Prov}\\rangle$，degenerate branch 記為 $\\Theta_n^{div}$，作為 True ETN／無限維張力場的本輪更新。",
      "definition_en": "Section 28 records the uniform branch's state as $\\Theta_n^{motif}=\\langle K_\\ast,\\gamma_0,\\lambda_2(K_\\ast),\\{\\mathfrak F_i\\},d_{SV},K_H^\\ast,\\operatorname{Prov}\\rangle$ and the degenerate branch's as $\\Theta_n^{div}$, this round's update to the True ETN / infinite-dimensional tension field.",
      "notes": "True ETN（無限維張力場）是跨輪次的公共狀態物件，列於 Internal dependencies；本輪只新增這兩個元組欄位，不重新定義整體框架。"
    },
    {
      "id": "ns.c3.c3t.branch_survivor_map",
      "latex": "\\text{T-U}:\\gamma_n\\ge\\gamma_0;\\quad \\text{T-D}:\\gamma_n\\to0",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "存活分支圖：T-U 與 T-D",
      "label_en": "Survivor-map branches T-U and T-D",
      "definition_zh": "第25節把全篇結論整理成兩條存活分支：T-U（uniform cone，剩餘 debt 為 mean-to-pointwise 與算子波動）與 T-D（cone degeneration，剩餘 debt 為 enstrophy 補償或 pressure-support diversification，外加 heredity 缺口）。",
      "definition_en": "Section 25 organizes the round's conclusions into two surviving branches: T-U (uniform cone, remaining debt = mean-to-pointwise plus operator fluctuation) and T-D (cone degeneration, remaining debt = enstrophy compensation or pressure-support diversification, plus the heredity gap)."
    },
    {
      "id": "ns.c3.c3t.x_integration_guards",
      "latex": "\\text{G-MEAN/PT, G-EIGGAP, G-FLUCT, G-PPOOR, G-HERED, G-MEANOP}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "X-Integration 六項防護準則",
      "label_en": "X-Integration guard conditions",
      "definition_zh": "第27節列出六條 X-Integration guards，分別要求不得混淆均值與逐點應變、窄錐須檢查特徵值間隙、須保存波動比 $\\mathfrak F_i$、pressure-poor 見證不可自動視為遺傳、因果射線需要遺傳性、以及均值 motif 不得用來控制算子波動。",
      "definition_en": "Section 27 lists six X-Integration guards requiring, respectively: not conflating mean and pointwise strain; checking the eigenvalue gap for narrow cones; tracking the fluctuation ratio $\\mathfrak F_i$; never auto-promoting a pressure-poor witness to hereditary status; requiring heredity/bounded-gap inheritance for a causal ray; and forbidding the mean motif from controlling operator fluctuation."
    },
    {
      "id": "ns.c3.c3t.c3u_frontier",
      "latex": "\\text{C3-U}",
      "series": "NS",
      "first_appearance": "C3-T",
      "label_zh": "C3-U：下一輪前沿",
      "label_en": "C3-U next-round frontier",
      "definition_zh": "第29–30節把本輪未閉合缺口正式命名為 C3-U——Hereditary Pressure-Poor Ancestry and Mean-to-Pointwise Strain Rigidity，並展開為 U1 至 U8 共八項 proof obligations。",
      "definition_en": "Sections 29–30 formally name this round's open gaps C3-U — Hereditary Pressure-Poor Ancestry and Mean-to-Pointwise Strain Rigidity — and expand it into eight proof obligations, U1 through U8.",
      "notes": "八項 obligations（U1 pressure-poor heredity、U2 bounded-generation heredity、U3 far-pressure matrix transport、U4 mean-strain transport、U5 mean fluctuation control、U6 narrow-cone pointwise branch、U7 operator fluctuation branch、U8 pressure-poor ray closure）逐一對應本輪各 no-go 缺口，是本輪對後續輪次的直接交付。"
    },
    {
      "id": "ns.c3.c3u.pressure_source_f",
      "latex": "f = \\operatorname{tr}((\\nabla u)^2)",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "壓力源項",
      "label_en": "Pressure source term",
      "definition_zh": "第1節將壓力源定義為 f = tr((∇u)²)，滿足 -Δp = f。",
      "definition_en": "Section 1 defines the pressure source as f = tr((∇u)²), satisfying -Δp = f.",
      "defining_relation": "-\\Delta p = f = \\operatorname{tr}((\\nabla u)^2)"
    },
    {
      "id": "ns.c3.c3u.pressure_hessian_kernel",
      "latex": "H_p(x,t)=\\nabla^2p(x,t)=\\int_{\\mathbb R^3}K(x-y)f(y,t)\\,dy,\\quad |\\nabla^mK(z)|\\le C_m|z|^{-3-m}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "壓力 Hessian 核",
      "label_en": "Pressure Hessian kernel",
      "definition_zh": "第1節將壓力 Hessian 場 H_p(x,t)=∇²p(x,t) 寫成核 K 對壓力源 f 的卷積，並給出核滿足 |z|^(-3-m) 的衰減估計。",
      "definition_en": "Section 1 expresses the pressure Hessian field H_p(x,t)=∇²p(x,t) as the convolution of kernel K against the pressure source f, with a |z|^(-3-m) decay bound on the kernel.",
      "notes": "同一符號 H_p 在第3節被重新賦義為 parent 的遠場截斷矩陣，與此處「壓力 Hessian 場」原意不同，下文各定理實際採用第3節之義。"
    },
    {
      "id": "ns.c3.c3u.parent_child_ancestry",
      "latex": "P=(x_p,t_p,R_p),\\quad C=(x_c,t_c,R_c),\\quad t_p<t_c,\\quad c_LR_p\\le R_c\\le C_LR_p",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "Parent-Child 祖裔對",
      "label_en": "Parent-child ancestry pair",
      "definition_zh": "第2節將一步 ancestry 表示成 parent 時空球 P=(x_p,t_p,R_p) 與 child C=(x_c,t_c,R_c)，並要求半徑之間為 bounded scale jump。",
      "definition_en": "Section 2 represents one ancestry step as a parent spacetime ball P=(x_p,t_p,R_p) and a child C=(x_c,t_c,R_c), with a bounded scale jump between radii."
    },
    {
      "id": "ns.c3.c3u.displacement_dpc",
      "latex": "d_{pc}=|x_c-x_p|",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "Parent-Child 位移",
      "label_en": "Parent-child spatial displacement",
      "definition_zh": "第2節定義 d_pc = |x_c - x_p| 作為 ancestry route 中心之間的空間位移量。",
      "definition_en": "Section 2 defines d_pc = |x_c - x_p| as the spatial displacement between ancestry-route centers."
    },
    {
      "id": "ns.c3.c3u.far_cutoff_kappa_psi",
      "latex": "\\kappa\\gg1,\\quad \\psi_p=0\\text{ on }B_{\\kappa R_p}(x_p),\\ \\psi_p=1\\text{ outside }B_{2\\kappa R_p}(x_p)",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "遠場截斷尺度與截斷函數",
      "label_en": "Far cutoff scale and cutoff functions",
      "definition_zh": "第3節固定大參數 κ 並取光滑截斷 ψ_p、ψ_c，在半徑 κR_p 內為零、2κR_p 外為一，藉此界定「遠場」。",
      "definition_en": "Section 3 fixes a large parameter κ and takes smooth cutoffs ψ_p, ψ_c vanishing inside radius κR_p and equal to one outside 2κR_p, delineating the far region."
    },
    {
      "id": "ns.c3.c3u.far_pressure_operator_and_matrices",
      "latex": "T_x(g)=\\int K(x-y)g(y)\\,dy,\\quad H_p=T_{x_p}(\\psi_pf_p),\\quad H_c=T_{x_c}(\\psi_cf_c)",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "遠場壓力算子與遠場矩陣",
      "label_en": "Far-pressure operator and far matrices",
      "definition_zh": "第3節用算子 T_x(g)=∫K(x-y)g(y)dy 把 H_p、H_c 重新定義成 parent／child 各自的遠場截斷壓力矩陣，是之後所有 heredity 估計的操作對象。",
      "definition_en": "Section 3 uses the operator T_x(g)=∫K(x-y)g(y)dy to redefine H_p, H_c as the far-cutoff pressure matrices at the parent and child, the operative objects for every later heredity estimate."
    },
    {
      "id": "ns.c3.c3u.c3u1_pressure_heredity_decomposition",
      "latex": "H_c-H_p=\\Delta H_{\\rm recl}+\\Delta H_{\\rm time}+\\Delta H_{\\rm space}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "C3-U.1：精確壓力遺傳分解",
      "label_en": "C3-U.1: Exact Pressure-Heredity Decomposition",
      "definition_zh": "第4節的核心定理，藉插入兩個中介項將遠場壓力矩陣變化 H_c - H_p 精確拆成 reclassification、temporal turnover、spatial shift 三項的代數恆等式。",
      "definition_en": "Section 4's central theorem, which inserts two intermediate terms to split the far-pressure matrix change H_c - H_p exactly into reclassification, temporal-turnover, and spatial-shift pieces.",
      "defining_relation": "H_c-H_p=\\Delta H_{\\rm recl}+\\Delta H_{\\rm time}+\\Delta H_{\\rm space}"
    },
    {
      "id": "ns.c3.c3u.delta_H_recl",
      "latex": "\\Delta H_{\\rm recl}=T_{x_c}[(\\psi_c-\\psi_p)f_c]",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "重新分類項",
      "label_en": "Reclassification term",
      "definition_zh": "第4節定義 ΔH_recl 為 parent／child 遠近分類不同所貢獻的壓力矩陣差，第7節由環域擾流量界定其大小。",
      "definition_en": "Section 4 defines ΔH_recl as the pressure-matrix contribution from the parent/child near-far reclassification, bounded in Section 7 by the annular enstrophy."
    },
    {
      "id": "ns.c3.c3u.delta_H_time",
      "latex": "\\Delta H_{\\rm time}=T_{x_c}[\\psi_p(f_c-f_p)]",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "時間源轉換項",
      "label_en": "Temporal source turnover term",
      "definition_zh": "第4節定義 ΔH_time 為固定 parent 遠近分類下壓力源隨時間變化所貢獻的矩陣差，第8-9節指出 energy inequality 本身不控制它。",
      "definition_en": "Section 4 defines ΔH_time as the matrix contribution from the pressure source's change in time under the fixed parent classification; Sections 8-9 show the energy inequality alone does not control it."
    },
    {
      "id": "ns.c3.c3u.delta_H_space",
      "latex": "\\Delta H_{\\rm space}=T_{x_c}(\\psi_pf_p)-T_{x_p}(\\psi_pf_p)",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "空間中心位移項",
      "label_en": "Spatial center-shift term",
      "definition_zh": "第4節定義 ΔH_space 為僅移動求值中心所致的矩陣差，第5節定理5.1證明它比遠場 Hessian 振幅本身多壓一個 κ^(-1) 因子。",
      "definition_en": "Section 4 defines ΔH_space as the matrix difference from moving the evaluation center alone; Theorem 5.1 in Section 5 shows it carries an extra κ^(-1) suppression beyond the far-Hessian amplitude itself.",
      "defining_relation": "|\\Delta H_{\\rm space}|\\le Cd_{pc}(\\kappa R_p)^{-4}\\|f_p\\|_1"
    },
    {
      "id": "ns.c3.c3u.normalized_debt_Ep_Hhat",
      "latex": "\\mathfrak E_p=\\frac{R_p\\|\\nabla u(t_p)\\|_2^2}{\\nu^2},\\quad \\widehat H=\\frac{R_p^4}{\\nu^2}H",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "正規化擾流債與正規化遠場矩陣",
      "label_en": "Normalized enstrophy debt and normalized far matrix",
      "definition_zh": "第6節定義無量綱擾流量 E_p 與依 parent 尺度正規化的矩陣 Ĥ，使空間位移項僅付出 O(κ^(-4) E_p)。",
      "definition_en": "Section 6 defines the dimensionless enstrophy quantity E_p and the parent-scale normalized matrix Ĥ, showing the spatial-shift term costs only O(κ^(-4) E_p)."
    },
    {
      "id": "ns.c3.c3u.reclassification_annulus_Eann",
      "latex": "\\mathcal A_{pc}=\\operatorname{supp}(\\psi_c-\\psi_p),\\quad \\mathfrak E_{pc}^{ann}=\\frac{R_p}{\\nu^2}\\int_{\\mathcal A_{pc}}|\\nabla u(y,t_c)|^2dy",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "重分類環域與環域擾流債（U-H1）",
      "label_en": "Reclassification annulus and annular enstrophy debt (U-H1)",
      "definition_zh": "第7節將 ψ_c - ψ_p 的支撐集記為環域 A_pc，並用其上重整化擾流量 E_pc^ann（定理7.1）界定 ΔH_recl。",
      "definition_en": "Section 7 denotes the support of ψ_c - ψ_p as the annulus A_pc and uses the rescaled enstrophy E_pc^ann over it (Theorem 7.1) to bound ΔH_recl.",
      "defining_relation": "|\\Delta\\widehat H_{\\rm recl}|\\le C\\kappa^{-3}\\mathfrak E_{pc}^{ann}",
      "notes": "第20節將此量標記為 heredity 三債之首 U-H1。"
    },
    {
      "id": "ns.c3.c3u.temporal_turnover_debt_Tfar",
      "latex": "\\mathfrak T_{pc}^{far}=\\frac{R_p}{\\nu^2}\\|\\psi_p(f_c-f_p)\\|_1",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "遠場壓力源時間轉換債（U-H2）",
      "label_en": "Far pressure-source temporal turnover debt (U-H2)",
      "definition_zh": "第8節定義無量綱時間轉換量 T_pc^far 以界定 ΔH_time（定理8.1），第9節指出它可寫成 ∂_t f 的時間積分且非 energy-level 有限量。",
      "definition_en": "Section 8 defines the dimensionless temporal-turnover quantity T_pc^far bounding ΔH_time (Theorem 8.1); Section 9 shows it is a time integral of ∂_t f and is not controlled at the energy level.",
      "defining_relation": "|\\Delta\\widehat H_{\\rm time}|\\le C\\kappa^{-3}\\mathfrak T_{pc}^{far}",
      "notes": "第20節標記為 heredity 三債之二 U-H2。"
    },
    {
      "id": "ns.c3.c3u.pressure_heredity_debt_theorem",
      "latex": "|\\Delta\\widehat H_{pc}|\\le C\\left[\\frac{d_{pc}}{R_p}\\kappa^{-4}\\mathfrak E_p+\\kappa^{-3}\\mathfrak E_{pc}^{ann}+\\kappa^{-3}\\mathfrak T_{pc}^{far}\\right]",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "壓力遺傳債定理（定理10.1）",
      "label_en": "Pressure-Heredity Debt Theorem (Theorem 10.1)",
      "definition_zh": "第10節綜合三個 carrier 給出遠場壓力矩陣正規化變化量的總估計，確立 spatial shift、reclassification、temporal source turnover 是 parent→child 變化的真正來源。",
      "definition_en": "Section 10 combines the three carriers into one total estimate for the normalized far-pressure matrix variation, identifying spatial shift, reclassification, and temporal source turnover as the true carriers of parent-to-child change.",
      "defining_relation": "|\\Delta\\widehat H_{pc}|\\le C\\left[\\frac{d_{pc}}{R_p}\\kappa^{-4}\\mathfrak E_p+\\kappa^{-3}\\mathfrak E_{pc}^{ann}+\\kappa^{-3}\\mathfrak T_{pc}^{far}\\right]"
    },
    {
      "id": "ns.c3.c3u.pressure_support_direction_KH",
      "latex": "K_p^H=-\\frac{H_p}{|H_p|},\\quad K_c^H=-\\frac{H_c}{|H_c|}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "壓力支撐方向",
      "label_en": "Pressure support direction",
      "definition_zh": "第12節在 H_p、H_c 非零下定義正規化方向 K_p^H、K_c^H，並證明遠場矩陣相對變化小則其方向亦穩定。",
      "definition_en": "Section 12 defines the normalized directions K_p^H, K_c^H for nonzero far matrices, showing that a small relative change in the far matrix implies a stable direction."
    },
    {
      "id": "ns.c3.c3u.adjoint_cutoff_and_mean_strain_numerator",
      "latex": "\\partial_t\\chi+u\\cdot\\nabla\\chi+\\nu\\Delta\\chi=0,\\quad M_\\chi(t)=\\int\\chi(x,t)S(x,t)\\,dx",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "伴隨截斷與平均應變分子",
      "label_en": "Adjoint cutoff and mean-strain numerator",
      "definition_zh": "第13節取滿足伴隨（反向）輸運方程的截斷函數 χ，並以其加權定義局部平均應變的分子 M_χ(t) = ∫χS dx。",
      "definition_en": "Section 13 takes an adjoint cutoff χ solving a backward transport equation and uses it to weight the numerator M_χ(t) = ∫χS dx of the local mean strain."
    },
    {
      "id": "ns.c3.c3u.c3u2_adjoint_mean_strain_transport",
      "latex": "\\frac{d}{dt}M_\\chi(t)=-\\int\\chi\\left[S^2+\\frac14\\omega\\otimes\\omega-\\frac14|\\omega|^2I+\\nabla^2p\\right]dx",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "C3-U.2：精確伴隨平均應變輸運恆等式",
      "label_en": "C3-U.2: Exact Adjoint Mean-Strain Transport Identity",
      "definition_zh": "第14節定理14.1利用伴隨截斷方程與應變方程消去 diffusion 及 advection 項，得到 M_χ 的精確時間導數恆等式。",
      "definition_en": "Theorem 14.1 in Section 14 uses the adjoint cutoff equation and the strain equation to cancel the diffusion and advection terms, yielding an exact identity for the time derivative of M_χ.",
      "defining_relation": "\\frac{d}{dt}M_\\chi(t)=-\\int\\chi\\left[S^2+\\frac14\\omega\\otimes\\omega-\\frac14|\\omega|^2I+\\nabla^2p\\right]dx"
    },
    {
      "id": "ns.c3.c3u.mean_strain_residual_RS",
      "latex": "\\mathcal R_S=S^2+\\frac14\\omega\\otimes\\omega-\\frac14|\\omega|^2I+\\nabla^2p",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "平均應變轉換餘量",
      "label_en": "Mean-strain turnover residual",
      "definition_zh": "第15節將定理14.1中被積式命名為 R_S，其時間積分給出 M_χ(t_c) - M_χ(t_p) 的精確表達與界。",
      "definition_en": "Section 15 names the integrand of Theorem 14.1 as R_S, whose time integral gives an exact expression and bound for M_χ(t_c) - M_χ(t_p).",
      "defining_relation": "M_\\chi(t_c)-M_\\chi(t_p)=-\\int_{t_p}^{t_c}\\int\\chi\\mathcal R_S\\,dxdt",
      "notes": "第20節之 U-H3（heredity 三債之三，mean-strain rotation debt）即由 R_S 的正規化積分 (1/|M_p|)∫∫χ|R_S| 構成。"
    },
    {
      "id": "ns.c3.c3u.mean_strain_direction_v",
      "latex": "v_p=\\frac{M_p}{|M_p|},\\quad v_c=\\frac{M_c}{|M_c|}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "平均應變方向",
      "label_en": "Mean-strain direction",
      "definition_zh": "第16節在 M_p 非零下定義正規化方向 v_p、v_c，證明伴隨矩陣轉換相對 |M_p| 小則方向穩定。",
      "definition_en": "Section 16 defines normalized directions v_p, v_c for nonzero M_p, showing small adjoint matrix turnover relative to |M_p| implies a stable direction."
    },
    {
      "id": "ns.c3.c3u.pressure_support_efficiency_eta",
      "latex": "\\eta_p=K_p^H:v_p,\\quad \\eta_c=K_c^H:v_c",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "壓力支撐效率",
      "label_en": "Pressure support efficiency",
      "definition_zh": "第17節將壓力支撐方向與平均應變方向的內積定義為效率 η，η 小即代表 pressure-poor。",
      "definition_en": "Section 17 defines the efficiency η as the inner product of the pressure support direction and the mean-strain direction; small η signals a pressure-poor state.",
      "defining_relation": "\\eta=K^H:v"
    },
    {
      "id": "ns.c3.c3u.c3u3_conditional_pressure_poor_heredity",
      "latex": "\\eta_c\\le\\eta_0+\\delta_H+\\delta_M",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "C3-U.3：條件壓力貧乏遺傳定理",
      "label_en": "C3-U.3: Conditional Pressure-Poor Heredity Theorem",
      "definition_zh": "第18節定理18.1證明若 parent 效率 η_p ≤ η_0 且方向轉換 |K_c^H - K_p^H| ≤ δ_H、|v_c - v_p| ≤ δ_M，則 child 效率同型上界成立；第22節將此條件沿有界代數的 ancestry chain延伸為 bounded-generation heredity criterion。",
      "definition_en": "Theorem 18.1 in Section 18 shows that if the parent efficiency satisfies η_p ≤ η_0 and the direction turnovers satisfy |K_c^H - K_p^H| ≤ δ_H, |v_c - v_p| ≤ δ_M, then the child efficiency obeys the same-form bound; Section 22 extends this along bounded-generation ancestry chains as the bounded-generation heredity criterion.",
      "defining_relation": "\\eta_c-\\eta_p=(K_c^H-K_p^H):v_c+K_p^H:(v_c-v_p)\\ \\Rightarrow\\ \\eta_c\\le\\eta_0+\\delta_H+\\delta_M"
    },
    {
      "id": "ns.c3.c3u.c3u4_mean_to_pointwise_theorem",
      "latex": "\\bar S_R=\\fint_{B_R}S(x)\\,dx,\\quad \\lambda_2(\\bar S_R)>C_pR^{1-3/p}\\|\\nabla S\\|_{L^p(B_R)}\\ \\Rightarrow\\ \\lambda_2(S(x))>0\\text{ on }B_{R/2}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "C3-U.4：平均到逐點中間特徵值定理",
      "label_en": "C3-U.4: Mean-to-Pointwise Middle-Eigenvalue Theorem",
      "definition_zh": "第23節定義球平均應變 S̄_R 並引入 p>3 之 Morrey–Poincaré 波動界，第24節定理24.1用 Weyl 不等式證明只要平均中間特徵值超過此波動界，逐點 λ_2 在內核 B_{R/2} 上即為正。",
      "definition_en": "Section 23 defines the ball-average strain S̄_R and introduces the p>3 Morrey-Poincaré fluctuation bound; Theorem 24.1 in Section 24 uses Weyl's inequality to show that once the averaged middle eigenvalue exceeds this fluctuation bound, the pointwise λ_2 is positive throughout the inner core B_{R/2}.",
      "defining_relation": "\\lambda_2(S(x))\\ge\\lambda_2(\\bar S_R)-\\|S(x)-\\bar S_R\\|_{\\rm op}\\ge\\lambda_2(\\bar S_R)-C_pR^{1-3/p}\\|\\nabla S\\|_{L^p(B_R)}"
    },
    {
      "id": "ns.c3.c3u.oscillation_ratio_and_middle_gap",
      "latex": "\\mathfrak O_{p,R}=\\frac{C_pR^{1-3/p}\\|\\nabla S\\|_{L^p(B_R)}}{|\\bar S_R|},\\quad \\delta_{2,R}=\\frac{\\lambda_2(\\bar S_R)}{|\\bar S_R|}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "尺度不變波動比與正規化中間間隙",
      "label_en": "Scale-invariant oscillation ratio and normalized middle gap",
      "definition_zh": "第26節把定理24.1的門檻無量綱化為波動比 O_{p,R} 與中間間隙比 δ_{2,R}，O_{p,R} < δ_{2,R} 保證逐點 λ_2 > 0 且在 N–S scaling 下尺度不變。",
      "definition_en": "Section 26 nondimensionalizes Theorem 24.1's threshold into the oscillation ratio O_{p,R} and the middle-gap ratio δ_{2,R}; the condition O_{p,R} < δ_{2,R} guarantees pointwise λ_2 > 0 and is scale-invariant under N-S scaling.",
      "defining_relation": "\\mathfrak O_{p,R}<\\delta_{2,R}\\ \\Rightarrow\\ \\lambda_2(S(x))>0"
    },
    {
      "id": "ns.c3.c3u.endpoint_barrier_p3",
      "latex": "p=3:\\ W^{1,3}\\not\\hookrightarrow L^\\infty",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "端點障礙 p=3",
      "label_en": "Endpoint barrier at p=3",
      "definition_zh": "第27節指出 Morrey 逐點升級機制在 p=3 因 W^{1,3} 不嵌入 L^∞ 而失效，是 mean→pointwise 路線一個真正的端點障礙。",
      "definition_en": "Section 27 shows the Morrey pointwise-upgrade mechanism fails precisely at p=3 because W^{1,3} does not embed in L^∞, a genuine endpoint barrier for the mean-to-pointwise route.",
      "notes": "對應第40節 X-Integration guard G-ENDPOINT：p=3 不能靜默使用 L^∞ Morrey embedding。"
    },
    {
      "id": "ns.c3.c3u.band_limited_strain_shell_Sq",
      "latex": "S_q=\\Delta_qS,\\quad \\|\\nabla S_q\\|_\\infty\\le C\\lambda_q\\|S_q\\|_\\infty,\\quad R=\\lambda_q^{-1}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "頻帶限制應變殼層",
      "label_en": "Band-limited strain shell",
      "definition_zh": "第29節取 Littlewood–Paley 殼層 S_q = Δ_q S 並用 Bernstein 不等式將其梯度控制在殼層自身振幅之下，作為 mean-to-pointwise 的第二條路線起點。",
      "definition_en": "Section 29 takes the Littlewood-Paley shell S_q = Δ_q S and uses a Bernstein inequality to control its gradient by the shell's own amplitude, launching the second mean-to-pointwise route."
    },
    {
      "id": "ns.c3.c3u.c3u5_band_limited_persistence",
      "latex": "\\lambda_2(S_q(x_0))\\ge\\delta\\|S_q\\|_\\infty\\ \\Rightarrow\\ \\lambda_2(S_q(x))\\ge\\frac{\\delta}{2}\\|S_q\\|_\\infty\\ \\text{for }|x-x_0|\\le c\\delta R",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "C3-U.5：頻帶中間特徵值持續性",
      "label_en": "C3-U.5: Band-Limited Middle-Eigenvalue Persistence",
      "definition_zh": "第30節定理30.1證明若殼層在一點有中間特徵值間隙 δ，則此正號在半徑 cδR 的鄰域內持續，由 Bernstein 配合 Weyl 不等式得出。",
      "definition_en": "Theorem 30.1 in Section 30 shows that a middle-eigenvalue gap δ at one point of the shell persists in sign over a neighborhood of radius cδR, via Bernstein plus Weyl's inequality.",
      "defining_relation": "|x-x_0|\\le c\\delta R\\ \\Rightarrow\\ \\lambda_2(S_q(x))\\ge\\frac{\\delta}{2}\\|S_q\\|_\\infty"
    },
    {
      "id": "ns.c3.c3u.strain_remainder_RqS",
      "latex": "R_q^S=S-S_q,\\quad \\|R_q^S\\|_{L^\\infty,\\rm op}\\le\\frac{\\delta}{4}\\|S_q\\|_\\infty\\ \\Rightarrow\\ \\lambda_2(S(x))\\ge\\frac{\\delta}{4}\\|S_q\\|_\\infty",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "應變殼層餘量",
      "label_en": "Strain shell remainder",
      "definition_zh": "第31節定義餘量 R_q^S = S - S_q，證明只要其算子範數在 subcore 上小於殼層振幅之 δ/4，逐點 λ_2(S) 仍為正，是 band-limited 路線的第二個 proof obligation。",
      "definition_en": "Section 31 defines the remainder R_q^S = S - S_q and shows that once its operator norm on the subcore is smaller than δ/4 of the shell amplitude, the pointwise λ_2(S) remains positive, the band-limited route's second proof obligation."
    },
    {
      "id": "ns.c3.c3u.miller_operator_QSV",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "Miller 算子活躍核",
      "label_en": "Miller operator-active core",
      "definition_zh": "第33節指出由 Q_SV 與 ΔS 局部比值選出的 operator-active core雖更貼近應變導數，但 ‖ΔS‖_2 大仍不推出 λ_2(S) 為正。",
      "definition_en": "Section 33 notes that the operator-active core selected by the local ratio of Q_SV to ΔS sits closer to strain derivatives, yet a large ‖ΔS‖_2 still does not imply λ_2(S) is positive.",
      "notes": "承接內部依賴文件 NS_C3P_OperatorEscape_FarPressureMatrix_v0.1.md 中引入的 operator-escape 構造。"
    },
    {
      "id": "ns.c3.c3u.c3u6_parabolic_middle_strain_toll",
      "latex": "\\lambda_2^+(S)\\ge c_0R^{-2}\\text{ on }B_{cR}\\times[t_n-cR^2,t_n]\\ \\Rightarrow\\ \\int_{t_n-cR^2}^{t_n}\\|\\lambda_2^+(t)\\|_3^2dt\\ge c_1>0",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "C3-U.6：拋物型中間應變事件稅",
      "label_en": "C3-U.6: Parabolic Middle-Strain Toll",
      "definition_zh": "第34-35節證明一個振幅 R^(-2)、體積 R^3、時長 R^2 的相干正中間特徵值事件必支付 O(1) 的 scale-critical L_t^2 L_x^3 稅，故無限多此類事件蘊含此範數發散，恰與 Miller blow-up necessity相容而非矛盾。",
      "definition_en": "Sections 34-35 show a coherent positive-middle-eigenvalue event of amplitude R^(-2), volume R^3, duration R^2 must pay an O(1) scale-critical L_t^2 L_x^3 toll, so infinitely many such events force this norm to diverge — consistent with, not contradicting, Miller's blow-up necessity.",
      "defining_relation": "\\int_{t_n-cR^2}^{t_n}\\|\\lambda_2^+(t)\\|_3^2dt\\ge c_1>0"
    },
    {
      "id": "ns.c3.c3u.etn_state_and_guards",
      "latex": "\\Theta_{pc}^{press}=\\langle H_p,H_c,\\Delta H_{\\rm space},\\mathfrak E_{pc}^{ann},\\mathfrak T_{pc}^{far},K_p^H,K_c^H\\rangle,\\ \\Theta_{pc}^{mean}=\\langle M_p,M_c,\\int\\chi\\mathcal R_S,v_p,v_c\\rangle,\\ \\Theta_R^{pt}=\\langle\\bar S_R,\\delta_{2,R},\\mathfrak O_{p,R},S_q,R_q^S\\rangle",
      "series": "NS",
      "first_appearance": "C3-U",
      "label_zh": "ETN 狀態元組與 X-Integration guards",
      "label_en": "ETN state tuples and X-Integration guards",
      "definition_zh": "第41節將本輪量打包成三個 True ETN 狀態元組（壓力、平均應變、逐點升級），第40節同步更新九條 X-Integration guards（G-PHD、G-ANN、G-PTURN、G-MTURN、G-HDIR、G-MORREY、G-ENDPOINT、G-SHELLGAP、G-SHELLREM）供後續回合遵守。",
      "definition_en": "Section 41 packages this round's quantities into three True-ETN state tuples (pressure, mean-strain, pointwise-upgrade); Section 40 correspondingly updates nine X-Integration guards (G-PHD, G-ANN, G-PTURN, G-MTURN, G-HDIR, G-MORREY, G-ENDPOINT, G-SHELLGAP, G-SHELLREM) for later rounds to respect."
    },
    {
      "id": "ns.c3.c3v.ancestry_pair",
      "latex": "P=(x_p,t_p,R_p),\\ C=(x_c,t_c,R_c)",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "母代／子代三元組",
      "label_en": "Parent/child triple",
      "definition_zh": "第2節沿用C3-U記號，以三元組P=(x_p,t_p,R_p)與C=(x_c,t_c,R_c)標記母代與子代的中心、時刻與尺度，並假設R_c≍R_p。",
      "definition_en": "Section 2 reuses the C3-U notation in which triples P=(x_p,t_p,R_p) and C=(x_c,t_c,R_c) record the center, time, and scale of a parent and child ancestry node, with R_c≍R_p assumed.",
      "notes": "Carried over from C3-U; every subsequent turnover/enstrophy quantity in this round is built on this pair."
    },
    {
      "id": "ns.c3.c3v.pressure_source_scalar",
      "latex": "f(t,x)=\\operatorname{tr}((\\nabla u(t,x))^2)",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "壓力源純量",
      "label_en": "Pressure source scalar",
      "definition_zh": "第1節沿用的壓力源純量f=tr((∇u)^2)滿足逐點界|f|≤|∇u|^2，是本輪端點遞轉論證的起點。",
      "definition_en": "Section 1 recalls the pressure source scalar f=tr((∇u)^2), which obeys the pointwise bound |f|≤|∇u|^2 and anchors this round's endpoint turnover argument."
    },
    {
      "id": "ns.c3.c3v.far_pressure_turnover",
      "latex": "\\mathfrak T_{pc}^{far}=\\frac{R_p}{\\nu^2}\\left\\|\\psi_p(f_c-f_p)\\right\\|_1",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "遠場壓力遞轉量",
      "label_en": "Far-pressure turnover",
      "definition_zh": "第2節回顧C3-U對母子代遠場壓力源差異的加權L^1遞轉量定義𝔗_{pc}^{far}。",
      "definition_en": "Section 2 recalls C3-U's definition of the far-pressure turnover 𝔗_{pc}^{far}, a weighted L^1 measure of the pressure-source difference between parent and child.",
      "defining_relation": "\\mathfrak T_{pc}^{far}=\\frac{R_p}{\\nu^2}\\left\\|\\psi_p(f_c-f_p)\\right\\|_1",
      "notes": "Inherited from C3-U, where it was bounded via ∫_{t_p}^{t_c}∂_tf; C3-V's Theorem 3.1 replaces that with a cheaper endpoint bound."
    },
    {
      "id": "ns.c3.c3v.thm_endpoint_pressure_turnover_bound",
      "latex": "\\mathfrak T_{pc}^{far}\\le\\frac{R_p}{\\nu^2}\\left[\\|\\nabla u(t_c)\\|_2^2+\\|\\nabla u(t_p)\\|_2^2\\right]",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.1（端點壓力遞轉界）",
      "label_en": "Theorem C3-V.1 (Endpoint Pressure-Turnover Bound)",
      "definition_zh": "第3節定理3.1證明遠場壓力遞轉量可由母子代端點enstrophy直接界定，無需∂_tf可積性假設。",
      "definition_en": "Theorem 3.1 in Section 3 shows the far-pressure turnover is bounded directly by the parent and child endpoint enstrophies, avoiding any integrability assumption on ∂_tf.",
      "defining_relation": "\\mathfrak T_{pc}^{far}\\le\\frac{R_p}{\\nu^2}\\left[\\|\\nabla u(t_c)\\|_2^2+\\|\\nabla u(t_p)\\|_2^2\\right]"
    },
    {
      "id": "ns.c3.c3v.rescaled_enstrophy",
      "latex": "\\mathfrak E_p=\\frac{R_p\\|\\nabla u(t_p)\\|_2^2}{\\nu^2},\\quad \\mathfrak E_{c|p}=\\frac{R_p\\|\\nabla u(t_c)\\|_2^2}{\\nu^2}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "（母代尺度下）重標化enstrophy",
      "label_en": "Parent-scale rescaled enstrophy",
      "definition_zh": "第4節定義母代重標化enstrophy 𝔈_p與以母代尺度R_p計算的子代enstrophy 𝔈_{c|p}，並得到𝔗_{pc}^{far}≤𝔈_p+𝔈_{c|p}。",
      "definition_en": "Section 4 defines the parent rescaled enstrophy 𝔈_p and the child enstrophy 𝔈_{c|p} evaluated at the parent scale R_p, yielding 𝔗_{pc}^{far}≤𝔈_p+𝔈_{c|p}.",
      "defining_relation": "\\mathfrak E_p=\\frac{R_p\\|\\nabla u(t_p)\\|_2^2}{\\nu^2},\\ \\ \\mathfrak E_{c|p}=\\frac{R_p\\|\\nabla u(t_c)\\|_2^2}{\\nu^2},\\ \\ \\mathfrak T_{pc}^{far}\\le\\mathfrak E_p+\\mathfrak E_{c|p}",
      "notes": "𝔈_{c|p}≍𝔈_c since R_c≍R_p; this pair is the load-bearing scalar behind every later boxed bound in Sections 5-11."
    },
    {
      "id": "ns.c3.c3v.far_matrix_variation",
      "latex": "\\Delta\\widehat H_{pc},\\ \\Delta\\widehat H_{\\rm space},\\ \\Delta\\widehat H_{\\rm recl}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "遠矩陣變化量（空間位移／再分類分量）",
      "label_en": "Far-matrix variation (spatial-shift / reclassification components)",
      "definition_zh": "承第0節C3-U對ΔH的空間、再分類、時間三分解，第5-7節將其正規化為ΔĤ_{recl}、ΔĤ_{space}並合併為第7節定理7.1所界定的ΔĤ_{pc}。",
      "definition_en": "Building on C3-U's three-way split of ΔH (spatial, reclassification, temporal) recalled in Section 0, Sections 5-7 normalize the spatial and reclassification pieces and combine them into ΔĤ_{pc}, bounded in Theorem 7.1 of Section 7.",
      "notes": "Section 0's ΔH=ΔH_space+ΔH_recl+ΔH_time is the C3-U decomposition this round refines on the space/reclassification pieces only."
    },
    {
      "id": "ns.c3.c3v.thm_endpoint_far_matrix_variation",
      "latex": "|\\Delta\\widehat H_{pc}|\\le C\\left[\\kappa^{-3}(\\mathfrak E_p+\\mathfrak E_{c|p})+\\frac{d_{pc}}{R_p}\\kappa^{-4}\\mathfrak E_p\\right]",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.2（端點遠矩陣變化定理）",
      "label_en": "Theorem C3-V.2 (Endpoint Far-Matrix Variation Theorem)",
      "definition_zh": "第7節定理7.1在R_c≍R_p下，把遠矩陣變化量完全用端點重標化enstrophy與壓力視界κ界定，說明壓力方向遺傳不需先證∂_tf可積。",
      "definition_en": "Theorem 7.1 in Section 7 bounds the far-matrix variation entirely by endpoint rescaled enstrophy and the pressure horizon κ under R_c≍R_p, showing pressure-direction heredity need not first establish integrability of ∂_tf.",
      "defining_relation": "|\\Delta\\widehat H_{pc}|\\le C\\left[\\kappa^{-3}(\\mathfrak E_p+\\mathfrak E_{c|p})+\\frac{d_{pc}}{R_p}\\kappa^{-4}\\mathfrak E_p\\right]"
    },
    {
      "id": "ns.c3.c3v.bounding_constants",
      "latex": "\\mathfrak E_p,\\mathfrak E_{c|p}\\le E_*,\\quad \\frac{d_{pc}}{R_p}\\le D_*",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "有界enstrophy／尺度比常數",
      "label_en": "Bounded-enstrophy / scale-ratio constants",
      "definition_zh": "第8節引入常數E_*與D_*分別界定重標化enstrophy上界及母子中心距與尺度比，作為後續條件版遺傳定理的假設。",
      "definition_en": "Section 8 introduces the constants E_* and D_*, bounding the rescaled enstrophies and the center-distance-to-scale ratio respectively, as the hypotheses feeding the conditional heredity theorems that follow."
    },
    {
      "id": "ns.c3.c3v.far_pressure_unit_direction",
      "latex": "K_p^H=-\\frac{\\widehat H_p}{|\\widehat H_p|},\\quad K_c^H=-\\frac{\\widehat H_c}{|\\widehat H_c|}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "遠壓力單位方向",
      "label_en": "Far-pressure unit direction",
      "definition_zh": "第9節在非退化假設|Ĥ_p|≥h_*>0下，把母子代遠壓力矩陣正規化為單位方向K_p^H、K_c^H，證明|Ĥ_c-Ĥ_p|小則|K_c^H-K_p^H|≤4ε。",
      "definition_en": "Under the nondegeneracy hypothesis |Ĥ_p|≥h_*>0, Section 9 normalizes the parent and child far-pressure matrices into unit directions K_p^H, K_c^H and shows a small |Ĥ_c-Ĥ_p| forces |K_c^H-K_p^H|≤4ε."
    },
    {
      "id": "ns.c3.c3v.thm_direction_stability",
      "latex": "\\forall\\epsilon_H>0\\ \\exists\\kappa_0(E_*,D_*,h_*,\\epsilon_H):\\ \\kappa\\ge\\kappa_0\\Rightarrow|K_c^H-K_p^H|\\le\\epsilon_H",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.3（有界enstrophy下遠壓力方向穩定性）",
      "label_en": "Theorem C3-V.3 (Bounded-Enstrophy Far-Pressure Direction Stability)",
      "definition_zh": "第10節定理10.1證明在E_*,D_*,h_*固定下，存在門檻κ_0使κ≥κ_0時遠壓力方向差可任意小，是真正的條件式壓力方向遺傳定理。",
      "definition_en": "Theorem 10.1 in Section 10 shows that for fixed E_*, D_*, h_* there is a threshold κ_0 beyond which |K_c^H-K_p^H| can be made arbitrarily small, giving a genuine conditional pressure-direction heredity theorem."
    },
    {
      "id": "ns.c3.c3v.pressure_failure_branch_trichotomies",
      "latex": "\\text{V-P1, V-P2, V-P3};\\quad \\text{H-F1, H-F2, H-F3}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "壓力方向／壓力遺傳失敗分支",
      "label_en": "Pressure-direction / pressure-heredity failure branches",
      "definition_zh": "第11節列出遠壓力方向無法遺傳時必居其一的三分支：V-P1重標化enstrophy發散、V-P2遠矩陣退化、V-P3壓力視界不足。",
      "definition_en": "Section 11 lists the trichotomy V-P1 (rescaled-enstrophy escape), V-P2 (far-matrix degeneracy), V-P3 (insufficient pressure horizon) that must contain the cause whenever far-pressure direction fails to be hereditary.",
      "notes": "Section 42 restates an analogous overall trichotomy H-F1/H-F2/H-F3 for pressure-poor-to-rich recovery, adding mean-strain rotation (H-F3) as the branch this round isolates."
    },
    {
      "id": "ns.c3.c3v.pressure_efficiency",
      "latex": "\\eta_p=K_p^H:v_p,\\quad \\eta_c=K_c^H:v_c",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "壓力效率",
      "label_en": "Pressure efficiency",
      "definition_zh": "第12節定義壓力效率η=K^H:v，即遠壓力單位方向與平均應變方向的內積，用以量測pressure-poor heredity是否恢復。",
      "definition_en": "Section 12 defines the pressure efficiency η=K^H:v as the inner product of the far-pressure unit direction with the mean-strain direction, used to detect recovery from pressure-poor heredity."
    },
    {
      "id": "ns.c3.c3v.mean_strain_vector",
      "latex": "M_p=av_p,\\ M_c=bv_c,\\quad |M_p|,|M_c|\\ge\\mu_*\\nu R_p",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "局部平均應變向量",
      "label_en": "Local mean-strain vector",
      "definition_zh": "第13節以M_p=av_p、M_c=bv_c記局部平均應變向量並要求非退化下界|M_p|,|M_c|≥μ_*νR_p，由此得|M_c-M_p|≥μ_*νR_p|v_c-v_p|。",
      "definition_en": "Section 13 writes the local mean-strain vectors as M_p=av_p, M_c=bv_c with nondegeneracy lower bound |M_p|,|M_c|≥μ_*νR_p, giving |M_c-M_p|≥μ_*νR_p|v_c-v_p|."
    },
    {
      "id": "ns.c3.c3v.thm_mean_rotation_toll",
      "latex": "\\frac{|M_c-M_p|}{\\nu R_p}\\ge\\frac{\\mu_*\\delta}{2}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.4（壓力效率恢復需付平均旋轉代價）",
      "label_en": "Theorem C3-V.4 (Pressure-Efficiency Recovery Requires Mean-Rotation Toll)",
      "definition_zh": "第14節證明若壓力效率回升η_c-η_p≥δ且遠壓力方向穩定，則局部平均應變向量差必須付出正規化下界μ_*δ/2的旋轉代價。",
      "definition_en": "Section 14 shows that if pressure efficiency recovers by η_c-η_p≥δ while the far-pressure direction stays stable, the mean-strain vector must pay a normalized rotation toll of at least μ_*δ/2.",
      "defining_relation": "\\frac{|M_c-M_p|}{\\nu R_p}\\ge\\frac{\\mu_*\\delta}{2}"
    },
    {
      "id": "ns.c3.c3v.quadratic_strain_vorticity_tensor",
      "latex": "Q_S=S^2+\\frac14\\omega\\otimes\\omega-\\frac14|\\omega|^2I",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "應變-渦度二次張量",
      "label_en": "Strain-vorticity quadratic tensor",
      "definition_zh": "第15節由C3-U的伴隨恆等式引入二次張量Q_S=S^2+¼ω⊗ω-¼|ω|^2I，作為平均應變遞轉的局部核心項。",
      "definition_en": "Section 15 introduces the quadratic tensor Q_S=S^2+¼ω⊗ω-¼|ω|^2I from C3-U's adjoint identity as the local kernel driving mean-strain turnover.",
      "defining_relation": "Q_S=S^2+\\frac14\\omega\\otimes\\omega-\\frac14|\\omega|^2I"
    },
    {
      "id": "ns.c3.c3v.normalized_turnover_functionals",
      "latex": "\\mathfrak R_{pc}^{Q}=\\frac{1}{\\nu R_p}\\int_{I_{pc}}\\!\\!\\int\\chi|Q_S|\\,dxdt,\\quad \\mathfrak R_{pc}^{P}=\\frac{1}{\\nu R_p}\\int_{I_{pc}}\\!\\!\\int\\chi|\\nabla^2p|\\,dxdt",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "正規化二次／壓力Hessian遞轉量",
      "label_en": "Normalized quadratic / pressure-Hessian turnover",
      "definition_zh": "第15節定義正規化局部應變-渦度二次遞轉量𝔯_{pc}^Q與局部壓力Hessian遞轉量𝔯_{pc}^P，兩者之和界定平均應變向量差|M_c-M_p|/(νR_p)。",
      "definition_en": "Section 15 defines the normalized local quadratic turnover 𝔯_{pc}^Q and the normalized local pressure-Hessian turnover 𝔯_{pc}^P, whose sum bounds the mean-strain difference |M_c-M_p|/(νR_p).",
      "defining_relation": "\\mathfrak R_{pc}^{Q}=\\frac{1}{\\nu R_p}\\int_{I_{pc}}\\int\\chi|Q_S|\\,dxdt,\\ \\ \\mathfrak R_{pc}^{P}=\\frac{1}{\\nu R_p}\\int_{I_{pc}}\\int\\chi|\\nabla^2p|\\,dxdt,\\ \\ \\frac{|M_c-M_p|}{\\nu R_p}\\le\\mathfrak R_{pc}^{Q}+\\mathfrak R_{pc}^{P}",
      "notes": "Section 16's mean-rotation carrier dichotomy shows at least one of 𝔯_{pc}^Q≥r_0/2 or 𝔯_{pc}^P≥r_0/2 must hold whenever the mean-strain difference is ≥r_0."
    },
    {
      "id": "ns.c3.c3v.thm_weighted_quadratic_packing",
      "latex": "\\sum_nR_n\\mathfrak R_n^{Q}\\le\\frac{C}{\\nu}\\int_0^{T_*}\\|\\nabla u(t)\\|_2^2dt\\le\\frac{C\\|u_0\\|_2^2}{\\nu^2}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.5（加權二次遞轉封裝定理）",
      "label_en": "Theorem C3-V.5 (Weighted Quadratic-Turnover Packing)",
      "definition_zh": "第18節定理18.1證明對兩兩不相交的祖先窗口，尺度加權的二次遞轉量總和∑R_n𝔯_n^Q由初始動能界定，是R-加權而非未加權的封裝律。",
      "definition_en": "Theorem 18.1 in Section 18 shows that over pairwise-disjoint ancestry windows, the scale-weighted sum ∑R_n𝔯_n^Q of quadratic turnovers is bounded by the initial kinetic energy — an R-weighted, not unweighted, packing law.",
      "defining_relation": "\\sum_nR_n\\mathfrak R_n^{Q}\\le\\frac{C}{\\nu}\\int_0^{T_*}\\|\\nabla u(t)\\|_2^2dt\\le\\frac{C\\|u_0\\|_2^2}{\\nu^2}",
      "notes": "Section 19 stresses this does NOT give ∑𝔯_n^Q<∞; under geometric R_n=R_0r^{-n} it permits 𝔯_n^Q~1 every generation, which Section 20's Zeno No-Go and Section 43 use to declare energy-only Pressure-Poor Heredity a NO-GO."
    },
    {
      "id": "ns.c3.c3v.thm_turnover_zeno_nogo",
      "latex": "\\sum_n\\nu\\int_{I_n}\\|\\nabla u\\|_2^2dt<\\infty\\ \\text{yet}\\ \\mathfrak R_n^Q\\sim E_*\\ \\forall n",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.6（遞轉芝諾不可行定理）",
      "label_en": "Theorem C3-V.6 (Turnover Zeno No-Go)",
      "definition_zh": "第20節用幾何尺度R_n=2^{-n}R_0的抽象記帳構造，展示有限動能耗散並不蘊含有限的總平均方向變化，𝔯_n^Q每代可保持O(1)。",
      "definition_en": "Section 20 constructs an abstract geometric-ledger bookkeeping with R_n=2^{-n}R_0 showing finite kinetic-energy dissipation does not imply finite total mean-direction variation, since 𝔯_n^Q can stay O(1) every generation.",
      "notes": "Explicitly flagged in the text as a scaling ledger, not a Navier-Stokes blow-up construction."
    },
    {
      "id": "ns.c3.c3v.persistent_window_hypothesis",
      "latex": "J_n\\subset I_n,\\ |J_n|\\ge\\theta\\frac{R_n^2}{\\nu},\\quad \\mathfrak E_{R_n}(t)\\ge cb_0^{2/3}\\kappa_n^2\\gamma_n^{-2/3}\\ \\text{on}\\ J_n",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "持續黏性窗口假設",
      "label_en": "Persistent viscous-window hypothesis",
      "definition_zh": "第23節加入明確條件：每個祖先窗口I_n存在測度≥θR_n^2/ν的固定比例子集J_n，其上共同遠壓力支持的錐退化不等式（第22節b_0常數）持續成立。",
      "definition_en": "Section 23 adds an explicit condition that each ancestry window I_n contains a fixed-fraction subset J_n of measure ≥θR_n^2/ν on which the cone-degeneration inequality (with the b_0 constant from Section 22) persists."
    },
    {
      "id": "ns.c3.c3v.thm_persistent_cone_degeneration_packing",
      "latex": "\\sum_nR_n\\kappa_n^2\\gamma_n^{-2/3}\\le C\\frac{\\|u_0\\|_2^2}{\\theta\\nu^2b_0^{2/3}}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.7（持續錐退化封裝定理）",
      "label_en": "Theorem C3-V.7 (Persistent Cone-Degeneration Packing Theorem)",
      "definition_zh": "第24節定理24.1在持續黏性窗口假設下，證明尺度加權的錐退化量∑R_nκ_n^2γ_n^{-2/3}由初始動能界定，是首個直接接上global energy budget的幾何衰減速率限制。",
      "definition_en": "Theorem 24.1 in Section 24 proves that under the persistent-window hypothesis, the scale-weighted cone-degeneration sum ∑R_nκ_n^2γ_n^{-2/3} is bounded by the initial energy — the first geometric decay-rate restriction tied directly to the global energy budget.",
      "defining_relation": "\\sum_nR_n\\kappa_n^2\\gamma_n^{-2/3}\\le C\\frac{\\|u_0\\|_2^2}{\\theta\\nu^2b_0^{2/3}}"
    },
    {
      "id": "ns.c3.c3v.cone_collapse_rate_exponent",
      "latex": "\\gamma_n\\asymp(R_n/R_0)^\\alpha,\\quad \\alpha<\\frac32",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "錐塌縮速率指數",
      "label_en": "Cone collapse rate exponent",
      "definition_zh": "第25節在幾何尺度R_n=R_0r^{-n}下引入錐邊界指數α（γ_n≍(R_n/R_0)^α），證明持續支持分支需要α<3/2，否則與定理24.1矛盾。",
      "definition_en": "Section 25 introduces the cone-margin exponent α via γ_n≍(R_n/R_0)^α under geometric scales R_n=R_0r^{-n}, showing the persistent-support branch forces α<3/2 on pain of contradicting Theorem 24.1."
    },
    {
      "id": "ns.c3.c3v.morrey_obstruction",
      "latex": "\\mathfrak O_{p,R}=\\frac{C_pR^{1-3/p}\\|\\nabla S\\|_{L^p(B_R)}}{|\\bar S_R|}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "Morrey型平均-逐點障礙量",
      "label_en": "Morrey mean-to-pointwise obstruction",
      "definition_zh": "第27節對p>3定義障礙量𝔒_{p,R}，其若小於正規化中間特徵值間隙即可把平均應變符號升為逐點符號。",
      "definition_en": "For p>3, Section 27 defines the obstruction 𝔒_{p,R}, which, when smaller than the normalized middle-eigenvalue gap, allows the mean-strain sign to be promoted to a pointwise sign.",
      "defining_relation": "\\mathfrak O_{p,R}=\\frac{C_pR^{1-3/p}\\|\\nabla S\\|_{L^p(B_R)}}{|\\bar S_R|}"
    },
    {
      "id": "ns.c3.c3v.normalized_mean_strain",
      "latex": "\\mu_R=\\frac{R^2|\\bar S_R|}{\\nu}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "正規化平均應變",
      "label_en": "Normalized mean strain",
      "definition_zh": "第28節定義尺度不變量μ_R=R^2|S̄_R|/ν，作為量測局部平均應變非退化程度的正規化量。",
      "definition_en": "Section 28 defines the scale-invariant quantity μ_R=R^2|S̄_R|/ν, normalizing the local mean-strain magnitude for nondegeneracy comparisons.",
      "defining_relation": "\\mu_R=\\frac{R^2|\\bar S_R|}{\\nu}"
    },
    {
      "id": "ns.c3.c3v.active_volume_and_fraction",
      "latex": "\\mathcal V_p(g)=\\left(\\frac{\\|g\\|_2}{\\|g\\|_p}\\right)^{1/a_p},\\quad \\phi_{p,R}=\\frac{\\mathcal V_p(g)}{R^3},\\quad a_p=\\frac12-\\frac1p",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "有效活躍體積與正規化活躍體積分率",
      "label_en": "Effective active volume and normalized active-volume fraction",
      "definition_zh": "第29節對g=∇S|_{B_R}與p>2定義具體積量綱的局部有效活躍體積𝒱_p(g)及其正規化分率φ_{p,R}=𝒱_p(g)/R^3，作為本專案自定的局部集中度診斷量。",
      "definition_en": "For g=∇S restricted to B_R and p>2, Section 29 defines the volume-dimensioned local effective active volume 𝒱_p(g) and its normalized fraction φ_{p,R}=𝒱_p(g)/R^3, a project-specific local-concentration diagnostic.",
      "defining_relation": "\\mathcal V_p(g)=\\left(\\frac{\\|g\\|_2}{\\|g\\|_p}\\right)^{1/a_p},\\ \\ \\phi_{p,R}=\\frac{\\mathcal V_p(g)}{R^3},\\ \\ a_p=\\frac12-\\frac1p",
      "notes": "Section 29 explicitly cautions φ_{p,R} is not identical to the full Cheskidov-Shvydkoy volumetric intermittency machinery, only a project-local analogue."
    },
    {
      "id": "ns.c3.c3v.higher_derivative_stock",
      "latex": "\\mathfrak H_R=\\frac{R^3}{\\nu^2}\\|\\nabla S\\|_{L^2(B_R)}^2",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "高階導數應變儲量",
      "label_en": "Higher-derivative strain stock",
      "definition_zh": "第30節定義尺度不變的瞬時局部H^2-型應變梯度儲量𝔥_R=(R^3/ν^2)‖∇S‖_{L^2(B_R)}^2。",
      "definition_en": "Section 30 defines the scale-invariant instantaneous local H^2-type strain-gradient stock 𝔥_R=(R^3/ν^2)‖∇S‖_{L^2(B_R)}^2.",
      "defining_relation": "\\mathfrak H_R=\\frac{R^3}{\\nu^2}\\|\\nabla S\\|_{L^2(B_R)}^2"
    },
    {
      "id": "ns.c3.c3v.thm_fluctuation_intermittency_identity",
      "latex": "\\mathfrak O_{p,R}=C_p\\frac{\\mathfrak H_R^{1/2}}{\\mu_R\\phi_{p,R}^{\\,1/2-1/p}}",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.8（漲落-間歇性恆等式）",
      "label_en": "Theorem C3-V.8 (Fluctuation-Intermittency Identity)",
      "definition_zh": "第31節定理31.1證明在∇S≠0、S̄_R≠0下，Morrey障礙量、高階導數儲量、正規化平均應變與活躍體積分率間存在精確恆等式，所有R冪次完全抵消。",
      "definition_en": "Theorem 31.1 in Section 31 proves that whenever ∇S≠0 and S̄_R≠0, the Morrey obstruction, higher-derivative stock, normalized mean strain, and active-volume fraction satisfy an exact identity with all powers of R canceling.",
      "defining_relation": "\\mathfrak O_{p,R}=C_p\\frac{\\mathfrak H_R^{1/2}}{\\mu_R\\phi_{p,R}^{\\,1/2-1/p}}"
    },
    {
      "id": "ns.c3.c3v.thm_strain_fluctuation_escape_dichotomy",
      "latex": "\\phi_{p,R}\\le\\theta\\ \\ (\\text{V-F1})\\quad\\vee\\quad \\mathfrak H_R\\ge c\\delta^2\\mu_0^2\\theta^{\\,1-2/p}\\ \\ (\\text{V-F2})",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.9（應變漲落逃逸二分定理）",
      "label_en": "Theorem C3-V.9 (Strain-Fluctuation Escape Dichotomy)",
      "definition_zh": "第32節在μ_R≥μ_0>0且障礙量𝔒_{p,R}≥δ>0下，證明對任意0<θ<1必居其一：V-F1間歇集中φ_{p,R}≤θ，或V-F2高階導數儲量下界𝔥_R≥cδ^2μ_0^2θ^{1-2/p}。",
      "definition_en": "Under μ_R≥μ_0>0 and obstruction 𝔒_{p,R}≥δ>0, Section 32 proves that for any 0<θ<1, either V-F1 intermittent concentration φ_{p,R}≤θ holds, or V-F2 the higher-derivative stock satisfies 𝔥_R≥cδ^2μ_0^2θ^{1-2/p}.",
      "defining_relation": "\\phi_{p,R}\\le\\theta\\quad\\vee\\quad \\mathfrak H_R\\ge c\\delta^2\\mu_0^2\\theta^{\\,1-2/p}"
    },
    {
      "id": "ns.c3.c3v.frontier_rescaling_family",
      "latex": "V_Q(y,0)=\\frac{1}{\\nu\\lambda_Q}u\\left(x_Q+\\frac{y}{\\lambda_Q},T_Q\\right),\\quad \\Sigma_Q=\\nabla_{sym}V_Q,\\quad 2^{-j}\\|\\Delta_jV_Q(0)\\|_\\infty\\le\\beta_*",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "前緣重標化速度場與應變殼層",
      "label_en": "First-frontier rescaled velocity and strain shell",
      "definition_zh": "第36節回顧C3-I的前緣重標化速度場V_Q及其UV上界常數β_*，第37節由此定義重標化應變殼層Σ_Q=∇_{sym}V_Q。",
      "definition_en": "Section 36 recalls the C3-I first-frontier rescaled velocity field V_Q and its UV cap constant β_*, and Section 37 defines from it the rescaled strain shell Σ_Q=∇_{sym}V_Q.",
      "notes": "Inherited from C3-I (NS_C3I_FrontierUVCap_DefectTrichotomy); Sections 36-38 of this round track whether the velocity UV cap propagates to Σ_Q."
    },
    {
      "id": "ns.c3.c3v.thm_derivative_amplification_barrier",
      "latex": "\\|\\Delta_j\\Sigma_Q\\|_\\infty\\le C\\beta_*2^{2j}\\ \\ \\not\\Rightarrow\\ \\ \\sum_{j>M}\\|\\Delta_j\\Sigma_Q\\|_\\infty\\to0",
      "series": "NS",
      "first_appearance": "C3-V",
      "label_zh": "定理C3-V.10（導數放大障壁）",
      "label_en": "Theorem C3-V.10 (Derivative Amplification Barrier)",
      "definition_zh": "第38節證明前緣速度UV上界僅給出‖Δ_jΣ_Q‖_∞≤Cβ_*2^{2j}，允許高頻增長，因此速度UV上界不蘊含應變UV餘項變小，更不推出Morrey障礙量𝔒_{p,R}變小。",
      "definition_en": "Section 38 shows the first-frontier velocity UV cap only yields ‖Δ_jΣ_Q‖_∞≤Cβ_*2^{2j}, which permits high-frequency growth, so the velocity UV cap does not imply smallness of the strain UV remainder, let alone smallness of the Morrey obstruction 𝔒_{p,R}."
    },
    {
      "id": "ns.c3.c3w.riesz_pressure",
      "latex": "p = R_iR_j(u_iu_j)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "Riesz 壓力表示",
      "label_en": "Riesz pressure representation",
      "definition_zh": "第1節採用標準 Riesz transform 壓力規範 $p=R_iR_j(u_iu_j)$，由 Riesz 有界性得到臨界估計 $\\|p(t)\\|_{3/2}\\le C\\|u(t)\\|_3^2$。",
      "definition_en": "Section 1 fixes the standard Riesz-transform pressure gauge $p=R_iR_j(u_iu_j)$, giving the critical bound $\\|p(t)\\|_{3/2}\\le C\\|u(t)\\|_3^2$ via Riesz boundedness."
    },
    {
      "id": "ns.c3.c3w.cutoff_function",
      "latex": "\\chi_R(x) = \\chi_0\\!\\left(\\frac{x-x_0}{R}\\right)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "局部截斷函數",
      "label_en": "Local cutoff function",
      "definition_zh": "第2節固定標準球狀截斷 $\\chi_R$（於 $B_R$ 上為1、支撐於 $B_{2R}$），滿足 $\\|\\nabla^2\\chi_R\\|_\\infty\\le CR^{-2}$，是後續壓力矩量的基本工具。",
      "definition_en": "Section 2 fixes the standard radial cutoff $\\chi_R$ (equal to 1 on $B_R$, supported in $B_{2R}$) with $\\|\\nabla^2\\chi_R\\|_\\infty\\le CR^{-2}$, the basic tool for the pressure-moment quantities that follow."
    },
    {
      "id": "ns.c3.c3w.pressure_matrix_contribution",
      "latex": "P_{\\chi,R}(t) = \\int \\chi_R(x)\\,\\nabla^2 p(x,t)\\,dx",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "壓力矩陣貢獻（帶號平均應變強迫）",
      "label_en": "Signed pressure matrix contribution to mean strain",
      "definition_zh": "第3節定義 $P_{\\chi,R}(t)=\\int\\chi_R\\nabla^2p\\,dx$，作為 adjoint mean-strain transport $M_\\chi'$ 中壓力那一項的帶號、張量值平均強迫，刻意不取絕對值。",
      "definition_en": "Section 3 defines $P_{\\chi,R}(t)=\\int\\chi_R\\nabla^2p\\,dx$, the signed, tensor-valued pressure contribution to the adjoint mean-strain transport $M_\\chi'$, deliberately not absolute-valued.",
      "defining_relation": "P_{\\chi,R}(t) = \\int \\chi_R(x)\\,\\nabla^2 p(x,t)\\,dx",
      "notes": "與 Q_S（quadratic strain/vorticity 項）共同組成 M_chi' = -integral chi[Q_S+grad^2 p]dx；C3-W 只處理壓力項，quadratic 項承接自 C3-V。"
    },
    {
      "id": "ns.c3.c3w.thm_critical_pressure_forcing_bound",
      "latex": "\\text{C3-W.1}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.1：臨界壓力平均強迫界",
      "label_en": "Theorem C3-W.1: Critical Pressure Mean-Forcing Bound",
      "definition_zh": "第4節定理4.1證明 $|P_{\\chi,R}(t)|\\le CR^{-1}\\|p(t)-c(t)\\|_{L^{3/2}(B_{2R})}$，把壓力強迫由需要 $L^1$ Hessian降到只需臨界 $L^{3/2}$ 壓力振盪。",
      "definition_en": "Theorem 4.1 in Section 4 proves $|P_{\\chi,R}(t)|\\le CR^{-1}\\|p(t)-c(t)\\|_{L^{3/2}(B_{2R})}$, reducing the pressure forcing from an $L^1$ Hessian requirement to critical $L^{3/2}$ pressure oscillation only.",
      "defining_relation": "|P_{\\chi,R}(t)| \\le CR^{-1}\\|p(t)-c(t)\\|_{L^{3/2}(B_{2R}(x_0))}"
    },
    {
      "id": "ns.c3.c3w.normalized_pressure_oscillation",
      "latex": "\\Pi_R(t)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "尺度臨界局部壓力振盪",
      "label_en": "Scale-critical local pressure oscillation",
      "definition_zh": "第5節定義 $\\Pi_R(t)=\\nu^{-2}\\inf_c\\|p(t)-c\\|_{L^{3/2}(B_{2R}(x_0))}$，在 Navier–Stokes scaling下為無因次量，並滿足 $\\frac{R}{\\nu^2}|P_{\\chi,R}(t)|\\le C\\Pi_R(t)$。",
      "definition_en": "Section 5 defines $\\Pi_R(t)=\\nu^{-2}\\inf_c\\|p(t)-c\\|_{L^{3/2}(B_{2R}(x_0))}$, dimensionless under Navier–Stokes scaling and controlling $P_{\\chi,R}$ via $\\frac{R}{\\nu^2}|P_{\\chi,R}(t)|\\le C\\Pi_R(t)$."
    },
    {
      "id": "ns.c3.c3w.normalized_pressure_mean_rotation",
      "latex": "\\mathfrak R_I^P",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "標準化壓力平均旋轉量",
      "label_en": "Normalized pressure mean-rotation magnitude",
      "definition_zh": "第6節在滿足 $|I|\\le\\Theta R^2/\\nu$ 的黏性時間窗 $I$ 上定義 $\\mathfrak R_I^P=\\frac1{\\nu R}\\int_I|P_{\\chi,R}(t)|dt$，作為壓力驅動的平均應變轉動量。",
      "definition_en": "Section 6 defines, on a viscous window $I$ with $|I|\\le\\Theta R^2/\\nu$, the quantity $\\mathfrak R_I^P=\\frac1{\\nu R}\\int_I|P_{\\chi,R}(t)|dt$ measuring pressure-driven mean-strain rotation.",
      "defining_relation": "\\mathfrak R_I^P = \\frac{1}{\\nu R}\\int_I |P_{\\chi,R}(t)|\\,dt",
      "notes": "與 C3-V 的 quadratic 版本 R_n^Q 在第0、34節直接對照，兩者在 geometric scale 下有相同的 alpha<1 Zeno frontier，但 weighted budget 不同（R^2(R^P)^2 vs R·R^Q）。"
    },
    {
      "id": "ns.c3.c3w.thm_global_pressure_square_budget",
      "latex": "\\text{Thm 7.1}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理7.1：全域臨界壓力平方預算",
      "label_en": "Theorem 7.1: Global critical pressure square budget",
      "definition_zh": "第7節由 $\\|p\\|_{3/2}\\le C\\|u\\|_3^2$ 與能量不等式導出 $\\int_0^{T_*}\\|p(t)\\|_{3/2}^2dt\\le C\\|u_0\\|_2^4/\\nu$，是後續兩個 packing 定理的能量來源。",
      "definition_en": "Section 7 derives $\\int_0^{T_*}\\|p(t)\\|_{3/2}^2dt\\le C\\|u_0\\|_2^4/\\nu$ from $\\|p\\|_{3/2}\\le C\\|u\\|_3^2$ and the energy inequality, the energy source feeding the two packing theorems that follow."
    },
    {
      "id": "ns.c3.c3w.thm_pressure_rotation_r2_packing",
      "latex": "\\text{C3-W.2}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.2：壓力旋轉的 $R^2$ 加權 packing",
      "label_en": "Theorem C3-W.2: Pressure-Rotation $R^2$-Weighted Packing",
      "definition_zh": "第8節定理8.1證明對 pairwise disjoint 黏性窗口有 $\\sum_nR_n^2(\\mathfrak R_{I_n}^P)^2\\le C_\\Theta\\|u_0\\|_2^4/\\nu^4$，但第9節指出此界仍允許每代 $O(1)$ 旋轉的 Zeno-packing。",
      "definition_en": "Theorem 8.1 in Section 8 proves $\\sum_nR_n^2(\\mathfrak R_{I_n}^P)^2\\le C_\\Theta\\|u_0\\|_2^4/\\nu^4$ over pairwise disjoint viscous windows, though Section 9 shows this still permits Zeno-packing at $O(1)$ rotation per generation.",
      "defining_relation": "\\sum_n R_n^2\\left(\\mathfrak R_{I_n}^{P}\\right)^2 \\le C_\\Theta\\frac{\\|u_0\\|_2^4}{\\nu^4}"
    },
    {
      "id": "ns.c3.c3w.instantaneous_pressure_forcing",
      "latex": "\\pi_i(t)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "標準化瞬時壓力強迫",
      "label_en": "Normalized instantaneous pressure forcing",
      "definition_zh": "第10節在同尺度 disjoint 球 $B_{2R}(x_i)$ 上定義 $\\pi_i(t)=\\frac{R}{\\nu^2}\\left|\\int\\chi_{i,R}\\nabla^2p\\,dx\\right|$，作為瞬時、逐核心的壓力強迫強度。",
      "definition_en": "Section 10 defines $\\pi_i(t)=\\frac{R}{\\nu^2}\\left|\\int\\chi_{i,R}\\nabla^2p\\,dx\\right|$ on same-scale disjoint balls $B_{2R}(x_i)$ as the instantaneous per-core pressure-forcing strength."
    },
    {
      "id": "ns.c3.c3w.b_pressure_active_core",
      "latex": "\\pi_i(t) \\ge b",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "$b$-壓力活躍核心",
      "label_en": "$b$-pressure-active core",
      "definition_zh": "第10節將滿足 $\\pi_i(t)\\ge b$ 的核心稱為 $b$-pressure-active core，是第11至12節兩個 packing／時間預算定理的計數對象。",
      "definition_en": "Section 10 calls a core satisfying $\\pi_i(t)\\ge b$ a $b$-pressure-active core, the object counted by the two packing and time-budget theorems of Sections 11-12.",
      "defining_relation": "\\pi_i(t) \\ge b > 0"
    },
    {
      "id": "ns.c3.c3w.thm_pressure_active_core_packing",
      "latex": "m_b(t)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.3：壓力活躍核心 packing（$m_b$）",
      "label_en": "Theorem C3-W.3: Pressure-Active Core Packing ($m_b$)",
      "definition_zh": "第11節定理11.1證明同一瞬間 $b$-pressure-active disjoint 核心數 $m_b(t)$ 滿足 $m_b(t)\\le Cb^{-3/2}\\nu^{-3}\\|p(t)\\|_{3/2}^{3/2}\\le Cb^{-3/2}(\\|u(t)\\|_3/\\nu)^3$。",
      "definition_en": "Theorem 11.1 in Section 11 shows the count $m_b(t)$ of instantaneous disjoint $b$-pressure-active cores obeys $m_b(t)\\le Cb^{-3/2}\\nu^{-3}\\|p(t)\\|_{3/2}^{3/2}\\le Cb^{-3/2}(\\|u(t)\\|_3/\\nu)^3$.",
      "defining_relation": "m_b(t) \\le Cb^{-3/2}\\nu^{-3}\\|p(t)\\|_{3/2}^{3/2} \\le Cb^{-3/2}\\left(\\frac{\\|u(t)\\|_3}{\\nu}\\right)^3",
      "notes": "第31節與 C3-R 的 frontier core packing m_R^frontier ~< R^-1 並列比較，合併得 m_R ~< min{C R^-1, C b^{-3/2}(||u||_3/nu)^3}。"
    },
    {
      "id": "ns.c3.c3w.thm_pressure_active_multiplicity_time_budget",
      "latex": "\\text{C3-W.4}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.4：壓力活躍多重度時間預算",
      "label_en": "Theorem C3-W.4: Pressure-Active Multiplicity Time Budget",
      "definition_zh": "第12節由定理11.1導出尺度無關的時間預算 $\\int_0^{T_*}m_b(t)^{4/3}dt\\le Cb^{-2}\\|u_0\\|_2^4/\\nu^5$，但第13節指出此界仍與 multi-core cascade相容。",
      "definition_en": "Section 12 derives the scale-independent time budget $\\int_0^{T_*}m_b(t)^{4/3}dt\\le Cb^{-2}\\|u_0\\|_2^4/\\nu^5$ from Theorem 11.1, though Section 13 shows it remains compatible with multi-core cascade.",
      "defining_relation": "\\int_0^{T_\\ast} m_b(t)^{4/3}\\,dt \\le Cb^{-2}\\frac{\\|u_0\\|_2^4}{\\nu^5}",
      "notes": "第32節指出持續型 pressure-driven multiplicity 只給 alpha<3/2（m_n ~ R_n^{-alpha}），弱於 C3-R energy packing 已有的 alpha<=1，形式上是一個 no-go：pressure multiplicity 不改善 packing exponent。"
    },
    {
      "id": "ns.c3.c3w.constantin_pressure_regularity_theorem",
      "latex": "|p(x,t)|^{3/2}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "Constantin 壓力正則性定理（外部）",
      "label_en": "Constantin's pressure regularity theorem (external)",
      "definition_zh": "第14節引用 Constantin 的外部定理：若 $|p|^{3/2}$ 在小 Lebesgue 集上具有足夠強的一致可積性則正則性得以維持，故本輪將 pressure-turnover escape等同於 critical concentration branch。",
      "definition_en": "Section 14 invokes Constantin's external theorem — sufficiently strong uniform integrability of $|p|^{3/2}$ on small sets forces regularity — so this round identifies pressure-turnover escape with a critical concentration branch.",
      "notes": "外部結果（References #1, Constantin, arXiv:2301.04489），本輪不證明，只建立 interface（guard G-PUI）。"
    },
    {
      "id": "ns.c3.c3w.effective_volume",
      "latex": "\\mathcal V_p(g)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "有效體積",
      "label_en": "Effective volume",
      "definition_zh": "第15節回顧 C3-V 對 $g=\\nabla S$（於 $B_R$ 上）定義的有效體積 $\\mathcal V_p(g)=(\\|g\\|_2/\\|g\\|_p)^{1/(1/2-1/p)}$，是本輪 strain sparseness 論證的出發量。",
      "definition_en": "Section 15 recalls the C3-V effective volume $\\mathcal V_p(g)=(\\|g\\|_2/\\|g\\|_p)^{1/(1/2-1/p)}$ for $g=\\nabla S$ on $B_R$, the starting quantity of this round's strain-sparseness argument.",
      "notes": "承接自 C3-V（第15節標明「C3-V定義」）；本輪在此基礎上新增 A_eff、Omega_c 與 sparseness 升級鏈（C3-W.5–W.7）。"
    },
    {
      "id": "ns.c3.c3w.effective_volume_fraction",
      "latex": "\\phi_{p,R}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "標準化有效體積比（間歇度參數）",
      "label_en": "Normalized effective-volume fraction (intermittency parameter)",
      "definition_zh": "第15節定義 $\\phi_{p,R}=\\mathcal V_p(g)/R^3$，是貫穿全輪 strain intermittency 論證與 sparseness scale $r_{\\rm sp}\\sim\\phi_{p,R}^{1/3}R$ 的核心參數。",
      "definition_en": "Section 15 defines $\\phi_{p,R}=\\mathcal V_p(g)/R^3$, the central parameter running through the strain-intermittency argument and the sparseness scale $r_{\\rm sp}\\sim\\phi_{p,R}^{1/3}R$."
    },
    {
      "id": "ns.c3.c3w.effective_amplitude",
      "latex": "A_{\\rm eff}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "有效振幅",
      "label_en": "Effective amplitude",
      "definition_zh": "第16節定義 $A_{\\rm eff}=\\|g\\|_2/\\mathcal V_p(g)^{1/2}$，並由 $\\|g\\|_p^p\\le\\|g\\|_\\infty^{p-2}\\|g\\|_2^2$ 導出 $A_{\\rm eff}\\le\\|g\\|_\\infty$。",
      "definition_en": "Section 16 defines $A_{\\rm eff}=\\|g\\|_2/\\mathcal V_p(g)^{1/2}$ and derives $A_{\\rm eff}\\le\\|g\\|_\\infty$ from $\\|g\\|_p^p\\le\\|g\\|_\\infty^{p-2}\\|g\\|_2^2$."
    },
    {
      "id": "ns.c3.c3w.high_gradient_active_set",
      "latex": "\\Omega_c(g)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "高梯度活躍集",
      "label_en": "High-gradient active set",
      "definition_zh": "第17節定義 $\\Omega_c(g)=\\{x\\in B_R:|g(x)|>c\\|g\\|_\\infty\\}$，因 $A_{\\rm eff}\\le\\|g\\|_\\infty$ 而含於 $\\{|g|>cA_{\\rm eff}\\}$。",
      "definition_en": "Section 17 defines $\\Omega_c(g)=\\{x\\in B_R:|g(x)|>c\\|g\\|_\\infty\\}$, contained in $\\{|g|>cA_{\\rm eff}\\}$ since $A_{\\rm eff}\\le\\|g\\|_\\infty$.",
      "defining_relation": "\\Omega_c(g) = \\{x\\in B_R : |g(x)| > c\\|g\\|_\\infty\\}"
    },
    {
      "id": "ns.c3.c3w.thm_effective_volume_superlevel_bound",
      "latex": "\\text{C3-W.5}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.5：有效體積超水平集界",
      "label_en": "Theorem C3-W.5: Effective-Volume Superlevel Bound",
      "definition_zh": "第18節定理18.1由 Chebyshev 不等式證明 $|\\Omega_c(g)|\\le c^{-p}\\mathcal V_p(g)=c^{-p}\\phi_{p,R}R^3$，把體積間歇性直接轉成超水平集小體積。",
      "definition_en": "Theorem 18.1 in Section 18 uses Chebyshev's inequality to prove $|\\Omega_c(g)|\\le c^{-p}\\mathcal V_p(g)=c^{-p}\\phi_{p,R}R^3$, translating volumetric intermittency directly into small superlevel-set volume.",
      "defining_relation": "|\\Omega_c(g)| \\le c^{-p}\\mathcal V_p(g) = c^{-p}\\phi_{p,R}R^3"
    },
    {
      "id": "ns.c3.c3w.line_occupancy_fraction",
      "latex": "\\theta_A(x_0,r,d)",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "直線佔有比例",
      "label_en": "Line occupancy fraction",
      "definition_zh": "第19節對可測集 $A\\subset B_r(x_0)$ 與單位方向 $d\\in S^2/\\{\\pm1\\}$ 定義 $\\theta_A(x_0,r,d)=|A\\cap(x_0-rd,x_0+rd)|_1/(2r)$，量測 $A$ 沿該直線的一維佔有比例。",
      "definition_en": "Section 19 defines, for measurable $A\\subset B_r(x_0)$ and unit direction $d\\in S^2/\\{\\pm1\\}$, the line occupancy fraction $\\theta_A(x_0,r,d)=|A\\cap(x_0-rd,x_0+rd)|_1/(2r)$."
    },
    {
      "id": "ns.c3.c3w.thm_volume_to_1d_sparseness_lemma",
      "latex": "\\text{C3-W.6}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.6：體積轉一維稀疏性引理",
      "label_en": "Theorem C3-W.6: Volume-to-One-Dimensional-Sparseness Lemma",
      "definition_zh": "第20節定理20.1證明純幾何結果：若 $|A\\cap B_r(x_0)|<\\delta^3|B_r|$，則存在方向 $d$ 使 $\\theta_A(x_0,r,d)\\le\\delta$，以 hemisphere 上 $a^3+b^3$ 的凸性反證得到。",
      "definition_en": "Theorem 20.1 in Section 20 proves the purely geometric fact that $|A\\cap B_r(x_0)|<\\delta^3|B_r|$ forces some direction $d$ with $\\theta_A(x_0,r,d)\\le\\delta$, via a convexity argument on $a^3+b^3$ over the hemisphere.",
      "defining_relation": "|A\\cap B_r(x_0)| < \\delta^3|B_r| \\implies \\exists\\, d:\\ \\theta_A(x_0,r,d)\\le\\delta"
    },
    {
      "id": "ns.c3.c3w.sparseness_scale",
      "latex": "r_{\\rm sp}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "稀疏化尺度",
      "label_en": "Sparseness scale",
      "definition_zh": "第22節定義 $r_{\\rm sp}=Cc^{-p/3}\\delta^{-1}\\phi_{p,R}^{1/3}R\\asymp\\phi_{p,R}^{1/3}R$，是高梯度活躍集轉為一維 $\\delta$-稀疏所在的尺度。",
      "definition_en": "Section 22 defines $r_{\\rm sp}=Cc^{-p/3}\\delta^{-1}\\phi_{p,R}^{1/3}R\\asymp\\phi_{p,R}^{1/3}R$, the scale at which the high-gradient active set becomes linearly $\\delta$-sparse.",
      "defining_relation": "r_{\\rm sp} = Cc^{-p/3}\\delta^{-1}\\phi_{p,R}^{1/3}R \\asymp \\phi_{p,R}^{1/3}R"
    },
    {
      "id": "ns.c3.c3w.thm_strain_intermittency_to_sparseness",
      "latex": "\\text{C3-W.7}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.7：應變間歇性轉稀疏性定理",
      "label_en": "Theorem C3-W.7: Strain-Intermittency-to-Sparseness Theorem",
      "definition_zh": "第22節定理22.1證明若 $\\phi_{p,R}$ 足夠小使 $r_{\\rm sp}\\le R/2$，則 $\\Omega_c(\\nabla S)$ 在 $B_{R/2}$ 每點於 scale $r_{\\rm sp}\\asymp\\phi_{p,R}^{1/3}R$ 上沿某方向線性 $\\delta$-稀疏，是本輪把體積坍縮升級為幾何稀疏性的核心定理。",
      "definition_en": "Theorem 22.1 in Section 22 proves that if $\\phi_{p,R}$ is small enough that $r_{\\rm sp}\\le R/2$, then $\\Omega_c(\\nabla S)$ is linearly $\\delta$-sparse along some direction at scale $r_{\\rm sp}\\asymp\\phi_{p,R}^{1/3}R$ at every point of $B_{R/2}$, the round's central upgrade from volume collapse to geometric sparseness.",
      "defining_relation": "r_{\\rm sp} = Cc^{-p/3}\\delta^{-1}\\phi_{p,R}^{1/3}R \\le R/2 \\implies \\Omega_c(\\nabla S)\\ \\text{linearly}\\ \\delta\\text{-sparse at}\\ r_{\\rm sp}",
      "notes": "此定理是下一輪 C3-X 的兩大出發點之一（另一為壓力集中），見第38–39節。"
    },
    {
      "id": "ns.c3.c3w.nabla_s_d2u_equivalence",
      "latex": "|\\nabla S| \\asymp |D^2u|",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "$\\nabla S$ 與 $D^2u$ 的逐點線性等價",
      "label_en": "Pointwise linear equivalence of $\\nabla S$ and $D^2u$",
      "definition_zh": "第23節由恆等式 $\\partial_j\\partial_ku_i=\\partial_jS_{ik}+\\partial_kS_{ij}-\\partial_iS_{jk}$ 得到 $|\\nabla S|\\asymp|D^2u|$（up to universal constants），使 strain-gradient間歇性與二階速度導數稀疏性落在同一 derivative order。",
      "definition_en": "Section 23 derives $|\\nabla S|\\asymp|D^2u|$ (up to universal constants) from the identity $\\partial_j\\partial_ku_i=\\partial_jS_{ik}+\\partial_kS_{ij}-\\partial_iS_{jk}$, placing strain-gradient intermittency and second-order velocity-derivative sparseness at the same derivative order.",
      "defining_relation": "\\partial_j\\partial_k u_i = \\partial_jS_{ik}+\\partial_kS_{ij}-\\partial_iS_{jk}, \\qquad |\\nabla S|\\asymp|D^2u|"
    },
    {
      "id": "ns.c3.c3w.grujic_xu_higher_derivative_framework",
      "latex": "D^k u",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "Grujić–Xu 高階導數稀疏性框架（外部）",
      "label_en": "Grujić–Xu higher-derivative sparseness framework (external)",
      "definition_zh": "第24、25節引用 Grujić–Xu 對 $D^ku$ 分量正負超水平集稀疏性的外部框架（2025最終版指出 $k\\to\\infty$ 時 scaling gap趨於消失），第24節並指出這要求 component/sign threshold 的額外對齊，C3-W 定理22.1只給 magnitude 版本。",
      "definition_en": "Sections 24-25 invoke the external Grujić–Xu framework tracking positive/negative superlevel-set sparseness of components of $D^ku$ (whose final 2025 version shows the scaling gap vanishing as $k\\to\\infty$); Section 24 notes this needs extra component/sign threshold alignment, since C3-W's Theorem 22.1 only gives the magnitude version.",
      "notes": "外部結果（References #4, Grujić–Xu, arXiv:1911.00974，2025 final版），本輪只建立 interface（guard G-COMPDER），不聲稱直接套用成立。"
    },
    {
      "id": "ns.c3.c3w.grujic_geometric_measure_theorem",
      "latex": "\\text{Grujić geometric-measure criterion}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "Grujić 幾何測度正則性定理（外部）",
      "label_en": "Grujić's geometric measure regularity theorem (external)",
      "definition_zh": "第25節引用 Grujić 的外部定理：在潛在奇異時刻附近，若強速度／渦度區域在相關 spatial analyticity scale上對每個空間點皆存在方向呈一維稀疏，則可阻止 finite-time singularity。",
      "definition_en": "Section 25 invokes Grujić's external theorem: if, near a potential singular time, the intense velocity/vorticity region is linearly sparse along some direction at every point at the relevant spatial analyticity scale, finite-time singularity is precluded.",
      "notes": "外部結果（References #3, Grujić, arXiv:1111.0217），是第26節 analyticity-scale barrier 與 C3-W.8 的直接動機。"
    },
    {
      "id": "ns.c3.c3w.analyticity_scale",
      "latex": "\\rho_{\\rm an}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "解析尺度",
      "label_en": "Spatial analyticity scale",
      "definition_zh": "第26節將 $\\rho_{\\rm an}$ 定義為對應 derivative formulation／時間切片可用的 spatial analyticity scale，是判斷 sparseness能否啟動 Grujić 判據的比較基準。",
      "definition_en": "Section 26 introduces $\\rho_{\\rm an}$, the spatial analyticity scale available to the relevant derivative formulation/time slice, the benchmark against which sparseness is measured to trigger Grujić's criterion."
    },
    {
      "id": "ns.c3.c3w.analyticity_scale_ratio",
      "latex": "\\mathfrak A_R",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "解析尺度比",
      "label_en": "Analyticity-scale ratio",
      "definition_zh": "第26節定義 $\\mathfrak A_R=r_{\\rm sp}/\\rho_{\\rm an}\\asymp\\phi_{p,R}^{1/3}R/\\rho_{\\rm an}$，當 $\\mathfrak A_R\\lesssim1$ 且其餘條件對齊時，稀疏性落在可用解析尺度內，可能啟動 known regularity mechanism。",
      "definition_en": "Section 26 defines $\\mathfrak A_R=r_{\\rm sp}/\\rho_{\\rm an}\\asymp\\phi_{p,R}^{1/3}R/\\rho_{\\rm an}$; when $\\mathfrak A_R\\lesssim1$ and the remaining conditions align, the sparseness lies within the admissible analyticity scale and may trigger a known regularity mechanism.",
      "defining_relation": "\\mathfrak A_R = \\frac{r_{\\rm sp}}{\\rho_{\\rm an}} \\asymp \\frac{\\phi_{p,R}^{1/3}R}{\\rho_{\\rm an}}"
    },
    {
      "id": "ns.c3.c3w.thm_intermittency_survivor_scale_ordering",
      "latex": "\\text{C3-W.8}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "定理 C3-W.8：間歇性倖存者尺度排序（W-I1/I2/I3）",
      "label_en": "Theorem C3-W.8: Intermittency Survivor Scale Ordering (W-I1/I2/I3)",
      "definition_zh": "第27節證明應變間歇性要持續逃避正則性，至少須保留三者之一：解析尺度縮得更快（W-I1：$\\rho_{\\rm an}\\ll\\phi_{p,R}^{1/3}R$）、分量／正負號門檻不匹配（W-I2）、或時間選取不匹配（W-I3）。",
      "definition_en": "Section 27 proves that for strain intermittency to keep evading regularity, at least one of three conditions must persist: faster analyticity-scale collapse (W-I1: $\\rho_{\\rm an}\\ll\\phi_{p,R}^{1/3}R$), component/sign threshold mismatch (W-I2), or time-selection mismatch (W-I3).",
      "defining_relation": "\\rho_{\\rm an} \\ll \\phi_{p,R}^{1/3}R \\quad \\text{(W-I1)}"
    },
    {
      "id": "ns.c3.c3w.pressure_rotation_state",
      "latex": "\\Theta_R^{P}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "壓力旋轉狀態（True ETN）",
      "label_en": "Pressure rotation state (True ETN)",
      "definition_zh": "第37節把壓力側狀態彙整成 $\\Theta_R^P=\\langle\\Pi_R,\\mathfrak R_R^P,m_b,\\text{pressure mass concentration},\\operatorname{Prov}\\rangle$，納入 True ETN／無限維張力場框架。",
      "definition_en": "Section 37 assembles the pressure-side state into $\\Theta_R^P=\\langle\\Pi_R,\\mathfrak R_R^P,m_b,\\text{pressure mass concentration},\\operatorname{Prov}\\rangle$ within the True ETN / infinite-dimensional tension field framework.",
      "notes": "True ETN 為跨輪次的統一狀態記帳框架（見 Internal dependencies 中的「True ETN／無限維張力場」）；本輪新增壓力與間歇性兩個分支狀態。"
    },
    {
      "id": "ns.c3.c3w.strain_intermittency_state",
      "latex": "\\Theta_R^{I}",
      "series": "NS",
      "first_appearance": "C3-W",
      "label_zh": "應變間歇性狀態（True ETN）",
      "label_en": "Strain intermittency state (True ETN)",
      "definition_zh": "第37節把應變間歇性側狀態彙整成 $\\Theta_R^I=\\langle\\phi_{p,R},A_{\\rm eff},\\Omega_c,r_{\\rm sp},\\rho_{\\rm an},\\mathfrak A_R\\rangle$，與 $\\Theta_R^P$ 共同構成「壓力集中 vs 解析尺度稀疏性」的新 bifurcation。",
      "definition_en": "Section 37 assembles the strain-intermittency state into $\\Theta_R^I=\\langle\\phi_{p,R},A_{\\rm eff},\\Omega_c,r_{\\rm sp},\\rho_{\\rm an},\\mathfrak A_R\\rangle$, forming with $\\Theta_R^P$ the new bifurcation \"pressure concentration vs. analyticity-scale sparseness.\""
    },
    {
      "id": "ns.c3.c3x.pressure_hessian_forcing",
      "latex": "P_{\\chi,R}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "局部平均應變壓力驅動",
      "label_en": "Local mean-strain pressure forcing",
      "definition_zh": "第0節與第1節引入的截斷壓力Hessian積分，量測壓力二階導數在球$B_R$上被$\\chi_R$加權後的局部驅動量。",
      "definition_en": "A cutoff-weighted integral of the pressure Hessian introduced in Sections 0-1, measuring the local forcing exerted by $\\nabla^2p$ on the ball $B_R$.",
      "defining_relation": "P_{\\chi,R}=\\int \\chi_R\\nabla^2p\\,dx"
    },
    {
      "id": "ns.c3.c3x.hessian_pressure_oscillation",
      "latex": "\\Pi_R^{(2)}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "二階（Hessian敏感）壓力振盪",
      "label_en": "Hessian-sensitive second-order pressure oscillation",
      "definition_zh": "第1節定義的量，將常見的對常數取下確界改為對仿射函數類$\\mathcal A_1$取下確界，使其對純線性壓力場不敏感。",
      "definition_en": "Defined in Section 1, it replaces the usual infimum over constants with an infimum over the affine function class $\\mathcal A_1$, rendering the oscillation insensitive to purely affine pressure fields.",
      "defining_relation": "\\Pi_R^{(2)}(t)=\\frac{1}{\\nu^2}\\inf_{\\ell\\in\\mathcal A_1}\\|p(t)-\\ell\\|_{L^{3/2}(B_{2R})},\\quad \\mathcal A_1=\\{a+b\\cdot x\\}",
      "notes": "精煉自C3-W用來推導$\\int\\chi_R\\partial_i\\partial_jp=\\int(p-c)\\partial_i\\partial_j\\chi_R$的常數扣除識別式，本輪改為扣除最佳仿射逼近。"
    },
    {
      "id": "ns.c3.c3x.thm_hessian_pressure_bound",
      "latex": "\\frac{R}{\\nu^2}\\left|\\int\\chi_R\\nabla^2p\\,dx\\right|\\le C\\Pi_R^{(2)}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.1：Hessian敏感壓力界",
      "label_en": "C3-X.1: Hessian-Sensitive Pressure Bound",
      "definition_zh": "第2節定理2.1，藉由對仿射函數的兩次分部積分與Hölder不等式，證明壓力Hessian驅動力可被$\\Pi_R^{(2)}$控制。",
      "definition_en": "Theorem 2.1 in Section 2, proved via a double integration by parts against an affine function and Hölder's inequality, showing the pressure-Hessian forcing is controlled by $\\Pi_R^{(2)}$.",
      "notes": "此不等式左側即為第3節定義的$\\pi_R$，故本定理等價於$\\pi_R\\le C\\Pi_R^{(2)}$。"
    },
    {
      "id": "ns.c3.c3x.normalized_pressure_forcing",
      "latex": "\\pi_R",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "正規化壓力驅動量",
      "label_en": "Normalized pressure forcing",
      "definition_zh": "第3節定義的無因次化壓力Hessian驅動量，用以刻劃壓力活躍核心的強度閾值。",
      "definition_en": "The dimensionless pressure-Hessian forcing defined in Section 3, used to set the intensity threshold for a pressure-active core.",
      "defining_relation": "\\pi_R=\\frac{R}{\\nu^2}\\left|\\int\\chi_R\\nabla^2p\\,dx\\right|"
    },
    {
      "id": "ns.c3.c3x.pressure_active_core",
      "latex": "\\pi_R\\ge b>0",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "b-壓力活躍核心",
      "label_en": "$b$-pressure-active core",
      "definition_zh": "第3節命名的條件，指正規化壓力驅動量$\\pi_R$在尺度$R$上不低於正閾值$b$的區域。",
      "definition_en": "A named condition from Section 3 requiring the normalized forcing $\\pi_R$ to stay at or above a positive threshold $b$ at scale $R$."
    },
    {
      "id": "ns.c3.c3x.thm_critical_pressure_mass",
      "latex": "\\int_{B_{2R}}|p|^{3/2}dx\\ge cb^{3/2}\\nu^3",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.2：臨界壓力質量證書",
      "label_en": "C3-X.2: Critical Pressure-Mass Certificate",
      "definition_zh": "第4節定理4.1，證明$b$-壓力活躍核心必然攜帶與尺度$R$無關的臨界$L^{3/2}$壓力質量下界。",
      "definition_en": "Theorem 4.1 in Section 4, proving that a $b$-pressure-active core necessarily carries a scale-invariant lower bound on its critical $L^{3/2}$ pressure mass.",
      "defining_relation": "\\pi_R\\ge b \\ \\Longrightarrow\\ \\inf_{\\ell\\in\\mathcal A_1}\\|p-\\ell\\|_{L^{3/2}(B_{2R})}\\ge cb\\nu^2 \\ \\Longrightarrow\\ \\int_{B_{2R}}|p|^{3/2}dx\\ge cb^{3/2}\\nu^3"
    },
    {
      "id": "ns.c3.c3x.thm_small_volume_certificate",
      "latex": "|U_n|\\to0,\\ \\ \\inf_n\\int_{U_n}|p(t_n)|^{3/2}dx>0",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.3：小體積壓力集中證書",
      "label_en": "C3-X.3: Small-Volume Pressure Concentration Certificate",
      "definition_zh": "第6節（另稱定理6.1）證明若壓力活躍核心數$m_n$與尺度$R_n$滿足$m_nR_n^3\\to0$但$\\inf_n m_nb_n^{3/2}>0$，則存在測度趨零卻壓力質量不趨零的集合序列$U_n$。",
      "definition_en": "Section 6 (also cited as Theorem 6.1) shows that if core count $m_n$ and scale $R_n$ satisfy $m_nR_n^3\\to0$ while $\\inf_n m_nb_n^{3/2}>0$, there exist sets $U_n$ with vanishing measure but non-vanishing pressure mass.",
      "notes": "此證書即第7-8節中被接到Constantin壓力一致可積性判準外部介面的機制，說明$|p|^{3/2}$沿該序列失去一致可積性。"
    },
    {
      "id": "ns.c3.c3x.strain_gradient_tensor",
      "latex": "g=\\nabla S\\asymp D^2u",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "應變梯度張量",
      "label_en": "Strain-gradient tensor",
      "definition_zh": "第0節與第9節沿用C3-W記號，將應變張量的梯度$g=\\nabla S$等同於速度場二階導數$D^2u$的量級。",
      "definition_en": "Carried over from C3-W's notation in Sections 0 and 9, identifying the gradient of the strain tensor $g=\\nabla S$ with the second derivative $D^2u$ up to constants.",
      "notes": "沿用自前一輪C3-W；本輪未重新定義，僅延續使用以建立$\\Omega_c$。"
    },
    {
      "id": "ns.c3.c3x.strain_active_set",
      "latex": "\\Omega_c",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "應變活躍集",
      "label_en": "Strain-active set",
      "definition_zh": "第9節沿用C3-W的定義，指應變梯度超過其上確界某比例$c$的超水平集。",
      "definition_en": "Section 9 reuses C3-W's definition of the superlevel set where the strain gradient exceeds a fixed fraction $c$ of its supremum.",
      "defining_relation": "\\Omega_c=\\{x\\in B_R: |g(x)|>c\\|g\\|_\\infty\\}",
      "notes": "沿用自C3-W；本輪在其上疊加壓力測度$\\mu_p$以定義第10節的$\\Theta_{P/S}$。"
    },
    {
      "id": "ns.c3.c3x.overlap_coefficient",
      "latex": "\\Theta_{P/S}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "壓力–應變重疊係數",
      "label_en": "Pressure-strain overlap coefficient",
      "definition_zh": "第10節定義，以壓力測度$d\\mu_p=|p|^{3/2}dx$計算應變活躍集$\\Omega_c$承載的壓力質量佔全域比例，值域在$[0,1]$。",
      "definition_en": "Defined in Section 10 using the pressure measure $d\\mu_p=|p|^{3/2}dx$, it is the fraction of pressure mass carried by the strain-active set $\\Omega_c$, taking values in $[0,1]$.",
      "defining_relation": "d\\mu_p=|p|^{3/2}dx,\\qquad \\Theta_{P/S}=\\frac{\\mu_p(\\Omega_c)}{\\mu_p(B_{2R})}\\in[0,1]"
    },
    {
      "id": "ns.c3.c3x.colocated_segregated_dichotomy",
      "latex": "\\text{X-J1: }\\Theta_{P/S}\\ge\\theta_0>0;\\quad \\text{X-J2: }\\Theta_{P/S}\\to0",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "共位／分離集中二分法（X-J1／X-J2）",
      "label_en": "Co-located / segregated concentration dichotomy (X-J1 / X-J2)",
      "definition_zh": "第11節提出的二分條件，區分壓力臨界質量是否以固定比例落在應變活躍集上（共位X-J1）或趨於落在其外（分離X-J2）。",
      "definition_en": "A dichotomy introduced in Section 11 distinguishing whether critical pressure mass retains a fixed fraction on the strain-active set (co-located, X-J1) or drifts outside it (segregated, X-J2)."
    },
    {
      "id": "ns.c3.c3x.thm_colocated_certificate",
      "latex": "\\int_{\\Omega_c}|p|^{3/2}dx\\ge c\\theta_0b^{3/2}\\nu^3",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.4：共位聯合集中證書",
      "label_en": "C3-X.4: Co-Located Joint Concentration Certificate",
      "definition_zh": "第12節結果，證明當核心為$b$-壓力活躍且$\\Theta_{P/S}\\ge\\theta_0$時，壓力質量集中在體積至多與$\\phi_{p,R}R^3$同階的稀疏應變活躍集$\\Omega_c$上。",
      "definition_en": "A result in Section 12 showing that when a core is $b$-pressure-active with $\\Theta_{P/S}\\ge\\theta_0$, pressure mass concentrates on the sparse strain-active set $\\Omega_c$, whose volume is at most of order $\\phi_{p,R}R^3$."
    },
    {
      "id": "ns.c3.c3x.active_volume_fraction",
      "latex": "\\phi_{p,R}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "有效活躍體積分率",
      "label_en": "Effective active-volume fraction",
      "definition_zh": "第9節沿用C3-W記號，刻劃應變活躍集$\\Omega_c$相對球$B_R$體積的比例上界。",
      "definition_en": "Reused from C3-W in Section 9, it bounds the relative volume of the strain-active set via a fraction of $R^3$.",
      "notes": "定義關係為$|\\Omega_c|\\le C_c\\phi_{p,R}R^3$；沿用自C3-W，第16節起推廣為對任意導數階數$k$索引的$\\phi_k$。"
    },
    {
      "id": "ns.c3.c3x.sparseness_scale",
      "latex": "r_{\\rm sp}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "一維稀疏化尺度",
      "label_en": "One-dimensional sparseness scale",
      "definition_zh": "第9節沿用C3-W volume-to-line定理的結論，指高梯度集在此尺度上退化為近似一維的稀疏結構。",
      "definition_en": "Reused in Section 9 from C3-W's volume-to-line theorem, this is the scale at which the high-gradient set degenerates into an approximately one-dimensional sparse structure.",
      "defining_relation": "r_{\\rm sp}\\sim \\phi_{p,R}^{1/3}R",
      "notes": "沿用自C3-W；第24節推廣為帶階數上標的$r_{\\rm sp}^{(k)}$並用於定義$\\mathfrak A_k$。"
    },
    {
      "id": "ns.c3.c3x.regularity_scale",
      "latex": "\\ell_{\\rm reg}^{(k)}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "k階正則性尺度",
      "label_en": "Regularity-class sparseness scale (level $k$)",
      "definition_zh": "第14節引入，取自Grujić–Xu高階導數稀疏化層級中，第$k$階正則性判準所需要的尺度。",
      "definition_en": "Introduced in Section 14, the scale required by the level-$k$ regularity criterion within the Grujić–Xu higher-derivative sparseness hierarchy.",
      "defining_relation": "\\ell_{\\rm reg}^{(k)}\\sim A_k^{-1/(k+1)}"
    },
    {
      "id": "ns.c3.c3x.apriori_scale",
      "latex": "\\ell_{\\rm apr}^{(k)}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "k階先驗尺度",
      "label_en": "Energy-level a-priori sparseness scale (level $k$)",
      "definition_zh": "第14節引入，是由能量估計自然給出、與正則性尺度$\\ell_{\\rm reg}^{(k)}$相差一個代數因子的先驗尺度。",
      "definition_en": "Introduced in Section 14, the a-priori scale naturally supplied by energy estimates, differing from the regularity scale $\\ell_{\\rm reg}^{(k)}$ by an algebraic factor.",
      "defining_relation": "\\ell_{\\rm apr}^{(k)}\\sim A_k^{-1/(k+3/2)}"
    },
    {
      "id": "ns.c3.c3x.derivative_norm",
      "latex": "A_k",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "k階導數上確界範數",
      "label_en": "$k$-th derivative sup-norm",
      "definition_zh": "第14節定義，為速度場第$k$階導數的上確界範數，是本節起所有尺度指數公式的基準量。",
      "definition_en": "Defined in Section 14 as the sup-norm of the $k$-th velocity derivative, serving as the reference quantity for every scaling-exponent formula from this section onward.",
      "defining_relation": "A_k=\\|D^ku\\|_\\infty"
    },
    {
      "id": "ns.c3.c3x.level_k_active_volume",
      "latex": "\\phi_k",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "k階活躍體積分率",
      "label_en": "Level-$k$ active-volume fraction",
      "definition_zh": "第16節將$\\phi_{p,R}$推廣到任意導數階數$k$的高強度集活躍體積分率。",
      "definition_en": "Section 16 generalizes $\\phi_{p,R}$ to the active-volume fraction of the intense set at an arbitrary derivative level $k$.",
      "notes": "$k=2$特例即第18-19節討論的$\\phi_2$；第40節Y3提議將其進一步推廣到$D^ku$的volume-to-line稀疏化。"
    },
    {
      "id": "ns.c3.c3x.volume_sparseness",
      "latex": "r_{\\rm vol}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "體積誘導稀疏化尺度",
      "label_en": "Volume-induced sparseness scale",
      "definition_zh": "第16節由C3-W volume-to-line定理搭配$R=\\ell_{\\rm apr}^{(k)}$推得，是額外體積收縮所能達到的一維稀疏尺度。",
      "definition_en": "Derived in Section 16 by combining C3-W's volume-to-line theorem with $R=\\ell_{\\rm apr}^{(k)}$, this is the one-dimensional sparseness attainable through extra volume shrinkage.",
      "notes": "定義關係為$r_{\\rm vol}\\lesssim\\phi_k^{1/3}A_k^{-1/(k+3/2)}$；第42節結論將其重新併回$r_{\\rm sp}$記號（$r_{\\rm sp}\\sim\\phi_k^{1/3}\\ell_{\\rm apr}^{(k)}$）。"
    },
    {
      "id": "ns.c3.c3x.thm_gap_closure_lemma",
      "latex": "\\phi_k\\lesssim A_k^{-\\vartheta_k}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.5：有限k體積缺口封閉引理",
      "label_en": "C3-X.5: Finite-$k$ Volume Gap-Closure Lemma",
      "definition_zh": "第17節定理17.1，證明只要活躍體積分率$\\phi_k$滿足代數門檻$\\phi_k\\lesssim A_k^{-\\vartheta_k}$，體積誘導稀疏尺度便能壓到正則性尺度$\\ell_{\\rm reg}^{(k)}$。",
      "definition_en": "Theorem 17.1 in Section 17, proving that once the active-volume fraction $\\phi_k$ meets the algebraic threshold $\\phi_k\\lesssim A_k^{-\\vartheta_k}$, the volume-induced sparseness scale collapses onto the regularity scale $\\ell_{\\rm reg}^{(k)}$.",
      "defining_relation": "\\phi_k\\le CA_k^{-\\vartheta_k}\\ \\Longrightarrow\\ r_{\\rm vol}\\lesssim A_k^{-1/(k+1)}=\\ell_{\\rm reg}^{(k)}"
    },
    {
      "id": "ns.c3.c3x.gap_exponent",
      "latex": "\\vartheta_k",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "缺口封閉指數",
      "label_en": "Gap-closure exponent",
      "definition_zh": "第0節預告、第17節證明的代數指數，量化在導數階數$k$上補齊正則性尺度與先驗尺度之間差距所需的額外活躍體積冪次。",
      "definition_en": "Previewed in Section 0 and proved in Section 17, this algebraic exponent quantifies the extra active-volume power needed to close the gap between the regularity and a-priori scales at derivative level $k$.",
      "defining_relation": "\\vartheta_k=\\frac{3}{2(k+1)(k+3/2)},\\qquad \\vartheta_2=\\frac17,\\qquad \\vartheta_k\\sim\\frac{3}{2k^2}\\to0",
      "notes": "$k=2$特例$\\vartheta_2=1/7$於第18節算出，並在第19節命名為Second-Derivative Intermittency Gap-Closing Threshold。"
    },
    {
      "id": "ns.c3.c3x.second_derivative_threshold",
      "latex": "\\phi_2\\sim A_2^{-1/7}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "二階導數間歇性缺口封閉閾值",
      "label_en": "Second-Derivative Intermittency Gap-Closing Threshold",
      "definition_zh": "第19節命名的具體閾值，是$k=2$情形下區分「體積收縮過強」與「稀疏化仍不足」兩種情況的自然分界。",
      "definition_en": "A named threshold from Section 19, marking the natural dividing line at $k=2$ between volume collapse that overshoots the gap-closure requirement and collapse that still falls short of it.",
      "notes": "第22節指出，若要避免自動進入k=2正則尺度，倖存者需維持反向不等式$\\phi_2\\gtrsim A_2^{-1/7}$。"
    },
    {
      "id": "ns.c3.c3x.gap_load",
      "latex": "\\mathfrak G_k",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "間歇性缺口負載",
      "label_en": "Intermittency gap-load",
      "definition_zh": "第21節定義的無因次量，其值$\\lesssim1$表示尺度已封閉（scale-closed），$\\gg1$表示仍未封閉（scale-open）。",
      "definition_en": "A dimensionless quantity defined in Section 21, where $\\mathfrak G_k\\lesssim1$ signals the scale-closed regime and $\\mathfrak G_k\\gg1$ signals the scale-open regime.",
      "defining_relation": "\\mathfrak G_k=\\phi_kA_k^{\\vartheta_k};\\quad \\mathfrak G_k\\lesssim1\\ (\\text{closed}),\\ \\ \\mathfrak G_k\\gg1\\ (\\text{open})",
      "notes": "第35節推廣為雙變數版本$\\mathfrak G(k,\\phi,A)=\\phi A^{\\vartheta_k}$，並在第36節指出$\\mathfrak G_k\\ll1$反而不利倖存者。"
    },
    {
      "id": "ns.c3.c3x.analyticity_radius",
      "latex": "\\rho_{\\rm an}^{(k)}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "k階解析半徑",
      "label_en": "Analyticity radius (level $k$)",
      "definition_zh": "第24節引入，表示在選定時間切片與導數鏈分支中可用的空間解析半徑，屬外部harmonic-measure理論的量。",
      "definition_en": "Introduced in Section 24 as the available spatial analyticity radius on a selected time slice and derivative-chain branch, drawn from the external harmonic-measure framework.",
      "notes": "第33節指出其非純形式量，最新refined analyticity文獻持續改善其下界估計，但不能直接當任意奇異祖先的無條件下界。"
    },
    {
      "id": "ns.c3.c3x.analyticity_scale_ratio",
      "latex": "\\mathfrak A_k",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "解析尺度比",
      "label_en": "Analyticity-scale ratio",
      "definition_zh": "第24節定義，比較稀疏化尺度與解析半徑；當$\\mathfrak A_k\\lesssim1$且其餘介面條件對齊時，幾何正則性機制可被啟動。",
      "definition_en": "Defined in Section 24, this ratio compares the sparseness scale to the analyticity radius; when $\\mathfrak A_k\\lesssim1$ and the remaining interface hypotheses align, the geometric regularity mechanism can activate.",
      "defining_relation": "\\mathfrak A_k=\\frac{r_{\\rm sp}^{(k)}}{\\rho_{\\rm an}^{(k)}}"
    },
    {
      "id": "ns.c3.c3x.thm_analyticity_escape_necessity",
      "latex": "\\rho_{\\rm an}^{(k)}\\ll r_{\\rm sp}^{(k)}\\ \\text{(X-A1)}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.6：解析尺度逃逸必要性",
      "label_en": "C3-X.6: Analyticity-Scale Escape Necessity",
      "definition_zh": "第25節的條件式介面定理，要求假設性奇異祖先至少滿足三者之一：解析半徑塌縮（X-A1）、閾值/分量錯配（X-A2）或時間/鏈錯配（X-A3），合稱需支付「解析/介面逃逸債」。",
      "definition_en": "A conditional interface theorem in Section 25 requiring a hypothetical singular ancestry to satisfy at least one of three escape routes — analyticity-radius collapse (X-A1), threshold/component mismatch (X-A2), or time/chain mismatch (X-A3) — jointly termed the Analyticity/Interface Escape Debt.",
      "notes": "三條件之首X-A1給出唯一明確不等式$\\rho_{\\rm an}^{(k)}\\ll r_{\\rm sp}^{(k)}$；X-A2、X-A3為定性描述，無獨立公式。"
    },
    {
      "id": "ns.c3.c3x.nogo_pressure_rescue",
      "latex": "\\text{pressure concentration} \\not\\Rightarrow \\text{singularity rescue}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "No-Go 26.1：壓力集中無法解除幾何正則性",
      "label_en": "No-Go 26.1: Pressure Concentration Cannot Rescue Geometric Regularity",
      "definition_zh": "第26節的邏輯性結果，指出若某時間切片已完全滿足Grujić型幾何正則性判準的全部假設，壓力再怎麼集中也不能將其救回奇異。",
      "definition_en": "A logical result in Section 26 establishing that once a time slice fully satisfies a Grujić-type geometric regularity criterion's hypotheses, no degree of pressure concentration can rescue it back into singularity.",
      "notes": "此No-Go是第27節聯合倖存交集$\\mathcal S_{\\rm joint}$要求兩分支同時失敗的邏輯依據。"
    },
    {
      "id": "ns.c3.c3x.joint_survivor_intersection",
      "latex": "\\mathcal S_{\\rm joint}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "C3-X.7：聯合倖存交集",
      "label_en": "C3-X.7: Joint Survivor Intersection",
      "definition_zh": "第27節定義，證明假設性奇異點必須同時落在「臨界壓力一致可積性控制失效」與「解析性/稀疏化幾何封閉失效」兩集合的交集中，且此交集為平行必要條件而非已證蘊涵關係。",
      "definition_en": "Defined in Section 27, this establishes that a hypothetical singularity must lie in the intersection of \"critical pressure uniform-integrability control fails\" and \"analyticity/sparseness geometric closure fails,\" as a parallel necessary condition rather than a proven implication either way.",
      "defining_relation": "\\mathcal S_{\\rm pressure}=\\{\\text{critical pressure UI control fails}\\},\\ \\ \\mathcal S_{\\rm strain}=\\{\\text{analyticity/sparseness closure fails}\\},\\ \\ \\mathcal S_{\\rm joint}=\\mathcal S_{\\rm pressure}\\cap\\mathcal S_{\\rm strain}",
      "notes": "第42節結論以「Pressure Concentration ∩ Analyticity-Scale Escape」重述此交集，為全文標題級結論，並開啟下一輪C3-Y。"
    },
    {
      "id": "ns.c3.c3x.diversification_debt",
      "latex": "\\Theta_{P/S}\\to0",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "聯合集中多樣化債",
      "label_en": "Joint-Concentration Diversification Debt",
      "definition_zh": "第30節命名，指當$\\Theta_{P/S}\\to0$（分離情形）時，單一核心圖像必須拆成壓力載體與應變擾動載體至少兩種局部角色來承擔奇異性。",
      "definition_en": "Named in Section 30, this is the requirement that when $\\Theta_{P/S}\\to0$ (the segregated branch), the single-core picture must split into at least two local roles — a pressure carrier and a strain-fluctuation carrier — to sustain the hypothetical singularity.",
      "notes": "與第11節X-J2（Segregated）分支對應，說明分離本身不構成矛盾而是額外負擔。"
    },
    {
      "id": "ns.c3.c3x.true_etn_state",
      "latex": "\\Theta_R^{joint}",
      "series": "NS",
      "first_appearance": "C3-X",
      "label_zh": "True ETN聯合狀態向量",
      "label_en": "True ETN joint state vector",
      "definition_zh": "第38節將本輪結果寫入的狀態元組，把壓力狀態$\\Theta_R^{press}$、應變狀態$\\Theta_R^{strain}$與重疊係數$\\Theta_{P/S}$打包為聯合狀態$\\Theta_R^{joint}$。",
      "definition_en": "Section 38 packages this round's results into a state tuple, combining the pressure state $\\Theta_R^{press}$, strain state $\\Theta_R^{strain}$, and overlap coefficient $\\Theta_{P/S}$ into the joint state $\\Theta_R^{joint}$.",
      "notes": "隸屬跨輪持續追蹤的True ETN／無限維張力場物件（見文末Internal dependencies），非本輪獨有的一次性記號。"
    },
    {
      "id": "ns.c3.c3y.a_k",
      "latex": "A_k(s) = \\|D^k u(s)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "k階導數上確界範數",
      "label_en": "k-th derivative sup-norm",
      "definition_zh": "第1節在 $\\nu=1$ normalization（與 Grujić–Xu 一致）下定義 $A_k(s)=\\|D^ku(s)\\|_\\infty$，是全篇三個尺度與所有 threshold exponent 的共同基礎量。",
      "definition_en": "Section 1 defines $A_k(s)=\\|D^ku(s)\\|_\\infty$ under the $\\nu=1$ normalization matching Grujić--Xu, the base quantity underlying every scale and exponent in this round.",
      "defining_relation": "A_k(s) = \\|D^k u(s)\\|_\\infty"
    },
    {
      "id": "ns.c3.c3y.r_apr",
      "latex": "R_{\\rm apr}^{(k)} = C_{\\rm apr}(u_0,k)\\,A_k^{-1/(k+3/2)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "能量先驗尺度",
      "label_en": "energy a-priori scale",
      "definition_zh": "第2.1節由 Grujić–Xu Theorem 3.7 給出三維能量先驗尺度 $R_{\\rm apr}^{(k)}$，其常數主要由 $\\|u_0\\|_2$ 等固定資料決定。",
      "definition_en": "Section 2.1 gives the three-dimensional energy a-priori scale $R_{\\rm apr}^{(k)}$ from Grujić--Xu Theorem 3.7, with constants determined mainly by $\\|u_0\\|_2$ and other fixed data.",
      "defining_relation": "R_{\\rm apr}^{(k)} = C_{\\rm apr}(u_0,k)\\,A_k^{-1/(k+3/2)}",
      "notes": "三尺度中最粗，第3節證明 R_dir < R_chain < R_apr。"
    },
    {
      "id": "ns.c3.c3y.a_apr_exponent",
      "latex": "a_{\\rm apr}(k) = \\dfrac{1}{k+3/2}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "先驗尺度指數",
      "label_en": "a-priori scaling exponent",
      "definition_zh": "第2.1節定義 $R_{\\rm apr}^{(k)}$ 的 scaling exponent $a_{\\rm apr}(k)=1/(k+3/2)$，用於第3節的三尺度排序與第4節的兩個 scaling gap。",
      "definition_en": "Section 2.1 defines the scaling exponent $a_{\\rm apr}(k)=1/(k+3/2)$ of $R_{\\rm apr}^{(k)}$, used in Section 3's scale ordering and Section 4's two scaling gaps."
    },
    {
      "id": "ns.c3.c3y.r_dir",
      "latex": "R_{\\rm dir}^{(k)} = C_{\\rm dir}(k,M,u_0)\\,A_k^{-\\frac{3/2}{k+3/2}}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "直接有限k正則性尺度",
      "label_en": "direct finite-k regularity scale",
      "definition_zh": "第2.2節給出 Grujić–Xu Theorem 3.5 要求 component/sign superlevel set 需具備 local 1D sparseness 的尺度 $R_{\\rm dir}^{(k)}$。",
      "definition_en": "Section 2.2 gives the scale $R_{\\rm dir}^{(k)}$ at which Grujić--Xu Theorem 3.5 requires the component/sign superlevel set to have local 1D sparseness.",
      "defining_relation": "R_{\\rm dir}^{(k)} = C_{\\rm dir}(k,M,u_0)\\,A_k^{-\\frac{3/2}{k+3/2}}",
      "notes": "三尺度中最細，是 direct route 的目標尺度。"
    },
    {
      "id": "ns.c3.c3y.a_dir_exponent",
      "latex": "a_{\\rm dir}(k) = \\dfrac{3/2}{k+3/2}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "直接尺度指數",
      "label_en": "direct scaling exponent",
      "definition_zh": "第2.2節定義 $R_{\\rm dir}^{(k)}$ 的 scaling exponent $a_{\\rm dir}(k)=(3/2)/(k+3/2)$，是三個指數中最大者。",
      "definition_en": "Section 2.2 defines the scaling exponent $a_{\\rm dir}(k)=(3/2)/(k+3/2)$ of $R_{\\rm dir}^{(k)}$, the largest of the three exponents."
    },
    {
      "id": "ns.c3.c3y.r_chain",
      "latex": "R_{\\rm chain}^{(k)} = C_{\\rm chain}(\\ell,k,u_0)\\,A_k^{-1/(k+1)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "鏈輔助正則性尺度",
      "label_en": "chain-assisted regularity scale",
      "definition_zh": "第2.3節給出當 level $k$ 位於 admissible ascending derivative chain 且時間/常數條件成立時，Grujić–Xu Theorem 3.14 把目標尺度改善為 $R_{\\rm chain}^{(k)}$。",
      "definition_en": "Section 2.3 gives $R_{\\rm chain}^{(k)}$, the improved target scale Grujić--Xu Theorem 3.14 provides when level $k$ lies on an admissible ascending derivative chain and the time/constant hypotheses hold.",
      "defining_relation": "R_{\\rm chain}^{(k)} = C_{\\rm chain}(\\ell,k,u_0)\\,A_k^{-1/(k+1)}",
      "notes": "介於 R_dir 與 R_apr 之間，是 derivative-chain dynamics 把 required scale 拉回的中間尺度。"
    },
    {
      "id": "ns.c3.c3y.a_chain_exponent",
      "latex": "a_{\\rm chain}(k) = \\dfrac{1}{k+1}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "鏈尺度指數",
      "label_en": "chain scaling exponent",
      "definition_zh": "第2.3節定義 $R_{\\rm chain}^{(k)}$ 的 scaling exponent $a_{\\rm chain}(k)=1/(k+1)$，介於 $a_{\\rm apr}$ 與 $a_{\\rm dir}$ 之間。",
      "definition_en": "Section 2.3 defines the scaling exponent $a_{\\rm chain}(k)=1/(k+1)$ of $R_{\\rm chain}^{(k)}$, lying strictly between $a_{\\rm apr}$ and $a_{\\rm dir}$."
    },
    {
      "id": "ns.c3.c3y.delta_a_dir",
      "latex": "\\Delta a_{\\rm dir} = a_{\\rm dir}-a_{\\rm apr} = \\dfrac{1}{2(k+3/2)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "直接尺度差",
      "label_en": "direct scaling gap",
      "definition_zh": "第4節（C3-Y.1）定義 $\\Delta a_{\\rm dir}=a_{\\rm dir}-a_{\\rm apr}=1/(2(k+3/2))$，量化從先驗尺度推進到直接尺度所需的指數差距。",
      "definition_en": "Section 4 (C3-Y.1) defines $\\Delta a_{\\rm dir}=a_{\\rm dir}-a_{\\rm apr}=1/(2(k+3/2))$, quantifying the exponent gap needed to push from the a-priori scale to the direct scale."
    },
    {
      "id": "ns.c3.c3y.delta_a_chain",
      "latex": "\\Delta a_{\\rm chain} = a_{\\rm chain}-a_{\\rm apr} = \\dfrac{1}{2(k+1)(k+3/2)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "鏈尺度差",
      "label_en": "chain scaling gap",
      "definition_zh": "第4節（C3-Y.1）定義 $\\Delta a_{\\rm chain}=a_{\\rm chain}-a_{\\rm apr}=1/(2(k+1)(k+3/2))$，並得出 $\\Delta a_{\\rm dir}/\\Delta a_{\\rm chain}=k+1$，是本輪第一個核心 identity。",
      "definition_en": "Section 4 (C3-Y.1) defines $\\Delta a_{\\rm chain}=a_{\\rm chain}-a_{\\rm apr}=1/(2(k+1)(k+3/2))$ and shows $\\Delta a_{\\rm dir}/\\Delta a_{\\rm chain}=k+1$, the round's first core identity."
    },
    {
      "id": "ns.c3.c3y.omega_kc",
      "latex": "\\Omega_{k,c}(s) = \\{x : |D^k u(x,s)| > cA_k(s)\\}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "量級高值集",
      "label_en": "magnitude high set",
      "definition_zh": "第5節對固定 $0<c<1$ 定義 magnitude high set $\\Omega_{k,c}(s)$，為建立 uniformly-local intermittency condition 的基礎集合。",
      "definition_en": "Section 5 defines the magnitude high set $\\Omega_{k,c}(s)$ for a fixed constant $0<c<1$, the underlying set used to build the uniformly-local intermittency condition.",
      "defining_relation": "\\Omega_{k,c}(s) = \\{x : |D^k u(x,s)| > cA_k(s)\\}",
      "notes": "第6節證明所有 component/sign superlevel set S_{k,λ}^{i,±}（當 c≤λ）皆為其子集。"
    },
    {
      "id": "ns.c3.c3y.phi_kc",
      "latex": "\\Phi_{k,c}(s;R) = \\sup_{x_0\\in\\mathbb R^3}\\dfrac{|\\Omega_{k,c}(s)\\cap B_R(x_0)|}{|B_R|}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "均勻局部活躍體積因子",
      "label_en": "uniform-local active-volume factor",
      "definition_zh": "第5節定義 $\\Phi_{k,c}(s;R)$ 為對所有空間點取 sup 的均勻局部活躍體積因子，第20節指出這才是真正 theorem-ready 的量，而非 ancestry-local 的 $\\phi_{p,R}$。",
      "definition_en": "Section 5 defines $\\Phi_{k,c}(s;R)$ as the uniform-local active-volume factor taken as a sup over all spatial points; Section 20 identifies this, not the ancestry-local $\\phi_{p,R}$, as the true theorem-ready quantity.",
      "defining_relation": "\\Phi_{k,c}(s;R) = \\sup_{x_0\\in\\mathbb R^3}\\dfrac{|\\Omega_{k,c}(s)\\cap B_R(x_0)|}{|B_R|}",
      "notes": "與 ancestry-local φ_{p,R}（第19節）形成本輪核心對比；兩者間的 upper bound 仍是開放問題，留給 C3-Z 的 Z1。"
    },
    {
      "id": "ns.c3.c3y.s_klambda",
      "latex": "S_{k,\\lambda}^{i,\\pm} = \\{x : (D^k u)_i^\\pm(x) > \\lambda A_k\\}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "分量/正負號超水平集",
      "label_en": "component/sign superlevel set",
      "definition_zh": "第6節引入 Grujić–Xu geometric criteria 所用的 component/sign superlevel set $S_{k,\\lambda}^{i,\\pm}$，當 $c\\le\\lambda$ 時它是 $\\Omega_{k,c}$ 的子集。",
      "definition_en": "Section 6 introduces the component/sign superlevel set $S_{k,\\lambda}^{i,\\pm}$ used by the Grujić--Xu geometric criteria, a subset of $\\Omega_{k,c}$ whenever $c\\le\\lambda$."
    },
    {
      "id": "ns.c3.c3y.r_vol",
      "latex": "r_{\\rm vol} = C_\\delta\\,\\Phi^{1/3}\\,R",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "體積轉一維尺度",
      "label_en": "volume-to-line transfer scale",
      "definition_zh": "第7節由 C3-W 的 volume-to-one-dimensional-sparseness lemma 導出尺度 $r_{\\rm vol}=C_\\delta\\Phi^{1/3}R$，在此尺度上 $S_{k,\\lambda}^{i,\\pm}$ 於某方向具 1D $\\delta$-sparse 性質。",
      "definition_en": "Section 7 derives, via the C3-W volume-to-one-dimensional-sparseness lemma, the scale $r_{\\rm vol}=C_\\delta\\Phi^{1/3}R$ at which $S_{k,\\lambda}^{i,\\pm}$ is 1D $\\delta$-sparse along some direction.",
      "notes": "承接自 C3-W 引理；第8、10節分別以 r_vol ≤ R_dir^(k) 或 R_chain^(k) 推出 θ_k^dir、θ_k^chain。"
    },
    {
      "id": "ns.c3.c3y.theta_k_dir",
      "latex": "\\theta_k^{\\rm dir} = 3\\Delta a_{\\rm dir} = \\dfrac{3}{2(k+3/2)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "直接體積門檻指數",
      "label_en": "direct volume threshold exponent",
      "definition_zh": "第8節導出使 $r_{\\rm vol}\\le R_{\\rm dir}^{(k)}$ 所需的活躍體積門檻指數 $\\theta_k^{\\rm dir}=3/(2(k+3/2))$，是真正 theorem-ready 的 finite-$k$ direct threshold。",
      "definition_en": "Section 8 derives the active-volume threshold exponent $\\theta_k^{\\rm dir}=3/(2(k+3/2))$ needed for $r_{\\rm vol}\\le R_{\\rm dir}^{(k)}$, the genuine theorem-ready finite-$k$ direct threshold.",
      "defining_relation": "\\Phi \\le C_k^{\\rm dir}A_k^{-\\theta_k^{\\rm dir}},\\qquad \\theta_k^{\\rm dir}=3\\Delta a_{\\rm dir}=\\dfrac{3}{2(k+3/2)}",
      "notes": "第14節給出大 k 漸近行為 θ_k^dir ~ 3/(2k) = O(k^-1)；第16節於 k=2 給出數值 3/7。"
    },
    {
      "id": "ns.c3.c3y.direct_bridge_theorem",
      "latex": "\\text{Theorem 9.1 (C3-Y.2)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "直接間歇性橋接定理",
      "label_en": "Direct Intermittency Bridge Theorem",
      "definition_zh": "第9節的定理9.1（C3-Y.2）給出條件式應用模板：若在 Grujić–Xu Theorem 3.5 全部 hypotheses 下 $\\Phi_{k,c}(s;R_{\\rm apr}^{(k)})\\le C_k^{\\rm dir}A_k(s)^{-\\theta_k^{\\rm dir}}$ 成立，則 $T_\\ast$ 不是奇異時間。",
      "definition_en": "Theorem 9.1 (C3-Y.2) in Section 9 is a conditional application template: under the full hypotheses of Grujić--Xu Theorem 3.5, if $\\Phi_{k,c}(s;R_{\\rm apr}^{(k)})\\le C_k^{\\rm dir}A_k(s)^{-\\theta_k^{\\rm dir}}$ holds, then $T_\\ast$ is not a singular time.",
      "notes": "不需要 derivative-chain hypothesis，是本輪唯一不涉及 chain gate 的 closure 定理。"
    },
    {
      "id": "ns.c3.c3y.theta_k_chain",
      "latex": "\\theta_k^{\\rm chain} = 3\\Delta a_{\\rm chain} = \\dfrac{3}{2(k+1)(k+3/2)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "鏈輔助體積門檻指數",
      "label_en": "chain-assisted volume threshold exponent",
      "definition_zh": "第10節導出使 $r_{\\rm vol}\\le R_{\\rm chain}^{(k)}$ 所需的活躍體積門檻指數 $\\theta_k^{\\rm chain}=3/(2(k+1)(k+3/2))$，須配合 derivative-chain gate 方能使用。",
      "definition_en": "Section 10 derives the active-volume threshold exponent $\\theta_k^{\\rm chain}=3/(2(k+1)(k+3/2))$ needed for $r_{\\rm vol}\\le R_{\\rm chain}^{(k)}$, usable only in conjunction with the derivative-chain gate.",
      "defining_relation": "\\Phi \\le C_{\\ell,k}^{\\rm chain}A_k^{-\\theta_k^{\\rm chain}},\\qquad \\theta_k^{\\rm chain}=3\\Delta a_{\\rm chain}=\\dfrac{3}{2(k+1)(k+3/2)}",
      "notes": "第14節給出大 k 漸近行為 θ_k^chain ~ 3/(2k^2) = O(k^-2)；第17節於 k=2 給出數值 1/7，即 C3-X 原 φ_2 ≲ A_2^{-1/7} 的正式身分。"
    },
    {
      "id": "ns.c3.c3y.chain_bridge_theorem",
      "latex": "\\text{Theorem 11.1 (C3-Y.3)}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "鏈輔助間歇性橋接定理",
      "label_en": "Chain-Assisted Intermittency Bridge Theorem",
      "definition_zh": "第11節的定理11.1（C3-Y.3）給出條件式應用模板：在 Grujić–Xu Theorem 3.14 全部 hypotheses（含 ascending-chain condition）下，若 $\\Phi_{k,c}(s;R_{\\rm apr}^{(k)})\\le C_{\\ell,k}^{\\rm chain}A_k(s)^{-\\theta_k^{\\rm chain}}$ 成立，則 $T_\\ast$ 不是奇異時間。",
      "definition_en": "Theorem 11.1 (C3-Y.3) in Section 11 is a conditional application template: under the full hypotheses of Grujić--Xu Theorem 3.14 (including the ascending-chain condition), if $\\Phi_{k,c}(s;R_{\\rm apr}^{(k)})\\le C_{\\ell,k}^{\\rm chain}A_k(s)^{-\\theta_k^{\\rm chain}}$ holds, then $T_\\ast$ is not a singular time."
    },
    {
      "id": "ns.c3.c3y.ascending_chain_condition",
      "latex": "\\dfrac{\\|D^ju(t)\\|_\\infty^{1/(j+1)}}{c^{j/(j+1)}(j!)^{1/(j+1)}} \\le \\dfrac{\\|D^ku(t)\\|_\\infty^{1/(k+1)}}{c^{k/(k+1)}(k!)^{1/(k+1)}},\\quad \\ell\\le j\\le k",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "上升鏈條件",
      "label_en": "ascending-chain condition",
      "definition_zh": "第11節定理11.1的 hypothesis 2 引用 Grujić–Xu Theorem 3.8 的 ascending-chain condition，要求 factorial-normalized derivative magnitude 在 $\\ell\\le j\\le k$ 範圍內由 level $k$ 主導所有較低 level。",
      "definition_en": "Hypothesis 2 of Theorem 11.1 in Section 11 cites the ascending-chain condition from Grujić--Xu Theorem 3.8, requiring the factorial-normalized derivative magnitude at level $k$ to dominate all lower levels $\\ell\\le j\\le k$.",
      "notes": "第39節將此與 Theorem 3.8（ascending）、Theorem 3.9（descending）並列並聲明本輪不重新證明這些外部定理；第40節 guard G-CHAINLOAD 要求必須保存 (3.8)、(3.9) 兩式。"
    },
    {
      "id": "ns.c3.c3y.tradeoff_identity",
      "latex": "\\dfrac{\\theta_k^{\\rm dir}}{\\theta_k^{\\rm chain}} = k+1",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "導數鏈／間歇性權衡恆等式",
      "label_en": "Derivative-Chain / Intermittency Tradeoff Identity",
      "definition_zh": "第12節（C3-Y.4）證明 $\\theta_k^{\\rm dir}/\\theta_k^{\\rm chain}=k+1$，即 derivative-chain dynamics 把空間間歇性所需的 power-law exponent 負擔精確降低 $k+1$ 倍，是本輪最主要的新 structural result。",
      "definition_en": "Section 12 (C3-Y.4) proves $\\theta_k^{\\rm dir}/\\theta_k^{\\rm chain}=k+1$, showing derivative-chain dynamics reduces the power-law exponent burden of spatial intermittency by exactly a factor of $k+1$ -- the main new structural result of this round.",
      "defining_relation": "\\dfrac{\\theta_k^{\\rm dir}}{\\theta_k^{\\rm chain}} = k+1",
      "notes": "與第4節 Δa_dir/Δa_chain = k+1 為同一比值（因 θ=3Δa）；第18節於 k=2 給出具體版本比值 3。"
    },
    {
      "id": "ns.c3.c3y.k2_status_correction",
      "latex": "\\phi_2\\lesssim A_2^{-1/7} \\;=\\; \\theta_2^{\\rm chain}\\text{-scaling},\\qquad \\theta_2^{\\rm dir}=\\dfrac37 \\ne \\theta_2^{\\rm chain}=\\dfrac17",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "k=2結果之定位修正",
      "label_en": "k=2 exponent status correction",
      "definition_zh": "第0節與第15–18、35節將 C3-X 的 $\\phi_2\\lesssim A_2^{-1/7}$ 正式修正定位為 chain-assisted scale-gap closure exponent（即 $\\theta_2^{\\rm chain}=1/7$）而非 standalone $k=2$ regularity theorem，真正 theorem-ready 的 direct 門檻是 $\\theta_2^{\\rm dir}=3/7$。",
      "definition_en": "Section 0 and Sections 15--18, 35 formally reclassify C3-X's $\\phi_2\\lesssim A_2^{-1/7}$ as the chain-assisted scale-gap closure exponent ($\\theta_2^{\\rm chain}=1/7$) rather than a standalone $k=2$ regularity theorem, with the genuine theorem-ready direct threshold being $\\theta_2^{\\rm dir}=3/7$.",
      "notes": "NG-Y1（第43節）明確陳述 φ_2 ≲ A_2^{-1/7} ⇒ regularity 為 FALSE in general；差值 3/7-1/7=2/7 即 derivative-chain dynamics 補上的部分（第35節）。"
    },
    {
      "id": "ns.c3.c3y.ancestry_local_phi",
      "latex": "\\phi_{p,R}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "祖系局部有效體積因子",
      "label_en": "ancestry-local effective-volume factor",
      "definition_zh": "第19節指出 $\\phi_{p,R}$ 承接自 C3-W/X，是單一 ancestry core 的 local effective-volume factor，與本輪均勻局部的 $\\Phi_{k,c}$ 形成對比。",
      "definition_en": "Section 19 identifies $\\phi_{p,R}$, carried over from C3-W/X, as the local effective-volume factor of a single ancestry core, contrasted in this round with the uniform-local $\\Phi_{k,c}$.",
      "notes": "No-Go 19.1：φ_{p,R}(x_ancestry) ≪ 1 不推出 Φ_{k,c} ≪ 1，是第20節 Local Ancestry / Global Criterion Separation 的直接依據。"
    },
    {
      "id": "ns.c3.c3y.local_global_separation",
      "latex": "\\text{Local Ancestry / Global Criterion Separation}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "局部祖系與全域判準分離",
      "label_en": "Local Ancestry / Global Criterion Separation",
      "definition_zh": "第20節命名此原則：derivative/intermittency bridge 真正需要追蹤的是均勻局部量 $\\Phi_{k,c}(s;R_{\\rm apr})$，而非單一 ancestry core 的 $\\phi_{p,R}(x_n)$。",
      "definition_en": "Section 20 names this principle: the derivative/intermittency bridge must track the uniform-local quantity $\\Phi_{k,c}(s;R_{\\rm apr})$, not the single ancestry core's $\\phi_{p,R}(x_n)$.",
      "notes": "第21節列出三條未完成的 globalization route（Y-G1單一主導cluster、Y-G2 multi-core cover、Y-G3 localized geometric theorem），留待 C3-Z 的 Z1、Z2 處理。"
    },
    {
      "id": "ns.c3.c3y.temporal_gate",
      "latex": "s-t \\asymp A_k(t)^{-\\frac{3}{k+3/2}}\\ (\\text{direct}),\\qquad s-t \\asymp A_k(t)^{-\\frac{2}{k+1}}\\ (\\text{chain})",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "時間門檻排程",
      "label_en": "temporal gate schematic",
      "definition_zh": "第23節指出 direct 與 chain theorem 皆非同時刻判準，需 escape time $t$ 之後某 later time $s=s(t)$ 落在指定 analytic window，並給出兩者 $s-t$ 的 schematic scaling。",
      "definition_en": "Section 23 notes that neither the direct nor chain theorem is a same-time criterion -- both require a later time $s=s(t)$ after the escape time $t$ to land in a specified analytic window, and gives the schematic scaling of $s-t$ for each route.",
      "notes": "第24節（Time-gate no-go）指出 escape time 的小 Φ_k(t) 不保證 later slice 的 Φ_k(s(t)) 仍小，spatial gap closure 與 temporal gate closure 是兩個不同 proof obligations。"
    },
    {
      "id": "ns.c3.c3y.load_dir",
      "latex": "\\mathfrak{L}_k^{\\rm dir} = \\dfrac{\\Phi_{k,c}(s;R_{\\rm apr})}{C_k^{\\rm dir}A_k(s)^{-\\theta_k^{\\rm dir}}}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "直接封閉負載",
      "label_en": "direct closure load",
      "definition_zh": "第25節定義 direct closure load $\\mathfrak{L}_k^{\\rm dir}$，當其 $\\le1$ 且 Theorem 3.5 全部時間/分量 hypotheses 成立時可推出 regularity extension。",
      "definition_en": "Section 25 defines the direct closure load $\\mathfrak{L}_k^{\\rm dir}$; when it is $\\le1$ together with the full time/component hypotheses of Theorem 3.5, regularity extension follows.",
      "defining_relation": "\\mathfrak{L}_k^{\\rm dir} = \\dfrac{\\Phi_{k,c}(s;R_{\\rm apr})}{C_k^{\\rm dir}A_k(s)^{-\\theta_k^{\\rm dir}}}"
    },
    {
      "id": "ns.c3.c3y.load_chain",
      "latex": "\\mathfrak{L}_{\\ell,k}^{\\rm chain} = \\dfrac{\\Phi_{k,c}(s;R_{\\rm apr})}{C_{\\ell,k}^{\\rm chain}A_k(s)^{-\\theta_k^{\\rm chain}}}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "鏈輔助封閉負載",
      "label_en": "chain closure load",
      "definition_zh": "第26節定義 chain closure load $\\mathfrak{L}_{\\ell,k}^{\\rm chain}$，當其 $\\le1$ 且 ascending-chain condition、time gate 與 theorem 常數條件皆成立時可推出 Theorem 3.14 closure。",
      "definition_en": "Section 26 defines the chain closure load $\\mathfrak{L}_{\\ell,k}^{\\rm chain}$; when it is $\\le1$ together with the ascending-chain condition, time gate, and theorem constant conditions, Theorem 3.14 closure follows.",
      "defining_relation": "\\mathfrak{L}_{\\ell,k}^{\\rm chain} = \\dfrac{\\Phi_{k,c}(s;R_{\\rm apr})}{C_{\\ell,k}^{\\rm chain}A_k(s)^{-\\theta_k^{\\rm chain}}}"
    },
    {
      "id": "ns.c3.c3y.load_best",
      "latex": "\\mathfrak{L}_k^{best} = \\min\\{\\mathfrak{L}_k^{\\rm dir},\\ \\mathfrak{L}_{\\ell,k}^{\\rm chain}\\text{ if admissible}\\}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "最佳路由負載",
      "label_en": "adaptive routing score",
      "definition_zh": "第27節定義 formal routing score $\\mathfrak{L}_k^{best}$，取 direct 與（若 chain gate 開啟）chain load 之最小值；若存在 admissible $k$ 使其 $\\le1$，regularity route 即封閉。",
      "definition_en": "Section 27 defines the formal routing score $\\mathfrak{L}_k^{best}$ as the minimum of the direct load and (if the chain gate is open) the chain load; if some admissible $k$ achieves $\\mathfrak{L}_k^{best}\\le1$, the regularity route closes.",
      "defining_relation": "\\mathfrak{L}_k^{best} = \\min\\{\\mathfrak{L}_k^{\\rm dir},\\ \\mathfrak{L}_{\\ell,k}^{\\rm chain}\\text{ if admissible}\\}",
      "notes": "第38節強調最佳 derivative level 不能只比較指數 θ_k，必須比較 𝔏_k^best 本身，因高階導數與 theorem 常數也可能快速增長。"
    },
    {
      "id": "ns.c3.c3y.joint_routing_principle",
      "latex": "\\textbf{Pressure Concentration}\\cap\\textbf{Derivative-Bridge Failure at Every Admissible Gate}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "聯合集中路由原則",
      "label_en": "Joint-Concentration Routing Principle",
      "definition_zh": "第32節（C3-Y.5）證明若 pressure concentration branch 成立，則 derivative side 必須對每個 theorem-admissible $k$ 同時發生 Y-J1至Y-J5 中至少一項失敗，假設性奇異性才能維持。",
      "definition_en": "Section 32 (C3-Y.5) proves that if the pressure concentration branch holds, the derivative side must simultaneously fail via at least one of Y-J1 through Y-J5 at every theorem-admissible $k$ for a hypothetical singularity to survive.",
      "notes": "第42節重述為本輪後的 main survivor；五個子失敗模式為 Y-J1（𝔏_k^best>1）、Y-J2（globalization failure）、Y-J3（temporal gate failure）、Y-J4（chain gate failure）、Y-J5（threshold/interface failure）。"
    },
    {
      "id": "ns.c3.c3y.theta_ps",
      "latex": "\\Theta_{P/S}",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "壓力／應變共位參數",
      "label_en": "pressure-strain co-location parameter",
      "definition_zh": "第33節回顧 C3-X 曾區分 $\\Theta_{P/S}>0$ 與 $\\Theta_{P/S}\\to0$ 兩種情形，並指出即使 pressure 與 strain active sets 完全空間分離，只要 derivative geometric theorem 的 global hypotheses 閉合，regularity 仍成立。",
      "definition_en": "Section 33 recalls C3-X's distinction between $\\Theta_{P/S}>0$ and $\\Theta_{P/S}\\to0$, and shows regularity still holds even when pressure and strain active sets are fully spatially segregated, provided the derivative geometric theorem's global hypotheses close.",
      "notes": "承接自 C3-X；本輪結論是 co-location 並非最本質的邏輯介面，真正本質是 pressure singular branch 交 derivative-geometry theorem failure。"
    },
    {
      "id": "ns.c3.c3y.theta_k_di",
      "latex": "\\Theta_k^{DI} = \\left\\langle A_k, R_{\\rm apr}^{(k)}, R_{\\rm chain}^{(k)}, R_{\\rm dir}^{(k)}, \\Phi_k, \\theta_k^{\\rm dir}, \\theta_k^{\\rm chain}, \\mathsf{ChainGate}_k, \\mathsf{TimeGate}_k, \\mathfrak{L}_k^{best} \\right\\rangle",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "導數／間歇性狀態",
      "label_en": "derivative/intermittency state tuple",
      "definition_zh": "第41節在 True ETN 架構下把本輪所有量打包成狀態元組 $\\Theta_k^{DI}$，供全系列狀態簿記使用。",
      "definition_en": "Section 41 packages this round's quantities into the state tuple $\\Theta_k^{DI}$ within the True ETN bookkeeping framework used across the series."
    },
    {
      "id": "ns.c3.c3y.theta_k_joint",
      "latex": "\\Theta_k^{joint} = \\left\\langle \\Theta_k^{DI}, \\text{pressure concentration certificate}, \\text{pressure provenance} \\right\\rangle",
      "series": "NS",
      "first_appearance": "C3-Y",
      "label_zh": "聯合壓力狀態",
      "label_en": "joint pressure state tuple",
      "definition_zh": "第41節把 $\\Theta_k^{DI}$ 擴充為聯合壓力狀態 $\\Theta_k^{joint}$，額外納入 pressure concentration certificate 與 pressure provenance，對應第32節的 Joint-Concentration Routing。",
      "definition_en": "Section 41 extends $\\Theta_k^{DI}$ into the joint pressure state $\\Theta_k^{joint}$, adding a pressure concentration certificate and pressure provenance, corresponding to Section 32's Joint-Concentration Routing.",
      "notes": "延續 True ETN 記法，是全系列狀態簿記的最新一層。"
    },
    {
      "id": "ns.c4.c4a.ancestry_sequence",
      "latex": "\\Gamma_n=(t_n,x_n,R_n,q_n,\\sigma_n)",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "祖先事件序列",
      "label_en": "Ancestry event sequence",
      "definition_zh": "第4節定義的候選 blow-up 祖先序列，記錄時間、位置、尺度、頻率殼層與手性符號五個座標。",
      "definition_en": "A candidate blow-up ancestry sequence introduced in §4, recording time, location, scale, frequency shell, and chirality sign.",
      "defining_relation": "\\Gamma_n=(t_n,x_n,R_n,q_n,\\sigma_n),\\quad t_n\\uparrow T_\\ast,\\quad R_n\\downarrow0,\\quad R_n\\asymp\\lambda_{q_n}^{-1}"
    },
    {
      "id": "ns.c4.c4a.viscous_time_window",
      "latex": "I_n=\\left[t_n-\\theta\\dfrac{R_n^2}{\\nu},\\,t_n\\right]",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "黏性尺度時間窗",
      "label_en": "Viscous-scale time window",
      "definition_zh": "第4節以 parabolic／黏性尺度 $R_n^2/\\nu$ 為長度定義的時間窗，作為各 channel 是否「同時活躍」的共同比較基準。",
      "definition_en": "The parabolic (viscous-scale) time window of length $\\sim R_n^2/\\nu$ defined in §4, serving as the common clock against which channel activity is compared.",
      "defining_relation": "I_n=\\left[t_n-\\theta\\dfrac{R_n^2}{\\nu},\\,t_n\\right]"
    },
    {
      "id": "ns.c4.c4a.ancestry_core_ball",
      "latex": "B_n=B(x_n,cR_n)",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "祖先核心球",
      "label_en": "Ancestry core ball",
      "definition_zh": "第4節定義以 $x_n$ 為心、半徑正比於 $R_n$ 的空間球，作為 carrier label 中「core」的參考區域。",
      "definition_en": "The spatial ball centered at $x_n$ with radius proportional to $R_n$, defined in §4 as the reference region for the carrier label \"core\"."
    },
    {
      "id": "ns.c4.c4a.phase_space_event_window",
      "latex": "W_n=I_n\\times B_n\\times[q_n-C,q_n+C]\\times\\{\\pm\\}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "相空間事件窗",
      "label_en": "Phase-space event window",
      "definition_zh": "第4節將時間窗、核心球、頻率殼層鄰域與手性符號合併成的相空間窗，提供所有 channel 共同的 ancestry 標記。",
      "definition_en": "The phase-space window combining the time window, core ball, frequency-shell neighborhood, and chirality sign, defined in §4 to give every channel a common ancestry tag.",
      "defining_relation": "W_n=I_n\\times B_n\\times[q_n-C,q_n+C]\\times\\{\\pm\\}"
    },
    {
      "id": "ns.c4.c4a.unified_survivor_state",
      "latex": "\\mathfrak S_n=\\left\\langle\\Gamma_n,\\mathbf L_n,\\mathbf C_n,\\mathbf G_n,\\mathbf D_n\\right\\rangle",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "統一倖存者狀態",
      "label_en": "Unified Survivor State",
      "definition_zh": "第5節定義的核心物件，把 ancestry、load、carrier、gate、defect 五個向量打包成單一狀態，是全篇 state-transition 架構的基本單位。",
      "definition_en": "The central object of §5, packaging the ancestry tuple with the load, carrier, gate, and defect vectors into a single state — the basic unit of the paper's state-transition architecture.",
      "defining_relation": "\\mathfrak S_n=\\left\\langle\\Gamma_n,\\mathbf L_n,\\mathbf C_n,\\mathbf G_n,\\mathbf D_n\\right\\rangle"
    },
    {
      "id": "ns.c4.c4a.load_vector",
      "latex": "\\mathbf L_n=\\left(L_n^{UV},L_n^{Hel},L_n^{Str},L_n^{Op},L_n^{Pr},L_n^{Der}\\right)",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "負載向量",
      "label_en": "Load vector",
      "definition_zh": "第6節定義的六分量向量，分別記錄 UV、螺旋度、應變、算子、壓力、導數六個 survivor channel 各自的臨界負載。",
      "definition_en": "The six-component vector defined in §6, recording the critical load carried by each of the UV, helicity, strain, operator, pressure, and derivative survivor channels."
    },
    {
      "id": "ns.c4.c4a.miller_operator_defect",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "Miller 應變—渦度算子缺陷",
      "label_en": "Miller strain-vorticity operator defect",
      "definition_zh": "第2節（E4）引自 Miller 的算子，量化 full Navier–Stokes 與其正則 strain–vorticity 交互模型之間的偏差，並在第6節作為 operator load 的核心量。",
      "definition_en": "The operator cited from Miller in §2 (E4), quantifying the defect between full Navier–Stokes and its regular strain-vorticity interaction model, used in §6 as the core quantity of the operator load.",
      "notes": "External operator carried in from Miller's paper (arXiv:2407.02691); reused as the Operator load component of $\\mathbf L_n$ and referenced again in the §27 cross-channel coupling table."
    },
    {
      "id": "ns.c4.c4a.carrier_label",
      "latex": "C_n^a\\in\\{\\mathrm{core},\\mathrm{near},\\mathrm{far},\\mathrm{exterior},\\mathrm{async},\\mathrm{unknown}\\}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "承載標籤",
      "label_en": "Carrier label",
      "definition_zh": "第7節為每個 channel $a$ 定義的六值標籤，標示該 channel 的負載主要在哪種空間／時間關係中被觀測到或支付。",
      "definition_en": "The six-valued label defined in §7 for each channel $a$, indicating where — core, near, far, exterior, async, or unknown — its load is observed or paid."
    },
    {
      "id": "ns.c4.c4a.gate_vector",
      "latex": "\\mathbf G_n=\\left(G_n^{FC},G_n^{Adj},G_n^{PUI},G_n^{Miller},G_n^{Mid},G_n^{Dir},G_n^{Chain}\\right)",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "閘門向量",
      "label_en": "Gate vector",
      "definition_zh": "第8節定義的七分量向量，每個分量對應一個已知 sufficient regularity 判準是否真正關閉；依約定 $G=0$ 表示閘門仍開，$G=1$ 表示已閉合並迫使 singular chain 終止。",
      "definition_en": "The seven-component vector defined in §8, each entry tracking whether a known sufficient-regularity criterion has actually closed, with $G=0$ meaning the gate is still open and $G=1$ meaning it has closed and forces the singular chain to terminate.",
      "defining_relation": "\\mathbf G_n=\\left(G_n^{FC},G_n^{Adj},G_n^{PUI},G_n^{Miller},G_n^{Mid},G_n^{Dir},G_n^{Chain}\\right),\\qquad G=0\\ (\\text{open}),\\quad G=1\\ (\\text{closed})"
    },
    {
      "id": "ns.c4.c4a.defect_vector",
      "latex": "\\mathbf D_n=\\left(D_n^{IR},D_n^{UV},D_n^{Sp},D_n^{Tm},D_n^{Pr},D_n^{Op},D_n^{Fl}\\right)",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "缺陷向量",
      "label_en": "Defect vector",
      "definition_zh": "第9節定義的七分量向量，收納任何未被 local state 吸收、必須保留而不得憑空消失的 channel debt。",
      "definition_en": "The seven-component vector defined in §9 that absorbs any channel debt not captured by the local state, so that the debt is preserved rather than allowed to vanish."
    },
    {
      "id": "ns.c4.c4a.debt_preservation_identity",
      "latex": "\\mu_a=\\mu_a|_{W_n}+\\mu_a|_{W_n^c}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "債務保存恆等式（No-Deletion Rule）",
      "label_en": "Debt preservation identity (No-Deletion Rule)",
      "definition_zh": "第26節給出的精確分割恆等式，呼應第9節的 source-preservation rule，禁止把 local window 外未觀測到的 channel 負載直接當作零。",
      "definition_en": "The exact splitting identity of §26, echoing the source-preservation rule of §9, which forbids treating a channel's load outside the local window as if it were zero.",
      "defining_relation": "\\mu_a=\\mu_a|_{W_n}+\\mu_a|_{W_n^c},\\qquad \\mu_a(W_n)<\\tau\\ \\Rightarrow\\ \\text{missing debt}=\\mu_a(W_n^c)"
    },
    {
      "id": "ns.c4.c4a.mandatory_channel_set",
      "latex": "\\mathcal M\\subset\\{UV,Hel,Str,Op,Pr,Der\\}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "必要 channel 集合",
      "label_en": "Mandatory channel set",
      "definition_zh": "第10節為特定 branch 固定的一組必須被檢驗同步的 channel 子集，是定義 active-time set 與後續同步準則的前提。",
      "definition_en": "The subset of channels fixed as mandatory for a given branch in §10, the premise against which active-time sets and later synchronization criteria are defined."
    },
    {
      "id": "ns.c4.c4a.active_time_set",
      "latex": "E_{a,n}=\\{t\\in I_n: L_a(t;\\Gamma_n)\\ge\\tau_a\\}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "活躍時間集",
      "label_en": "Active-time set",
      "definition_zh": "第10節定義的集合，記錄 channel $a$ 的負載在時間窗 $I_n$ 內超過門檻 $\\tau_a$ 的所有時刻，是整個同步理論的基本觀測量。",
      "definition_en": "The set defined in §10 of times within $I_n$ at which channel $a$'s load exceeds threshold $\\tau_a$ — the basic observable underlying the entire synchronization theory.",
      "defining_relation": "E_{a,n}=\\left\\{t\\in I_n: L_a(t;\\Gamma_n)\\ge\\tau_a\\right\\}"
    },
    {
      "id": "ns.c4.c4a.strong_synchronization",
      "latex": "\\bigcap_{a\\in\\mathcal M}E_{a,n}\\ne\\varnothing",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "強同步",
      "label_en": "Strong synchronization",
      "definition_zh": "第11節定義的狀態：mandatory channels 的 active-time set 交集非空，且交集中存在某刻使所有 channel 的 carrier 皆為 core 或 near，才合法宣稱單一聯合倖存事件。",
      "definition_en": "The condition defined in §11 — the mandatory channels' active-time sets intersect nonemptily, and at some point in that intersection every channel's carrier is core or near — that legally licenses the claim of one joint survivor event.",
      "defining_relation": "\\bigcap_{a\\in\\mathcal M}E_{a,n}\\ne\\varnothing\\ \\text{ and }\\ \\exists\\,s_n\\in\\bigcap_aE_{a,n}:\\ \\text{all carriers}\\in\\{\\mathrm{core},\\mathrm{near}\\}"
    },
    {
      "id": "ns.c4.c4a.synchronization_hierarchy",
      "latex": "\\text{Sync-0}\\prec\\text{Sync-1}\\prec\\text{Sync-2}\\prec\\text{Sync-3}\\prec\\text{Sync-4}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "同步層級",
      "label_en": "Synchronization hierarchy",
      "definition_zh": "第12節劃分的五層架構（Marginal、Temporal、Scale、Spatial、Causal），把「同步」從單純邊際必要條件收斂為可合法轉移的因果狀態，Sync-4 為 C4 的終極目標。",
      "definition_en": "The five-level hierarchy (Marginal, Temporal, Scale, Spatial, Causal) laid out in §12, sharpening \"synchronization\" from mere marginal necessity into a causally transitionable state, with Sync-4 as C4's ultimate target."
    },
    {
      "id": "ns.c4.c4a.marginal_divergence_nogo",
      "latex": "\\int f=\\int g=\\infty\\ \\not\\Rightarrow\\ \\{f>0\\}\\cap\\{g>0\\}\\ne\\varnothing",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "邊際發散同步不可行定理（C4-A.1）",
      "label_en": "Marginal Divergence Synchronization No-Go (C4-A.1)",
      "definition_zh": "第13節的定理13.1，以錯開的 disjoint time windows 顯式構造兩個積分皆發散卻乘積恆零的密度，證明邊際發散並不蘊含 pointwise 交集。",
      "definition_en": "Theorem 13.1 in §13, which explicitly constructs two densities on staggered disjoint time windows whose integrals both diverge yet whose product vanishes identically, proving marginal divergence does not imply pointwise intersection.",
      "defining_relation": "f=\\tfrac{2^n}{n}1_{I_n^L},\\ g=\\tfrac{2^n}{n}1_{I_n^R};\\quad \\int f=\\int g=\\infty,\\quad fg\\equiv0",
      "notes": "§14 extends this to m channels staggered into m disjoint sub-blocks per window, showing finite-time divergent tolls can be perfectly staggered for any finite channel family."
    },
    {
      "id": "ns.c4.c4a.g_sync_hard_guard",
      "latex": "\\text{G-SYNC}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "G-SYNC 硬性防護",
      "label_en": "G-SYNC hard guard",
      "definition_zh": "第15節依定理13.1設立的硬性規則，禁止直接由「channel A 發散＋channel B 發散」推出「A 與 B 同時偏大」，任何交集宣稱都須額外提供 persistence／overlap／heredity／turnover 之一。",
      "definition_en": "The hard rule set in §15 on the strength of Theorem 13.1, forbidding the inference \"channel A diverges + channel B diverges ⇒ A and B are simultaneously large\" unless persistence, overlap, heredity, or turnover is separately supplied.",
      "defining_relation": "\\text{Channel A diverges}+\\text{Channel B diverges}\\ \\boxed{\\not\\Rightarrow}\\ \\text{A and B simultaneously large}",
      "notes": "Reprised in §40 as one of eight named guards (G-SYNC, G-DEBT, G-CARRIER, G-TIME, G-SPATIAL, G-HERED, G-TERM, G-REC) forming the interface contract for X-Integration."
    },
    {
      "id": "ns.c4.c4a.persistence_to_sync_lemma",
      "latex": "\\left|\\bigcap_{a=1}^mE_a\\right|\\ge\\left(1-\\sum_{a=1}^m\\varepsilon_a\\right)|I|",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "持續性至同步引理（C4-A.2）",
      "label_en": "Persistence-to-Synchronization Lemma (C4-A.2)",
      "definition_zh": "第16節定理16.1，以 union bound 證明若每個 channel 的非活躍測度比例不超過 $\\varepsilon_a$，則所有 channel 活躍集的交集測度至少為 $(1-\\sum\\varepsilon_a)|I|$。",
      "definition_en": "Theorem 16.1 in §16, proved via a union bound, showing that if each channel's inactive-measure fraction is at most $\\varepsilon_a$, the intersection of all active sets has measure at least $(1-\\sum\\varepsilon_a)|I|$.",
      "defining_relation": "|I\\setminus E_a|\\le\\varepsilon_a|I|\\ \\Longrightarrow\\ \\left|\\bigcap_{a=1}^mE_a\\right|\\ge\\left(1-\\sum_{a=1}^m\\varepsilon_a\\right)|I|"
    },
    {
      "id": "ns.c4.c4a.temporal_desync_debt",
      "latex": "\\sum_{a=1}^m\\varepsilon_a\\ge1",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "時間去同步債務（C4-A.3）",
      "label_en": "Temporal Desynchronization Debt (C4-A.3)",
      "definition_zh": "第18節由定理16.1的逆否命題得出：若 mandatory channels 的活躍集交集為空，則各 channel 非活躍比例 $\\varepsilon_a$ 之和必須至少為1，代表必須支付相當份量的 channel switching。",
      "definition_en": "Derived in §18 as the contrapositive of Theorem 16.1 — if the mandatory channels' active sets have empty intersection, their inactive fractions $\\varepsilon_a$ must sum to at least 1, forcing a substantial amount of channel switching to be paid.",
      "defining_relation": "\\bigcap_{a=1}^mE_a=\\varnothing\\ \\Longrightarrow\\ \\sum_{a=1}^m\\varepsilon_a\\ge1,\\qquad \\varepsilon_a=\\frac{|I\\setminus E_a|}{|I|}"
    },
    {
      "id": "ns.c4.c4a.recurrent_desynchronizer_lemma",
      "latex": "\\exists\\,a_\\ast:\\ \\varepsilon_{a_\\ast,n_j}\\ge\\tfrac1m\\ \\text{along a subsequence }n_j",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "復發去同步者引理（C4-A.4）",
      "label_en": "Recurrent Desynchronizer Lemma (C4-A.4)",
      "definition_zh": "第19節由鴿籠原理證明：若每個窗口都未達成完全同步，則有限 channel 族中必存在某一 channel $a_\\ast$，在無窮多子序列窗口上反覆貢獻至少 $1/m$ 的去同步比例。",
      "definition_en": "Proved by a pigeonhole argument in §19: if no window achieves full synchronization, some channel $a_\\ast$ among the finite family must repeatedly contribute a desynchronization fraction of at least $1/m$ along an infinite subsequence of windows.",
      "defining_relation": "\\sum_{a=1}^m\\varepsilon_{a,n}\\ge1\\ \\Longrightarrow\\ \\exists\\,a_\\ast,\\{n_j\\}:\\ \\varepsilon_{a_\\ast,n_j}\\ge\\tfrac1m",
      "notes": "Introduces the recurrently desynchronizing channel $a_\\ast$, which §20 reframes as a turnover-cost question carried forward as the announced topic of C4-B."
    },
    {
      "id": "ns.c4.c4a.spatial_carrier_synchronization",
      "latex": "X_{a,n}(t),\\qquad \\operatorname{dist}(X_{a,n}(t),x_n)\\lesssim R_n",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "空間承載同步",
      "label_en": "Spatial carrier synchronization",
      "definition_zh": "第21節定義 channel $a$ 的承載區域函數 $X_{a,n}(t)$，並規定強 Sync-3 須使所有 mandatory channel 的承載區域與 ancestry 中心 $x_n$ 保持 $O(R_n)$ 距離內，否則債務須標記進 $D_n^{Sp},D_n^{Pr},D_n^{Op}$。",
      "definition_en": "§21 defines each channel's carrier-region function $X_{a,n}(t)$ and requires, for strong Sync-3, that every mandatory channel's carrier stay within $O(R_n)$ of the ancestry center $x_n$, failing which the debt must be flagged into $D_n^{Sp}, D_n^{Pr}, D_n^{Op}$.",
      "defining_relation": "\\operatorname{dist}(X_{a,n}(t),x_n)\\lesssim R_n\\quad\\text{for all mandatory }a"
    },
    {
      "id": "ns.c4.c4a.legal_transition",
      "latex": "\\mathfrak S_n\\rightsquigarrow\\mathfrak S_{n+1}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "合法奇異轉移",
      "label_en": "Legal singular transition",
      "definition_zh": "第23節定義的合法狀態轉移，須同時滿足 T1–T8 八條規則，涵蓋時序、尺度逃逸、因果父證、缺陷保存、規範保存、壓力與算子溯源、以及閘門終止。",
      "definition_en": "The legal state transition defined in §23, required to jointly satisfy eight conditions T1–T8 covering time ordering, scale escape, the causal parent certificate, defect preservation, gauge preservation, pressure and operator provenance, and gate termination."
    },
    {
      "id": "ns.c4.c4a.singular_survivor_chain",
      "latex": "\\mathfrak S_0\\rightsquigarrow\\mathfrak S_1\\rightsquigarrow\\mathfrak S_2\\rightsquigarrow\\cdots",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "C4 奇異倖存者鏈",
      "label_en": "C4 Singular Survivor Chain",
      "definition_zh": "第24節定義的無窮合法轉移序列，要求 $t_n\\uparrow T_\\ast$、$R_n\\downarrow0$ 且所有 regularity gate 永不關閉；此序列是否存在即為 C4 的終極問題。",
      "definition_en": "The infinite legal-transition sequence defined in §24, requiring $t_n\\uparrow T_\\ast$, $R_n\\downarrow0$, and that no regularity gate ever closes; whether such a chain exists is C4's ultimate question."
    },
    {
      "id": "ns.c4.c4a.asynchronous_survivor_bundle",
      "latex": "\\mathfrak B_n=\\{\\mathfrak S_n^{UV},\\mathfrak S_n^{Str},\\mathfrak S_n^{Op},\\mathfrak S_n^{Pr},\\mathfrak S_n^{Der}\\}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "非同步倖存者束",
      "label_en": "Asynchronous Survivor Bundle",
      "definition_zh": "第25節在尚未證明同步前定義的較弱物件，僅要求各 channel 分量共享世代標籤與大致 blow-up 時間，不假設同時、同心或同一因果分支。",
      "definition_en": "The weaker object defined in §25 for use before synchronization is established, requiring only that each channel's component share a generation tag and approximate blow-up time, without assuming the same time, center, or causal branch."
    },
    {
      "id": "ns.c4.c4a.finite_recurrent_gate_failure_reduction",
      "latex": "\\mathcal F=\\{F_1,\\ldots,F_M\\}\\ \\Longrightarrow\\ \\exists F_\\ast\\ \\text{recurrent}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "有限復發閘門失敗化約（C4-A.5）",
      "label_en": "Finite Recurrent Gate-Failure Reduction (C4-A.5)",
      "definition_zh": "第30節以鴿籠原理證明，若每個奇異轉移都須使有限 gate 族 $\\mathcal F$ 中至少一個失敗，則任何無窮奇異鏈必存在某個 $F_\\ast$ 在無窮子序列上反覆失敗，使 C4 只需追蹤單一復發障礙類。",
      "definition_en": "§30 proves by pigeonhole that if every singular transition must fail at least one gate from the finite family $\\mathcal F$, an infinite singular chain must contain some $F_\\ast$ that fails recurrently along a subsequence, letting C4 track just one recurrent obstruction class.",
      "notes": "§31 catalogues six candidate recurrent-failure branches RF-1 through RF-6: synchronization failure, spatial carrier separation, pressure concentration escape, derivative globalization failure, chain-gate failure, and mean/pointwise fluctuation failure."
    },
    {
      "id": "ns.c4.c4a.temporal_sync_deficit",
      "latex": "\\Delta_{\\rm sync}(I)=\\sum_{a=1}^m\\frac{|I\\setminus E_a|}{|I|}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "時間同步赤字",
      "label_en": "Temporal synchronization deficit",
      "definition_zh": "第35節定義（並於第42節以窗口索引版本 $\\Delta_{\\rm sync,n}$ 重述）的量化門檻：$\\Delta_{\\rm sync}<1$ 保證存在聯合同步時刻，唯有 $\\Delta_{\\rm sync}\\ge1$ 才允許 route 保持非同步。",
      "definition_en": "The quantitative threshold defined in §35 (restated windowed as $\\Delta_{\\rm sync,n}$ in §42): $\\Delta_{\\rm sync}<1$ guarantees a joint synchronized instant exists, while only $\\Delta_{\\rm sync}\\ge1$ permits the route to remain asynchronous.",
      "defining_relation": "\\Delta_{\\rm sync}(I)=\\sum_{a=1}^m\\frac{|I\\setminus E_a|}{|I|},\\qquad \\Delta_{\\rm sync}<1\\iff\\text{full temporal synchronization exists}"
    },
    {
      "id": "ns.c4.c4a.spatial_sync_deficit",
      "latex": "\\Delta_{\\rm sp,n}=\\max_{a,b\\in\\mathcal M}\\frac{|x_{a,n}-x_{b,n}|}{R_n}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "空間同步赤字",
      "label_en": "Spatial synchronization deficit",
      "definition_zh": "第36節定義的量，以 carrier 中心點兩兩距離除以 $R_n$ 衡量空間分散程度；$O(1)$ 可併入 bounded rescaled cluster，發散則形成空間非同步的 survivor bundle。",
      "definition_en": "The quantity defined in §36 measuring spatial spread as the pairwise carrier-center distance normalized by $R_n$; an $O(1)$ value allows merging into a bounded rescaled cluster, while divergence produces a spatially asynchronous survivor bundle."
    },
    {
      "id": "ns.c4.c4a.causal_sync_deficit",
      "latex": "\\Delta_{\\rm her,n}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "因果同步赤字",
      "label_en": "Causal synchronization deficit",
      "definition_zh": "第37節定義的量，彙總 parent→child 轉移中壓力效率、平均應變方向、相位效率、算子承載與導數閘門等正規化轉移缺陷，目前尚無統一有限預算，是 C4 的終極缺口。",
      "definition_en": "The quantity defined in §37 aggregating the normalized parent-to-child transition defect across pressure efficiency, mean-strain direction, phase efficiency, operator carrier, and derivative gate; it currently has no unified finite budget and remains C4's ultimate gap."
    },
    {
      "id": "ns.c4.c4a.unified_closure_principle",
      "latex": "\\text{UCP-1}\\ \\text{through}\\ \\text{UCP-7}",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "C4 統一閉合原則",
      "label_en": "C4 Unified Closure Principle",
      "definition_zh": "第38節列出的七條原則，陳述任何假設性無窮奇異鏈必須同時滿足的結構性後果；UCP-1至UCP-6已建立為精確結果，UCP-7仍是下一前沿。",
      "definition_en": "The seven principles listed in §38 stating the structural consequences any hypothetical infinite singular chain must simultaneously satisfy; UCP-1 through UCP-6 are established as exact results, while UCP-7 remains the next frontier."
    },
    {
      "id": "ns.c4.c4a.etn_c4",
      "latex": "\\mathfrak T^{C4}=\\left(\\mathfrak S_n,\\mathfrak S_{n+1},\\operatorname{Transition},\\operatorname{DebtFlow},\\operatorname{GateStatus}\\right)",
      "series": "NS",
      "first_appearance": "C4-A",
      "label_zh": "C4 版擴展張力網絡",
      "label_en": "C4-version Extended Tension Network (ETN)",
      "definition_zh": "第39節把 True ETN 從單一時間切片的張力向量重新詮釋為涵蓋轉移、債務流與閘門狀態的五元組，核心問題由「哪個張力最大」轉為所有必要張力能否在狀態轉移中合法共同傳遞。",
      "definition_en": "§39 reinterprets the True ETN from a single-time-slice tension vector into a five-tuple spanning transition, debt flow, and gate status, shifting the central question from \"which tension is largest\" to whether all mandatory tensions can be jointly and legally carried through the state transition.",
      "notes": "Reuses the \"True ETN / 無限維張力場\" object listed under Internal dependencies from prior rounds, reframed here for the C4 state-transition setting."
    },
    {
      "id": "ns.c4.c4b.asynchronous_survivor_bundle",
      "latex": "\\textbf{Asynchronous Survivor Bundle}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "非同步倖存者叢",
      "label_en": "Asynchronous Survivor Bundle",
      "definition_zh": "C4-A 建立、C4-B 第0節重述的物件：mandatory channels 逐窗 inactive fractions 之和小於 1 時共同活動集合非空，否則其和必須不小於 1。",
      "definition_en": "The object recalled in C4-B Section 0 from C4-A: mandatory channels whose per-window inactive fractions sum below 1 have a nonempty common-active set, and otherwise the sum must be at least 1.",
      "notes": "直接繼承自 NS_C4A_UnifiedSurvivorState_SynchronizationClosure_v0.1.md；C4-B 全篇檢驗這個叢集能否被 generic turnover cost 逼出 temporal synchronization。"
    },
    {
      "id": "ns.c4.c4b.inactive_fraction",
      "latex": "\\varepsilon_{a,n}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "通道不活躍比例",
      "label_en": "channel inactive fraction",
      "definition_zh": "第0節定義為 $|I_n\\setminus E_{a,n}|/|I_n|$，用以量化 mandatory channel $a$ 在 viscous window $I_n$ 中未活動的比例。",
      "definition_en": "Defined in Section 0 as $|I_n\\setminus E_{a,n}|/|I_n|$, measuring the fraction of viscous window $I_n$ during which mandatory channel $a$ is inactive."
    },
    {
      "id": "ns.c4.c4b.recurrent_desynchronizer",
      "latex": "\\textbf{recurrent desynchronizer}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "週期性去同步者",
      "label_en": "recurrent desynchronizer",
      "definition_zh": "C4-A 保證有限 channel family 中至少存在一個的物件，第0節將其重新定義為本輪核心待決問題：它能否在不超出 C3 turnover budgets 下無限次關閉／重啟。",
      "definition_en": "An object guaranteed to exist among a finite channel family by C4-A; Section 0 reframes it as this round's central open question — whether it can switch off and on infinitely often without exceeding the C3 turnover budgets.",
      "notes": "本輪結論：generic turnover budgets 無法排除它的存在，見第23、38節。"
    },
    {
      "id": "ns.c4.c4b.qsv_strain_vorticity",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "Miller應變渦度交互量",
      "label_en": "Miller strain-vorticity interaction quantity",
      "definition_zh": "第1.2節引入，指 Miller operator-level regularity 模型中的應變–渦度交互量，其在所有 late viscous windows 持續偏大並非該模型的必然結論。",
      "definition_en": "Introduced in Section 1.2 as the strain-vorticity interaction quantity from Miller's operator-level regularity model, whose persistent largeness across all late viscous windows is not entailed by that model.",
      "notes": "於第40節 C3 proof obligation 中重新出現，作為 C4-C 待證的 lower-bound 目標。"
    },
    {
      "id": "ns.c4.c4b.channel_density",
      "latex": "F(t)",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "通道密度",
      "label_en": "channel density",
      "definition_zh": "第2節引入的非負可測函數，代表某個 channel 在區間 $I$ 上隨時間變化的活動強度。",
      "definition_en": "A nonnegative measurable function introduced in Section 2 representing a channel's time-varying activity intensity over an interval $I$."
    },
    {
      "id": "ns.c4.c4b.threshold_and_peak_capacity",
      "latex": "0\\le\\theta<M,\\quad M=\\operatorname*{ess\\,sup}_{t\\in I}F(t)",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "活動門檻與峰值容量",
      "label_en": "activity threshold and peak capacity",
      "definition_zh": "第2節定義：$\\theta$ 為判定活動的門檻（$0\\le\\theta<M$），$M$ 為 $F$ 在 $I$ 上的本質上界，即峰值容量。",
      "definition_en": "Defined in Section 2: $\\theta$ is the activity-determining threshold ($0\\le\\theta<M$) and $M$ is the essential supremum of $F$ on $I$, i.e. its peak capacity."
    },
    {
      "id": "ns.c4.c4b.active_set_theta",
      "latex": "E_\\theta",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "門檻活動集",
      "label_en": "threshold active set",
      "definition_zh": "第2節定義為 $\\{t\\in I: F(t)\\ge\\theta\\}$，即通道密度達到門檻 $\\theta$ 以上的時刻集合。",
      "definition_en": "Defined in Section 2 as $\\{t\\in I: F(t)\\ge\\theta\\}$, the set of times at which the channel density reaches or exceeds the threshold $\\theta$.",
      "defining_relation": "E_\\theta=\\{t\\in I: F(t)\\ge\\theta\\}"
    },
    {
      "id": "ns.c4.c4b.integrated_toll",
      "latex": "T",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "積分代價",
      "label_en": "integrated toll",
      "definition_zh": "第2節定義為 $T=\\int_IF(t)\\,dt$，即通道密度在整個窗口上的總積分。",
      "definition_en": "Defined in Section 2 as $T=\\int_IF(t)\\,dt$, the total integral of the channel density over the whole window."
    },
    {
      "id": "ns.c4.c4b.pulse_to_persistence_lemma",
      "latex": "\\textbf{Pulse-to-Persistence Lemma}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "脈衝–持續性引理",
      "label_en": "Pulse-to-Persistence Lemma",
      "definition_zh": "第3節定理3.1：當 $T>\\theta|I|$ 時，門檻活動集滿足 $|E_\\theta|\\ge(T-\\theta|I|)/(M-\\theta)$，是本輪最核心的新證定理。",
      "definition_en": "Theorem 3.1 in Section 3: whenever $T>\\theta|I|$, the threshold active set satisfies $|E_\\theta|\\ge(T-\\theta|I|)/(M-\\theta)$ — this round's single most central proved result.",
      "defining_relation": "|E_\\theta|\\ \\ge\\ \\frac{T-\\theta|I|}{M-\\theta}\\qquad(T>\\theta|I|)",
      "notes": "第6節以此引理解釋 C4-A 中 $\\int f=\\infty$、$\\int g=\\infty$ 但 $fg=0$ 構造的機制：peak amplitude 隨 generation 增大讓 divergent integral 由 vanishing duty cycle 支付。"
    },
    {
      "id": "ns.c4.c4b.duty_cycle_average_load",
      "latex": "d_\\theta,\\ \\bar F",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "占空比與平均負載",
      "label_en": "duty cycle and average load",
      "definition_zh": "第4節將 Pulse-to-Persistence Lemma 改寫為占空比形式：$d_\\theta=|E_\\theta|/|I|$、平均負載 $\\bar F=T/|I|$，得下界 $d_\\theta\\ge(\\bar F-\\theta)/(M-\\theta)$。",
      "definition_en": "Section 4 recasts the Pulse-to-Persistence Lemma in duty-cycle form via $d_\\theta=|E_\\theta|/|I|$ and average load $\\bar F=T/|I|$, giving the bound $d_\\theta\\ge(\\bar F-\\theta)/(M-\\theta)$.",
      "defining_relation": "d_\\theta=\\frac{|E_\\theta|}{|I|},\\quad \\bar F=\\frac{T}{|I|}\\quad\\Longrightarrow\\quad d_\\theta\\ge\\frac{\\bar F-\\theta}{M-\\theta}"
    },
    {
      "id": "ns.c4.c4b.pulse_capacity_escape",
      "latex": "\\textbf{Pulse-Capacity Escape}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "脈衝容量逃逸",
      "label_en": "Pulse-Capacity Escape",
      "definition_zh": "第5節命名的機制：當峰值容量 $M_n\\to\\infty$ 而平均負載 $\\bar F_n$ 增長較慢時，占空比下界趨於零，使 channel 得以振幅更高、占空比更低來逃避 persistence。",
      "definition_en": "A mechanism named in Section 5: when peak capacity $M_n\\to\\infty$ while average load $\\bar F_n$ grows more slowly, the duty-cycle lower bound tends to zero, letting a channel escape persistence via higher amplitude and lower duty cycle.",
      "notes": "本輪四大 structural escape 之一（見第0、38、42節總結）。"
    },
    {
      "id": "ns.c4.c4b.synchronization_duty_threshold",
      "latex": "\\sum_{a=1}^{m}d_a>m-1",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "同步占空比門檻",
      "label_en": "synchronization duty threshold",
      "definition_zh": "第7節導出：$m$ 個 mandatory channels 的共同活動集合非空，等價於其占空比之和跨過 $m-1$，而目前理論未證此門檻可達。",
      "definition_en": "Derived in Section 7: a nonempty common-active set for $m$ mandatory channels is equivalent to their duty cycles summing past $m-1$, a threshold currently unproved.",
      "defining_relation": "\\bigcap_aE_a\\ne\\varnothing \\iff \\sum_{a=1}^{m}d_a>m-1"
    },
    {
      "id": "ns.c4.c4b.hysteresis_thresholds",
      "latex": "\\alpha<\\beta",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "遲滯門檻對",
      "label_en": "hysteresis thresholds",
      "definition_zh": "第8節引入的一對門檻，供連續 scalar observable $z(t)$ 定義完整 upcrossing $z\\le\\alpha\\to z\\ge\\beta$，每次至少耗費變差 $\\beta-\\alpha$。",
      "definition_en": "A pair of thresholds introduced in Section 8 defining a complete upcrossing $z\\le\\alpha\\to z\\ge\\beta$ for a continuous scalar observable $z(t)$, each costing at least variation $\\beta-\\alpha$."
    },
    {
      "id": "ns.c4.c4b.finite_variation_switching_lemma",
      "latex": "\\textbf{Finite-Variation Switching Lemma}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "有限變差切換引理",
      "label_en": "Finite-Variation Switching Lemma",
      "definition_zh": "第9節定理9.1：若 $N$ 個 disjoint windows 中 observable $z$ 各完成一次 $\\alpha\\to\\beta$ upcrossing，則總變差滿足 $\\operatorname{Var}(z)\\ge N(\\beta-\\alpha)$，故有限變差只允許有限次固定間距切換。",
      "definition_en": "Theorem 9.1 in Section 9: if $z$ completes one $\\alpha\\to\\beta$ upcrossing in each of $N$ disjoint windows, its total variation satisfies $\\operatorname{Var}(z)\\ge N(\\beta-\\alpha)$, so finite variation permits only finitely many fixed-gap switches.",
      "defining_relation": "\\operatorname{Var}_{\\cup I_n}(z)\\ \\ge\\ N(\\beta-\\alpha)",
      "notes": "第10-12節指出此引理雖能限制單一固定 carrier 的切換次數，卻被第12節的 Carrier Relay 構造繞過。"
    },
    {
      "id": "ns.c4.c4b.channel_type_vs_carrier_identity",
      "latex": "\\text{channel type}\\ \\ne\\ \\text{carrier identity}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "通道類型與載體身分",
      "label_en": "channel type vs. carrier identity",
      "definition_zh": "第11節提出的關鍵區分：「通道類型」（如 UV 或 pressure）僅標示種類，「載體身分」（如 $(q,\\sigma,x,\\text{packet})$ 或特定 pressure core）才標示具體實體，同一通道類型反覆出現不代表同一載體反覆切換。",
      "definition_en": "A key distinction from Section 11: \"channel type\" (e.g. UV or pressure) names only a category, while \"carrier identity\" (e.g. $(q,\\sigma,x,\\text{packet})$ or a specific pressure core) names the concrete entity — a recurring channel type does not imply the same carrier is repeatedly switching.",
      "notes": "是 Carrier Relay 構造（第12節）與 $G_{\\rm RELAY}$ hard guard（第27節）的概念基礎。"
    },
    {
      "id": "ns.c4.c4b.carrier_relay",
      "latex": "\\textbf{Carrier Relay}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "載體接力",
      "label_en": "Carrier Relay",
      "definition_zh": "第12節（C4-B.3）構造：每個 disjoint window 建立一個只用一次的新載體 $z_n(t)$ 完成單次固定間距脈衝，使 channel type 每代都活動，卻使任何 per-carrier 有限變差論證失效。",
      "definition_en": "The Section 12 (C4-B.3) construction: each disjoint window gets a fresh, single-use carrier $z_n(t)$ completing one fixed-gap pulse, so the channel type is active every generation while defeating any per-carrier finite-variation argument.",
      "notes": "本輪四大 structural escape 之一；第13-14節以 UV shell 為例，證明它在既有 C3-K weighted hysteretic count 中確實存活。"
    },
    {
      "id": "ns.c4.c4b.weighted_hysteretic_count",
      "latex": "N_{q,\\sigma}^{up}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "加權遲滯上穿次數",
      "label_en": "weighted hysteretic up-crossing count",
      "definition_zh": "第14節回顧 C3-K 已證的 $\\sum_{q,\\sigma}(\\lambda_q/L_q)N_{q,\\sigma}^{up}<\\infty$，因高頻權重 $\\lambda_q/L_q\\sim\\lambda_q^{-2}$ 遞減，容許每個新 shell 只需 $N_q^{up}=1$，使 carrier relay 得以存活於此既有界限之中。",
      "definition_en": "Section 14 recalls C3-K's proved bound $\\sum_{q,\\sigma}(\\lambda_q/L_q)N_{q,\\sigma}^{up}<\\infty$; since the high-frequency weight $\\lambda_q/L_q\\sim\\lambda_q^{-2}$ decays, allowing $N_q^{up}=1$ per fresh shell, carrier relay survives inside this existing global count.",
      "defining_relation": "\\sum_{q,\\sigma}\\frac{\\lambda_q}{L_q}N_{q,\\sigma}^{up}<\\infty",
      "notes": "第13節以 $q_1<q_2<q_3<\\cdots$、每 shell 只啟動一次的構造具體展示 UV carrier relay，並引用 C3-J 的 fixed-shell hysteresis rigidity 作對照。"
    },
    {
      "id": "ns.c4.c4b.active_generations_inter_gen_routing",
      "latex": "\\mathcal N_a",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "活動世代集／世代間路由",
      "label_en": "active generations / Inter-Generation Routing",
      "definition_zh": "第15節定義 $\\mathcal N_a\\subset\\mathbb N$ 為通道 $a$ 活動的世代集合，並舉例顯示兩通道皆無限常返但交集為空，即世代間路由。",
      "definition_en": "Section 15 defines $\\mathcal N_a\\subset\\mathbb N$ as channel $a$'s set of active generations, exhibiting an example where both channels recur infinitely often yet their intersection is empty — Inter-Generation Routing.",
      "defining_relation": "\\mathcal N_A=\\{2,4,6,\\ldots\\},\\ \\ \\mathcal N_B=\\{1,3,5,\\ldots\\}\\ \\ \\Rightarrow\\ \\ |\\mathcal N_A|=|\\mathcal N_B|=\\infty\\ \\text{but}\\ \\mathcal N_A\\cap\\mathcal N_B=\\varnothing",
      "notes": "第16節稱此為 Generation Desynchronization No-Go：逐通道無限常返不推出共同世代無限多，比 C4-A 的 intra-window asynchrony 更強。"
    },
    {
      "id": "ns.c4.c4b.generation_block_persistence",
      "latex": "B_N,\\ \\delta_a",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "世代區塊持續性",
      "label_en": "generation block persistence",
      "definition_zh": "第17節取區塊 $B_N=\\{N,\\ldots,N+L-1\\}$、缺席比例 $\\delta_a$ 滿足 $\\#(B_N\\setminus\\mathcal N_a)\\le\\delta_aL$ 時得共同世代下界，第18節稱要跨過此界須有 cofinite／高區塊密度常返，即 Generation Persistence Debt。",
      "definition_en": "Section 17 takes block $B_N=\\{N,\\ldots,N+L-1\\}$ with miss fraction $\\delta_a$ satisfying $\\#(B_N\\setminus\\mathcal N_a)\\le\\delta_aL$ to bound common generations, and Section 18 names the resulting requirement — cofinite or high block-density recurrence — the Generation Persistence Debt.",
      "defining_relation": "B_N=\\{N,\\ldots,N+L-1\\},\\quad \\#(B_N\\setminus\\mathcal N_a)\\le\\delta_aL\\ \\Longrightarrow\\ \\#\\left(B_N\\cap\\bigcap_a\\mathcal N_a\\right)\\ge L\\left(1-\\sum_a\\delta_a\\right)"
    },
    {
      "id": "ns.c4.c4b.geometric_ancestry_radius",
      "latex": "R_n",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "幾何祖源半徑",
      "label_en": "geometric ancestry radius",
      "definition_zh": "第21節指出在幾何祖源 $R_n=R_0\\rho^n$（$0<\\rho<1$）下，任意 $\\alpha>0$ 均有 $\\sum_nR_n^\\alpha<\\infty$，解釋了為何一般有限預算只控制 $R_n^\\alpha\\times$事件代價卻仍允許每代 $O(1)$ 切換。",
      "definition_en": "Section 21 shows that under geometric ancestry $R_n=R_0\\rho^n$ ($0<\\rho<1$), $\\sum_nR_n^\\alpha<\\infty$ holds for every $\\alpha>0$, explaining why generic finite budgets that only control $R_n^\\alpha\\times$event-cost still permit an $O(1)$ switch per generation.",
      "defining_relation": "R_n=R_0\\rho^n\\ (0<\\rho<1)\\ \\Longrightarrow\\ \\sum_nR_n^\\alpha<\\infty\\ \\ (\\forall\\alpha>0)",
      "notes": "承自 C3 rounds 的核心遞減幾何量，是第19-25節 Summable-Weight Barrier 的具體實現，並貫穿第41-42節總結。"
    },
    {
      "id": "ns.c4.c4b.summable_weight_budget",
      "latex": "w_n,\\ C_n,\\ B",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "可加總權重預算",
      "label_en": "summable-weight budget",
      "definition_zh": "第19節引入一般化有限預算形式：正權重 $w_n>0$ 與非負代價 $C_n\\ge0$ 滿足 $\\sum_nw_nC_n\\le B$，涵蓋 C3 已證各種 turnover budgets 的共同骨架。",
      "definition_en": "Section 19 introduces the general finite-budget form: positive weights $w_n>0$ and nonnegative costs $C_n\\ge0$ with $\\sum_nw_nC_n\\le B$, the common skeleton underlying the various turnover budgets already proved in C3.",
      "defining_relation": "\\sum_nw_nC_n\\le B,\\quad w_n>0,\\ C_n\\ge0"
    },
    {
      "id": "ns.c4.c4b.summable_weight_nogo",
      "latex": "\\textbf{Summable-Weight No-Go}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "可加總權重不可行定理",
      "label_en": "Summable-Weight No-Go",
      "definition_zh": "第20節定理20.1：若 $\\sum_nw_n<\\infty$，則有限預算 $\\sum_nw_nC_n<\\infty$ 無法排除 $C_n\\ge c_0>0$ 對所有 $n$ 成立，證明只需取 $C_n$ 為常數 $c_0$。",
      "definition_en": "Theorem 20.1 in Section 20: if $\\sum_nw_n<\\infty$, a finite budget $\\sum_nw_nC_n<\\infty$ cannot rule out $C_n\\ge c_0>0$ for all $n$, proved simply by taking $C_n$ constant at $c_0$.",
      "defining_relation": "\\sum_nw_n<\\infty,\\quad C_n\\equiv c_0>0\\ \\Longrightarrow\\ \\sum_nw_nC_n=c_0\\sum_nw_n<\\infty",
      "notes": "第21節證明幾何祖源 $R_n=R_0\\rho^n$ 正是此不可行定理最典型的 friendly 案例。"
    },
    {
      "id": "ns.c4.c4b.synchronization_subcritical_budgets",
      "latex": "\\textbf{Synchronization-Subcritical Budgets}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "同步次臨界預算",
      "label_en": "Synchronization-Subcritical Budgets",
      "definition_zh": "第23節（C4-B.5）之裁決：目前所有已證、可跨世代加總的 C3 finite turnover/occupancy budgets，其世代權重在幾何祖源下皆可加總，故都不能單獨排除每代一個 $O(1)$ 切換/旋轉/激活事件。",
      "definition_en": "The verdict of Section 23 (C4-B.5): every proved, cross-generation-summable C3 finite turnover/occupancy budget has a generation weight that is summable under geometric ancestry, so none alone can rule out one $O(1)$ switching/rotation/activation event per generation.",
      "notes": "整合了第22節逐一審查的五類 C3 budgets：absolute active-shell worldvolume、quadratic mean-strain turnover、pressure mean rotation、fixed-shell hysteresis、cone-degeneration pressure debt。"
    },
    {
      "id": "ns.c4.c4b.finite_critical_budget_dichotomy",
      "latex": "\\text{Type F},\\ \\text{Type C}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "有限／臨界預算二分法",
      "label_en": "Finite-Budget/Critical-Budget Dichotomy",
      "definition_zh": "第25節（C4-B.6）將 C3 scalar budgets 分為 Type F（scale-weighted 有限，不足以禁止 $C_n=O(1)$）與 Type C（unweighted critical 發散，是 blow-up 必然性但同樣無法給矛盾），指出目前不存在 finite unweighted positive switching budget。",
      "definition_en": "Section 25 (C4-B.6) splits C3 scalar budgets into Type F (finite but scale-weighted, insufficient to forbid $C_n=O(1)$) and Type C (unweighted critical divergence, a blow-up necessity that likewise yields no contradiction), noting no finite unweighted positive switching budget currently exists.",
      "defining_relation": "\\text{Type F: }\\sum_nR_n^\\alpha C_n<\\infty\\ (\\alpha>0)\\qquad \\text{Type C: }\\sum_nC_n=\\infty"
    },
    {
      "id": "ns.c4.c4b.sufficient_routes_b1_b4",
      "latex": "\\text{Route B1--B4}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "充分路線 B1–B4",
      "label_en": "Sufficient Routes B1-B4",
      "definition_zh": "第26節列出四條若成立即可純靠 switching cost 逼出 synchronization 的路線：B1 unweighted finite variation、B2 nonsummable generation weights、B3 carrier recurrence、B4 shared-event coupling，本輪判定 B4 最值得攻。",
      "definition_en": "Section 26 lists four routes that would suffice to force synchronization from switching cost alone: B1 unweighted finite variation, B2 nonsummable generation weights, B3 carrier recurrence, B4 shared-event coupling — this round judges B4 most worth pursuing.",
      "notes": "B4 直接導向第35節 Shared-Event Synchronization 與下一輪 C4-C 的主題。"
    },
    {
      "id": "ns.c4.c4b.hard_guards",
      "latex": "G_{\\rm RELAY},\\ G_{\\rm PULSE},\\ G_{\\rm GEN},\\ G_{\\rm WEIGHT}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "C4 強制防護條款",
      "label_en": "C4 hard guards",
      "definition_zh": "第27–30節新增四條防護：$G_{\\rm RELAY}$ 要求先證載體無法遷移才能用固定載體變差論證、$G_{\\rm PULSE}$ 要求提供振幅上界才能由積分發散推占空比大、$G_{\\rm GEN}$ 禁止由逐通道無限常返直接推共同常返世代、$G_{\\rm WEIGHT}$ 禁止由可加總權重預算聲稱代價趨零。",
      "definition_en": "Sections 27-30 add four guards: $G_{\\rm RELAY}$ requires first proving carrier identity cannot migrate before invoking fixed-carrier variation, $G_{\\rm PULSE}$ requires an amplitude upper bound before inferring large duty cycle from divergent integrals, $G_{\\rm GEN}$ forbids inferring common recurrent generations directly from per-channel infinite recurrence, and $G_{\\rm WEIGHT}$ forbids claiming vanishing cost from a summable-weight budget.",
      "notes": "這四條防護分別對應本輪四大 structural escape（pulse、carrier relay、generation routing、summable weights），供未來 C4 rounds 遵守。"
    },
    {
      "id": "ns.c4.c4b.synchronization_failure_classification",
      "latex": "\\text{B-SF1--B-SF5}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "同步失敗分類",
      "label_en": "Synchronization Failure Classification",
      "definition_zh": "第32節將 temporal synchronization 失敗歸為五類：B-SF1 pulse desynchronization、B-SF2 carrier relay、B-SF3 generation routing、B-SF4 spatial relay、B-SF5 gate routing。",
      "definition_en": "Section 32 classifies temporal-synchronization failure into five modes: B-SF1 pulse desynchronization, B-SF2 carrier relay, B-SF3 generation routing, B-SF4 spatial relay, and B-SF5 gate routing."
    },
    {
      "id": "ns.c4.c4b.shared_event_coupling",
      "latex": "\\textbf{Shared-Event Coupling}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "共享事件耦合",
      "label_en": "Shared-Event Coupling",
      "definition_zh": "第0節命名為下一步主攻方向，第35節形式化為 Shared-Event Synchronization：若同一事件 $\\mathcal E_n$ 在同窗口／尺度／空間核同時強迫 $L_n^A\\ge a_0$ 與 $L_n^B\\ge b_0$，則 A、B 的 temporal synchronization 不再需要 persistence 論證。",
      "definition_en": "Named in Section 0 as the next attack direction and formalized in Section 35 as Shared-Event Synchronization: if one event $\\mathcal E_n$ forces both $L_n^A\\ge a_0$ and $L_n^B\\ge b_0$ in the same time window, scale, and spatial core, then A/B temporal synchronization no longer requires a persistence argument.",
      "defining_relation": "\\mathcal E_n\\Rightarrow L_n^A\\ge a_0\\quad\\text{and}\\quad \\mathcal E_n\\Rightarrow L_n^B\\ge b_0\\ \\ (\\text{same window/scale/core})",
      "notes": "是本輪整體策略結論，並構成下一輪 C4-C 的標題主題（Carrier Relay and Shared-Event Coupling Rigidity）。"
    },
    {
      "id": "ns.c4.c4b.common_source_certificate",
      "latex": "\\mathcal N_n,\\ \\mathcal F_A,\\ \\mathcal F_B",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "共同源證書",
      "label_en": "common source certificate",
      "definition_zh": "第36節之強化版本：若事件 $\\mathcal E_n$ 本身由共同 source term $\\mathcal N_n$ 產生且 $L_n^A=\\mathcal F_A(\\mathcal N_n)$、$L_n^B=\\mathcal F_B(\\mathcal N_n)$，則可望建立 $L_n^A+L_n^B\\ge c\\,\\mathcal C(\\mathcal N_n)$ 甚至乘積型下界。",
      "definition_en": "The strengthened version from Section 36: if event $\\mathcal E_n$ arises from a common source term $\\mathcal N_n$ with $L_n^A=\\mathcal F_A(\\mathcal N_n)$ and $L_n^B=\\mathcal F_B(\\mathcal N_n)$, one may hope to establish $L_n^A+L_n^B\\ge c\\,\\mathcal C(\\mathcal N_n)$ or even a product-type lower bound.",
      "defining_relation": "L_n^A=\\mathcal F_A(\\mathcal N_n),\\ \\ L_n^B=\\mathcal F_B(\\mathcal N_n)\\ \\Longrightarrow\\ L_n^A+L_n^B\\ge c\\,\\mathcal C(\\mathcal N_n)\\ \\ (\\text{or } L_n^AL_n^B\\ge c\\,\\mathcal C(\\mathcal N_n)^2)",
      "notes": "$\\mathcal N_n$ 於第40節 C1 進一步具體化為 high-frequency Duhamel 型積分 $\\int e^{\\nu(t_n-s)\\Delta}P_{>J_n}\\mathbb P\\nabla\\cdot(u\\otimes u)\\,ds$，留給 C4-C 估計。"
    },
    {
      "id": "ns.c4.c4b.temporal_sync_1",
      "latex": "\\text{Temporal Sync-1}",
      "series": "NS",
      "first_appearance": "C4-B",
      "label_zh": "時間同步命題一",
      "label_en": "Temporal Sync-1",
      "definition_zh": "第38節（定理/結論38.1）之目標命題名稱：現有 C3 finite turnover/occupancy budgets 加上目前 external necessary criteria，均不足以單獨推出此命題對整個 mandatory survivor family 成立。",
      "definition_en": "The name given in Section 38 (Theorem/Conclusion 38.1) to the target statement that current C3 finite turnover/occupancy budgets, together with present external necessary criteria, are insufficient alone to derive for the full mandatory survivor family.",
      "notes": "是本輪 main no-go 的正式標的，其失敗歸因於四個 structural escape：pulse、carrier relay、generation routing、summable weights。"
    },
    {
      "id": "ns.c4.c4c.shared_event_coupling",
      "latex": "\\mathcal E,\\ L^A,\\ L^B,\\quad \\mathcal E\\Rightarrow B_1\\vee\\cdots\\vee B_m",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "共享事件耦合",
      "label_en": "Shared-Event Coupling",
      "definition_zh": "第2-3節提出本輪的策略轉向：單一真正 N–S 事件 $\\mathcal E$ 若同時強迫兩通道負載 $L^A\\ge a_0$ 與 $L^B\\ge b_0$ 稱強共享事件耦合，若只強迫若干結果之一 $B_1\\vee\\cdots\\vee B_m$ 則稱分支邊。",
      "definition_en": "The round's strategic pivot from Sections 2-3: a single genuine Navier-Stokes event $\\mathcal E$ forcing both $L^A\\ge a_0$ and $L^B\\ge b_0$ at once is a strong shared-event coupling, while forcing only one of several outcomes $B_1\\vee\\cdots\\vee B_m$ is a (weaker) branching edge.",
      "defining_relation": "\\mathcal E \\Rightarrow L^A\\ge a_0,\\ \\ \\mathcal E\\Rightarrow L^B\\ge b_0 \\ \\Longrightarrow\\ A\\stackrel{\\mathcal E}{\\Longleftrightarrow}B;\\qquad \\mathcal E \\Rightarrow B_1\\vee\\cdots\\vee B_m",
      "notes": "Replaces C4-B's question of independently-recurring channels; Guard G-SHARED (Section 50) forbids all consequent branches vanishing once the antecedent holds under carrier relay."
    },
    {
      "id": "ns.c4.c4c.helical_triad",
      "latex": "k\\le p\\le q,\\quad e_k,e_p,e_q,\\quad s_k,s_p,s_q\\in\\{\\pm1\\}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "螺旋三元組",
      "label_en": "Helical Triad",
      "definition_zh": "第4節設定的單一 Fourier 三元組，波數 $k\\le p\\le q$、模態能量 $e_k,e_p,e_q$ 與螺旋號 $s_k,s_p,s_q\\in\\{\\pm1\\}$，同時滿足能量守恆與螺旋守恆。",
      "definition_en": "The single Fourier triad set up in Section 4, with wavenumbers $k\\le p\\le q$, modal energies $e_k,e_p,e_q$, and helicity signs $s_k,s_p,s_q\\in\\{\\pm1\\}$, subject to joint energy and helicity conservation."
    },
    {
      "id": "ns.c4.c4c.theta_tau",
      "latex": "\\Theta_\\tau",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "三元組導數係數",
      "label_en": "Triad Derivative Coefficient",
      "definition_zh": "第4節由單一三元組的能量與螺旋守恆聯立解出的唯一比例係數，決定三個模態能量導數所構成的向量。",
      "definition_en": "The unique proportionality coefficient solved in Section 4 from a single triad's energy- and helicity-conservation constraints, fixing the vector of the three modal energy derivatives.",
      "defining_relation": "(\\dot e_k,\\dot e_p,\\dot e_q) = \\Theta_\\tau\\left(s_pp-s_qq,\\ s_qq-s_kk,\\ s_kk-s_pp\\right)"
    },
    {
      "id": "ns.c4.c4c.helical_classes",
      "latex": "\\mathrm I:(+++),\\ \\mathrm{II}:(+--),\\ \\mathrm{III}:(+-+),\\ \\mathrm{IV}:(++-)",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "螺旋類別 I–IV",
      "label_en": "Helical Classes I-IV",
      "definition_zh": "第5節依最小模態螺旋號固定為正後，把三元組螺旋號組合分成同手性 Class I 與三種異手性 Class II、III、IV。",
      "definition_en": "Section 5's four-way classification of triad helicity-sign patterns, with the smallest-wavenumber mode fixed positive: homochiral Class I versus heterochiral Classes II, III, IV.",
      "notes": "Class IV is the exact critical case with $\\mathcal R_{IV}=G_{IV}^q$ (Section 8), unlike II/III which can degenerate (Section 10)."
    },
    {
      "id": "ns.c4.c4c.pair_production",
      "latex": "\\mathcal R_\\tau",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "臨界對產生量",
      "label_en": "Critical Pair Production",
      "definition_zh": "第6節沿用 C3-B 定義的三元組臨界螺旋對產生量，同手性時恆為零，異手性時等於 unique-sign 模態能量導數乘上係數 $r_\\tau$。",
      "definition_en": "The triad's critical helical pair-production quantity, carried over from C3-B and defined in Section 6: identically zero for homochiral triads, and equal to $r_\\tau$ times the unique-sign modal energy derivative for heterochiral ones.",
      "defining_relation": "\\mathcal R_\\tau=0\\ (\\text{homochiral});\\qquad \\mathcal R_\\tau=r_\\tau\\dot e_{\\rm uniq}\\ (\\text{heterochiral})",
      "notes": "Concretely $\\mathcal R_{II}=k(q-p)\\Theta$, $\\mathcal R_{III}=p(q-k)\\Theta$, $\\mathcal R_{IV}=q(k-p)\\Theta$ (Section 6)."
    },
    {
      "id": "ns.c4.c4c.high_mode_gain",
      "latex": "G_\\tau^q",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "最高模能量增益",
      "label_en": "Highest-Mode Energy Gain",
      "definition_zh": "第7節定義的三元組臨界加權最高模能量增益，只計入 $\\dot e_q>0$ 的部分並以波數 $q$ 加權。",
      "definition_en": "Defined in Section 7 as the triad's critical-weighted highest-mode energy gain, counting only the positive part of $\\dot e_q$ and weighted by wavenumber $q$.",
      "defining_relation": "G_\\tau^q=q[\\dot e_q]_+"
    },
    {
      "id": "ns.c4.c4c.c4c2_heterochiral_gain_theorem",
      "latex": "\\text{C4-C.2}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "異手性增益蘊含正對產生",
      "label_en": "C4-C.2: Heterochiral Gain Is Positive Pair Production",
      "definition_zh": "第9節證明的定理：對所有異手性 Class II–IV，只要最高模能量增益 $\\dot e_q>0$ 即得正臨界對產生 $\\mathcal R_\\tau>0$，第56節稱其為本輪最重要的單三元組螺旋結果。",
      "definition_en": "The theorem proved in Section 9: for every heterochiral Class II-IV, positive highest-mode energy gain $\\dot e_q>0$ forces positive critical pair production $\\mathcal R_\\tau>0$; flagged in Section 56 as the round's most important single-triad helical result.",
      "defining_relation": "\\text{heterochiral},\\ \\dot e_q>0\\ \\Longrightarrow\\ \\mathcal R_\\tau>0"
    },
    {
      "id": "ns.c4.c4c.coupling_ratio",
      "latex": "\\kappa_\\tau=\\dfrac{\\mathcal R_\\tau}{G_\\tau^q}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "對產生／增益耦合比",
      "label_en": "Pair-Production/Gain Coupling Ratio",
      "definition_zh": "第8-12節定義的耦合比 $\\kappa_\\tau=\\mathcal R_\\tau/G_\\tau^q$：Class IV 恆為1（C4-C.3），Class II、III 在局部可比 $k,p\\ge c_Lq$ 且徑向間隙不退化（$\\ge\\delta q$）時有下界 $c_L\\delta/2$，但間隙趨零時退化至0（第10節）。",
      "definition_en": "The coupling ratio $\\kappa_\\tau=\\mathcal R_\\tau/G_\\tau^q$ from Sections 8-12: identically 1 for Class IV (C4-C.3), bounded below by $c_L\\delta/2$ for Class II/III under local comparability $k,p\\ge c_Lq$ and a non-degenerate radial gap ($\\ge\\delta q$), but degenerating to 0 as the gap closes (Section 10).",
      "defining_relation": "\\kappa_\\tau=\\frac{\\mathcal R_\\tau}{G_\\tau^q};\\qquad \\kappa_{IV}=1;\\qquad \\kappa_{II},\\kappa_{III}\\ge\\frac{c_L\\delta}{2}\\ \\text{(robust regime)}"
    },
    {
      "id": "ns.c4.c4c.aggregate_gain",
      "latex": "G^+=G_{\\rm hom}+G_{\\rm deg}+G_{\\rm rob}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "聚合正增益分解",
      "label_en": "Aggregate Positive-Gain Decomposition",
      "definition_zh": "第13節在有限 Galerkin 截斷中定義的聚合正增益 $G^+=G_{\\rm hom}+G_{\\rm deg}+G_{\\rm rob}$，把所有 $\\dot e_q>0$ 三元組的高模增益依同手性、退化異手性、穩健異手性三類求和。",
      "definition_en": "The aggregate positive gain $G^+=G_{\\rm hom}+G_{\\rm deg}+G_{\\rm rob}$ defined in Section 13 within a finite Galerkin truncation, summing the high-mode gain of every $\\dot e_q>0$ triad into homochiral, gap-degenerate heterochiral, and robust heterochiral parts."
    },
    {
      "id": "ns.c4.c4c.positive_variation",
      "latex": "P_+=\\sum_\\tau[\\mathcal R_\\tau]_+",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "三元組正螺旋變化量",
      "label_en": "Triadwise Positive Helical Variation",
      "definition_zh": "第14節定義的所有三元組正對產生量總和 $P_+$，由穩健耦合可得下界 $P_+\\ge c(c_L,\\delta)G_{\\rm rob}$。",
      "definition_en": "Defined in Section 14 as the sum $P_+$ of positive pair production over all triads, bounded below by $P_+\\ge c(c_L,\\delta)G_{\\rm rob}$ via the robust coupling bound."
    },
    {
      "id": "ns.c4.c4c.net_production",
      "latex": "P_-=\\sum_\\tau[-\\mathcal R_\\tau]_+,\\quad \\mathcal R_{\\rm net}=P_+-P_-",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "淨螺旋對產生",
      "label_en": "Negative Variation & Net Pair Production",
      "definition_zh": "第15節定義的負向三元組對產生總和 $P_-$ 與全域淨螺旋對產生 $\\mathcal R_{\\rm net}=P_+-P_-$，說明穩健正對產生仍可能被同時發生的負三元組完全抵消。",
      "definition_en": "Section 15's sum of negative-triad pair production $P_-$ and the global net helical pair production $\\mathcal R_{\\rm net}=P_+-P_-$, showing robust positive production can still be entirely offset by simultaneous negative triads."
    },
    {
      "id": "ns.c4.c4c.c4c4_cancellation_dichotomy",
      "latex": "\\text{C4-C.4}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "穩健增益抵消二分法",
      "label_en": "C4-C.4: Robust-Gain Cancellation Dichotomy",
      "definition_zh": "第16節證明：固定 $0<\\eta<1$，若 $G_{\\rm rob}>0$，則淨螺旋生產分支 $[\\mathcal R_{\\rm net}]_+\\ge\\eta cG_{\\rm rob}$ 與抵消分支 $P_-\\ge(1-\\eta)cG_{\\rm rob}$ 至少一者成立。",
      "definition_en": "Proved in Section 16: for fixed $0<\\eta<1$, if $G_{\\rm rob}>0$ then either the net-helicity branch $[\\mathcal R_{\\rm net}]_+\\ge\\eta cG_{\\rm rob}$ or the cancellation branch $P_-\\ge(1-\\eta)cG_{\\rm rob}$ must hold."
    },
    {
      "id": "ns.c4.c4c.uv_amplitude",
      "latex": "a_q^\\sigma=\\dfrac{\\|u_q^\\sigma\\|_\\infty}{\\nu\\lambda_q}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "UV 臨界振幅",
      "label_en": "UV Critical Amplitude",
      "definition_zh": "第19節重新強調的 C3-G first-crossing 振幅比 $a_q^\\sigma$，本質是振幅／範數事件而非模態能量導數／通量事件，是本輪關鍵區分的起點。",
      "definition_en": "The C3-G first-crossing amplitude ratio $a_q^\\sigma$, re-centered in Section 19 as an amplitude/norm event rather than a modal energy-derivative/flux event — the starting point of the round's key distinction.",
      "defining_relation": "a_q^\\sigma=\\frac{\\|u_q^\\sigma\\|_\\infty}{\\nu\\lambda_q}",
      "notes": "Carried over from NS_C3G_FirstCrossing_CausalAncestry_DepletionNoGo_v0.1; this round shows it does NOT by itself force positive energy gain (no-go NG-C1, Section 49)."
    },
    {
      "id": "ns.c4.c4c.c4c5_phase_rearrangement_nogo",
      "latex": "u_\\theta(x)=\\sum_{m=1}^{N}a_mh_me^{i(k_m\\cdot x+\\theta_m)}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "相位重排無go定理",
      "label_en": "C4-C.5: Phase-Rearrangement No-Go",
      "definition_zh": "第20節構造同一 dyadic shell內固定振幅 $|a_m|$、只變相位 $\\theta_m$ 的場 $u_\\theta$，證明 $\\|u_\\theta\\|_2$ 與相位無關但 $\\|u_\\theta\\|_\\infty$ 可隨相位路徑增加，說明振幅資訊本身無法代數決定 shell 能量通量正負號。",
      "definition_en": "Section 20 constructs a field $u_\\theta$ on one dyadic shell with fixed amplitudes $|a_m|$ and varying phases $\\theta_m$, showing $\\|u_\\theta\\|_2$ is phase-independent while $\\|u_\\theta\\|_\\infty$ can grow along a smooth phase path, so amplitude information alone cannot algebraically fix the sign of shell energy flux.",
      "defining_relation": "\\frac{d}{dt}\\|u_{\\theta(t)}\\|_2^2=0\\quad\\text{while}\\quad \\|u_{\\theta(t)}\\|_\\infty\\ \\text{increases}",
      "notes": "Not a Navier-Stokes solution construction; it is the proof vehicle for the Amplitude-to-Flux Barrier (Section 21)."
    },
    {
      "id": "ns.c4.c4c.amplitude_flux_barrier",
      "latex": "a_q^\\sigma\\uparrow\\ \\not\\Rightarrow\\ \\dot e_q>0\\ \\text{or}\\ \\Phi_q>0",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "振幅—通量障礙",
      "label_en": "Amplitude-to-Flux Barrier",
      "definition_zh": "第21節命名（第44節重申）的本輪最關鍵缺口：UV 振幅 $a_q^\\sigma$ 或 $\\|P_{>J}u\\|_3$ 增大無法單靠範數代數推出模態能量導數 $\\dot e_q>0$ 或 shell 通量 $\\Phi_q>0$，因此 C1 first-crossing 錨點尚不能直接接上 C4-C 螺旋三元組能量轉移代數。",
      "definition_en": "The round's headline gap, named in Section 21 and restated in Section 44: growth of the UV amplitude $a_q^\\sigma$ or $\\|P_{>J}u\\|_3$ cannot, by norm algebra alone, yield $\\dot e_q>0$ or positive shell flux $\\Phi_q>0$, so the C1 first-crossing anchor cannot yet attach to C4-C's helical triad transfer algebra.",
      "notes": "Guard G-AMPFLUX (Section 50); paired with Helical Cancellation Packing as the two priority targets for the next round C4-D (Sections 52-53)."
    },
    {
      "id": "ns.c4.c4c.uv_critical_stock",
      "latex": "H_{q,\\sigma}=\\lambda_q\\|u_q^\\sigma\\|_2^2;\\quad \\|S_q^\\sigma\\|_2^2,\\ \\|\\omega_q^\\sigma\\|_2^2",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "UV 臨界庫存",
      "label_en": "UV Critical Stock",
      "definition_zh": "第22-25節由 UV 振幅 $a_q^\\sigma\\ge\\beta$ 經 Bernstein 不等式推出的同時刻庫存下界：螺旋臨界 shell 庫存 $H_{q,\\sigma}$（C4-C.6，第23節）與應變、渦度 shell 庫存（C4-C.7，第24節）皆滿足 $\\gtrsim\\nu^2\\beta^2$ 型下界，屬 stock synchronization 而非 production synchronization。",
      "definition_en": "The same-time stock lower bounds derived in Sections 22-25 from $a_q^\\sigma\\ge\\beta$ via the Bernstein inequality: helical critical shell stock $H_{q,\\sigma}$ (C4-C.6, Section 23) and strain/vorticity shell stock (C4-C.7, Section 24) all satisfy $\\gtrsim\\nu^2\\beta^2$-type lower bounds, i.e. stock synchronization rather than production synchronization.",
      "defining_relation": "H_{q,\\sigma}=\\lambda_q\\|u_q^\\sigma\\|_2^2;\\qquad a_q^\\sigma\\ge\\beta\\ \\Longrightarrow\\ \\frac{H_{q,\\sigma}}{\\nu^2}\\ge c\\beta^2",
      "notes": "Forms Sync subset C1 (Section 48); each new carrier pays an O(1) normalized stock toll but only a summable O(lambda_qn^{-1}) energy cost (Section 26), so carrier relay still survives."
    },
    {
      "id": "ns.c4.c4c.stock_to_production_barrier",
      "latex": "\\text{critical stock}\\ \\not\\Rightarrow\\ \\text{production}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "庫存—生產障礙",
      "label_en": "Stock-to-Production Barrier",
      "definition_zh": "第45節命名的第二個缺口：儘管 UV 振幅同步了螺旋、應變、渦度庫存，庫存可以靜態存在，目前尚無法由庫存推出正的生產、伸展或算子逃逸。",
      "definition_en": "The second named gap, from Section 45: although UV amplitude synchronizes helical, strain, and vorticity stock, stock can sit statically, and no bridge yet forces positive production, stretching, or operator escape from it.",
      "notes": "Corresponds to no-go NG-C3 (Section 49) and Guard G-STOCKPROD (Section 50)."
    },
    {
      "id": "ns.c4.c4c.local_strain_growth",
      "latex": "G_\\chi=E_\\chi'+D_\\chi",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "局部應變增長率",
      "label_en": "Local Strain-Growth Rate",
      "definition_zh": "第27節沿用 C3-O adjoint cutoff 定義的局部量 $G_\\chi=E_\\chi'+D_\\chi$，等於三個邊界／自放大項之和 $A_\\chi+B_\\chi^B+B_\\chi^P$。",
      "definition_en": "The local quantity $G_\\chi=E_\\chi'+D_\\chi$ from Section 27, carried over from the C3-O adjoint cutoff, equal to the sum of three self-amplification/boundary terms $A_\\chi+B_\\chi^B+B_\\chi^P$."
    },
    {
      "id": "ns.c4.c4c.strain_self_amplification",
      "latex": "A_\\chi=-2\\int\\chi\\det S",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "應變自放大項",
      "label_en": "Strain Self-Amplification Term",
      "definition_zh": "第27節定義的局部應變自放大項 $A_\\chi=-2\\int\\chi\\det S$，是局部應變增長分解 $G_\\chi$ 的第一支。",
      "definition_en": "Defined in Section 27 as $A_\\chi=-2\\int\\chi\\det S$, the self-amplification branch of the local strain-growth decomposition $G_\\chi$."
    },
    {
      "id": "ns.c4.c4c.betchov_current",
      "latex": "B_\\chi^B=\\tfrac13\\int\\nabla\\chi\\cdot F_B",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "Betchov 邊界流",
      "label_en": "Betchov Boundary Current",
      "definition_zh": "第27節定義、第29節由精確局部 Betchov 恆等式連結的邊界流項 $B_\\chi^B$，滿足 $V_\\chi=2A_\\chi-4B_\\chi^B$。",
      "definition_en": "The boundary current $B_\\chi^B$ defined in Section 27 and tied to the exact local Betchov identity in Section 29 via $V_\\chi=2A_\\chi-4B_\\chi^B$.",
      "notes": "C4-C.9 (Section 30) shows $A_\\chi\\ge a>0$ forces $|B_\\chi^B|\\ge a/4$ or $V_\\chi\\ge a$."
    },
    {
      "id": "ns.c4.c4c.pressure_current",
      "latex": "B_\\chi^P=\\int\\nabla\\chi\\cdot F_p",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "局部壓力流",
      "label_en": "Local Pressure Current",
      "definition_zh": "第27節定義的局部壓力邊界流項 $B_\\chi^P=\\int\\nabla\\chi\\cdot F_p$，是局部應變增長分解 $G_\\chi$ 的第三支。",
      "definition_en": "Defined in Section 27 as $B_\\chi^P=\\int\\nabla\\chi\\cdot F_p$, the pressure branch of the local strain-growth decomposition $G_\\chi$."
    },
    {
      "id": "ns.c4.c4c.c4c8_strain_growth_trichotomy",
      "latex": "\\text{C4-C.8}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "應變增長三分法",
      "label_en": "C4-C.8: Local Strain-Growth Trichotomy",
      "definition_zh": "第28節定理28.1：若 $G_\\chi\\ge g>0$，則自放大項、Betchov流、壓力流三者至少一者達 $g/3$，證法僅用三項總和等於 $G_\\chi$ 的三角不等式。",
      "definition_en": "Theorem 28.1 in Section 28: if $G_\\chi\\ge g>0$ then at least one of the self-amplification, Betchov-current, and pressure-current terms reaches $g/3$, proved by a bare triangle-inequality argument on the three-term sum."
    },
    {
      "id": "ns.c4.c4c.vortex_stretching_local",
      "latex": "V_\\chi=\\int\\chi\\,\\omega\\cdot S\\omega",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "局部渦度伸展量",
      "label_en": "Local Vortex Stretching",
      "definition_zh": "第29節定義的局部渦度伸展量 $V_\\chi$，經精確局部 Betchov 關係滿足 $V_\\chi=2A_\\chi-4B_\\chi^B$，並在第32節被 middle-strain 與 principal-alignment 兩支加權渦度控制。",
      "definition_en": "The local vortex-stretching quantity $V_\\chi$ from Section 29, satisfying the exact local Betchov relation $V_\\chi=2A_\\chi-4B_\\chi^B$, and bounded in Section 32 by a middle-strain-weighted plus a principal-alignment-weighted vorticity branch.",
      "defining_relation": "V_\\chi=2A_\\chi-4B_\\chi^B",
      "notes": "C4-C.10 (Section 31) combines this with C4-C.8/C4-C.9 to force strain growth into pressure, Betchov, or vortex stretching; C4-C.11 (Section 33) further splits the vortex-stretching branch by geometry."
    },
    {
      "id": "ns.c4.c4c.mean_strain_pressure_decomp",
      "latex": "M_\\chi'=-Q_\\chi-P_\\chi",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "平均應變—壓力分解",
      "label_en": "Mean-Strain/Pressure Decomposition",
      "definition_zh": "第34節定義的 adjoint mean-strain transport 分解 $M_\\chi'=-Q_\\chi-P_\\chi$，其中 $Q_\\chi$ 為應變—渦度二次型、$P_\\chi$ 為局部壓力 Hessian 積分，C4-C.12（第35節）證 $|P_\\chi|\\ge p_0$ 強迫 $|M_\\chi'|\\ge p_0/2$ 或 $|Q_\\chi|\\ge p_0/2$。",
      "definition_en": "The adjoint mean-strain-transport decomposition $M_\\chi'=-Q_\\chi-P_\\chi$ from Section 34, where $Q_\\chi$ is a quadratic strain/vorticity functional and $P_\\chi$ a local pressure-Hessian integral; C4-C.12 (Section 35) shows $|P_\\chi|\\ge p_0$ forces $|M_\\chi'|\\ge p_0/2$ or $|Q_\\chi|\\ge p_0/2$."
    },
    {
      "id": "ns.c4.c4c.miller_operator",
      "latex": "\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\tfrac34\\omega\\otimes\\omega\\right)",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "Miller 應變—渦度算子",
      "label_en": "Miller Strain-Vorticity Operator",
      "definition_zh": "第36節在 $\\nu=1$ 正規化下引入的 Miller 算子 $\\mathcal Q_{SV}$，將 strain-vorticity interaction 投影到 $P_{st}$，第38節之 Miller escape 版本要求 $\\|\\mathcal Q_{SV}\\|_2\\ge c\\|-\\Delta S\\|_2$。",
      "definition_en": "The Miller operator $\\mathcal Q_{SV}$ introduced in Section 36 under $\\nu=1$ normalization, projecting the strain-vorticity interaction through $P_{st}$; the Miller-escape version in Section 38 requires $\\|\\mathcal Q_{SV}\\|_2\\ge c\\|-\\Delta S\\|_2$.",
      "defining_relation": "\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\tfrac34\\omega\\otimes\\omega\\right)=\\mathcal A_{adv}+\\mathcal A_{S^2}+\\mathcal A_{\\omega^2}"
    },
    {
      "id": "ns.c4.c4c.operator_sources_debt",
      "latex": "\\mathcal A_{adv},\\ \\mathcal A_{S^2},\\ \\mathcal A_{\\omega^2}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "算子來源與抵消債",
      "label_en": "Operator Sources & Cancellation Debt",
      "definition_zh": "第36節分解 Miller 算子的三個投影來源（平流、應變平方、渦度二次），C4-C.13（第37節）證任一者達 $d/3$，第39節指出若 $\\mathcal A_{\\omega^2}$ 大但 $\\mathcal Q_{SV}$ 小，則平流／應變平方分量必須同步抵消該差額，稱算子抵消債。",
      "definition_en": "The three projected sources of the Miller operator from Section 36 (advection, strain-square, vorticity-quadratic); C4-C.13 (Section 37) shows at least one reaches $d/3$, and Section 39 shows that if $\\mathcal A_{\\omega^2}$ is large while $\\mathcal Q_{SV}$ is small, the advection/strain-square components must absorb the difference, termed the Operator Cancellation Debt.",
      "defining_relation": "\\|\\mathcal A_{adv}+\\mathcal A_{S^2}\\|_2\\ \\ge\\ \\|\\mathcal A_{\\omega^2}\\|_2-\\|\\mathcal Q_{SV}\\|_2",
      "notes": "Guard G-OPCANCEL (Section 50) forbids inferring a large full Miller operator from one large projected component alone (no-go NG-C5, Section 49)."
    },
    {
      "id": "ns.c4.c4c.helical_cancellation_packing",
      "latex": "\\textbf{Helical Cancellation Packing}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "螺旋抵消堆積問題",
      "label_en": "Helical Cancellation Packing",
      "definition_zh": "第46節命名的第三個缺口：穩健異手性增益只給三元組正變化量 $P_+\\gtrsim G_{\\rm rob}$，並非全域淨生產 $[\\mathcal R_{\\rm net}]_+$，因仍可能被 $P_-$ 抵消，與 Amplitude-to-Flux Barrier 並列第52節「下一輪應直接攻」的兩個目標。",
      "definition_en": "The third named gap, from Section 46: robust heterochiral gain only gives triadwise positive variation $P_+\\gtrsim G_{\\rm rob}$, not global net production $[\\mathcal R_{\\rm net}]_+$, since $P_-$ can still cancel it; flagged in Section 52, alongside the Amplitude-to-Flux Barrier, as one of two targets the next round should attack directly.",
      "notes": "Directly feeds proof obligation D6 for the next round C4-D (Section 54)."
    },
    {
      "id": "ns.c4.c4c.local_to_operator_bridge",
      "latex": "\\textbf{Local-to-Operator Bridge}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "局部—算子橋接問題",
      "label_en": "Local-to-Operator Bridge",
      "definition_zh": "第47節命名的第四個缺口：局部渦度伸展量 $V_\\chi$ 大只給局部幾何與局部渦度二次負荷，因投影、局部化與抵消，尚不能直接推出全域 Miller 算子逃逸 $\\|\\mathcal Q_{SV}\\|_2/\\|\\Delta S\\|_2$ 大。",
      "definition_en": "The fourth named gap, from Section 47: a large local vortex-stretching quantity $V_\\chi$ only yields local geometry and local vorticity-quadratic load; projection, localization, and cancellation still block a direct implication to a large global Miller operator escape ratio $\\|\\mathcal Q_{SV}\\|_2/\\|\\Delta S\\|_2$.",
      "notes": "Section 40 gives the one directional bound that does hold: $\\|P_{st}(\\omega\\otimes\\omega)\\|_2\\ge|\\int\\omega\\cdot S\\omega|/\\|S\\|_2$."
    },
    {
      "id": "ns.c4.c4c.closure_graph",
      "latex": "\\text{High-mode gain}\\to\\text{hom}\\vee\\text{deg}\\vee\\text{net}\\vee\\text{cancel};\\ \\ \\text{Strain growth}\\to\\text{pressure}\\vee\\text{Betchov}\\vee\\text{stretch}",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "共享事件封閉圖",
      "label_en": "Shared-Event Closure Graph v0.1",
      "definition_zh": "第41節彙整本輪七條 exact／conditional 共享事件邊（振幅→庫存、穩健異手性增益→正螺旋變化、高模增益→四分支、應變增長→三分支、渦度伸展→兩分支、壓力活躍→兩分支、Miller 逃逸→三分支）而成的封閉圖，第55節列為尚 OPEN 的整體閉合。",
      "definition_en": "The synthesis object from Section 41 collecting the round's seven exact/conditional shared-event edges (amplitude to stock; robust heterochiral gain to positive helical variation; high-mode gain to a four-way branch; strain growth to a three-way branch; vortex stretching to a two-way branch; pressure-active to a two-way branch; Miller escape to a three-way branch); Section 55 lists full graph closure as still OPEN.",
      "notes": "Sync subsets C1-C4 (Section 48) are the graph's current seed nodes; Section 49's five no-gos (NG-C1 to NG-C5) mark edges the graph does NOT yet contain."
    },
    {
      "id": "ns.c4.c4c.true_etn_shared",
      "latex": "\\Theta^{shared}=\\left\\langle \\mathcal E,\\operatorname{CarrierID},\\operatorname{LoadVector},\\operatorname{BranchSet},\\operatorname{CancellationDebt},\\operatorname{Prov}\\right\\rangle",
      "series": "NS",
      "first_appearance": "C4-C",
      "label_zh": "True ETN 共享事件狀態",
      "label_en": "True ETN Shared-Event State",
      "definition_zh": "第51節更新的 True ETN 狀態元組 $\\Theta^{shared}$，把事件、carrier識別、負載向量、分支集合、抵消債與 provenance 打包，並給出螺旋三元組的具體實例 $\\Theta_\\tau^{gain}$。",
      "definition_en": "The updated True ETN state tuple from Section 51, packaging the event, carrier identity, load vector, branch set, cancellation debt, and provenance, together with a concrete helical-triad instance $\\Theta_\\tau^{gain}$.",
      "defining_relation": "\\Theta_\\tau^{gain}=\\left\\langle (k,p,q),(s_k,s_p,s_q),G_\\tau^q,\\mathcal R_\\tau,\\kappa_\\tau,\\operatorname{Gap}\\right\\rangle",
      "notes": "Extends the True ETN / infinite-dimensional tension field framework listed among this document's internal dependencies."
    },
    {
      "id": "ns.c4.c4d.hereditary_uv_amplitude",
      "latex": "a_q^\\sigma(t) = \\dfrac{\\|u_q^\\sigma(t)\\|_\\infty}{\\nu\\lambda_q}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "遺傳性UV振幅",
      "label_en": "Hereditary UV amplitude",
      "definition_zh": "第0節重申此為 C4-C 已建立的核心 hereditary UV anchor：shell-helical 速度場的 sup 範數除以黏性尺度 $\\nu\\lambda_q$。",
      "definition_en": "Section 0 restates the hereditary UV anchor already established in C4-C: the sup-norm of the shell-helical velocity field normalized by the viscous scale $\\nu\\lambda_q$.",
      "notes": "Carries over from C4-C (NS_C4C_SharedEventCoupling_AmplitudeFluxBarrier_v0.1); C4-D works mainly with its single-shell simplification $a(t)$ from section 3."
    },
    {
      "id": "ns.c4.c4d.shell_helical_field",
      "latex": "f(t,x) = u_q^\\sigma(t,x) = \\Delta_q P^\\sigma u(t,x)",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "shell-helical 速度場",
      "label_en": "Shell-helical velocity field",
      "definition_zh": "第2節將 $f$ 定義為速度場 $u$ 經 Littlewood–Paley 投影 $\\Delta_q$ 與 helicity 投影 $P^\\sigma$ 後所得、支撐於固定 dyadic annulus 的分量。",
      "definition_en": "Section 2 defines $f$ as the component of the velocity field $u$ obtained via the Littlewood–Paley projector $\\Delta_q$ and helicity projector $P^\\sigma$, supported in a fixed dyadic annulus."
    },
    {
      "id": "ns.c4.c4d.shell_nonlinear_source",
      "latex": "N(t,x) = \\Delta_q P^\\sigma \\mathbb P \\nabla\\cdot(u\\otimes u)",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "shell 非線性源項",
      "label_en": "Shell nonlinear source",
      "definition_zh": "第2節定義 $N$ 為 Navier–Stokes 非線性項經 Leray 投影 $\\mathbb P$ 與 shell/helicity 投影後的量，是驅動 $f$ 演化的來源。",
      "definition_en": "Section 2 defines $N$ as the Navier–Stokes nonlinear term after Leray projection $\\mathbb P$ and shell/helicity projection, the source driving the evolution of $f$."
    },
    {
      "id": "ns.c4.c4d.shell_equation",
      "latex": "\\partial_t f - \\nu\\Delta f + N = 0",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "shell 演化方程",
      "label_en": "Shell evolution equation",
      "definition_zh": "第2節給出 $f$ 在固定 shell 上滿足的精確 PDE，是全篇 amplitude-source 耦合分析的出發點。",
      "definition_en": "Section 2 states the exact PDE satisfied by $f$ on the fixed shell, the starting point for the amplitude-source coupling analysis carried through the round.",
      "defining_relation": "\\partial_t f - \\nu\\Delta f + N = 0"
    },
    {
      "id": "ns.c4.c4d.critical_amplitude",
      "latex": "a(t) = \\dfrac{\\|f(t)\\|_\\infty}{\\nu\\lambda}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "臨界振幅",
      "label_en": "Critical amplitude",
      "definition_zh": "第3節引入單一 shell 上的局部記法 $a(t)$，作為 hereditary 振幅的簡化版本，用來定義 crossing 事件。",
      "definition_en": "Section 3 introduces the local single-shell notation $a(t)$, a simplified form of the hereditary amplitude used to define crossing events throughout the round.",
      "defining_relation": "a(t) = \\dfrac{\\|f(t)\\|_\\infty}{\\nu\\lambda}"
    },
    {
      "id": "ns.c4.c4d.crossing_parameters",
      "latex": "0<\\beta_0<\\beta_1;\\quad a(t_0)=\\beta_0,\\ a(t_1)=\\beta_1,\\ t_0<t_1",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "交叉門檻與時刻",
      "label_en": "Crossing thresholds and times",
      "definition_zh": "第3節固定門檻 $\\beta_0<\\beta_1$，並令 $t_1$ 為 $a$ 達到 $\\beta_1$ 的 first/hysteretic crossing 時刻、$t_0$ 為其前最後一次 $a=\\beta_0$ 的時刻。",
      "definition_en": "Section 3 fixes thresholds $\\beta_0<\\beta_1$ and lets $t_1$ be the first/hysteretic crossing time where $a=\\beta_1$, with $t_0$ the last preceding time where $a=\\beta_0$."
    },
    {
      "id": "ns.c4.c4d.viscous_window_scale",
      "latex": "\\tau_\\lambda = \\dfrac{\\theta}{\\nu\\lambda^2}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "黏性視窗尺度",
      "label_en": "Viscous window timescale",
      "definition_zh": "第4節定義 $\\tau_\\lambda$ 為以固定常數 $\\theta$ 加權的 shell 黏性時間尺度，用來設定 backward window 的長度。",
      "definition_en": "Section 4 defines $\\tau_\\lambda$ as the shell viscous timescale weighted by a fixed constant $\\theta$, setting the length of the backward window."
    },
    {
      "id": "ns.c4.c4d.backward_viscous_window",
      "latex": "I_\\lambda = [t_1-\\tau_\\lambda,\\ t_1]",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "反向黏性視窗",
      "label_en": "Backward viscous window",
      "definition_zh": "第4節定義 $I_\\lambda$ 為緊接在 crossing 時刻 $t_1$ 之前、長度為 $\\tau_\\lambda$ 的時間區間，是定理5.1二分法的作用範圍。",
      "definition_en": "Section 4 defines $I_\\lambda$ as the length-$\\tau_\\lambda$ interval immediately preceding the crossing time $t_1$, the domain on which the Theorem 5.1 dichotomy acts."
    },
    {
      "id": "ns.c4.c4d.persistence_fast_dichotomy",
      "latex": "\\text{D-PERSIST}\\ \\vee\\ \\text{D-FAST}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "持續或快速交叉二分定理（C4-D.1）",
      "label_en": "Persistence-or-Fast-Crossing Dichotomy (C4-D.1)",
      "definition_zh": "第5節定理5.1證明任一 $\\beta_0\\to\\beta_1$ crossing 必落入 D-PERSIST（整個 backward window 內 $a>\\beta_0$）或 D-FAST（crossing 在 $\\le\\tau_\\lambda$ 時間內完成）之一。",
      "definition_en": "Theorem 5.1 in section 5 shows that any $\\beta_0\\to\\beta_1$ crossing must fall into D-PERSIST (amplitude stays above $\\beta_0$ throughout the backward window) or D-FAST (the crossing completes within time $\\tau_\\lambda$).",
      "notes": "D-PERSIST routes directly back to the C4-A persistence-to-synchronization machinery (section 6), so C4-D's own analysis concerns only D-FAST."
    },
    {
      "id": "ns.c4.c4d.branch_d_fast",
      "latex": "t_1-t_0\\le\\dfrac{\\theta}{\\nu\\lambda^2}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "D-FAST 分支",
      "label_en": "D-FAST branch",
      "definition_zh": "第5-6節指出，唯一需要 C4-D 深入研究的分支，即振幅在受黏性尺度界定的短時間內完成 $\\beta_0\\to\\beta_1$ 的 crossing。",
      "definition_en": "Sections 5-6 identify D-FAST — the branch where amplitude completes the $\\beta_0\\to\\beta_1$ crossing within a viscously-bounded short time — as the sole branch C4-D goes on to analyze."
    },
    {
      "id": "ns.c4.c4d.sup_envelope",
      "latex": "M(t) = \\|f(t)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "sup範數包絡",
      "label_en": "Sup-norm envelope",
      "definition_zh": "第7節定義 $M(t)$ 為 $f$ 的 sup 範數，在 band-limited 條件下為 locally Lipschitz 函數且於某點 $x_t$ 達到。",
      "definition_en": "Section 7 defines $M(t)$ as the sup-norm of $f$, locally Lipschitz under the band-limited condition and attained at some point $x_t$.",
      "defining_relation": "M(t) = \\|f(t)\\|_\\infty = f(t,x_t)\\cdot e_t"
    },
    {
      "id": "ns.c4.c4d.envelope_direction",
      "latex": "e_t = \\dfrac{f(t,x_t)}{M(t)}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "包絡方向向量",
      "label_en": "Envelope maximizer direction",
      "definition_zh": "第8節在 $M$ 可微的 a.e. 時刻，於最大化點 $x_t$ 定義單位方向 $e_t$，並給出標準 max-envelope/Danskin-type 恆等式 $M'=e_t\\cdot\\partial_tf(t,x_t)$。",
      "definition_en": "Section 8 defines the unit direction $e_t$ at the maximizing point $x_t$, giving the standard max-envelope/Danskin-type identity $M'=e_t\\cdot\\partial_t f(t,x_t)$.",
      "defining_relation": "M'(t) = e_t\\cdot\\partial_t f(t,x_t)"
    },
    {
      "id": "ns.c4.c4d.positive_source_theorem",
      "latex": "g(t) := -e_t\\cdot N(t,x_t) \\ge M'(t) > 0",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "正振幅變化需正非線性源定理（C4-D.2）",
      "label_en": "Positive Amplitude Variation Requires Positive Nonlinear Source (C4-D.2)",
      "definition_zh": "第9-10節用最大點處 $e_t\\cdot\\Delta f\\le0$ 證明，凡 $M'(t)>0$ 的可微時刻必有源投影 $g(t)\\ge M'(t)$，即振幅上升只能由非線性源驅動而非黏性。",
      "definition_en": "Sections 9-10 use $e_t\\cdot\\Delta f\\le 0$ at the maximizing point to show that whenever $M'(t)>0$, the source projection satisfies $g(t)\\ge M'(t)$ — amplitude growth can only be driven by the nonlinear source, never viscosity.",
      "defining_relation": "g(t) := -e_t\\cdot N(t,x_t) \\ge M'(t) > 0"
    },
    {
      "id": "ns.c4.c4d.source_efficiency",
      "latex": "\\eta(t) = \\dfrac{g(t)}{\\|N(t)\\|_\\infty} \\in (0,1]",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "源效率",
      "label_en": "Source efficiency",
      "definition_zh": "第12節定義 $\\eta(t)$ 為源投影 $g(t)$ 占源 sup-norm 的比例，並固定門檻 $\\eta_0\\in(0,1)$ 以分割後續的 good/bad source 時間集合。",
      "definition_en": "Section 12 defines $\\eta(t)$ as the fraction of the source sup-norm captured by the projection $g(t)$, with a fixed threshold $\\eta_0\\in(0,1)$ splitting the subsequent good/bad source time sets.",
      "defining_relation": "\\eta(t) = \\dfrac{g(t)}{\\|N(t)\\|_\\infty}"
    },
    {
      "id": "ns.c4.c4d.good_bad_sets",
      "latex": "G=\\{t: M'(t)>0,\\ \\eta(t)\\ge\\eta_0\\},\\quad B=\\{t: M'(t)>0,\\ \\eta(t)<\\eta_0\\}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "好/壞源效率時間集",
      "label_en": "Good/bad source-efficiency time sets",
      "definition_zh": "第12-13節依源效率是否達 $\\eta_0$ 把振幅上升時刻分成 $G$（導向 D-WORK 分支）與 $B$（導向 D-SRC 分支），且正變化總量至少一半集中在其中一個集合上。",
      "definition_en": "Sections 12-13 split amplitude-increasing times by whether source efficiency reaches $\\eta_0$ into $G$ (leading to the D-WORK branch) and $B$ (leading to the D-SRC branch), with at least half the total positive variation concentrated on one of the two."
    },
    {
      "id": "ns.c4.c4d.source_overcapacity_theorem",
      "latex": "\\dfrac{1}{\\nu\\lambda}\\int_B \\|N(t)\\|_\\infty\\,dt \\ge \\dfrac{\\Delta\\beta}{2\\eta_0}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "非線性源超載脈衝定理（定理14.1）",
      "label_en": "Nonlinear Source-Overcapacity Impulse (Theorem 14.1)",
      "definition_zh": "第14節證明若 D-SRC 成立，源項 sup-norm 的時間積分必超過以 $\\Delta\\beta/\\eta_0$ 定標的臨界量，本文稱之為 Nonlinear Source-Overcapacity Impulse。",
      "definition_en": "Section 14 proves that if D-SRC holds, the time-integrated source sup-norm must exceed a critical quantity scaled by $\\Delta\\beta/\\eta_0$, termed the Nonlinear Source-Overcapacity Impulse.",
      "defining_relation": "\\dfrac{1}{\\nu\\lambda}\\int_B \\|N(t)\\|_\\infty\\,dt \\ge \\dfrac{\\Delta\\beta}{2\\eta_0}"
    },
    {
      "id": "ns.c4.c4d.shell_work_density",
      "latex": "w(t,y) = -f(t,y)\\cdot N(t,y)",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "非線性 shell 功率密度",
      "label_en": "Nonlinear shell work density",
      "definition_zh": "第16節定義 $w(t,y)$ 為 $f$ 與源項 $N$ 的負內積，於最大化點 $x_t$ 恰等於 $M(t)g(t)>0$。",
      "definition_en": "Section 16 defines $w(t,y)$ as the negative inner product of $f$ and the source $N$, equal to $M(t)g(t)>0$ at the maximizing point $x_t$.",
      "defining_relation": "w(t,y) = -f(t,y)\\cdot N(t,y)"
    },
    {
      "id": "ns.c4.c4d.positive_work_ball_theorem",
      "latex": "w(t,y) \\ge c_\\ast M(t)g(t),\\quad y\\in B_t = B(x_t,\\,c_\\ast\\eta_0\\lambda^{-1})",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "帶限局部正功球定理（C4-D.3 / 定理17.1）",
      "label_en": "Band-Limited Local Positive-Work Ball (C4-D.3 / Theorem 17.1)",
      "definition_zh": "第17節用 Bernstein 估計把最大化點的 pointwise 正功率，擴展為半徑 $c_\\ast\\eta_0\\lambda^{-1}$ 的球 $B_t$ 上處處成立的正下界。",
      "definition_en": "Section 17 uses Bernstein estimates to extend the pointwise positive work rate at the maximizing point into a lower bound holding throughout a ball $B_t$ of radius $c_\\ast\\eta_0\\lambda^{-1}$.",
      "defining_relation": "w(t,y) \\ge c_\\ast M(t)g(t),\\quad y\\in B_t"
    },
    {
      "id": "ns.c4.c4d.local_work_rate",
      "latex": "L(t) := \\int_{B_t} w(t,y)\\,dy \\ge c\\,\\eta_0^3\\nu\\beta_0\\lambda^{-2}M'(t)",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "局部正功率",
      "label_en": "Local positive work rate",
      "definition_zh": "第18節將正功密度在 $B_t$ 上積分得 $L(t)$，並用 crossing 區間內 $M(t)\\ge\\nu\\lambda\\beta_0$ 化為以 $M'(t)$ 定標的下界。",
      "definition_en": "Section 18 integrates the positive work density over $B_t$ to obtain $L(t)$, then uses $M(t)\\ge\\nu\\lambda\\beta_0$ within the crossing interval to bound it below in terms of $M'(t)$."
    },
    {
      "id": "ns.c4.c4d.global_shell_work",
      "latex": "W_q^\\sigma(t) = -\\int_{\\mathbb R^3} f(t,x)\\cdot N(t,x)\\,dx",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "全域 shell 非線性功",
      "label_en": "Global shell nonlinear work",
      "definition_zh": "第20節定義 $W_q^\\sigma$ 為功密度 $w$ 的全空間積分，並在 shell energy balance 中指出它是未扣黏性耗散的非線性能量輸入項。",
      "definition_en": "Section 20 defines $W_q^\\sigma$ as the whole-space integral of the work density $w$, identified in the shell energy balance as the nonlinear energy input term prior to subtracting viscous dissipation.",
      "defining_relation": "\\tfrac12\\tfrac{d}{dt}\\|f\\|_2^2 + \\nu\\|\\nabla f\\|_2^2 = W_q^\\sigma"
    },
    {
      "id": "ns.c4.c4d.spatial_work_split",
      "latex": "W^+(t)=\\int[w]_+dx,\\quad W^-(t)=\\int[-w]_+dx,\\quad W_q^\\sigma=W^+-W^-",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "正/負空間功變化",
      "label_en": "Positive/negative spatial work variation",
      "definition_zh": "第21節把功密度 $w$ 拆成空間上的正、負部分積分 $W^+,W^-$，使全域 shell work 等於兩者之差。",
      "definition_en": "Section 21 splits the work density $w$ into positive and negative spatial integrals $W^+,W^-$, so the global shell work equals their difference."
    },
    {
      "id": "ns.c4.c4d.local_to_global_cancellation",
      "latex": "[W_q^\\sigma]_+ + W^- \\ge W^+ \\ge L\\ \\text{on } G",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "局部至全域功抵消恆等式（C4-D.5）",
      "label_en": "Local-to-Global Work Cancellation Identity (C4-D.5)",
      "definition_zh": "第22節證明正 shell work 與負功抵消之和必至少覆蓋 good-source 時刻上局部正功球提供的功率 $L$。",
      "definition_en": "Section 22 proves that positive shell work together with negative-work cancellation must jointly cover the local positive-work-ball contribution $L$ at good-source times."
    },
    {
      "id": "ns.c4.c4d.dimensionless_positive_work",
      "latex": "\\mathfrak F_q = \\dfrac{\\lambda}{\\nu^2}\\int_G [W_q^\\sigma(t)]_+\\,dt",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "無量綱正 shell 功",
      "label_en": "Dimensionless positive shell work",
      "definition_zh": "第23節定義 $\\mathfrak F_q$ 為正 shell work 在 $G$ 上以臨界權重 $\\lambda/\\nu^2$ 定標的無量綱時間積分，是 branching bridge 的其中一支。",
      "definition_en": "Section 23 defines $\\mathfrak F_q$ as the critically-weighted, dimensionless time-integral of positive shell work over $G$, gating one leg of the branching bridge.",
      "defining_relation": "\\mathfrak F_q = \\dfrac{\\lambda}{\\nu^2}\\int_G [W_q^\\sigma(t)]_+\\,dt \\ge c\\,\\eta_0^3\\beta_0\\Delta\\beta"
    },
    {
      "id": "ns.c4.c4d.dimensionless_work_cancellation",
      "latex": "\\mathfrak C_q^{sp} = \\dfrac{\\lambda}{\\nu^2}\\int_G W^-(t)\\,dt",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "無量綱空間功抵消量",
      "label_en": "Dimensionless spatial work-cancellation quantity",
      "definition_zh": "第23節定義 $\\mathfrak C_q^{sp}$ 為負空間功變化以相同臨界權重定標的無量綱積分，量化 spatial work cancellation 這條逃逸支。",
      "definition_en": "Section 23 defines $\\mathfrak C_q^{sp}$ as the critically-weighted dimensionless integral of negative spatial work variation, quantifying the spatial-work-cancellation escape leg.",
      "defining_relation": "\\mathfrak C_q^{sp} = \\dfrac{\\lambda}{\\nu^2}\\int_G W^-(t)\\,dt \\ge c\\,\\eta_0^3\\beta_0\\Delta\\beta"
    },
    {
      "id": "ns.c4.c4d.amplitude_to_work_bridge",
      "latex": "\\mathfrak S_q\\gtrsim1\\ \\vee\\ \\mathfrak F_q\\ge c\\,\\eta_0^3\\beta_0\\Delta\\beta\\ \\vee\\ \\mathfrak C_q^{sp}\\ge c\\,\\eta_0^3\\beta_0\\Delta\\beta",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "振幅至功分支橋（C4-D.6）",
      "label_en": "Amplitude-to-Work Branching Bridge (C4-D.6)",
      "definition_zh": "第23節提出的本輪核心結果：每次 D-FAST crossing 必落入 source-overcapacity、positive shell work、spatial work cancellation 三支之一。",
      "definition_en": "The round's central result, stated in section 23: every D-FAST crossing must fall into one of three legs — source overcapacity, positive shell work, or spatial work cancellation.",
      "defining_relation": "\\mathfrak S_q\\gtrsim1\\ \\vee\\ \\mathfrak F_q\\ge c\\,\\eta_0^3\\beta_0\\Delta\\beta\\ \\vee\\ \\mathfrak C_q^{sp}\\ge c\\,\\eta_0^3\\beta_0\\Delta\\beta",
      "notes": "Replaces the direct amplitude-to-flux implication already disproved in C4-C (NS_C4C_SharedEventCoupling_AmplitudeFluxBarrier_v0.1); this theorem gives section 35 (C4-D.8) its partial closure of the C4-C barrier."
    },
    {
      "id": "ns.c4.c4d.work_dipole_debt",
      "latex": "|\\Omega_-| \\ge c\\,\\eta_0^4\\lambda^{-3},\\quad \\Omega_- = \\{w<0\\}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "功偶極/功多重性債務",
      "label_en": "Work-Dipole / Work-Multiplicity Debt",
      "definition_zh": "第25節證明若空間功抵消要達到與局部正功球相當的規模，負功集合 $\\Omega_-$ 的體積必至少達到一個 shell-volume 級下界，本文稱之為 Work-Dipole / Work-Multiplicity Debt。",
      "definition_en": "Section 25 proves that for spatial work cancellation to reach a scale comparable to the local positive-work ball, the negative-work set $\\Omega_-$ must have volume at least a shell-volume-scale lower bound, termed the Work-Dipole / Work-Multiplicity Debt."
    },
    {
      "id": "ns.c4.c4d.robust_heterochiral_coefficient",
      "latex": "\\mathcal R_\\tau = \\kappa_\\tau q_\\tau \\dot e_{q_\\tau},\\quad c_\\ast\\le\\kappa_\\tau\\le1",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "穩健異手性三元組係數",
      "label_en": "Robust heterochiral triad coefficient",
      "definition_zh": "第30節引用 C4-C 已證結果，指出臨界 helical variation $\\mathcal R_\\tau$ 與最高模態非線性能量導數 $q_\\tau\\dot e_{q_\\tau}$ 成正比，係數 $\\kappa_\\tau\\in[c_\\ast,1]$ 對兩種正負號都成立。",
      "definition_en": "Section 30 recalls the C4-C result that the critical helical variation $\\mathcal R_\\tau$ is proportional to the highest-mode nonlinear energy derivative $q_\\tau\\dot e_{q_\\tau}$, with coefficient $\\kappa_\\tau\\in[c_\\ast,1]$ valid for either sign.",
      "defining_relation": "\\mathcal R_\\tau = \\kappa_\\tau q_\\tau \\dot e_{q_\\tau}",
      "notes": "The coefficient bound $c_\\ast\\le\\kappa_\\tau\\le1$ itself is proved in C4-C; C4-D reuses it as the mechanism behind Theorem C4-D.7."
    },
    {
      "id": "ns.c4.c4d.helical_forces_work_cancellation",
      "latex": "X_\\pm = \\sum_{\\tau\\in rob}[\\pm q_\\tau\\dot e_{q_\\tau}]_+,\\ P_\\pm=\\sum_{\\tau\\in rob}[\\pm\\mathcal R_\\tau]_+;\\quad X_- \\ge (1-\\eta)c_\\ast X_+",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "螺旋度抵消迫使高模態功抵消定理（C4-D.7）",
      "label_en": "Helical Cancellation Forces High-Mode Work Cancellation (C4-D.7)",
      "definition_zh": "第31節在正負部分和 $X_\\pm,P_\\pm$ 上證明，若臨界 helical production 大部分被 $P_-$ 抵消，則必存在相當規模的負最高模態非線性能量功 $X_-$，即 helical cancellation 並非獨立逃逸通道。",
      "definition_en": "Section 31 proves, via the positive/negative sums $X_\\pm,P_\\pm$, that substantial cancellation of positive critical helical production by $P_-$ forces a comparable negative highest-mode nonlinear energy work $X_-$ — helical cancellation is not an independent escape channel.",
      "defining_relation": "X_- \\ge (1-\\eta)c_\\ast X_+",
      "notes": "Section 32 notes this is strictly stronger than the corresponding C4-C result, which only forced net helicity or $P_-$-cancellation without pinning down $X_-$."
    },
    {
      "id": "ns.c4.c4d.rank_defect_branch",
      "latex": "G_{\\rm top}^+ + G_{\\rm nontop}^+ = G_q^+",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "秩虧/高頻參與分支",
      "label_en": "Rank-Defect / Higher-Frequency-Participation Branch",
      "definition_zh": "第27-28節將正 triad work $G_q^+$ 依 receiving mode 是否為 triad 最高波數分成 top-rank 與 non-top 兩部分，若 non-top 佔主導則表示仍有更高絕對頻率參與，本文稱為 Rank-Defect Branch。",
      "definition_en": "Sections 27-28 split the positive triad work $G_q^+$ by whether the receiving mode is the triad's highest wavenumber; when the non-top part dominates, still-higher absolute frequencies are shown to participate, termed the Rank-Defect Branch.",
      "notes": "Section 39 links this branch to the C1/C3-G absolute-frequency ancestry and carrier-relay machinery for future closure."
    },
    {
      "id": "ns.c4.c4d.barrier_partial_closure",
      "latex": "\\text{amplitude}\\Rightarrow\\text{flux}:\\ \\mathrm{FALSE};\\quad \\text{amplitude}\\Rightarrow\\text{finite branch set}:\\ \\mathrm{PROVED}",
      "series": "NS",
      "first_appearance": "C4-D",
      "label_zh": "振幅至通量障礙——部分閉合（C4-D.8）",
      "label_en": "Amplitude-to-Flux Barrier — Partial Closure (C4-D.8)",
      "definition_zh": "第35節總結本輪狀態：C4-C 的 direct amplitude-to-flux implication 仍為 FALSE，但 amplitude crossing 蘊含 finite structured branch set 現已 PROVED，是本輪最主要的 closure。",
      "definition_en": "Section 35 summarizes the round's status: the C4-C direct amplitude-to-flux implication remains FALSE, but amplitude crossing implying a finite structured branch set is now PROVED — the round's main closure.",
      "notes": "The 'Amplitude-to-Flux Barrier' itself is the named object left open by C4-C (NS_C4C_SharedEventCoupling_AmplitudeFluxBarrier_v0.1); C4-D closes it only in the branching sense, not the direct sense."
    },
    {
      "id": "ns.c4.c4e.transport_free_remainder",
      "latex": "R_q^\\sigma",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "去傳輸剩餘項",
      "label_en": "Transport-free remainder",
      "definition_zh": "第3節將 C4-D 的殼層非線性源 $N_q^\\sigma$ 減去低模傳輸項定義出 $R_q^\\sigma=N_q^\\sigma-v_q\\cdot\\nabla f$，使殼層方程變成 $\\partial_tf-\\nu\\Delta f+v_q\\cdot\\nabla f+R_q^\\sigma=0$。",
      "definition_en": "Section 3 defines $R_q^\\sigma=N_q^\\sigma-v_q\\cdot\\nabla f$ by subtracting the low-mode transport term from C4-D's shell nonlinear source $N_q^\\sigma$, turning the shell equation into $\\partial_tf-\\nu\\Delta f+v_q\\cdot\\nabla f+R_q^\\sigma=0$.",
      "defining_relation": "R_q^\\sigma = N_q^\\sigma - v_q\\cdot\\nabla f,\\qquad \\partial_t f - \\nu\\Delta f + v_q\\cdot\\nabla f + R_q^\\sigma = 0",
      "notes": "Supersedes C4-D's use of the full $N_q^\\sigma$ as source; every subsequent amplitude/work/overcapacity argument in this round is re-run on $R_q^\\sigma$ instead (formalized as guard G-TFREE in §47)."
    },
    {
      "id": "ns.c4.c4e.low_transport_velocity",
      "latex": "v_q = u_{\\le q-L_0}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "低模傳輸速度",
      "label_en": "Low transport velocity",
      "definition_zh": "第2節在固定支撐間隙 $L_0\\ge4$ 下，將無散度的低頻截斷速度場 $v_q=u_{\\le q-L_0}$ 定義為驅動殼層方程的傳輸速度。",
      "definition_en": "Section 2 defines the divergence-free low-frequency-truncated velocity $v_q=u_{\\le q-L_0}$, with fixed support gap $L_0\\ge4$, as the transport velocity in the shell equation."
    },
    {
      "id": "ns.c4.c4e.shell_nonlinear_source",
      "latex": "N_q^\\sigma = \\Delta_q P^\\sigma \\mathbb P \\nabla\\cdot(u\\otimes u)",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "殼層非線性源",
      "label_en": "Shell nonlinear source",
      "definition_zh": "第3節重述 C4-D 的殼層非線性源 $N_q^\\sigma=\\Delta_qP^\\sigma\\mathbb P\\nabla\\cdot(u\\otimes u)$，作為定義去傳輸剩餘項 $R_q^\\sigma$ 的起點。",
      "definition_en": "Section 3 restates C4-D's shell nonlinear source $N_q^\\sigma=\\Delta_qP^\\sigma\\mathbb P\\nabla\\cdot(u\\otimes u)$ as the starting point for defining the transport-free remainder $R_q^\\sigma$.",
      "notes": "Explicitly labeled in the text as the \"C4-D shell source\" (§3); C4-E's contribution is to split it via $v_q$, not to redefine it."
    },
    {
      "id": "ns.c4.c4e.sup_norm_maximum",
      "latex": "M(t) = \\|f(t)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "上確界振幅",
      "label_en": "Sup-norm maximum",
      "definition_zh": "第4節（C4-E.1）在可微時刻取 $M(t)=\\|f(t)\\|_\\infty$ 及其極值方向 $e_t$，證明純傳輸項在極值點恰好消失，故 $M'(t)\\le-e_t\\cdot R_q^\\sigma(t,x_t)$。",
      "definition_en": "Section 4 (C4-E.1) takes $M(t)=\\|f(t)\\|_\\infty$ at a differentiability time and shows the pure-transport term vanishes exactly at the maximizer, giving $M'(t)\\le-e_t\\cdot R_q^\\sigma(t,x_t)$.",
      "defining_relation": "M(t)=\\|f(t)\\|_\\infty,\\qquad M'(t)\\le -e_t\\cdot R_q^\\sigma(t,x_t)"
    },
    {
      "id": "ns.c4.c4e.max_point_direction",
      "latex": "e_t = f(t,x_t)/M(t)",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "極值方向向量",
      "label_en": "Maximizer direction",
      "definition_zh": "第4節在振幅取得最大值的點 $x_t$ 定義單位方向 $e_t=f(t,x_t)/M(t)$，用以使純傳輸項在該點的貢獻恰為零。",
      "definition_en": "Section 4 defines the unit direction $e_t=f(t,x_t)/M(t)$ at the maximizer $x_t$, used to show the pure-transport contribution vanishes there."
    },
    {
      "id": "ns.c4.c4e.shell_nonlinear_work",
      "latex": "W_q^\\sigma = -\\int f\\cdot N_q^\\sigma\\,dx",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "殼層非線性作功",
      "label_en": "Shell nonlinear work",
      "definition_zh": "第5節（C4-E.2）因 $\\nabla\\cdot v_q=0$，證明全域殼層作功 $W_q^\\sigma=-\\int f\\cdot N_q^\\sigma dx$ 精確等於 $-\\int f\\cdot R_q^\\sigma dx$，即傳輸項對作功亦無貢獻。",
      "definition_en": "Section 5 (C4-E.2) shows that since $\\nabla\\cdot v_q=0$, the global shell work $W_q^\\sigma=-\\int f\\cdot N_q^\\sigma dx$ equals exactly $-\\int f\\cdot R_q^\\sigma dx$, so transport contributes nothing to the work either.",
      "defining_relation": "W_q^\\sigma = -\\int f\\cdot N_q^\\sigma\\,dx = -\\int f\\cdot R_q^\\sigma\\,dx",
      "notes": "Together with C4-E.1 this establishes that amplitude growth and shell energy work are driven by the same transport-free remainder (§5 boxed conclusion)."
    },
    {
      "id": "ns.c4.c4e.refined_source_efficiency",
      "latex": "\\eta_R(t) = \\dfrac{-e_t\\cdot R_q^\\sigma(t,x_t)}{\\|R_q^\\sigma(t)\\|_\\infty}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "精化源效率",
      "label_en": "Refined source efficiency",
      "definition_zh": "第6節將 C4-D 的源效率改以 $R_q^\\sigma$ 取代 $N_q^\\sigma$ 重新定義為 $\\eta_R(t)$，在 $M'>0$ 時滿足 $0<\\eta_R\\le1$。",
      "definition_en": "Section 6 redefines C4-D's source efficiency as $\\eta_R(t)$, replacing $N_q^\\sigma$ with $R_q^\\sigma$, satisfying $0<\\eta_R\\le1$ whenever $M'>0$."
    },
    {
      "id": "ns.c4.c4e.transport_free_overcapacity_impulse",
      "latex": "\\mathfrak S_q^R := \\dfrac{1}{\\nu\\lambda_q}\\int_I \\|R_q^\\sigma(t)\\|_\\infty\\,dt \\ge s_0",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "去傳輸源超載脈衝",
      "label_en": "Transport-free source-overcapacity impulse",
      "definition_zh": "第7節將 C4-D 的 source-overcapacity impulse 升級為以 $R_q^\\sigma$ 定義的 $\\mathfrak S_q^R$，重新詮釋為「變形／跨尺度剩餘脈衝」而非純傳輸容量。",
      "definition_en": "Section 7 upgrades C4-D's source-overcapacity impulse to $\\mathfrak S_q^R$, defined via $R_q^\\sigma$, reinterpreted as a \"deformation/interscale remainder impulse\" rather than pure transport capacity.",
      "defining_relation": "\\mathfrak S_q^R := \\frac{1}{\\nu\\lambda_q}\\int_I \\|R_q^\\sigma(t)\\|_\\infty\\,dt \\ge s_0,\\qquad s_0\\asymp\\frac{\\beta_1-\\beta_0}{\\eta_0}"
    },
    {
      "id": "ns.c4.c4e.comparable_shell_envelope",
      "latex": "V_q = \\sum_{|p-q|\\le C_0} U_p,\\qquad U_p(t)=\\|u_p(t)\\|_\\infty",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "可比殼層振幅包絡",
      "label_en": "Comparable-shell amplitude envelope",
      "definition_zh": "第8節先定義單殼層振幅 $U_p(t)=\\|u_p(t)\\|_\\infty$，再以固定 LP cutoff 常數 $C_0$ 加總鄰近殼層得到可比包絡 $V_q=\\sum_{|p-q|\\le C_0}U_p$。",
      "definition_en": "Section 8 first defines the single-shell amplitude $U_p(t)=\\|u_p(t)\\|_\\infty$, then sums over neighboring shells within the fixed LP cutoff $C_0$ to get the comparable envelope $V_q=\\sum_{|p-q|\\le C_0}U_p$."
    },
    {
      "id": "ns.c4.c4e.low_mode_gradient_load",
      "latex": "G_{<q} = \\sum_{r\\le q-L_0} \\lambda_r U_r",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "低模梯度負載",
      "label_en": "Low-mode gradient load",
      "definition_zh": "第8節定義低模梯度負載 $G_{<q}=\\sum_{r\\le q-L_0}\\lambda_rU_r$，第14節證明它與渦量殼層範數 $\\|\\omega_r\\|_\\infty\\asymp\\lambda_rU_r$ 同級，是一個 critical vorticity/strain $L^\\infty$ 負載。",
      "definition_en": "Section 8 defines the low-mode gradient load $G_{<q}=\\sum_{r\\le q-L_0}\\lambda_rU_r$; Section 14 shows it is comparable to the vorticity shell norm $\\|\\omega_r\\|_\\infty\\asymp\\lambda_rU_r$, making it a critical vorticity/strain $L^\\infty$ load.",
      "defining_relation": "G_{<q} = \\sum_{r\\le q-L_0} \\lambda_r U_r",
      "notes": "Its $O(1)$ lower bound (E-SHEAR) is identified with the Cheskidov–Dai frequency-localized critical vorticity criterion cited in §1."
    },
    {
      "id": "ns.c4.c4e.high_high_pair_load",
      "latex": "H_q^{HH} = \\sum_{p\\ge q-C_0} U_p\\widetilde U_p,\\qquad \\widetilde U_p=\\sum_{|r-p|\\le C_0}U_r",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "高頻對載量",
      "label_en": "High-high pair load",
      "definition_zh": "第8節定義高頻對載量 $H_q^{HH}=\\sum_{p\\ge q-C_0}U_p\\widetilde U_p$，其中 $\\widetilde U_p=\\sum_{|r-p|\\le C_0}U_r$，捕捉 Bony 分解中高高交互作用對輸出殼層 $q$ 的貢獻。",
      "definition_en": "Section 8 defines the high-high pair load $H_q^{HH}=\\sum_{p\\ge q-C_0}U_p\\widetilde U_p$ with $\\widetilde U_p=\\sum_{|r-p|\\le C_0}U_r$, capturing the high-high Bony-decomposition contribution to output shell $q$."
    },
    {
      "id": "ns.c4.c4e.theorem_transport_free_remainder_estimate",
      "latex": "\\|R_q^\\sigma\\|_\\infty \\le C\\left[G_{<q}V_q + \\lambda_q H_q^{HH}\\right]",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "去傳輸剩餘項估計定理",
      "label_en": "Transport-Free Remainder Estimate",
      "definition_zh": "第9節定理9.1（C4-E.3）用 low-high commutator 估計與 Bony 分解證明 $\\|R_q^\\sigma\\|_\\infty\\le C[G_{<q}V_q+\\lambda_qH_q^{HH}]$，把去傳輸剩餘項精確控制在低模梯度負載與高高對載量之和。",
      "definition_en": "Section 9's Theorem 9.1 (C4-E.3) uses a low-high commutator estimate plus Bony decomposition to prove $\\|R_q^\\sigma\\|_\\infty\\le C[G_{<q}V_q+\\lambda_qH_q^{HH}]$, bounding the transport-free remainder by the low-mode gradient load and high-high pair load.",
      "defining_relation": "\\|R_q^\\sigma\\|_\\infty \\le C\\left[G_{<q} V_q + \\lambda_q H_q^{HH}\\right]",
      "notes": "Proved via $R_q^\\sigma=[T_q^\\sigma,v_q\\cdot\\nabla]u+T_q^\\sigma((u-v_q)\\cdot\\nabla u)$ with $T_q^\\sigma=\\Delta_qP^\\sigma\\mathbb P$ (§9 proof architecture)."
    },
    {
      "id": "ns.c4.c4e.critical_amplitude",
      "latex": "a_p = \\dfrac{U_p}{\\nu\\lambda_p}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "臨界振幅",
      "label_en": "Critical amplitude",
      "definition_zh": "第10節定義無量綱臨界振幅 $a_p=U_p/(\\nu\\lambda_p)$，是貫穿本輪、界定 crossing threshold $\\beta_0,\\beta_1$ 的核心變量。",
      "definition_en": "Section 10 defines the dimensionless critical amplitude $a_p=U_p/(\\nu\\lambda_p)$, the core variable running through the round that defines the crossing thresholds $\\beta_0,\\beta_1$.",
      "defining_relation": "a_p = \\frac{U_p}{\\nu\\lambda_p}",
      "notes": "The per-shell version of the $\\beta_0\\to\\beta_1$ crossing variable whose recurrence C4-D established (§0)."
    },
    {
      "id": "ns.c4.c4e.dimensionless_hh_congestion",
      "latex": "\\mathfrak h_q = \\dfrac{H_q^{HH}}{\\nu^2\\lambda_q^2}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "無量綱高頻對壅塞量",
      "label_en": "Dimensionless high-high congestion",
      "definition_zh": "第10節將高頻對載量無量綱化為 $\\mathfrak h_q=H_q^{HH}/(\\nu^2\\lambda_q^2)$，連同 $\\mathfrak g_q=G_{<q}/(\\nu\\lambda_q^2)$ 與 $\\mathfrak v_q=V_q/(\\nu\\lambda_q)$，把定理9.1改寫成臨界形式 $\\|R_q^\\sigma\\|_\\infty/(\\nu^2\\lambda_q^3)\\le C[\\mathfrak g_q\\mathfrak v_q+\\mathfrak h_q]$。",
      "definition_en": "Section 10 nondimensionalizes the high-high pair load as $\\mathfrak h_q=H_q^{HH}/(\\nu^2\\lambda_q^2)$, which together with $\\mathfrak g_q=G_{<q}/(\\nu\\lambda_q^2)$ and $\\mathfrak v_q=V_q/(\\nu\\lambda_q)$ recasts Theorem 9.1 into the critical form $\\|R_q^\\sigma\\|_\\infty/(\\nu^2\\lambda_q^3)\\le C[\\mathfrak g_q\\mathfrak v_q+\\mathfrak h_q]$.",
      "defining_relation": "\\mathfrak h_q = \\frac{H_q^{HH}}{\\nu^2\\lambda_q^2},\\qquad \\frac{\\|R_q^\\sigma\\|_\\infty}{\\nu^2\\lambda_q^3}\\le C\\left[\\mathfrak g_q\\mathfrak v_q+\\mathfrak h_q\\right]",
      "notes": "Later split into near/far parts (§15) and drives the Small-Threshold Far-Relay Theorem (§17)."
    },
    {
      "id": "ns.c4.c4e.viscous_normalized_time",
      "latex": "d\\tau = \\nu\\lambda_q^2\\,dt",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "黏性正規化時間",
      "label_en": "Viscous-normalized time",
      "definition_zh": "第11節引入黏性正規化時間 $d\\tau=\\nu\\lambda_q^2dt$，將 $\\mathfrak S_q^R$ 重寫成 $\\int[\\mathfrak g_q\\mathfrak v_q+\\mathfrak h_q]d\\tau$ 的形式。",
      "definition_en": "Section 11 introduces the viscous-normalized time $d\\tau=\\nu\\lambda_q^2dt$, rewriting $\\mathfrak S_q^R$ as $\\int[\\mathfrak g_q\\mathfrak v_q+\\mathfrak h_q]d\\tau$."
    },
    {
      "id": "ns.c4.c4e.frontier_cap",
      "latex": "q\\ge Q+C_0,\\qquad a_p(t)\\le\\beta_1\\ (p\\ge Q)",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "前沿上限條件",
      "label_en": "Frontier cap",
      "definition_zh": "第12節針對 first-frontier/frontier-safe crossing（$q\\ge Q+C_0$），假設 crossing 前所有相關高殼層滿足 $a_p(t)\\le\\beta_1$，由此得出 $\\mathfrak v_q\\le C_1\\beta_1$。",
      "definition_en": "Section 12, for a first-frontier/frontier-safe crossing ($q\\ge Q+C_0$), assumes $a_p(t)\\le\\beta_1$ on all relevant high shells before crossing, giving $\\mathfrak v_q\\le C_1\\beta_1$."
    },
    {
      "id": "ns.c4.c4e.theorem_source_overcapacity_routing",
      "latex": "\\mathfrak S_q^R\\ge s_0 \\;\\Rightarrow\\; \\text{E-SHEAR}\\ \\vee\\ \\text{E-HH}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "源超載路由定理",
      "label_en": "Source-Overcapacity Routing Theorem",
      "definition_zh": "第13節定理 C4-E.4 證明在 frontier cap 下，若 $\\mathfrak S_q^R\\ge s_0$，則必有 E-SHEAR（低模梯度負載時間積分達標）或 E-HH（高高壅塞量達標）成立。",
      "definition_en": "Section 13's Theorem C4-E.4 proves that under the frontier cap, $\\mathfrak S_q^R\\ge s_0$ forces either E-SHEAR (the time-integrated low-mode gradient load reaches threshold) or E-HH (the high-high congestion reaches threshold).",
      "defining_relation": "\\int_I G_{<q}(t)\\,dt \\ge c\\frac{s_0}{\\beta_1} \\ (\\text{E-SHEAR}) \\qquad \\vee \\qquad \\int_I \\mathfrak h_q(t)\\,d\\tau \\ge c\\,s_0 \\ (\\text{E-HH})"
    },
    {
      "id": "ns.c4.c4e.e_shear_condition",
      "latex": "\\text{E-SHEAR}:\\quad \\int_I G_{<q}(t)\\,dt \\gtrsim 1",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "E-SHEAR 條件",
      "label_en": "E-SHEAR condition",
      "definition_zh": "第13節首次引入 E-SHEAR，第14節在固定遲滯比 $\\beta_0=\\vartheta\\beta_1$ 下證明它給出真正 $O(1)$ 的臨界低模渦量／應變負荷，與 Cheskidov–Dai frequency-localized 判準同一導數層級。",
      "definition_en": "First introduced in Section 13, E-SHEAR is shown in Section 14 — under a fixed hysteresis ratio $\\beta_0=\\vartheta\\beta_1$ — to give a genuine $O(1)$ critical low-mode vorticity/strain toll, at the same derivative level as the Cheskidov–Dai frequency-localized criterion.",
      "defining_relation": "\\int_I G_{<q}(t)\\,dt \\ge c\\,\\frac{s_0}{\\beta_1} \\;\\xrightarrow{\\ \\beta_0=\\vartheta\\beta_1\\ }\\; \\int_I G_{<q}(t)\\,dt \\gtrsim 1",
      "notes": "Renamed the \"UV–Low-Strain Synchronization Motif\" $\\mathrm{M}_2$ later in the round (§35, §37)."
    },
    {
      "id": "ns.c4.c4e.near_far_hh_split",
      "latex": "\\mathfrak h_q = \\mathfrak h_q^{near,L} + \\mathfrak h_q^{far,L}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "高高壅塞近／遠分解",
      "label_en": "Near/far high-high split",
      "definition_zh": "第15節固定 $L\\ge C_0$，把高高壅塞量分解成近部 $\\mathfrak h_q^{near,L}$（$q-C_0\\le p\\le q+L$）與遠部 $\\mathfrak h_q^{far,L}$（$p>q+L$）。",
      "definition_en": "Section 15 fixes $L\\ge C_0$ and splits the high-high congestion into a near part $\\mathfrak h_q^{near,L}$ ($q-C_0\\le p\\le q+L$) and a far part $\\mathfrak h_q^{far,L}$ ($p>q+L$).",
      "notes": "Section 16 shows $\\mathfrak h_q^{near,L}\\le C_L\\beta_1^2$ under the frontier cap — the key input to the Far-Relay Theorem."
    },
    {
      "id": "ns.c4.c4e.theorem_small_threshold_far_relay",
      "latex": "\\beta_1\\le\\dfrac{c_\\vartheta}{2\\theta C_L} \\;\\Rightarrow\\; \\int_I \\mathfrak h_q^{far,L}\\,d\\tau \\ge c\\beta_1",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "小閾值遠距接力定理",
      "label_en": "Small-Threshold Far-Relay Theorem",
      "definition_zh": "第17節定理 C4-E.5 證明當閾值 $\\beta_1$ 足夠小時，E-HH 分支必進一步把高高源容量逼到嚴格更高頻的遠部 $p\\ge q+L$，本文稱之為 Strict Higher-Frequency Source Relay。",
      "definition_en": "Section 17's Theorem C4-E.5 proves that once the threshold $\\beta_1$ is sufficiently small, the E-HH branch must further push the high-high source capacity into the strictly-higher-frequency far part $p\\ge q+L$, termed the Strict Higher-Frequency Source Relay.",
      "defining_relation": "\\beta_1\\le\\frac{c_\\vartheta}{2\\theta C_L} \\;\\Rightarrow\\; \\int_I \\mathfrak h_q^{far,L}\\,d\\tau \\ge c\\beta_1",
      "notes": "Feeds directly into C4-E.6 (§20), where this far-relay mechanism is identified with C4-D's Rank Defect to form the Higher-Frequency Relay motif."
    },
    {
      "id": "ns.c4.c4e.higher_frequency_relay_motif",
      "latex": "\\mathrm{M}_4:\\ \\text{Higher-Frequency Relay}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "高頻接力母題",
      "label_en": "Higher-Frequency Relay motif",
      "definition_zh": "第20節（C4-E.6）證明 C4-D 的 Rank Defect 與 C4-E 的 Strict Higher-Frequency Source Relay 雖非同一數值 observable，卻屬同一結構母題，統一命名為 Higher-Frequency Relay（即 $\\mathrm{M}_4$）。",
      "definition_en": "Section 20 (C4-E.6) shows that C4-D's Rank Defect and C4-E's Strict Higher-Frequency Source Relay, while not the same numerical observable, share one structural motif, unified as the Higher-Frequency Relay ($\\mathrm{M}_4$).",
      "notes": "This is the round's first genuine branch merge (§44): Source Overcapacity + Rank Defect now share one \"higher-frequency provenance\" certificate (§20 boxed statement)."
    },
    {
      "id": "ns.c4.c4e.relay_to_active_parent_gap",
      "latex": "q\\longleftarrow p,\\quad p\\ge q+L \\;\\not\\Rightarrow\\; a_p\\ge\\beta",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "接力—活躍母代缺口",
      "label_en": "Relay-to-Active-Parent gap",
      "definition_zh": "第21節指出 Higher-Frequency Relay 只給出有向的絕對頻率邊 $q\\leftarrow p$（$p\\ge q+L$），目前無法直接推出 $a_p\\ge\\beta$，此缺口稱為 Relay-to-Active-Parent Bridge，第39節再稱為 Relay-to-Activity Gap。",
      "definition_en": "Section 21 notes that Higher-Frequency Relay only yields a directed absolute-frequency edge $q\\leftarrow p$ ($p\\ge q+L$), which cannot yet imply $a_p\\ge\\beta$; this gap is named the Relay-to-Active-Parent Bridge in §21 and the Relay-to-Activity Gap in §39.",
      "notes": "Listed as proof obligations F1/F2 for the next round C4-F (§51)."
    },
    {
      "id": "ns.c4.c4e.homochiral_triad_exact_split",
      "latex": "(\\dot e_k,\\dot e_p,\\dot e_q) = \\Theta(p-q,\\,q-k,\\,k-p)",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "同手性三元組精確分裂",
      "label_en": "Homochiral triad exact split",
      "definition_zh": "第22節對 Class I（$(+++)$）同手性三元組寫出精確能量導數 $(\\dot e_k,\\dot e_p,\\dot e_q)=\\Theta(p-q,q-k,k-p)$，當最高模 $q$ 得能（$\\Theta<0$）時同時給出 $g_q=(p-k)|\\Theta|>0$ 與 $g_k=(q-p)|\\Theta|>0$，且 $-\\dot e_p=g_q+g_k$。",
      "definition_en": "Section 22 writes the exact triad energy derivative $(\\dot e_k,\\dot e_p,\\dot e_q)=\\Theta(p-q,q-k,k-p)$ for the homochiral Class I ($(+++)$) triad; when the highest mode $q$ gains energy ($\\Theta<0$) this simultaneously gives $g_q=(p-k)|\\Theta|>0$ and $g_k=(q-p)|\\Theta|>0$, with $-\\dot e_p=g_q+g_k$.",
      "defining_relation": "g_q:=\\dot e_q=(p-k)|\\Theta|>0,\\qquad g_k:=\\dot e_k=(q-p)|\\Theta|>0,\\qquad -\\dot e_p=(q-k)|\\Theta|=g_q+g_k",
      "notes": "This exact algebra (from Waleffe) underlies §23's \"homochiral high-mode gain is bidirectional\" and is explicitly contrasted with Biferale–Titi's decimated-model regularity result in §46."
    },
    {
      "id": "ns.c4.c4e.lemma_homochiral_gap_or_reverse",
      "latex": "g_q>0 \\;\\Rightarrow\\; \\text{E-HGAP}\\ \\vee\\ \\text{E-HREV}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "同手性間隙或反向共得引理",
      "label_en": "Homochiral Gap-or-Reverse-Co-Gain Lemma",
      "definition_zh": "第24節引理 C4-E.7 證明對固定 $0<\\delta<1$，只要最高模得能 $g_q>0$，則必有 E-HGAP（$q-p<\\delta(p-k)$，徑向間隙退化）或 E-HREV（$g_k\\ge\\delta g_q$，最小模可比得能）成立，證明只需 $g_k/g_q=(q-p)/(p-k)$。",
      "definition_en": "Section 24's Lemma C4-E.7 proves that for fixed $0<\\delta<1$, whenever the highest mode gains energy ($g_q>0$), either E-HGAP ($q-p<\\delta(p-k)$, radial gap degeneration) or E-HREV ($g_k\\ge\\delta g_q$, comparable co-gain of the smallest mode) must hold, via the identity $g_k/g_q=(q-p)/(p-k)$.",
      "defining_relation": "\\text{E-HGAP}:\\ q-p<\\delta(p-k) \\qquad\\vee\\qquad \\text{E-HREV}:\\ g_k\\ge \\delta g_q",
      "notes": "E-HGAP/E-HREV labels are reused in §26's homochiral branch-compression trichotomy alongside E-HNONLOCAL ($k/q<c_L$)."
    },
    {
      "id": "ns.c4.c4e.bidirectional_critical_work_split",
      "latex": "kg_k \\ge c_L\\delta\\,qg_q",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "雙向臨界作功分裂",
      "label_en": "Bidirectional Critical Work Split",
      "definition_zh": "第25節在額外假設 $k\\ge c_Lq$ 下，由 E-HREV 導出臨界加權版本 $kg_k\\ge c_L\\delta\\,qg_q$，說明未退化的局部同手性 UV 得能必同時產生可比的臨界加權低模得能，本文稱之為 Bidirectional Critical Work Split。",
      "definition_en": "Section 25, under the additional assumption $k\\ge c_Lq$, upgrades E-HREV to the critical-weighted bound $kg_k\\ge c_L\\delta\\,qg_q$, showing that non-degenerate local homochiral UV gain necessarily produces comparable critical-weighted lower-mode gain — named the Bidirectional Critical Work Split."
    },
    {
      "id": "ns.c4.c4e.near_equilateral_radial_condensation",
      "latex": "\\text{Class III}:\\quad q-k<\\delta q \\;\\Rightarrow\\; (1-\\delta)q<k\\le p\\le q",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "近等邊徑向凝聚",
      "label_en": "Near-Equilateral Radial Condensation",
      "definition_zh": "第28節定義 Class III 徑向退化 $q-k<\\delta q$，由 $k\\le p\\le q$ 立即得三個徑向量級全落在相對厚度 $\\delta$ 內，本文稱為 Near-Equilateral Radial Condensation。",
      "definition_en": "Section 28 defines the Class III radial degeneration $q-k<\\delta q$; since $k\\le p\\le q$, this immediately forces all three radial magnitudes into a relative thickness $\\delta$, named Near-Equilateral Radial Condensation.",
      "notes": "Contrasted with Class II (§27, $q-p\\le k$, nonlocal two-high/one-low geometry); together with homochiral upper-gap collapse these feed the Spectral-Geometry Degeneration Motif (§29)."
    },
    {
      "id": "ns.c4.c4e.spectral_geometry_degeneration_motif",
      "latex": "\\mathrm{M}_6:\\ \\text{Spectral-Geometry Degeneration}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "頻譜幾何退化母題",
      "label_en": "Spectral-Geometry Degeneration motif",
      "definition_zh": "第29節把強非局部性、Class II 上間隙塌縮、Class III 近等邊徑向凝聚與同手性上間隙塌縮統一記為 Spectral-Geometry Degeneration（即 $\\mathrm{M}_6$），其共通角色是使螺旋共享事件耦合係數失去固定下界。",
      "definition_en": "Section 29 unifies strong nonlocality, Class II upper-gap collapse, Class III near-equilateral radial condensation, and homochiral upper-gap collapse as Spectral-Geometry Degeneration ($\\mathrm{M}_6$), whose common role is that the helical shared-event coupling coefficient loses its fixed lower bound.",
      "notes": "Also absorbs C4-D's \"Homochiral Dominance\" branch whenever E-HGAP/E-HNONLOCAL fires rather than E-HREV (§26)."
    },
    {
      "id": "ns.c4.c4e.critical_work_variation_motif",
      "latex": "\\mathfrak V_q^{work} = \\dfrac{\\lambda_q}{\\nu^2}\\int_I\\left(W_q^++W_q^-\\right)dt",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "臨界作功變異母題",
      "label_en": "Critical Work-Variation motif",
      "definition_zh": "第31節定義 $\\mathfrak V_q^{work}=\\frac{\\lambda_q}{\\nu^2}\\int_I(W_q^++W_q^-)dt$，第30節證明空間作功抵消、穩健螺旋抵消與同手性雙向反向得能三個 C4-D 分支都強迫 $\\mathfrak V_q^{work}\\gtrsim1$，統一稱為 Critical Work-Variation Motif（即 $\\mathrm{M}_5$）。",
      "definition_en": "Section 31 defines $\\mathfrak V_q^{work}=\\frac{\\lambda_q}{\\nu^2}\\int_I(W_q^++W_q^-)dt$; Section 30 shows spatial work cancellation, robust helical cancellation, and homochiral bidirectional reverse co-gain — three C4-D branches — all force $\\mathfrak V_q^{work}\\gtrsim1$, unified as the Critical Work-Variation Motif ($\\mathrm{M}_5$).",
      "defining_relation": "\\mathfrak V_q^{work} = \\frac{\\lambda_q}{\\nu^2}\\int_I\\left(W_q^++W_q^-\\right)dt \\gtrsim 1",
      "notes": "§32 stresses this is genuinely unconstrained: ordinary energy balance only controls $W_q^+-W_q^-$, not $W_q^++W_q^-$ — the \"signed balance ≠ total variation\" gap recurring from C3/C4."
    },
    {
      "id": "ns.c4.c4e.theorem_uv_motif_compression",
      "latex": "\\{\\mathrm{M}_1,\\ldots,\\mathrm{M}_6\\}",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "UV遞迴母題壓縮定理",
      "label_en": "UV Recurrent Motif Compression Theorem",
      "definition_zh": "第37節定理37.1（C4-E.8）在 frontier-safe、固定遲滯比與充分小閾值 $\\beta_1\\le\\beta_\\ast(L,\\theta,\\vartheta)$ 下，證明每個 critical UV shell crossing 必進六類母題之一：三個 closure-friendly（$\\mathrm{M}_1$ Persistence、$\\mathrm{M}_2$ Low-Strain Sync、$\\mathrm{M}_3$ Helical Sync）與三個真正未解的 escape 母題（$\\mathrm{M}_4$ Relay、$\\mathrm{M}_5$ Work Variation、$\\mathrm{M}_6$ Spectral Degeneration）。",
      "definition_en": "Section 37's Theorem 37.1 (C4-E.8) proves that under frontier-safety, a fixed hysteresis ratio, and a sufficiently small threshold $\\beta_1\\le\\beta_\\ast(L,\\theta,\\vartheta)$, every critical UV shell crossing falls into one of six motifs: three closure-friendly ($\\mathrm{M}_1$ Persistence, $\\mathrm{M}_2$ Low-Strain Sync, $\\mathrm{M}_3$ Helical Sync) and three genuinely unresolved escape motifs ($\\mathrm{M}_4$ Relay, $\\mathrm{M}_5$ Work Variation, $\\mathrm{M}_6$ Spectral Degeneration).",
      "defining_relation": "\\text{every crossing}\\in\\{\\mathrm{M}_1,\\mathrm{M}_2,\\mathrm{M}_3,\\mathrm{M}_4,\\mathrm{M}_5,\\mathrm{M}_6\\},\\quad \\beta_1\\le\\beta_\\ast(L,\\theta,\\vartheta)",
      "notes": "§38 shows an infinite blow-up sequence must recur in a fixed $\\mathrm{M}_*$; if $\\mathrm{M}_*\\in\\{\\mathrm{M}_1,\\mathrm{M}_2,\\mathrm{M}_3\\}$ this already yields new synchronization, so the genuinely unresolved recurrent case is $\\mathrm{M}_4\\vee\\mathrm{M}_5\\vee\\mathrm{M}_6$ (§38, §53)."
    },
    {
      "id": "ns.c4.c4e.true_etn_uv_state",
      "latex": "\\Theta_n^{UV} = \\langle q_n,\\beta_0,\\beta_1,R_{q_n}^\\sigma,G_{<q_n},\\mathfrak h_{q_n}^{far},\\mathfrak V_{q_n}^{work},\\operatorname{SpectralGeometry},\\operatorname{HelicalNet},\\operatorname{RelayEdge}\\rangle",
      "series": "NS",
      "first_appearance": "C4-E",
      "label_zh": "UV母題狀態元組",
      "label_en": "UV motif state tuple",
      "definition_zh": "第48節把 True ETN 框架更新為 UV 母題狀態元組 $\\Theta_n^{UV}$，紀錄第 $n$ 次 crossing 的殼層指標、閾值、去傳輸剩餘、低模負載、遠高高壅塞、作功變異與各幾何／螺旋／接力指標，並配上母題標籤 $\\mathsf M_n\\in\\{\\mathrm{M}_1,\\ldots,\\mathrm{M}_6\\}$。",
      "definition_en": "Section 48 updates the True ETN framework to the UV motif state tuple $\\Theta_n^{UV}$, recording the $n$-th crossing's shell index, thresholds, transport-free remainder, low-mode load, far high-high congestion, work variation, and geometric/helical/relay indicators, tagged with a motif label $\\mathsf M_n\\in\\{\\mathrm{M}_1,\\ldots,\\mathrm{M}_6\\}$.",
      "notes": "True ETN is the cross-round bookkeeping object referenced throughout the AMRAL NS series (see Internal Dependencies: \"True ETN / 無限維張力場\")."
    },
    {
      "id": "ns.c4.c4f.motif_m4_higher_frequency_relay",
      "latex": "M_4",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "高頻中繼",
      "label_en": "Higher-Frequency Relay",
      "definition_zh": "M4 是承襲自 C4-E、於 §0 重申的三個殘存 UV 逃逸機制之一，指高頻殼層透過 far high-high 交互作用向固定接收殼層 q 中繼能量的機制，並於 §2 起展開。",
      "definition_en": "M4 is one of the three residual unsynchronized UV-escape motifs carried over from C4-E and restated in §0, denoting the mechanism by which far high-high shell interactions relay energy into a fixed receiving shell q, developed from §2.",
      "notes": "Carried over from NS_C4E_RecurrentEscapeBranch_UVMotifCompression_v0.1.md; C4-F proves M4 forces Critical Far-UV Tail Stock + Effective Parent Multiplicity (§36)."
    },
    {
      "id": "ns.c4.c4f.motif_m5_critical_work_variation",
      "latex": "M_5",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "臨界作功變異",
      "label_en": "Critical Work Variation",
      "definition_zh": "M5 是三個殘存 UV 逃逸機制之一，於 §0 重申、§15 起展開，指殼層 q 上 transport-free 作功項的號誌變異在臨界尺度上不消失的機制。",
      "definition_en": "M5 is one of the three residual UV-escape motifs, restated in §0 and developed from §15, denoting the mechanism by which signed transport-free work on shell q fails to vanish at the critical scale.",
      "notes": "Carried over from C4-E; C4-F proves M5 forces a Fixed Strain/Deformation-Forcing Impulse routed through the Miller/vorticity/transport trichotomy (§36)."
    },
    {
      "id": "ns.c4.c4f.motif_m6_spectral_geometry_degeneration",
      "latex": "M_6",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "譜幾何退化",
      "label_en": "Spectral-Geometry Degeneration",
      "definition_zh": "M6 是三個殘存 UV 逃逸機制之一，於 §0 重申、§25 起展開，指 triad 幾何在正規化徑向座標下退化到 measure-zero 邊界集合的機制。",
      "definition_en": "M6 is one of the three residual UV-escape motifs, restated in §0 and developed from §25, denoting the mechanism by which triad geometry degenerates toward measure-zero boundary sets in normalized radial coordinates.",
      "notes": "Carried over from C4-E; C4-F proves M6 forces Radial Triad-Work Concentration (§36)."
    },
    {
      "id": "ns.c4.c4f.far_high_high_relay_source",
      "latex": "R_{q,L}^{far}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "遠場 high-high 中繼源",
      "label_en": "Far high-high relay source",
      "definition_zh": "於 §2 定義為固定接收殼層 q 上、由頻率間隔至少 L 的 high-high 乘積 $u_p\\otimes u_r$ 經 $T_q^\\sigma=\\Delta_qP^\\sigma\\mathbb P$ 投影取散度後所得的遠場來源項。",
      "definition_en": "Defined in §2 as the far-field source term on fixed receiving shell q obtained by applying $T_q^\\sigma=\\Delta_qP^\\sigma\\mathbb P$ and divergence to high-high products $u_p\\otimes u_r$ separated by at least dyadic distance L.",
      "defining_relation": "R_{q,L}^{far} = T_q^\\sigma \\nabla\\cdot \\sum_{\\substack{p\\ge q+L\\\\|r-p|\\le C_0}} u_p\\otimes u_r, \\qquad T_q^\\sigma=\\Delta_qP^\\sigma\\mathbb P",
      "notes": "Directly continues the M4/Higher-Frequency Relay construction inherited from C4-E."
    },
    {
      "id": "ns.c4.c4f.far_kinetic_energy_tail",
      "latex": "E_{>q+L-C_0}(t)",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "遠場動能尾",
      "label_en": "Far kinetic-energy tail",
      "definition_zh": "於 §3 定義為頻率指標 $p\\ge q+L-C_0$ 之上所有殼層動能 $\\|u_p(t)\\|_2^2$ 的總和，是 Theorem 4.1 low-output 上界的來源量。",
      "definition_en": "Defined in §3 as the sum of shell kinetic energies $\\|u_p(t)\\|_2^2$ over all $p\\ge q+L-C_0$, the source quantity bounding the low-output estimate of Theorem 4.1.",
      "defining_relation": "E_{>q+L-C_0}(t) = \\sum_{p\\ge q+L-C_0} \\|u_p(t)\\|_2^2"
    },
    {
      "id": "ns.c4.c4f.thm_low_output_high_high_bound",
      "latex": "\\text{C4-F.1 (Thm 4.1)}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "低輸出 high-high 能量尾界定理",
      "label_en": "Low-Output High-High Energy-Tail Bound",
      "definition_zh": "§4 定理 4.1（C4-F.1）利用 $\\Delta_q\\nabla$ 核的 $L^1\\to L^\\infty$ 界 $\\|K_q\\|_\\infty\\lesssim\\lambda_q^4$ 與 Cauchy–Schwarz，證明遠場中繼源可由遠場動能尾以因子 $\\lambda_q^4$ 控制。",
      "definition_en": "Theorem 4.1 (C4-F.1) in §4 uses the $L^1\\to L^\\infty$ kernel bound $\\|K_q\\|_\\infty\\lesssim\\lambda_q^4$ together with Cauchy–Schwarz to control the far relay source by the far kinetic-energy tail with factor $\\lambda_q^4$.",
      "defining_relation": "\\|R_{q,L}^{far}(t)\\|_\\infty \\le C\\lambda_q^4 E_{>q+L-C_0}(t)"
    },
    {
      "id": "ns.c4.c4f.normalized_relay_impulse",
      "latex": "\\mathfrak S_{q,L}^{relay}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "正規化中繼衝量",
      "label_en": "Normalized relay impulse",
      "definition_zh": "於 §5 以黏性時間 $d\\tau=\\nu\\lambda_q^2dt$ 正規化定義，量測遠場中繼源在窗口 I 上累積的無量綱衝量，recurrent relay event 假設其支付下界 $s_R>0$。",
      "definition_en": "Defined in §5 via the viscous time rescaling $d\\tau=\\nu\\lambda_q^2dt$ as the dimensionless accumulated impulse of the far relay source over window I, assumed under a recurrent relay event to pay a fixed toll $s_R>0$.",
      "defining_relation": "\\mathfrak S_{q,L}^{relay} = \\int_I \\frac{\\|R_{q,L}^{far}\\|_\\infty}{\\nu^2\\lambda_q^3}\\,d\\tau = \\frac{1}{\\nu\\lambda_q}\\int_I \\|R_{q,L}^{far}(t)\\|_\\infty\\,dt \\ge s_R>0"
    },
    {
      "id": "ns.c4.c4f.thm_relay_to_critical_tail_stock",
      "latex": "\\text{C4-F.2}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "中繼-臨界尾儲量定理",
      "label_en": "Relay-to-Critical-Tail-Stock Theorem",
      "definition_zh": "§7 之 C4-F.2 結合定理 4.1 與均值論證，證明存在 $t_\\ast\\in I$ 使遠場臨界 Sobolev 儲量被中繼衝量下界 $s_R$ 乘上 $2^L$ 因子強迫為非退化。",
      "definition_en": "C4-F.2 in §7 combines Theorem 4.1 with a mean-value argument to show there exists $t_\\ast\\in I$ at which the far critical Sobolev stock is forced nondegenerate, scaled by a factor $2^L$ against the relay toll $s_R$.",
      "defining_relation": "\\mathfrak H_{>q+L-C_0}(t_\\ast) \\ge c\\,2^L\\frac{s_R}{\\theta}",
      "notes": "§8 warns the admissible threshold $\\beta_\\ast(L)$ may shrink as L grows, so the $2^L$ factor is only used at fixed L, not to claim an arbitrarily large lower bound."
    },
    {
      "id": "ns.c4.c4f.far_critical_sobolev_stock",
      "latex": "\\mathfrak H_{>q+L-C_0}(t)",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "遠場臨界 Sobolev 儲量",
      "label_en": "Far critical Sobolev ($\\dot H^{1/2}$) stock",
      "definition_zh": "於 §7 定義為尾端殼層上 $\\lambda_p\\|u_p(t)\\|_2^2/\\nu^2$ 的總和，即 far-UV 部分的絕對臨界（$\\dot H^{1/2}$型）儲量。",
      "definition_en": "Defined in §7 as the sum of $\\lambda_p\\|u_p(t)\\|_2^2/\\nu^2$ over tail shells, the far-UV portion of the absolute critical ($\\dot H^{1/2}$-type) stock.",
      "defining_relation": "\\mathfrak H_{>q+L-C_0}(t) = \\frac{1}{\\nu^2}\\sum_{p\\ge q+L-C_0} \\lambda_p\\|u_p(t)\\|_2^2",
      "notes": "§41 identifies this with the far-UV portion of the absolute helical critical stock via $u_p=u_p^++u_p^-$, so Relay already establishes UV to far helical critical stock."
    },
    {
      "id": "ns.c4.c4f.frontier_subcriticality_ratio",
      "latex": "a_p(t)=\\dfrac{\\|u_p(t)\\|_\\infty}{\\nu\\lambda_p}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "前緣次臨界比",
      "label_en": "Frontier subcriticality ratio",
      "definition_zh": "於 §9 定義為殼層 p 的 $L^\\infty$ 振幅與黏性尺度 $\\nu\\lambda_p$ 之比，first-frontier safe state 假設所有 $p\\ge q+L-C_0$ 皆滿足 $a_p(t)\\le\\beta_1$。",
      "definition_en": "Defined in §9 as the ratio of shell-p $L^\\infty$ amplitude to the viscous scale $\\nu\\lambda_p$, with the first-frontier safe state hypothesis requiring $a_p(t)\\le\\beta_1$ for all $p\\ge q+L-C_0$.",
      "notes": "$\\beta_1$ recurs as the fixed threshold through §12 (fixed-ratio crossing, with $\\beta_0=\\vartheta\\beta_1$) and the final status table in §51."
    },
    {
      "id": "ns.c4.c4f.effective_shell_cell_multiplicity",
      "latex": "m_p^{eff}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "有效殼層多重度",
      "label_en": "Effective shell-cell multiplicity",
      "definition_zh": "於 §10 定義為 $\\lambda_p^3\\|u_p\\|_2^2/\\|u_p\\|_\\infty^2$，是一個無量綱有效體積診斷量，衡量殼層 p 能量分佈的離域程度而非字面封包數。",
      "definition_en": "Defined in §10 as $\\lambda_p^3\\|u_p\\|_2^2/\\|u_p\\|_\\infty^2$, a dimensionless effective-volume diagnostic measuring the delocalization of shell-p energy rather than a literal packet count.",
      "defining_relation": "m_p^{eff} = \\lambda_p^3\\,\\frac{\\|u_p\\|_2^2}{\\|u_p\\|_\\infty^2} \\quad (m_p^{eff}:=0 \\text{ if } u_p=0)",
      "notes": "Guard G-EFFCELL (§44) reiterates that $m_p^{eff}$ is an effective-volume/cell diagnostic, not a literal packet count."
    },
    {
      "id": "ns.c4.c4f.thm_subcritical_parent_multiplicity",
      "latex": "\\text{C4-F.3}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "次臨界母代多重度界定理",
      "label_en": "Subcritical Parent Multiplicity Bound",
      "definition_zh": "§11 之 C4-F.3 透過殼層臨界儲量 $h_p=\\lambda_p\\|u_p\\|_2^2/\\nu^2$ 與前緣次臨界假設 $\\|u_p\\|_\\infty\\le\\nu\\beta_1\\lambda_p$，證明尾端有效多重度總和被 $\\mathfrak H_{>q+L-C_0}/\\beta_1^2$ 由下方界定。",
      "definition_en": "C4-F.3 in §11 uses the shell critical stock $h_p=\\lambda_p\\|u_p\\|_2^2/\\nu^2$ together with the first-frontier bound $\\|u_p\\|_\\infty\\le\\nu\\beta_1\\lambda_p$ to show the tail sum of effective multiplicities is bounded below by $\\mathfrak H_{>q+L-C_0}/\\beta_1^2$.",
      "defining_relation": "\\sum_{p\\ge q+L-C_0} m_p^{eff}(t_\\ast) \\ge \\frac{\\mathfrak H_{>q+L-C_0}(t_\\ast)}{\\beta_1^2} \\ge c\\,\\frac{2^Ls_R}{\\theta\\beta_1^2}"
    },
    {
      "id": "ns.c4.c4f.tail_stock_to_active_parent_gap",
      "latex": "\\text{Tail-Stock-to-Active-Parent / Packetization Gap}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "尾儲量—活躍母代（封包化）缺口",
      "label_en": "Tail-Stock-to-Active-Parent / Packetization Gap",
      "definition_zh": "於 §13 將 C4-E 遺留的「higher-frequency participation ⇒ active parent」缺失引理重新命名，因 C4-F 只證得 Relay ⇒ Critical Tail Stock + Effective Parent Multiplicity，尚未證得單一活躍母代。",
      "definition_en": "Introduced in §13 as the renamed form of the missing C4-E lemma \"higher-frequency participation implies active parent,\" since C4-F only establishes Relay implies Critical Tail Stock + Effective Parent Multiplicity, not a single active parent.",
      "notes": "Reappears as proof obligation G1 in the C4-G roadmap (§49)."
    },
    {
      "id": "ns.c4.c4f.transport_free_remainder",
      "latex": "R_q^\\sigma",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "傳輸自由餘項",
      "label_en": "Transport-free remainder",
      "definition_zh": "沿用 C4-E 於 §15 定義為非線性項 $N_q^\\sigma$ 減去低頻傳輸項 $u_{\\le q-L_0}\\cdot\\nabla u_q^\\sigma$ 後的餘項，其 Fourier 支撐固定於殼層 q 附近（§17）。",
      "definition_en": "Carried from C4-E and defined in §15 as the nonlinear term $N_q^\\sigma$ minus the low-frequency transport term $u_{\\le q-L_0}\\cdot\\nabla u_q^\\sigma$, with Fourier support fixed near shell q (§17).",
      "notes": "§17 shows its Fourier support obeys $c\\lambda_q\\le|\\xi|\\le C\\lambda_q$, giving the Bernstein-type bound $\\|\\nabla R_q^\\sigma\\|_2\\ge c\\lambda_q\\|R_q^\\sigma\\|_2$ used to derive C4-F.5 in §19."
    },
    {
      "id": "ns.c4.c4f.absolute_transport_free_work",
      "latex": "A_q(t)",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "絕對傳輸自由作功",
      "label_en": "Absolute transport-free work",
      "definition_zh": "於 §15 定義為 $|f_q^\\sigma\\cdot R_q^\\sigma|$（其中 $f_q^\\sigma=u_q^\\sigma$）對空間積分，是作功變異 motif $\\mathfrak V_q^{work}$ 的被積量。",
      "definition_en": "Defined in §15 as the spatial integral of $|f_q^\\sigma\\cdot R_q^\\sigma|$ with $f_q^\\sigma=u_q^\\sigma$, the integrand feeding the work-variation motif $\\mathfrak V_q^{work}$."
    },
    {
      "id": "ns.c4.c4f.work_variation_motif",
      "latex": "\\mathfrak V_q^{work}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "作功變異量",
      "label_en": "Work-variation quantity",
      "definition_zh": "於 §15 定義為 $\\lambda_q/\\nu^2$ 乘上 $A_q(t)$ 在窗口 I 上的時間積分，是 motif M5 的核心量，假設保持下界 $v_0>0$。",
      "definition_en": "Defined in §15 as $\\lambda_q/\\nu^2$ times the time integral of $A_q(t)$ over window I, the central quantity of motif M5, assumed to satisfy a fixed lower bound $v_0>0$.",
      "defining_relation": "\\mathfrak V_q^{work} = \\frac{\\lambda_q}{\\nu^2}\\int_I A_q(t)\\,dt \\ge v_0>0"
    },
    {
      "id": "ns.c4.c4f.thm_work_variation_source_impulse",
      "latex": "\\text{C4-F.4}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "作功變異迫使非線性源衝量定理",
      "label_en": "Work Variation Forces a Nonlinear Source Impulse",
      "definition_zh": "§16 之 C4-F.4 利用 Cauchy–Schwarz 與能量不等式 $\\|f_q^\\sigma\\|_2\\le\\|u_0\\|_2$，把 $\\mathfrak V_q^{work}\\ge v_0$ 轉成餘項 $R_q^\\sigma$ 在 $L_t^1L_x^2$ 下的固定衝量下界。",
      "definition_en": "C4-F.4 in §16 uses Cauchy–Schwarz and the energy bound $\\|f_q^\\sigma\\|_2\\le\\|u_0\\|_2$ to convert $\\mathfrak V_q^{work}\\ge v_0$ into a fixed $L_t^1L_x^2$ impulse lower bound on the remainder $R_q^\\sigma$."
    },
    {
      "id": "ns.c4.c4f.symmetric_gradient_operator",
      "latex": "\\mathscr S R = \\tfrac12(\\nabla R+\\nabla R^T)",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "對稱梯度（應變）算子",
      "label_en": "Symmetric gradient (strain) operator",
      "definition_zh": "於 §18 定義為向量場 R 的對稱梯度，並以 Korn 型恆等式 $\\|\\mathscr SR\\|_2^2=\\frac12\\|\\nabla R\\|_2^2+\\frac12\\|\\nabla\\cdot R\\|_2^2$ 把作功衝量升級為應變衝量。",
      "definition_en": "Defined in §18 as the symmetric gradient of a vector field R, related to the full gradient by the Korn-type identity $\\|\\mathscr SR\\|_2^2=\\frac12\\|\\nabla R\\|_2^2+\\frac12\\|\\nabla\\cdot R\\|_2^2$, which upgrades the work impulse into a strain impulse."
    },
    {
      "id": "ns.c4.c4f.thm_fixed_deformation_forcing_impulse",
      "latex": "\\text{C4-F.5}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "固定形變迫力衝量定理",
      "label_en": "Fixed Deformation-Forcing Impulse",
      "definition_zh": "§19 之 C4-F.5 結合 §16–18，證明 $\\mathscr SR_q^\\sigma$ 在窗口 I 上的 $L^2$ 時間積分被固定下界 $cv_0\\nu^2/\\|u_0\\|_2$ 支配，且右側不含 $\\lambda_q^{-1}$ 或 Zeno 權重。",
      "definition_en": "C4-F.5 in §19 combines §16–18 to show the time integral of $\\|\\mathscr SR_q^\\sigma\\|_2$ over window I is bounded below by a fixed $cv_0\\nu^2/\\|u_0\\|_2$, with no $\\lambda_q^{-1}$ or Zeno weight on the right side.",
      "defining_relation": "\\int_I \\|\\mathscr S R_q^\\sigma(t)\\|_2\\,dt \\ge c\\,\\frac{v_0\\nu^2}{\\|u_0\\|_2}",
      "notes": "§19 stresses this is not yet a contradiction: no known global a-priori budget $\\int_0^{T_\\ast}\\|\\mathscr SR_q\\|_2dt<\\infty$ exists to clash with it."
    },
    {
      "id": "ns.c4.c4f.projected_nonlinear_strain_forcing",
      "latex": "\\mathcal N_{\\rm proj}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "投影非線性應變迫力",
      "label_en": "Projected nonlinear strain-forcing operator",
      "definition_zh": "於 §20 定義為滿秩非線性項 $\\mathcal N_u=\\mathbb P(u\\cdot\\nabla u)$ 經對稱梯度 $\\mathscr S$ 作用後之結果，依 C3-Q 等於 $P_{st}((u\\cdot\\nabla)S+S^2+\\frac14\\omega\\otimes\\omega)$。",
      "definition_en": "Defined in §20 as $\\mathscr S\\mathcal N_u$ applied to the full nonlinear term $\\mathcal N_u=\\mathbb P(u\\cdot\\nabla u)$, equal by C3-Q to $P_{st}((u\\cdot\\nabla)S+S^2+\\frac14\\omega\\otimes\\omega)$.",
      "defining_relation": "\\mathscr S\\mathcal N_u = \\mathcal N_{\\rm proj} = P_{st}\\!\\left((u\\cdot\\nabla)S+S^2+\\tfrac14\\omega\\otimes\\omega\\right)",
      "notes": "§20 gives the exact identity $\\mathcal N_{\\rm proj}=\\mathcal Q_{SV}-\\tfrac12P_{st}(\\omega\\otimes\\omega)$ linking it to the Miller operator, sourced from C3-Q (NS_C3Q_PressureProjection_OperatorLocalization_v0.1.md)."
    },
    {
      "id": "ns.c4.c4f.miller_strain_vorticity_operator",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "Miller 應變—渦度算子",
      "label_en": "Miller strain-vorticity operator",
      "definition_zh": "於 §1 primary-source audit 引入、§20 給出公式，是 Miller 論文中 strain–vorticity 算子分解所得的算子，源自 C3-P，作為 UV-to-operator interface 的中心物件。",
      "definition_en": "Introduced in the §1 primary-source audit and formulated in §20, this is Miller's strain-vorticity decomposition operator from C3-P, the central object of the operator-level UV regularity interface.",
      "defining_relation": "\\mathcal Q_{SV} = P_{st}\\!\\left((u\\cdot\\nabla)S+S^2+\\tfrac34\\omega\\otimes\\omega\\right)",
      "notes": "Sourced from E. Miller, arXiv:2407.02691, via C3-P (NS_C3P_OperatorEscape_FarPressureMatrix_v0.1.md); differs from $\\mathcal N_{\\rm proj}$ only in the coefficient of $\\omega\\otimes\\omega$ ($3/4$ vs $1/4$)."
    },
    {
      "id": "ns.c4.c4f.projected_vorticity_quadratic_operator",
      "latex": "P_{st}(\\omega\\otimes\\omega)",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "投影渦度二次算子",
      "label_en": "Projected vorticity-quadratic operator",
      "definition_zh": "於 §1 與 §20 引用，是渦度自身外積 $\\omega\\otimes\\omega$ 經 $P_{st}$ 投影後的算子，是 C4-F.6 trichotomy 中 F-VORT 分支的核心來源項。",
      "definition_en": "Referenced in §1 and §20, this is the $P_{st}$-projection of the vorticity outer product $\\omega\\otimes\\omega$, the core source term of the F-VORT branch in the C4-F.6 trichotomy.",
      "notes": "An external Miller/C3-P primitive rather than an object defined from scratch in this round; enters the exact identity $\\mathcal N_{\\rm proj}=\\mathcal Q_{SV}-\\tfrac12P_{st}(\\omega\\otimes\\omega)$ of §20."
    },
    {
      "id": "ns.c4.c4f.shell_helicity_strain_multiplier",
      "latex": "\\mathscr T_{q,\\sigma}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "殼層/螺旋度應變乘子算子",
      "label_en": "Shell/helicity strain multiplier operator",
      "definition_zh": "於 §21 定義為使 $\\mathscr SN_q^\\sigma=\\mathscr T_{q,\\sigma}\\mathcal N_{\\rm proj}$ 成立的有界算子，源於 $\\mathscr S$ 與 dyadic/Fourier 乘子在 order-zero 誤差內可交換。",
      "definition_en": "Defined in §21 as the bounded operator satisfying $\\mathscr SN_q^\\sigma=\\mathscr T_{q,\\sigma}\\mathcal N_{\\rm proj}$, arising because $\\mathscr S$ commutes with dyadic/Fourier multipliers up to an order-zero strain-space multiplier.",
      "notes": "Used to phrase both the F-OP branch ($\\mathscr T_{q,\\sigma}\\mathcal Q_{SV}$) and the F-VORT branch ($\\mathscr T_{q,\\sigma}P_{st}(\\omega\\otimes\\omega)$) of the C4-F.6 trichotomy in §22."
    },
    {
      "id": "ns.c4.c4f.thm_work_variation_trichotomy",
      "latex": "\\text{C4-F.6 (F-OP} \\vee \\text{F-VORT} \\vee \\text{F-TR)}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "作功變異算子源三分定理",
      "label_en": "Work-Variation Operator-Source Trichotomy",
      "definition_zh": "§22 之 C4-F.6 以 $D_0=cv_0\\nu^2/\\|u_0\\|_2$ 為單位，證明 C4-F.5 的形變衝量必分配到 F-OP（Miller 算子分支）、F-VORT（渦度二次分支）或 F-TR（低頻傳輸形變分支）三者之一。",
      "definition_en": "C4-F.6 in §22 takes $D_0=cv_0\\nu^2/\\|u_0\\|_2$ as the unit and shows the deformation impulse of C4-F.5 must be routed into at least one of F-OP (Miller-operator branch), F-VORT (vorticity-quadratic branch), or F-TR (low-transport deformation branch).",
      "notes": "This is the round's UV-to-Operator Bridge (§42); whether F-OP/F-VORT/F-TR connect respectively to Miller global escape ratio, vortex-stretching geometry, or low-mode strain toll is left open for C4 (§23)."
    },
    {
      "id": "ns.c4.c4f.normalized_radial_work_measure",
      "latex": "\\widehat\\mu_q \\text{ on } \\mathcal D=\\{(x,y):0<x\\le y\\le1,\\ x+y\\ge1\\}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "正規化徑向交互作功測度",
      "label_en": "Normalized radial interaction-work measure",
      "definition_zh": "於 §25–26 定義：正規化徑向座標 $x=k/q,y=p/q$ 落在單純形 $\\mathcal D$ 上，將正的臨界 triad 作功 $[q\\dot e_q]_+dt$ 推前得有限測度 $\\mu_q$，再除以 $\\mu_q(\\mathcal D)$ 正規化成機率測度 $\\widehat\\mu_q$。",
      "definition_en": "Defined in §25–26: normalized radial coordinates $x=k/q,y=p/q$ live on the simplex $\\mathcal D$, and pushing forward the positive critical triad work $[q\\dot e_q]_+dt$ gives a finite measure $\\mu_q$, normalized by $\\mu_q(\\mathcal D)$ into the probability measure $\\widehat\\mu_q$.",
      "defining_relation": "\\mathcal D=\\{(x,y):0<x\\le y\\le1,\\ x+y\\ge1\\}, \\qquad \\widehat\\mu_q = \\frac{\\mu_q}{\\mu_q(\\mathcal D)}",
      "notes": "§43 stresses $\\widehat\\mu_q$ is a triad-work measure on radial interaction geometry, distinct from physical-space strain intermittency, Fourier energy density, or pressure concentration, and must not be automatically merged with them."
    },
    {
      "id": "ns.c4.c4f.class_ii_upper_gap_set",
      "latex": "D_{II}(\\delta)=\\{(x,y)\\in\\mathcal D: 1-y\\le\\delta\\}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "第二類上端隙退化集",
      "label_en": "Class-II upper-gap degeneration set",
      "definition_zh": "於 §28 定義為 $\\mathcal D$ 中 $1-y\\le\\delta$ 的子集，其 Lebesgue 面積滿足 $|D_{II}(\\delta)|\\le C\\delta$，即 $m=1$ 型退化。",
      "definition_en": "Defined in §28 as the subset of $\\mathcal D$ with $1-y\\le\\delta$, whose Lebesgue area satisfies $|D_{II}(\\delta)|\\le C\\delta$, an $m=1$-type degeneration.",
      "notes": "Parallels the nonlocal set $D_{NL}(\\chi)=\\{x\\le\\chi\\}$ (§27, $|D_{NL}|\\le C\\chi$) and the homochiral gap set $D_H(\\delta)$ (§30, $|D_H(\\delta)\\cap\\{x\\ge c_L\\}|\\le C_{c_L}\\delta$); the name echoes the Class-II nonlocality of C3-C (NS_C3C_ClassII_Nonlocality_Tax_Radial_Congestion_v0.1.md), and §34 shows a summability guard $\\sum\\chi_n=\\infty$ is separately needed for the Class-II route to sustain an infinite genealogy."
    },
    {
      "id": "ns.c4.c4f.class_iii_near_equilateral_set",
      "latex": "D_{III}(\\delta)=\\{(x,y)\\in\\mathcal D: 1-x\\le\\delta\\}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "第三類近等邊集",
      "label_en": "Class-III near-equilateral set",
      "definition_zh": "於 §29 定義為 $\\mathcal D$ 中 $1-x\\le\\delta$ 的子集，因 $x\\le y\\le1$ 其面積對小 $\\delta$ 精確等於 $\\frac12\\delta^2$，即 codimension 更強的 $m=2$ 型退化。",
      "definition_en": "Defined in §29 as the subset of $\\mathcal D$ with $1-x\\le\\delta$; since $x\\le y\\le1$, its area equals exactly $\\frac12\\delta^2$ for small $\\delta$, a codimension-stronger $m=2$-type degeneration.",
      "defining_relation": "|D_{III}(\\delta)| = \\tfrac12\\delta^2 \\quad \\text{(small } \\delta\\text{)}",
      "notes": "§31 unifies all degeneration sets as $|D_\\varepsilon|\\lesssim\\varepsilon^m$ with $m=1$ for nonlocal/Class-II/local-homochiral-gap sets and $m=2$ only for Class III, the sharper exponent driving the concentration lemma of C4-F.7."
    },
    {
      "id": "ns.c4.c4f.thm_radial_work_concentration",
      "latex": "\\text{C4-F.7}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "徑向作功集中引理",
      "label_en": "Radial Work-Concentration Lemma",
      "definition_zh": "§32 之 C4-F.7 證明：若一列退化事件的測度份額 $\\widehat\\mu_n(D_{\\varepsilon_n})\\ge\\rho_0>0$ 而集合面積 $|D_{\\varepsilon_n}|\\le C\\varepsilon_n^m$ 隨 $\\varepsilon_n\\to0$ 縮小，則 $\\{\\widehat\\mu_n\\}$ 不可能對 Lebesgue 測度保持一致絕對連續，其密度（若存在）滿足 $\\|g_n\\|_\\infty\\gtrsim\\varepsilon_n^{-m}$。",
      "definition_en": "C4-F.7 in §32 shows that if a sequence of degeneration events carries measure share $\\widehat\\mu_n(D_{\\varepsilon_n})\\ge\\rho_0>0$ while $|D_{\\varepsilon_n}|\\le C\\varepsilon_n^m$ shrinks as $\\varepsilon_n\\to0$, then $\\{\\widehat\\mu_n\\}$ cannot maintain uniform absolute continuity with respect to Lebesgue measure, and its density (if it exists) obeys $\\|g_n\\|_\\infty\\gtrsim\\varepsilon_n^{-m}$.",
      "defining_relation": "\\widehat\\mu_n(D_{\\varepsilon_n})\\ge\\rho_0,\\ |D_{\\varepsilon_n}|\\le C\\varepsilon_n^m \\ \\Longrightarrow\\ \\|g_n\\|_\\infty \\ge c\\rho_0\\varepsilon_n^{-m},\\quad \\|g_n\\|_2\\ge c\\rho_0\\varepsilon_n^{-m/2}",
      "notes": "Guard G-ABSCont (§44) records this as a uniform-absolute-continuity failure, not to be directly called a singularity contradiction."
    },
    {
      "id": "ns.c4.c4f.thm_uv_congestion_trilemma",
      "latex": "\\text{C4-F.8 (Thm 37.1)}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "UV 壅塞三難定理",
      "label_en": "UV Congestion Trilemma",
      "definition_zh": "§37 定理 37.1（C4-F.8）是本輪主結果：在 C4-E 的 frontier/hysteresis/small-threshold 假設下，若無限次臨界 UV crossing 永久避開三個已同步機制，則必有無限子序列落入 C-F1（Tail/Packet Congestion）、C-F2（Deformation/Operator Congestion）或 C-F3（Radial Interaction Congestion）三種壅塞類別之一。",
      "definition_en": "Theorem 37.1 (C4-F.8) in §37 is this round's headline result: under C4-E's frontier/hysteresis/small-threshold hypotheses, if infinitely many critical UV crossings permanently avoid the three synchronized mechanisms, an infinite subsequence must fall into one of three congestion classes — C-F1 (Tail/Packet Congestion), C-F2 (Deformation/Operator Congestion), or C-F3 (Radial Interaction Congestion).",
      "defining_relation": "UV \\text{ escape} \\ \\Longrightarrow\\ \\text{C-F1}\\ \\vee\\ \\text{C-F2}\\ \\vee\\ \\text{C-F3}",
      "notes": "§39 stresses Congestion is NOT yet Contradiction (no known finite global measure closes any of the three classes); §48 names the follow-up round C4-G — Cross-Congestion Synchronization and Phase-Space Closure."
    },
    {
      "id": "ns.c4.c4f.congestion_state_etn_tuple",
      "latex": "\\Theta_n^{cong}",
      "series": "NS",
      "first_appearance": "C4-F",
      "label_zh": "壅塞狀態 ETN 元組",
      "label_en": "Congestion-state ETN tuple",
      "definition_zh": "於 §45 定義為 True ETN 架構下本輪的狀態元組，收納尾儲量 $\\mathfrak H_{tail,n}$、有效多重度總和 $\\mathfrak M_{eff,n}$、算子形變積分 $\\mathfrak D_{op,n}$、徑向測度 $\\widehat\\mu_n$、退化尺度 $\\varepsilon_n$ 與 carrier provenance。",
      "definition_en": "Defined in §45 as this round's state tuple within the True ETN framework, packaging the tail stock $\\mathfrak H_{tail,n}$, summed effective multiplicity $\\mathfrak M_{eff,n}$, operator deformation integral $\\mathfrak D_{op,n}$, radial measure $\\widehat\\mu_n$, degeneration scale $\\varepsilon_n$, and carrier provenance.",
      "defining_relation": "\\Theta_n^{cong} = \\left\\langle \\mathfrak H_{tail,n}, \\mathfrak M_{eff,n}, \\mathfrak D_{op,n}, \\widehat\\mu_n, \\varepsilon_n, \\operatorname{CarrierProv} \\right\\rangle, \\quad \\mathfrak M_{eff,n}=\\!\\!\\sum_{p\\in tail}\\!\\! m_p^{eff},\\ \\ \\mathfrak D_{op,n}=\\int_{I_n}\\|\\mathscr SR_{q_n}^{\\sigma_n}\\|_2dt",
      "notes": "Ties this round's findings back into the shared True ETN / 無限維張力場 formalism referenced in the Internal Dependencies list."
    },
    {
      "id": "ns.c4.c4g.c_tp",
      "latex": "C_{TP}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "尾端/封包壅塞",
      "label_en": "Tail/Packet Congestion",
      "definition_zh": "承襲自 C4-F 的三種壅塞類別之一，代表 frequency/spatial-stock 邊際的壅塞，§0 與 §36 重申其定義。",
      "definition_en": "One of the three congestion classes inherited from C4-F, representing the frequency/spatial-stock marginal congestion, restated in §0 and §36.",
      "notes": "C4-G shows this is no longer an independent exit from C_DO but an extra phase-space coordinate riding on operator/deformation forcing events (§16, §36)."
    },
    {
      "id": "ns.c4.c4g.c_do",
      "latex": "C_{DO}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "形變/算子壅塞",
      "label_en": "Deformation/Operator Congestion",
      "definition_zh": "三種壅塞類別之一，代表 PDE-source 邊際的壅塞，本輪核心結論為它是所有未解 motifs 匯聚的普遍迫力漏斗（§0, §15）。",
      "definition_en": "One of the three congestion classes, representing the PDE-source marginal congestion, shown in this round to be the universal forcing funnel that every unresolved motif collapses into (§0, §15).",
      "notes": "§15 informally names this collapse the 'Cross-Congestion Funnel'; §16 corrects C4-F's trilemma so that C_DO is the universal channel while C_TP and C_RI become extra congestion coordinates riding on it."
    },
    {
      "id": "ns.c4.c4g.c_ri",
      "latex": "C_{RI}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "徑向交互作用壅塞",
      "label_en": "Radial Interaction Congestion",
      "definition_zh": "三種壅塞類別之一，代表 Fourier-triad 幾何邊際的壅塞，§14 證明它伴隨 M6 但同樣匯入 C_DO。",
      "definition_en": "One of the three congestion classes, representing the Fourier-triad geometry marginal congestion, shown in §14 to accompany M6 while still funneling into C_DO."
    },
    {
      "id": "ns.c4.c4g.strain_tensor",
      "latex": "S",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "應變張量",
      "label_en": "Strain tensor",
      "definition_zh": "§1 引用 Miller 應變演化方程中的主要場，滿足 \\partial_tS-\\Delta S+\\mathcal Q_{SV}=0。",
      "definition_en": "The principal field in Miller's strain-evolution equation cited in §1, satisfying \\partial_tS-\\Delta S+\\mathcal Q_{SV}=0."
    },
    {
      "id": "ns.c4.c4g.q_sv",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "Miller 全 N–S 缺陷算子",
      "label_en": "Miller's full N–S defect operator",
      "definition_zh": "§1 引用 Miller 定義相對 globally regular 應變–渦度交互模型的 full Navier–Stokes defect。",
      "definition_en": "Miller's full Navier–Stokes defect relative to the globally-regular strain–vorticity interaction model, cited in §1.",
      "defining_relation": "\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)",
      "notes": "§25 splits it as \\mathcal N_{\\rm proj}=\\mathcal Q_{SV}-\\frac12P_{st}(\\omega\\otimes\\omega); throughout §27-31 it is the target of the G-O1 Miller-operator impulse branch."
    },
    {
      "id": "ns.c4.c4g.miller_ratio",
      "latex": "\\frac{\\|\\mathcal Q_{SV}(t)\\|_2}{\\|-\\Delta S(t)\\|_2}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "Miller 比值判準",
      "label_en": "Miller ratio criterion",
      "definition_zh": "§1 引用 Miller 的 finite-time blow-up 必要條件，§30 與 H1（§47）將其作為 UV forcing events 接入的比值座標。",
      "definition_en": "Miller's necessary condition for finite-time blow-up cited in §1, used in §30 and obligation H1 (§47) as the ratio coordinate that UV forcing events feed into.",
      "defining_relation": "\\limsup_{t\\uparrow T_\\ast}\\frac{\\|\\mathcal Q_{SV}(t)\\|_2}{\\|-\\Delta S(t)\\|_2}\\ge1",
      "notes": "NG-G2 (§42) blocks the converse reading that a large deformation impulse alone forces this ratio to exceed 1; §30 stresses C4-G does not reprove Miller's theorem."
    },
    {
      "id": "ns.c4.c4g.m4",
      "latex": "M_4",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "高頻中繼",
      "label_en": "Higher-Frequency Relay",
      "definition_zh": "承襲自 C4-E/F 的三個未解 motif 之一，§2 標記為仍未解決，§3 起追蹤其兩種來源。",
      "definition_en": "One of the three unresolved motifs carried over from C4-E/F, flagged still-open in §2, with its two origin provenances traced from §3 onward.",
      "notes": "§15 places it in M_4\\subset C_{TP}\\cap C_{DO}; §12 (Thm 12.1) proves M_4\\Rightarrow C_{DO} via both Relay-S and Relay-W; §18-19 further shows it carries near-antipodal Fourier geometry."
    },
    {
      "id": "ns.c4.c4g.m5",
      "latex": "M_5",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "臨界工作變化",
      "label_en": "Critical Work Variation",
      "definition_zh": "三個未解 motif 之一，§2 重申其未解狀態，§15 指出其本身即屬於 C_DO。",
      "definition_en": "One of the three unresolved motifs, restated as open in §2, shown directly in §15 to already lie inside C_DO.",
      "notes": "Unlike M4 and M6, this round gives it no dedicated provenance section — §15 asserts M_5\\subset C_{DO} directly from its C4-F definition as a work-variation phenomenon."
    },
    {
      "id": "ns.c4.c4g.m6",
      "latex": "M_6",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "頻譜幾何退化",
      "label_en": "Spectral-Geometry Degeneration",
      "definition_zh": "三個未解 motif 之一，§13–14 證明其為 positive-work branch 的子情形，因此自動帶有 C_DO。",
      "definition_en": "One of the three unresolved motifs, proved in §13-14 to be a subcase of the positive-work branch and hence automatically carry C_DO.",
      "notes": "§15 places it in M_6\\subset C_{RI}\\cap C_{DO}; §21 lists its congestion coordinates as radial work-measure concentration plus positive/absolute shell work."
    },
    {
      "id": "ns.c4.c4g.relay_s",
      "latex": "\\text{Relay-S}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "來源型中繼",
      "label_en": "Source Relay",
      "definition_zh": "§3 定義的 M4 兩個來源之一，源自 source-overcapacity 經 high-high 到 far high-high 的 Strict Higher-Frequency Source Relay。",
      "definition_en": "One of M4's two origins defined in §3, arising from source-overcapacity via a high-high to far high-high Strict Higher-Frequency Source Relay.",
      "notes": "§10 (Thm C4-G.2) proves Relay-S automatically forces C_DO while retaining its C_TP side certificate."
    },
    {
      "id": "ns.c4.c4g.relay_w",
      "latex": "\\text{Relay-W}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "工作型中繼",
      "label_en": "Work Relay",
      "definition_zh": "§3 定義的 M4 另一來源，源自 positive shell-work branch 中由 p>q+L 的 triads 承擔 current shell 正工作的 Rank Defect。",
      "definition_en": "M4's other origin defined in §3, arising from a Rank Defect in the positive shell-work branch where triads with p>q+L carry the current shell's positive work.",
      "notes": "§12 (Thm C4-G.3) proves Relay-W\\Rightarrow C_{DO}, inheriting its forcing certificate from the C4-D positive shell-work branch via C4-F."
    },
    {
      "id": "ns.c4.c4g.r_q_sigma",
      "latex": "R_q^\\sigma",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "無傳輸剩餘項",
      "label_en": "Transport-free remainder",
      "definition_zh": "§4 沿用的 transport-free remainder，定義為 shell 非線性項減去低模平流項。",
      "definition_en": "The transport-free remainder carried over and used starting §4, equal to the shell nonlinear term minus the low-mode advection term.",
      "defining_relation": "R_q^\\sigma=N_q^\\sigma-u_{\\le q-L_0}\\cdot\\nabla u_q^\\sigma",
      "notes": "Its symmetric-gradient time integral defines \\mathfrak D_q(I) (§22), the paper's central deformation-forcing observable; §26 further decomposes it via \\mathscr T_{q,\\sigma}."
    },
    {
      "id": "ns.c4.c4g.lambda_q",
      "latex": "\\lambda_q",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "二進頻率尺度",
      "label_en": "Dyadic frequency scale",
      "definition_zh": "貫穿全文的標準 Littlewood–Paley 二進頻率尺度，自 §4 起用於各 shell-indexed 量，並在 §5 annular Bernstein 估計中扮演核心角色。",
      "definition_en": "The standard Littlewood-Paley dyadic frequency scale used from §4 onward for every shell-indexed quantity, playing the central role in the §5 annular Bernstein estimates.",
      "notes": "Governs the \\lambda_q^{1/2} growth rate in the C4-G.1 deformation-impulse lower bound (§7-8) and the p\\ge q+L frequency separation in the triad geometry of §17-19."
    },
    {
      "id": "ns.c4.c4g.s_q_r",
      "latex": "\\mathfrak S_q^R",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "來源超容量憑證",
      "label_en": "Source-overcapacity certificate",
      "definition_zh": "§4 定義的 Source-Overcapacity branch 憑證，衡量 R_q^\\sigma 正規化後的 L_t^1L_x^\\infty 質量是否超過門檻 s_0。",
      "definition_en": "The Source-Overcapacity branch certificate defined in §4, measuring whether the normalized L_t^1L_x^\\infty mass of R_q^\\sigma exceeds the threshold s_0.",
      "defining_relation": "\\mathfrak S_q^R=\\frac{1}{\\nu\\lambda_q}\\int_I\\|R_q^\\sigma(t)\\|_\\infty dt\\ge s_0>0",
      "notes": "The hypothesis of Theorem C4-G.1 (§7); s_0 is the fixed positive threshold constant carried through §8-10."
    },
    {
      "id": "ns.c4.c4g.nabla_sym",
      "latex": "\\nabla_{\\rm sym}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "對稱梯度算子",
      "label_en": "Symmetric-gradient operator",
      "definition_zh": "§6 給出其與一般梯度及散度的恆等關係，作為連結剩餘項與形變迫力的核心算子。",
      "definition_en": "The symmetric-gradient operator, whose identity relating it to the ordinary gradient and divergence is given in §6, the core operator linking the remainder to deformation forcing.",
      "defining_relation": "\\|\\nabla_{\\rm sym}F\\|_2^2=\\frac12\\|\\nabla F\\|_2^2+\\frac12\\|\\nabla\\cdot F\\|_2^2"
    },
    {
      "id": "ns.c4.c4g.thm_g1",
      "latex": "\\text{C4-G.1}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "定理 C4-G.1：來源脈衝到形變脈衝增長定理",
      "label_en": "Theorem C4-G.1: Source-Impulse to Growing Deformation-Impulse",
      "definition_zh": "§7 證明的定理，指出 \\mathfrak S_q^R\\ge s_0 蘊含對稱梯度剩餘項的時間積分下界隨 \\lambda_q^{1/2} 增長。",
      "definition_en": "The theorem proved in §7, showing \\mathfrak S_q^R\\ge s_0 forces the symmetric-gradient remainder's time-integral lower bound to grow like \\lambda_q^{1/2}.",
      "defining_relation": "\\int_I\\|\\nabla_{\\rm sym}R_q^\\sigma(t)\\|_2dt\\ge c\\,\\nu\\,s_0\\,\\lambda_q^{1/2}",
      "notes": "§9 stresses this is a strong congestion certificate, not a regularity contradiction, since Leray energy theory gives no finite unweighted global budget for its left-hand side."
    },
    {
      "id": "ns.c4.c4g.v_q_work",
      "latex": "\\mathfrak V_q^{work}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "總絕對工作憑證",
      "label_en": "Total absolute work certificate",
      "definition_zh": "§11 由 C4-D 的 positive shell-work 下界導出的總絕對工作量，作為 Relay-W 與 M6 共享的來源憑證。",
      "definition_en": "The total absolute work quantity derived in §11 from the C4-D positive shell-work lower bound, the shared source certificate for both Relay-W and M6.",
      "defining_relation": "\\mathfrak V_q^{work}\\ge w_0",
      "notes": "Traces to \\frac{\\lambda_q}{\\nu^2}\\int_I[W_q^\\sigma]_+dt\\ge w_0 from the C4-D positive shell-work branch; §13 uses it to show M6 automatically carries C_DO."
    },
    {
      "id": "ns.c4.c4g.triad_geometry",
      "latex": "\\xi,\\ \\eta,\\ \\zeta=\\xi+\\eta,\\ \\theta=\\angle(\\xi,\\eta)",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "高高頻對之 Fourier 幾何（近反平行定理）",
      "label_en": "High-high pair Fourier geometry (near-antipodal theorem)",
      "definition_zh": "§17–18 引入的三元組：\\xi,\\eta 為高頻母頻率、\\zeta 為輸出頻率、\\theta 為母頻率夾角，用以陳述定理 C4-G.5（18.1）的近反平行幾何。",
      "definition_en": "The tuple introduced in §17-18 — parent high frequencies \\xi,\\eta, output frequency \\zeta, and parent angle \\theta — used to state Theorem C4-G.5 (18.1)'s near-antipodal geometry.",
      "defining_relation": "1+\\cos\\theta\\le C\\left(\\frac{\\lambda_q}{\\lambda_p}\\right)^2,\\qquad |\\pi-\\theta|\\le C\\lambda_q/\\lambda_p",
      "notes": "Preserved forward as the G-RELAYGEOM guard (§43), which requires tracking ||\\xi|-|\\eta|| and \\pi-\\angle(\\xi,\\eta) in any far-relay event."
    },
    {
      "id": "ns.c4.c4g.d_q",
      "latex": "\\mathfrak D_q(I)",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "共同形變迫力可觀測量",
      "label_en": "Common deformation-forcing observable",
      "definition_zh": "§22 定義的核心可觀測量，為對稱梯度剩餘項在時窗 I 上的時間積分，本輪所有未解 motifs 皆共享其正下界 d_0。",
      "definition_en": "The central observable defined in §22, the time integral over window I of the symmetric-gradient remainder, whose shared positive lower bound d_0 every unresolved motif in this round satisfies.",
      "defining_relation": "\\mathfrak D_q(I)=\\int_I\\|\\nabla_{\\rm sym}R_q^\\sigma(t)\\|_2dt\\ge d_0>0",
      "notes": "The hinge quantity for both the C4-G.6 Universal Deformation-Funnel Theorem (§23) and the C4-G.8 Cross-Congestion Synchronization Theorem (§38)."
    },
    {
      "id": "ns.c4.c4g.thm_g6",
      "latex": "\\text{C4-G.6}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "定理 C4-G.6：普遍形變漏斗定理",
      "label_en": "Theorem C4-G.6: Universal Deformation-Funnel Theorem",
      "definition_zh": "§23 陳述的定理，指出在 C4-E 假設下每次 critical UV crossing 至少落入 G-U1–G-U4 四通道之一，第四支保證 \\mathfrak D_q(I)\\ge d_0。",
      "definition_en": "The theorem stated in §23, asserting that under the C4-E hypotheses every critical UV crossing falls into at least one of the four G-U1-G-U4 channels, with the fourth guaranteeing \\mathfrak D_q(I)\\ge d_0.",
      "notes": "Compresses the program's progression 8 branches -> 6 motifs -> 3 congestion classes -> 4 synchronization channels (§23)."
    },
    {
      "id": "ns.c4.c4g.g_u_closure",
      "latex": "\\text{G-U1}\\vee\\text{G-U2}\\vee\\text{G-U3}\\vee\\text{G-U4}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "G-U1–G-U4：UV 相空間封閉四通道",
      "label_en": "G-U1-G-U4: UV phase-space closure channels",
      "definition_zh": "§23 命名的四個通道（UV Persistence、Low Strain/Vorticity Critical Toll、Positive Helical Production、Deformation/Operator Forcing），於 §39 正式寫成 UV Phase-Space Closure。",
      "definition_en": "The four channels named in §23 (UV Persistence, Low Strain/Vorticity Critical Toll, Positive Helical Production, Deformation/Operator Forcing), formally boxed as the UV Phase-Space Closure in §39.",
      "defining_relation": "\\text{UV Crossing}\\Rightarrow\\text{Persistence}\\vee\\text{Low Strain/Vorticity}\\vee\\text{Positive Helical Production}\\vee\\text{Deformation/Operator Forcing}",
      "notes": "§24 uses the finiteness of this four-channel family to force a recurrent channel along any infinite critical-crossing subsequence."
    },
    {
      "id": "ns.c4.c4g.operator_decomposition",
      "latex": "\\mathcal N_{\\rm proj},\\ \\mathscr T_{q,\\sigma}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "算子分解：投影算子與殼/螺旋度應變算子",
      "label_en": "Operator decomposition: projected operator and shell/helicity strain operator",
      "definition_zh": "§25–26 定義的一對算子：\\mathcal N_{\\rm proj} 為省略 \\frac12\\omega\\otimes\\omega 項後與 \\mathcal Q_{SV} 相關的投影算子，\\mathscr T_{q,\\sigma} 為將其映到 \\nabla_{\\rm sym}N_q^\\sigma 的 bounded order-zero 殼/螺旋度應變算子。",
      "definition_en": "The pair of operators defined in §25-26: \\mathcal N_{\\rm proj}, the projected operator related to \\mathcal Q_{SV} by dropping the \\frac12\\omega\\otimes\\omega term, and \\mathscr T_{q,\\sigma}, the bounded order-zero shell/helicity strain operator mapping it to \\nabla_{\\rm sym}N_q^\\sigma.",
      "defining_relation": "\\nabla_{\\rm sym}R_q^\\sigma=\\mathscr T_{q,\\sigma}\\mathcal Q_{SV}-\\frac12\\mathscr T_{q,\\sigma}P_{st}(\\omega\\otimes\\omega)-\\nabla_{\\rm sym}(v_q\\cdot\\nabla f_q^\\sigma)",
      "notes": "This decomposition (§26) generates the G-O1/G-O2/G-O3 trichotomy in §27; v_q and f_q^\\sigma denote the low-mode velocity u_{\\le q-L_0} and shell field u_q^\\sigma from §4's original R_q^\\sigma=N_q^\\sigma-u_{\\le q-L_0}\\cdot\\nabla u_q^\\sigma."
    },
    {
      "id": "ns.c4.c4g.g_o_trichotomy",
      "latex": "\\text{G-O1}\\vee\\text{G-O2}\\vee\\text{G-O3}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "G-O1–G-O3：統一迫力三分支",
      "label_en": "G-O1-G-O3: unified forcing trichotomy",
      "definition_zh": "§27 由三角不等式導出的三分支——Miller 算子脈衝、渦度二次脈衝、平流形變脈衝——當 \\mathfrak D_q(I)\\ge d_0 時至少一支成立。",
      "definition_en": "The three branches derived via the triangle inequality in §27 — Miller operator impulse, vorticity-quadratic impulse, advective/sweeping deformation impulse — at least one holding whenever \\mathfrak D_q(I)\\ge d_0.",
      "defining_relation": "\\int_I\\|\\mathscr T_{q,\\sigma}\\mathcal Q_{SV}\\|_2dt\\ge cd_0\\ \\vee\\ \\int_I\\|\\mathscr T_{q,\\sigma}P_{st}(\\omega\\otimes\\omega)\\|_2dt\\ge cd_0\\ \\vee\\ \\int_I\\|\\nabla_{\\rm sym}(v_q\\cdot\\nabla f_q^\\sigma)\\|_2dt\\ge cd_0",
      "notes": "G-O2 further reduces via Gagliardo-Nirenberg (§33) to an enstrophy-versus-higher-vorticity-derivative dichotomy; G-O3 is guarded by the C3-O Balance-Fixed-Point-\\neq-Dynamics-Fixed-Point caveat (§34) and the sweeping/gauge caveat (§35)."
    },
    {
      "id": "ns.c4.c4g.thm_g7",
      "latex": "\\text{C4-G.7}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "定理 C4-G.7：算子比值或高階導數脈衝二分性",
      "label_en": "Theorem C4-G.7: Operator-Ratio or Higher-Derivative Impulse dichotomy",
      "definition_zh": "§28–29 定義 D(t)=\\|\\mathscr T_{q,\\sigma}\\mathcal Q_{SV}(t)\\|_2、H(t)=\\|-\\Delta S(t)\\|_2 與比值優勢集 E_\\rho，證明 \\int_ID\\ge d_1 時 G-RATIO 或 G-HDER 至少一支成立。",
      "definition_en": "§28-29 define D(t)=\\|\\mathscr T_{q,\\sigma}\\mathcal Q_{SV}(t)\\|_2, H(t)=\\|-\\Delta S(t)\\|_2, and the ratio-dominant set E_\\rho, proving \\int_ID\\ge d_1 forces at least one of G-RATIO or G-HDER.",
      "defining_relation": "E_\\rho=\\{t\\in I:D(t)\\ge\\rho H(t)\\};\\quad \\int_{E_\\rho}D(t)dt\\ge\\frac{d_1}{2}\\ \\vee\\ \\int_I\\|-\\Delta S(t)\\|_2dt\\ge\\frac{d_1}{2\\rho}",
      "notes": "§31 notes the G-HDER branch feeds \\Delta S\\sim D^3u higher-derivative/derivative-chain geometry, connecting to C3-W/X/Y; §30 clarifies this does not reprove Miller's theorem."
    },
    {
      "id": "ns.c4.c4g.thm_g8",
      "latex": "\\text{C4-G.8}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "定理 C4-G.8：跨壅塞同步定理",
      "label_en": "Theorem C4-G.8: Cross-Congestion Synchronization Theorem",
      "definition_zh": "§38 陳述的定理，指出若 critical UV crossing 避開前三通道，則存在共同 crossing window 使 \\mathfrak D_{q_n}(I_n)\\ge d_0，並依事件類型附帶額外的 tail 或 radial 憑證。",
      "definition_en": "The theorem stated in §38, asserting that if a critical UV crossing avoids the first three channels, a common crossing window exists with \\mathfrak D_{q_n}(I_n)\\ge d_0, carrying additional tail or radial certificates depending on event type.",
      "notes": "For Higher-Frequency Relay events it adds \\mathfrak H_{tail,n}\\gtrsim1 plus near-antipodal geometry; for Spectral Degeneration events it adds radial work-measure concentration."
    },
    {
      "id": "ns.c4.c4g.xi_n",
      "latex": "\\Xi_n",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "相空間載子狀態",
      "label_en": "Phase-space carrier state",
      "definition_zh": "§37 定義的 source-preserving unified event record 元組，收錄 shell 指標、時窗、剩餘項與 tail/multiplicity/radial 憑證，明言並非 compactness theorem。",
      "definition_en": "The source-preserving unified event-record tuple defined in §37, collecting the shell index, time window, remainder, and tail/multiplicity/radial certificates — explicitly not a compactness theorem.",
      "defining_relation": "\\Xi_n=\\left\\langle q_n,I_n,R_{q_n}^{\\sigma_n},\\mathfrak H_{tail,n},\\mathfrak M_{eff,n},\\widehat\\mu_n^{rad},\\mathfrak D_n,\\operatorname{OperatorBranch}_n\\right\\rangle",
      "notes": "\\mathfrak H_{tail,n} is the effective-multiplicity/tail-stock certificate and \\widehat\\mu_n^{rad} the radial work-measure concentration, both drawn from M4/M6's congestion-coordinate lists in §20-21."
    },
    {
      "id": "ns.c4.c4g.theta_n_force",
      "latex": "\\Theta_n^{force}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "統一迫力狀態（True ETN 更新）",
      "label_en": "Unified forcing state (True ETN update)",
      "definition_zh": "§44 定義的 C4-G unified forcing state，作為 True ETN 的更新版本，彙整本輪建立的核心迫力可觀測量。",
      "definition_en": "The C4-G unified forcing state defined in §44, an update to the True ETN that aggregates every core forcing observable established in this round.",
      "defining_relation": "\\Theta_n^{force}=\\left\\langle\\mathfrak D_n,\\mathfrak H_{tail,n},\\mathfrak M_{eff,n},\\widehat\\mu_n^{rad},\\mathcal Q_{SV,n},\\omega\\otimes\\omega,\\mathcal A_{adv,n},\\Delta S_n\\right\\rangle",
      "notes": "\\mathcal A_{adv,n} denotes the advective/sweeping deformation component from the G-O3 branch."
    },
    {
      "id": "ns.c4.c4g.no_go_guards",
      "latex": "\\text{NG-G1}\\text{–}\\text{NG-G5}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "反向蘊含禁區 NG-G1–NG-G5",
      "label_en": "No-go guards NG-G1-NG-G5",
      "definition_zh": "§42 列出的五個反向蘊含，各自明確標記為 FALSE 或 OPEN，防止本輪的正向結果被誤讀成更強的等價或可逆結論。",
      "definition_en": "The five reverse implications listed in §42, each explicitly flagged FALSE or OPEN, blocking any misreading of this round's forward results as stronger equivalences or reversible conclusions.",
      "notes": "E.g. NG-G2 blocks \"deformation impulse => Miller ratio >=1\"; NG-G3 blocks \"vorticity quadratic large => positive vortex stretching\"."
    },
    {
      "id": "ns.c4.c4g.x_integration_guards",
      "latex": "\\text{G-CROSSFUNNEL, G-RELAYGEOM, G-OPRATIO, G-HDER, G-V4, G-SWEEP}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "X-Integration 更新守則",
      "label_en": "X-Integration guards update",
      "definition_zh": "§43 為 X_Integral_Unified_Program 新增的六條守則，要求未來各輪保存本輪建立的共同前提、幾何憑證與分支界線，不得任意升格。",
      "definition_en": "The six guard clauses added to the X_Integral_Unified_Program in §43, requiring future rounds to preserve this round's shared antecedents, geometric certificates, and branch boundaries without unwarranted upgrades.",
      "notes": "Includes G-CROSSFUNNEL (preserve the common deformation-forcing antecedent for M4/M5/M6) and G-SWEEP (preserve the sweeping/gauge distinction for advective deformation)."
    },
    {
      "id": "ns.c4.c4g.c4h_frontier",
      "latex": "\\text{C4-H}",
      "series": "NS",
      "first_appearance": "C4-G",
      "label_zh": "下一輪前沿：C4-H 算子到閘門封閉",
      "label_en": "Next frontier: C4-H Operator-to-Gate Closure",
      "definition_zh": "§41 提出、§46 正式命名的下一輪主題，處理 Miller Ratio、Middle-Strain 幾何、壓力與導數鏈之間的算子到 regularity-gate 封閉問題，§47 列出 H1–H8 八項證明義務。",
      "definition_en": "The next round's theme first raised in §41 and formally named in §46, addressing the operator-to-regularity-gate closure among the Miller ratio, middle-strain geometry, pressure, and derivative chains, with eight proof obligations H1-H8 listed in §47.",
      "notes": "§48's status table marks both \"operator-to-regularity-gate closure\" and \"global regularity\" OPEN, framing this as the program's unresolved successor problem."
    },
    {
      "id": "ns.c4.c4i.record_window",
      "latex": "J_j=(\\tau_j,\\tau_{j+1}),\\ J=(a,b)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "記錄窗口",
      "label_en": "Record window",
      "definition_zh": "承接C4-H的收縮式late-time記錄窗口J_j=(tau_j,tau_{j+1})（見§0），本輪§2將其寫成一般記法J=(a,b)作為容量-重疊分析的基本區間。",
      "definition_en": "The shrinking late-time record windows J_j=(tau_j,tau_{j+1}) inherited from C4-H (§0) are written generically as J=(a,b) in §2, the base interval for this round's capacity-overlap analysis.",
      "notes": "Carries over directly from C4-H's record ladder; |J_j|->0 as j->infinity toward the putative singular time T_*."
    },
    {
      "id": "ns.c4.c4i.middle_load_density",
      "latex": "m(t)=\\int_{\\mathbb R^3}\\lambda_2^+(x,t)|S(x,t)|^2\\,dx",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "中間應變負荷密度",
      "label_en": "Middle load density",
      "definition_zh": "§2定義的時間函數，把Miller中間特徵值正部λ_2^+與應變張量平方在空間上積分成單一非負的middle toll密度。",
      "definition_en": "A time-dependent quantity defined in §2 that integrates the positive part of Miller's middle eigenvalue λ2+ weighted by |S|^2 over space into a single nonnegative middle-toll density.",
      "defining_relation": "m(t)=\\int_{\\mathbb R^3}\\lambda_2^+(x,t)|S(x,t)|^2\\,dx,\\qquad m(t)\\ge0"
    },
    {
      "id": "ns.c4.c4i.operator_load_density",
      "latex": "o(t)=\\nu[\\zeta(t)r_\\nu(t)-1]_+\\|\\Delta S(t)\\|_2^2",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "算子負荷密度",
      "label_en": "Operator load density",
      "definition_zh": "§2定義的時間函數，用C4-H的閾值超額[ζr_ν-1]_+乘上‖ΔS‖²_2衡量算子項超出增長閾值的負荷。",
      "definition_en": "A time-dependent quantity defined in §2 measuring how far the operator term exceeds the growth threshold, via [ζ r_ν - 1]_+ times the squared norm of ΔS.",
      "defining_relation": "o(t)=\\nu[\\zeta(t)r_\\nu(t)-1]_+\\|\\Delta S(t)\\|_2^2,\\qquad o(t)\\ge0"
    },
    {
      "id": "ns.c4.c4i.peak_capacities",
      "latex": "M=\\operatorname*{ess\\,sup}_{t\\in J}m(t),\\quad O=\\operatorname*{ess\\,sup}_{t\\in J}o(t)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "峰值容量",
      "label_en": "Peak capacities",
      "definition_zh": "§3將m、o在記錄窗口J上的本質上確界分別記作M、O，作為之後duty-cycle下界公式的分母。",
      "definition_en": "§3 sets M and O as the essential suprema of m and o over the record window J, the denominators in the duty-cycle lower bounds that follow."
    },
    {
      "id": "ns.c4.c4i.threshold_active_sets",
      "latex": "E_m(\\mu)=\\{t\\in J:m(t)\\ge\\mu\\},\\quad E_o(\\omega)=\\{t\\in J:o(t)\\ge\\omega\\}",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "閾值活躍集",
      "label_en": "Threshold-active sets",
      "definition_zh": "§3定義的時間子集，分別收集middle load與operator load超過給定閾值μ、ω的時刻。",
      "definition_en": "Time subsets defined in §3 collecting the moments at which the middle load and operator load exceed given thresholds μ and ω respectively."
    },
    {
      "id": "ns.c4.c4i.capacity_to_overlap_theorem",
      "latex": "|E_m(\\mu)\\cap E_o(\\omega)|\\ge\\left[\\frac{A-\\mu|J|}{M-\\mu}+\\frac{B-\\omega|J|}{O-\\omega}-|J|\\right]_+",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "中間-算子容量至重疊定理",
      "label_en": "Middle–Operator Capacity-to-Overlap Theorem (C4-I.1)",
      "definition_zh": "§5定理5.1，由包含-排斥不等式從單通道duty-cycle下界推出E_m(μ)、E_o(ω)的same-time重疊測度下界，是本輪核心正面結果。",
      "definition_en": "Theorem 5.1 in §5, deriving a lower bound on the measure of the same-time overlap E_m(μ)∩E_o(ω) from single-channel duty-cycle bounds via inclusion-exclusion — the round's central positive result.",
      "defining_relation": "|E_m(\\mu)\\cap E_o(\\omega)|\\ge\\left[\\frac{A-\\mu|J|}{M-\\mu}+\\frac{B-\\omega|J|}{O-\\omega}-|J|\\right]_+",
      "notes": "§6-7 specialize this to a same-time overlap criterion and a zero-threshold version; §50 packages its right-hand side as the functional 𝔠_overlap inside the state tuple Θ_J^MO."
    },
    {
      "id": "ns.c4.c4i.desync_debt",
      "latex": "\\text{Middle–Operator Peak-Capacity Desynchronization Debt}",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "峰值容量去同步債務",
      "label_en": "Peak-Capacity Desynchronization Debt",
      "definition_zh": "§8將E_m(μ)∩E_o(ω)=∅時必然成立的容量比不等式命名為此債務，量化若無同時重疊峰值容量比需付出的代價。",
      "definition_en": "§8 names the capacity-ratio inequality that must hold whenever E_m(μ)∩E_o(ω)=∅ as this debt, quantifying the cost paid in peak-capacity ratios when same-time overlap fails."
    },
    {
      "id": "ns.c4.c4i.peak_average_ratio_theorem",
      "latex": "M\\le K_m\\frac{A}{|J|},\\ O\\le K_o\\frac{B}{|J|}\\ \\Rightarrow\\ \\tfrac1{K_m}+\\tfrac1{K_o}>1\\Rightarrow\\text{overlap forced}",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "有界峰值/平均比定理",
      "label_en": "Bounded Peak/Average Ratios Force Same-Time Overlap (C4-I.2)",
      "definition_zh": "§10的條件式定理：若峰值容量M、O被平均容量的常數倍K_m、K_o控制且1/K_m+1/K_o>1，則same-time overlap被迫成立，但目前未有已證的一致K_m、K_o上界。",
      "definition_en": "The conditional theorem of §10: if peak capacities M, O are bounded by constant multiples K_m, K_o of the averages and 1/K_m+1/K_o>1, same-time overlap is forced, though no uniform bound on K_m, K_o is yet proved.",
      "notes": "Explicitly logged as open (no proved uniform K_m, K_o upper bounds); becomes proof obligation J1 of the next round C4-J (§54)."
    },
    {
      "id": "ns.c4.c4i.growth_direction",
      "latex": "D=\\|\\Delta S\\|_2,\\qquad e_D=\\frac{-\\Delta S}{D}",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "增長方向與模長",
      "label_en": "Growth direction and magnitude",
      "definition_zh": "§12定義的‖ΔS‖_2模長D及對應單位方向e_D=-ΔS/D，作為之後算子平行/正交分解的參考軸。",
      "definition_en": "The magnitude D=‖ΔS‖_2 and its associated unit vector e_D=-ΔS/D, defined in §12 as the reference axis for the operator's parallel/orthogonal decomposition that follows."
    },
    {
      "id": "ns.c4.c4i.normalized_operator",
      "latex": "\\widehat Q=\\frac{\\mathcal Q_{SV}}{\\nu D},\\qquad \\|\\widehat Q\\|_2=r_\\nu",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "正規化算子與Miller比",
      "label_en": "Normalized operator and Miller ratio",
      "definition_zh": "§12把Miller strain/vorticity算子Q_SV（引自§1.2）除以νD正規化為Q̂，其模長即為C4-H沿用的Miller比r_ν。",
      "definition_en": "§12 normalizes Miller's strain/vorticity operator Q_SV (introduced in §1.2) by νD into Q̂, whose norm recovers the Miller ratio r_ν carried over from C4-H.",
      "notes": "Q_SV, r_ν and ζ are inherited notation from Miller's 2026 paper and C4-G/C4-H; this round's new contribution is the normalized Q̂ and its angle decomposition."
    },
    {
      "id": "ns.c4.c4i.gate_variable",
      "latex": "g=\\zeta r_\\nu,\\quad \\widehat Q=-g\\,e_D+Q_\\perp,\\quad \\tfrac12\\tfrac{d}{dt}\\|S\\|_{\\dot H^1}^2=\\nu(g-1)D^2",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "閘門變數g",
      "label_en": "Gate variable g",
      "definition_zh": "§13將Q̂沿e_D的投影係數命名為g=ζr_ν，並證明Ḣ¹增長率恰為ν(g-1)D²，使g取代單純算子範數成為§46所稱的真正gate variable。",
      "definition_en": "§13 names the projection coefficient of Q̂ along e_D as g=ζr_ν and shows the Ḣ¹ growth rate equals ν(g-1)D², making g — rather than the bare operator norm — the round's true gate variable, as stated explicitly in §46.",
      "defining_relation": "g=\\zeta r_\\nu,\\qquad \\widehat Q=-g\\,e_D+Q_\\perp,\\qquad \\langle Q_\\perp,e_D\\rangle=0,\\qquad \\tfrac12\\tfrac{d}{dt}\\|S\\|_{\\dot H^1}^2=\\nu(g-1)D^2",
      "notes": "§46 demotes r_ν alone and installs g as the operator-funnel gate variable refining C4-G; §45 tabulates the three branches g>1 (Growth-Aligned), -1≤g≤1 with r_ν≫1 (Orthogonal Congestion), g<-1 (Growth-Opposing)."
    },
    {
      "id": "ns.c4.c4i.orthogonal_operator_component",
      "latex": "Q_\\perp,\\qquad \\|Q_\\perp\\|_2^2=r_\\nu^2-g^2",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "算子正交分量",
      "label_en": "Orthogonal operator component",
      "definition_zh": "§13由畢氏定理得到的Q̂正交於e_D的殘餘分量，其模長平方等於r_ν²-g²。",
      "definition_en": "The component of Q̂ orthogonal to e_D obtained via the Pythagorean identity in §13, with squared norm r_ν²-g².",
      "notes": "Recurs as a coordinate of the operator-angle state tuple Θ^angle (§50) and is bounded below in the C4-I.3 dichotomy (§14)."
    },
    {
      "id": "ns.c4.c4i.large_ratio_dichotomy",
      "latex": "r_\\nu\\ge R>1,\\ g\\le1\\ \\Rightarrow\\ (g<-1)\\ \\vee\\ \\big(-1\\le g\\le1\\ \\wedge\\ \\|Q_\\perp\\|_2\\ge\\sqrt{R^2-1}\\big)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "大比值非增長路由定理",
      "label_en": "Large-Ratio Non-Growth Routing Theorem (C4-I.3)",
      "definition_zh": "§14定理，證明若Miller比大但g不大於1，則必落入I-OPPOSE（強烈對抗增長，g<-1）或I-ORTH（正交算子分量至少√(R²-1)）之一。",
      "definition_en": "The theorem of §14 proving that a large Miller ratio with g≤1 must fall into either I-OPPOSE (strongly growth-opposing, g<-1) or I-ORTH (orthogonal operator component at least √(R²-1)).",
      "defining_relation": "r_\\nu\\ge R>1,\\ g\\le1\\ \\Rightarrow\\ \\underbrace{g<-1}_{\\text{I-OPPOSE}}\\ \\vee\\ \\underbrace{\\big(-1\\le g\\le1\\ \\wedge\\ \\|Q_\\perp\\|_2\\ge\\sqrt{R^2-1}\\big)}_{\\text{I-ORTH}}",
      "notes": "§15 reframes this as a sharper replacement for the cruder 1-ζ depletion classification used before C4-I."
    },
    {
      "id": "ns.c4.c4i.vorticity_quadratic",
      "latex": "W=P_{st}(\\omega\\otimes\\omega),\\qquad \\langle W,-\\Delta S\\rangle=0,\\qquad W\\in\\{e_D\\}^\\perp",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "渦度平方項",
      "label_en": "Vorticity-quadratic term",
      "definition_zh": "§16定義W=P_st(ω⊗ω)，並引用Miller正交性⟨W,-ΔS⟩=0證明它恆屬於增長方向的正交補空間。",
      "definition_en": "§16 defines W=P_st(ω⊗ω) and invokes Miller's orthogonality ⟨W,-ΔS⟩=0 to show it always lies in the orthogonal complement of the growth direction.",
      "notes": "The operator-level analogue of the pressure-Hessian orthogonality of §24; both trace back to Miller's 2026 identity quoted in §1.2."
    },
    {
      "id": "ns.c4.c4i.advection_strain_square_operator",
      "latex": "A=P_{st}\\big((u\\cdot\\nabla)S+S^2\\big),\\qquad \\mathcal Q_{SV}=A+\\tfrac34W,\\qquad \\langle A,e_D\\rangle=\\langle\\mathcal Q_{SV},e_D\\rangle",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "平流/應變平方算子",
      "label_en": "Advection/strain-square operator",
      "definition_zh": "§17定義A=P_st((u·∇)S+S²)，證明所有growth-parallel資訊都活在A中，並給出Miller算子的分解Q_SV=A+¾W。",
      "definition_en": "§17 defines A=P_st((u·∇)S+S²), shows all growth-parallel information lives in A, and gives the decomposition Q_SV=A+(3/4)W of Miller's operator.",
      "defining_relation": "A=P_{st}\\big((u\\cdot\\nabla)S+S^2\\big),\\qquad \\mathcal Q_{SV}=A+\\tfrac34W,\\qquad \\langle A,e_D\\rangle=\\langle\\mathcal Q_{SV},e_D\\rangle"
    },
    {
      "id": "ns.c4.c4i.orthogonal_congestion_theorem",
      "latex": "\\|(\\mathcal Q_{SV})_\\perp\\|_2\\ge Q_0\\ \\Rightarrow\\ \\|A_\\perp\\|_2\\ge\\tfrac{Q_0}2\\ \\vee\\ \\|W\\|_2\\ge\\tfrac{2Q_0}3",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "正交壅塞分派定理",
      "label_en": "Orthogonal Congestion Split (Theorem 18.1)",
      "definition_zh": "§18定理，證明growth-orthogonal算子壅塞(Q_SV)_⊥若夠大，必由正交平流/應變平方分量A_⊥或渦度平方項W之一承擔。",
      "definition_en": "The theorem of §18 showing that if the growth-orthogonal operator congestion (Q_SV)_⊥ is large enough, it must be carried by either the orthogonal advection/strain-square component A_⊥ or the vorticity-quadratic term W."
    },
    {
      "id": "ns.c4.c4i.growth_aligned_source_dichotomy",
      "latex": "g>1\\ \\Rightarrow\\ \\big(-\\langle A_{adv},-\\Delta S\\rangle>\\tfrac{\\nu D^2}2\\big)\\ \\vee\\ \\big(-\\langle A_{S^2},-\\Delta S\\rangle>\\tfrac{\\nu D^2}2\\big)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "增長對齊來源二分定理",
      "label_en": "Growth-Aligned Operator Source Dichotomy (C4-I.4)",
      "definition_zh": "§19-20定理，把A分成平流部分A_adv與應變平方部分A_S²，證明g>1時增長必至少一半來自I-ADV或I-SSA其中之一，渦度項因Miller正交性不能直接驅動增長。",
      "definition_en": "The theorem of §19-20 splitting A into an advective part A_adv and a strain-square part A_S², proving that when g>1 the growth must come at least half from I-ADV or I-SSA, since Miller orthogonality forbids the vorticity term from directly driving growth.",
      "defining_relation": "A=A_{adv}+A_{S^2},\\ A_{adv}=P_{st}((u\\cdot\\nabla)S),\\ A_{S^2}=P_{st}(S^2),\\quad g>1\\Rightarrow \\text{I-ADV}\\vee\\text{I-SSA}"
    },
    {
      "id": "ns.c4.c4i.ssa_pointwise_nogo",
      "latex": "S=\\operatorname{diag}(-2,-1,3),\\ \\lambda_2(S)=-1<0,\\ B=e_1\\otimes e_1,\\ -\\operatorname{tr}(SB^2)=2>0",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "SSA增長不逼λ2+ pointwise的反例",
      "label_en": "SSA-growth-does-not-force-pointwise-λ2+ counterexample",
      "definition_zh": "§22用具體對角矩陣構造pointwise矩陣代數反例，證明strain-square-aligned的Ḣ¹增長本身不能推出同一點λ_2^+>0，即NG-I3。",
      "definition_en": "§22 constructs a pointwise matrix-algebra counterexample with an explicit diagonal matrix, showing that strain-square-aligned Ḣ¹ growth alone cannot force λ2+>0 at the same point — logged as NG-I3.",
      "notes": "A purely algebraic no-go, not a constructed Navier–Stokes solution (stated explicitly in §22); reinforced in §23 as a second obstruction to same-time middle/operator overlap."
    },
    {
      "id": "ns.c4.c4i.adjoint_cutoff",
      "latex": "\\partial_t\\chi+u\\cdot\\nabla\\chi+\\nu\\Delta\\chi=0",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "伴隨局部截斷函數χ",
      "label_en": "Adjoint local cutoff χ",
      "definition_zh": "§26引入滿足反向平流-擴散方程的局部測試函數χ，作為之後adjoint local core上pressure re-entry分析的權重。",
      "definition_en": "§26 introduces the local test function χ solving the backward advection-diffusion equation, used as the weight for the pressure-re-entry analysis on the adjoint local core that follows.",
      "notes": "Carried over from C3-U's pressure-heredity framework and C3-X's Hessian-sensitive pressure oscillation, which §29 explicitly says this round continues (\"沿用 C3-X\")."
    },
    {
      "id": "ns.c4.c4i.adjoint_mean_strain_balance",
      "latex": "M_\\chi(t)=\\int\\chi S\\,dx,\\quad M_\\chi'=-B_\\chi-P_\\chi,\\quad B_\\chi=\\int\\chi\\Big[S^2+\\tfrac14\\omega\\otimes\\omega-\\tfrac14|\\omega|^2I\\Big]dx,\\quad P_\\chi=\\int\\chi\\nabla^2p\\,dx",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "伴隨局部平均應變平衡式",
      "label_en": "Adjoint local mean-strain balance",
      "definition_zh": "§26給出C3-U的exact恆等式M_χ'=-B_χ-P_χ，將局部平均應變的時間變化率精確分解成二次平均強迫B_χ與壓力平均強迫P_χ兩項。",
      "definition_en": "§26 states C3-U's exact identity M_χ'=-B_χ-P_χ, exactly splitting the time-derivative of the local mean strain into a quadratic mean forcing term B_χ and a pressure mean forcing term P_χ.",
      "defining_relation": "M_\\chi(t)=\\int\\chi S\\,dx,\\qquad M_\\chi'=-B_\\chi-P_\\chi,\\qquad B_\\chi=\\int\\chi\\Big[S^2+\\tfrac14\\omega\\otimes\\omega-\\tfrac14|\\omega|^2I\\Big]dx,\\qquad P_\\chi=\\int\\chi\\nabla^2p\\,dx",
      "notes": "The pivot identity for the round's second half (§26 onward); B_χ and P_χ are the two compensators whose trade-off drives C4-I.5 through C4-I.7."
    },
    {
      "id": "ns.c4.c4i.normalized_scale_quantities",
      "latex": "b_\\chi=\\tfrac{R}{\\nu^2}|B_\\chi|,\\quad r_\\chi=\\tfrac{R}{\\nu^2}|M_\\chi'|,\\quad \\pi_\\chi=\\tfrac{R}{\\nu^2}|P_\\chi|,\\quad b_\\chi\\le r_\\chi+\\pi_\\chi",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "尺度正規化三角量",
      "label_en": "Scale-normalized triangle quantities",
      "definition_zh": "§27把B_χ、M_χ'、P_χ依半徑R與ν²正規化為無量綱量b_χ、r_χ、π_χ，並給出exact三角不等式b_χ≤r_χ+π_χ。",
      "definition_en": "§27 normalizes B_χ, M_χ', P_χ by radius R and ν² into the dimensionless quantities b_χ, r_χ, π_χ, and states the exact triangle inequality b_χ≤r_χ+π_χ.",
      "defining_relation": "b_\\chi=\\frac{R}{\\nu^2}|B_\\chi|,\\quad r_\\chi=\\frac{R}{\\nu^2}|M_\\chi'|,\\quad \\pi_\\chi=\\frac{R}{\\nu^2}|P_\\chi|,\\quad b_\\chi\\le r_\\chi+\\pi_\\chi"
    },
    {
      "id": "ns.c4.c4i.mean_rotation_pressure_dichotomy",
      "latex": "b_\\chi\\ge b_0\\ \\Rightarrow\\ (r_\\chi\\ge\\theta b_0)\\ \\vee\\ (\\pi_\\chi\\ge(1-\\theta)b_0)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "平均旋轉/壓力二分定理",
      "label_en": "Adjoint Mean-Rotation/Pressure Dichotomy (C4-I.5)",
      "definition_zh": "§28定理，若二次平均強迫b_χ不小於b_0，則必至少走I-MROT（平均旋轉r_χ夠大）或I-PRESS（壓力強迫π_χ夠大）之一。",
      "definition_en": "The theorem of §28: if the quadratic mean forcing b_χ is at least b_0, then either I-MROT (mean rotation r_χ large enough) or I-PRESS (pressure forcing π_χ large enough) must hold.",
      "defining_relation": "b_\\chi\\ge b_0>0\\ \\Rightarrow\\ \\underbrace{r_\\chi\\ge\\theta b_0}_{\\text{I-MROT}}\\ \\vee\\ \\underbrace{\\pi_\\chi\\ge(1-\\theta)b_0}_{\\text{I-PRESS}}"
    },
    {
      "id": "ns.c4.c4i.critical_pressure_oscillation",
      "latex": "\\Pi_R^{(2)}=\\frac1{\\nu^2}\\inf_{\\ell\\in\\mathcal A_1}\\|p-\\ell\\|_{L^{3/2}(B_{CR})},\\qquad \\Pi_R^{(2)}\\ge c\\,\\pi_\\chi",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "臨界局部壓力振盪",
      "label_en": "Critical local pressure oscillation Π_R^(2)",
      "definition_zh": "§30由§29的Hessian估計定義的正規化臨界L^{3/2}壓力振盪量，滿足Π_R^(2)≥cπ_χ，是本輪連接local pressure Hessian強迫與critical積分範數的橋樑。",
      "definition_en": "Defined in §30 from the Hessian estimate of §29 as the normalized critical L^{3/2} local pressure oscillation, satisfying Π_R^(2)≥cπ_χ — this round's bridge between local pressure-Hessian forcing and a critical integral norm.",
      "defining_relation": "\\Pi_R^{(2)}=\\frac1{\\nu^2}\\inf_{\\ell\\in\\mathcal A_1}\\|p-\\ell\\|_{L^{3/2}(B_{CR})},\\qquad \\Pi_R^{(2)}\\ge c\\,\\pi_\\chi",
      "notes": "Built on the affine-subtraction Hessian estimate of §29 (continuing C3-X) and legitimated by the Bradshaw–Tsai local pressure expansion cited in §1.3 and §25."
    },
    {
      "id": "ns.c4.c4i.mean_stability_forces_reentry",
      "latex": "b_\\chi\\ge b_0,\\ r_\\chi\\le\\varepsilon<b_0\\ \\Rightarrow\\ \\Pi_R^{(2)}\\ge c(b_0-\\varepsilon)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "平均穩定性逼壓力重入定理",
      "label_en": "Mean-Stability Forces Pressure Re-entry Theorem (C4-I.6)",
      "definition_zh": "§31定理，證明當二次強迫nondegenerate而局部平均應變旋轉被壓低到ε以下時，臨界壓力振盪必至少為c(b_0-ε)。",
      "definition_en": "The theorem of §31 proving that when the quadratic forcing is nondegenerate but the local mean-strain rotation is suppressed below ε, the critical pressure oscillation must be at least c(b_0-ε).",
      "defining_relation": "b_\\chi\\ge b_0,\\quad r_\\chi\\le\\varepsilon<b_0\\ \\Rightarrow\\ \\Pi_R^{(2)}\\ge c(b_0-\\varepsilon)"
    },
    {
      "id": "ns.c4.c4i.pressure_concentration_certificate",
      "latex": "R_n\\to0,\\ \\Pi_{R_n}^{(2)}\\ge\\pi_0>0\\ \\Rightarrow\\ \\int_{B_{CR_n}}|p|^{3/2}dx\\ge c\\,\\pi_0^{3/2}\\nu^3",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "壓力集中證書",
      "label_en": "Pressure Concentration Certificate",
      "definition_zh": "§32命名的結果，證明若收縮核上臨界壓力振盪保持≥π_0，則即使球體體積|B_CR_n|→0，對應的L^{3/2}壓力質量仍不消失。",
      "definition_en": "The result named in §32, showing that if the critical pressure oscillation on shrinking cores stays ≥π_0, the corresponding L^{3/2} pressure mass does not vanish even as the ball volume tends to zero.",
      "notes": "§48 links this to the Constantin pressure-regularity interface (§1.4): a hypothetical singularity on the pressure branch must exhibit exactly this concentration / loss of small-set control."
    },
    {
      "id": "ns.c4.c4i.quadratic_absolute_intensity",
      "latex": "A_\\chi^{quad}=\\int\\chi\\Big|S^2+\\tfrac14\\omega\\otimes\\omega-\\tfrac14|\\omega|^2I\\Big|dx,\\qquad a_\\chi^{quad}=\\tfrac{R}{\\nu^2}A_\\chi^{quad}",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "局部二次絕對強度",
      "label_en": "Local quadratic absolute intensity",
      "definition_zh": "§33定義，把B_χ被積式取絕對值再積分得到A_χ^quad，並給出正規化版本a_χ^quad，用以和signed平均B_χ對比。",
      "definition_en": "Defined in §33 by taking the absolute value of B_χ's integrand before integrating to get A_χ^quad, with its normalized version a_χ^quad, contrasted against the signed mean B_χ.",
      "defining_relation": "A_\\chi^{quad}=\\int\\chi\\Big|S^2+\\tfrac14\\omega\\otimes\\omega-\\tfrac14|\\omega|^2I\\Big|dx,\\qquad a_\\chi^{quad}=\\frac{R}{\\nu^2}A_\\chi^{quad}"
    },
    {
      "id": "ns.c4.c4i.quadratic_coherence_ratio",
      "latex": "\\kappa_\\chi^{quad}=\\frac{|B_\\chi|}{A_\\chi^{quad}}\\in[0,1]",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "局部二次相干比",
      "label_en": "Local quadratic coherence ratio κ_χ^quad",
      "definition_zh": "§34定義的比值，衡量signed平均強迫B_χ相對其絕對強度A_χ^quad保留了多少相干性，越接近0代表matrix/spatial cancellation越嚴重。",
      "definition_en": "The ratio defined in §34 measuring how much of the signed mean forcing B_χ survives relative to its absolute intensity A_χ^quad — values near 0 indicate severe matrix/spatial cancellation.",
      "defining_relation": "\\kappa_\\chi^{quad}=\\frac{|B_\\chi|}{A_\\chi^{quad}}\\in[0,1]\\quad(\\kappa_\\chi^{quad}:=0\\text{ if }A_\\chi^{quad}=0)"
    },
    {
      "id": "ns.c4.c4i.quadratic_three_way_reentry",
      "latex": "a_\\chi^{quad}\\ge a_0\\ \\Rightarrow\\ (\\kappa_\\chi^{quad}<\\kappa_0)\\ \\vee\\ (r_\\chi\\ge\\theta\\kappa_0a_0)\\ \\vee\\ (\\Pi_R^{(2)}\\ge c(1-\\theta)\\kappa_0a_0)",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "二次強迫三路重入定理",
      "label_en": "Quadratic Forcing Three-Way Re-entry Theorem (C4-I.7)",
      "definition_zh": "§35本輪最終定理，證明局部二次絕對強度夠大時，必至少走I-QCANCEL（相干比小）、I-MROT（平均旋轉大）、I-PRESS（臨界壓力振盪大）三者之一。",
      "definition_en": "The round's culminating theorem in §35, proving that when the local quadratic absolute intensity is large enough, at least one of I-QCANCEL (low coherence), I-MROT (large mean rotation), or I-PRESS (large critical pressure oscillation) must hold.",
      "defining_relation": "a_\\chi^{quad}\\ge a_0\\ \\Rightarrow\\ \\underbrace{\\kappa_\\chi^{quad}<\\kappa_0}_{\\text{I-QCANCEL}}\\ \\vee\\ \\underbrace{r_\\chi\\ge\\theta\\kappa_0a_0}_{\\text{I-MROT}}\\ \\vee\\ \\underbrace{\\Pi_R^{(2)}\\ge c(1-\\theta)\\kappa_0a_0}_{\\text{I-PRESS}}",
      "notes": "Proved from C4-I.5 (§28) plus §30 in §35's proof; restated in prose as the round's headline conclusion in §36 and §56."
    },
    {
      "id": "ns.c4.c4i.desynchronization_mechanisms",
      "latex": "\\text{I-D1: Temporal Gate Pulse Separation}\\quad\\vee\\quad\\text{I-D2: Local Compensation Separation}",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "兩種剩餘去同步機制",
      "label_en": "Two Remaining Desynchronization Mechanisms (C4-I.8)",
      "definition_zh": "§44總結C4-I後仍可逃避更強同步化的兩條路徑：I-D1時間脈衝分離（middle/operator在同一J_j內不同次時段支付）與I-D2局部補償分離（透過平均旋轉或二次矩陣消去避開壓力重新進場）。",
      "definition_en": "§44 summarizes the two remaining escape routes from stronger synchronization after C4-I: I-D1, temporal pulse separation (middle and operator growth paid at different sub-times within the same J_j), and I-D2, local compensation separation (evading pressure re-entry via mean rotation or quadratic matrix cancellation).",
      "notes": "These become the two compensator mechanisms framing the next round C4-J (§53-54), whose obligations J1-J4 target I-D1 and J5-J7 target I-D2."
    },
    {
      "id": "ns.c4.c4i.state_tuples",
      "latex": "\\Theta_J^{MO}=\\langle A,B,M,O,E_m,E_o,\\mathfrak C_{overlap}\\rangle,\\ \\Theta^{angle}=\\langle r_\\nu,g,Q_\\perp,A_\\perp,W\\rangle,\\ \\Theta_\\chi^{reentry}=\\langle a_\\chi^{quad},\\kappa_\\chi^{quad},r_\\chi,\\Pi_R^{(2)}\\rangle",
      "series": "NS",
      "first_appearance": "C4-I",
      "label_zh": "True ETN狀態元組",
      "label_en": "True ETN state tuples",
      "definition_zh": "§50為本輪三個子系統各登記一個狀態元組：記錄窗口的Θ_J^MO、算子角度的Θ^angle、與局部壓力重新進場的Θ_χ^reentry，作為True ETN（無限維張力場）框架下的bookkeeping更新。",
      "definition_en": "§50 registers one state tuple for each of the round's three subsystems — the record-window state Θ_J^MO, the operator-angle state Θ^angle, and the local pressure-reentry state Θ_χ^reentry — as this round's bookkeeping update to the True ETN (infinite-dimensional tension field) framework.",
      "notes": "True ETN is the running cross-round state-tracking framework listed among Internal dependencies; these three tuples are C4-I's specific contribution to it."
    },
    {
      "id": "ns.c4.c4j.middle_eigenvalue_lambda2",
      "latex": "\\lambda_2^+",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "中間應變特徵值正部",
      "label_en": "Middle strain eigenvalue (positive part)",
      "definition_zh": "應變張量三個特徵值中之中間特徵值的正部，第1.1節重申其scale-critical可積性失效是finite-time blow-up的必要正則性閘門（Miller判準）。",
      "definition_en": "The positive part of the middle eigenvalue of the strain tensor; Section 1.1 reaffirms that its failure of scale-critical integrability is a necessary regularity gate for finite-time blow-up (Miller's criterion).",
      "notes": "external anchor inherited from Miller (arXiv:1710.05569) and used throughout the C4 record-ladder framework since C4-H."
    },
    {
      "id": "ns.c4.c4j.miller_qsv_operator",
      "latex": "\\mathcal Q_{SV}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "Miller應變–渦度算子",
      "label_en": "Miller strain-vorticity operator",
      "definition_zh": "第1.2節引用的算子 $\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)$，連同恆等式 $\\langle-\\Delta S,\\omega\\otimes\\omega\\rangle=0$ 共同刻畫blow-up必逃離之perturbative operator regime。",
      "definition_en": "The operator $\\mathcal Q_{SV}=P_{st}\\left((u\\cdot\\nabla)S+S^2+\\frac34\\omega\\otimes\\omega\\right)$ cited in Section 1.2, which together with the identity $\\langle-\\Delta S,\\omega\\otimes\\omega\\rangle=0$ characterizes the perturbative operator regime any blow-up must escape.",
      "notes": "external anchor from Miller's arXiv:2407.02691v2 strain-vorticity framework, not original to C4-J."
    },
    {
      "id": "ns.c4.c4j.record_window",
      "latex": "J_j=(\\tau_j,\\tau_{j+1})",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "記錄窗口",
      "label_en": "Record window",
      "definition_zh": "第2節沿C4-H/I記錄梯次定義的收縮時間區間，滿足 $|J_j|\\to0$，是本輪所有compensation ledger的基本時間單元。",
      "definition_en": "The shrinking time interval along the C4-H/I record ladder introduced in Section 2, satisfying $|J_j|\\to0$, serving as the basic temporal unit for every compensation ledger in this round."
    },
    {
      "id": "ns.c4.c4j.middle_strain_toll_density",
      "latex": "m_j(t)=\\int\\lambda_2^+|S|^2dx",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "中間應變徵費密度",
      "label_en": "Middle-strain toll density",
      "definition_zh": "第2節定義為 $\\int\\lambda_2^+|S|^2dx$，其在窗口 $J_j$ 上的積分下界為 $A_j>0$。",
      "definition_en": "Defined in Section 2 as $\\int\\lambda_2^+|S|^2dx$, whose integral over $J_j$ is bounded below by $A_j>0$."
    },
    {
      "id": "ns.c4.c4j.operator_growth_toll_density",
      "latex": "o_j(t)=\\nu[\\zeta r_\\nu-1]_+\\|\\Delta S\\|_2^2",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "算子成長徵費密度",
      "label_en": "Operator-growth toll density",
      "definition_zh": "第2節定義為 $\\nu[\\zeta r_\\nu-1]_+\\|\\Delta S\\|_2^2$，其在窗口 $J_j$ 上的積分下界為 $B_j>0$。",
      "definition_en": "Defined in Section 2 as $\\nu[\\zeta r_\\nu-1]_+\\|\\Delta S\\|_2^2$, whose integral over $J_j$ is bounded below by $B_j>0$."
    },
    {
      "id": "ns.c4.c4j.record_window_integral_peak_quantities",
      "latex": "A_j,\\ B_j,\\ M_j,\\ O_j",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "窗口積分與峰值量",
      "label_en": "Record-window integral and peak quantities",
      "definition_zh": "第2-3節定義的輔助量：$m_j,o_j$ 的積分下界 $A_j,B_j$ 與 $L^\\infty(J_j)$ 峰值 $M_j,O_j$，共同組成capacity ratios的分子分母。",
      "definition_en": "The auxiliary quantities from Sections 2-3: the integral lower bounds $A_j,B_j$ of $m_j,o_j$ and their $L^\\infty(J_j)$ peak values $M_j,O_j$, together forming the numerators and denominators of the capacity ratios."
    },
    {
      "id": "ns.c4.c4j.capacity_ratios",
      "latex": "K_{m,j},\\ K_{o,j}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "峰均容量比",
      "label_en": "Peak-to-average capacity ratios",
      "definition_zh": "第3節定義的比值，恆滿足 $K_{m,j},K_{o,j}\\ge1$，是本輪判定middle-strain與operator-growth兩pulse是否重疊的核心量。",
      "definition_en": "The ratios defined in Section 3, always satisfying $K_{m,j},K_{o,j}\\ge1$, the central quantities governing whether the middle-strain and operator-growth pulses overlap in this round.",
      "defining_relation": "K_{m,j}=\\frac{M_j|J_j|}{A_j},\\qquad K_{o,j}=\\frac{O_j|J_j|}{B_j}"
    },
    {
      "id": "ns.c4.c4j.overlap_sufficient_condition",
      "latex": "\\frac{1}{K_{m,j}}+\\frac{1}{K_{o,j}}>1",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "重疊充分條件",
      "label_en": "Overlap sufficient condition",
      "definition_zh": "第4節回顧的C4-I條件，保證zero threshold下 $\\{m_j>0\\}\\cap\\{o_j>0\\}$ 測度非零，即middle strain與operator growth同時發生。",
      "definition_en": "The C4-I condition recalled in Section 4, guaranteeing nonzero measure for $\\{m_j>0\\}\\cap\\{o_j>0\\}$ at the zero threshold, i.e. simultaneous middle-strain and operator-growth occurrence.",
      "defining_relation": "|\\{m_j>0\\}\\cap\\{o_j>0\\}|\\ge\\left[\\frac{A_j}{M_j}+\\frac{B_j}{O_j}-|J_j|\\right]_+\\ \\Longleftarrow\\ \\frac{1}{K_{m,j}}+\\frac{1}{K_{o,j}}>1",
      "notes": "inherited from C4-I; C4-J.1 shows it is sharp in the symmetric case $K_m=K_o=2$."
    },
    {
      "id": "ns.c4.c4j.thm_bounded_peakiness_nogo",
      "latex": "K_{m,j}=K_{o,j}=2",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "有界尖峰脈衝分離不可行定理（C4-J.1）",
      "label_en": "Bounded-Peakiness Pulse-Separation No-Go Theorem (C4-J.1)",
      "definition_zh": "第5節定理5.1，以equal-halves顯式構造證明即使 $K_{m,j}=K_{o,j}=2$ 對所有 $j$ 成立，$m_j,o_j$ 仍可在 $J_j$ 上幾乎處處互斥。",
      "definition_en": "Theorem 5.1 of Section 5, using an explicit equal-halves construction to show that even with $K_{m,j}=K_{o,j}=2$ for every $j$, $m_j$ and $o_j$ can remain a.e. mutually exclusive on $J_j$.",
      "defining_relation": "K_{m,j}=K_{o,j}=2\\ \\text{for every }j,\\qquad m_j(t)o_j(t)=0\\ \\text{a.e. on }J_j",
      "notes": "shows the overlap sufficient condition is sharp at the purely measure-theoretic level; not an N-S solution construction."
    },
    {
      "id": "ns.c4.c4j.growth_variation_rate",
      "latex": "h(t)=\\nu(\\zeta r_\\nu-1)\\|\\Delta S\\|_2^2",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "成長對齊算子變化率",
      "label_en": "Growth-aligned operator variation rate",
      "definition_zh": "第7節定義為 $\\nu(\\zeta r_\\nu-1)\\|\\Delta S\\|_2^2$，恰為應變 $\\dot H^1$ 能量 $E_1$ 的時間導數。",
      "definition_en": "Defined in Section 7 as $\\nu(\\zeta r_\\nu-1)\\|\\Delta S\\|_2^2$, exactly the time derivative of the strain $\\dot H^1$ energy $E_1$.",
      "defining_relation": "h(t)=\\nu(\\zeta r_\\nu-1)\\|\\Delta S\\|_2^2,\\qquad E_1'(t)=h(t)"
    },
    {
      "id": "ns.c4.c4j.strain_h1_energy",
      "latex": "E_1=\\frac12\\|S\\|_{\\dot H^1}^2",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "應變 $\\dot H^1$ 能量",
      "label_en": "Strain $\\dot H^1$ energy",
      "definition_zh": "第7節定義為 $\\frac12\\|S\\|_{\\dot H^1}^2$，其在窗口間的增量 $\\Delta E_{1,j}$ 是C4-J.2成長補償恆等式的核心。",
      "definition_en": "Defined in Section 7 as $\\frac12\\|S\\|_{\\dot H^1}^2$; its inter-window increment $\\Delta E_{1,j}$ is central to the C4-J.2 growth-compensation identity."
    },
    {
      "id": "ns.c4.c4j.growth_compensation_parts",
      "latex": "P_j=\\int_{J_j}[h]_+dt,\\ N_j=\\int_{J_j}[-h]_+dt",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "正/負成長分量",
      "label_en": "Positive/negative growth parts",
      "definition_zh": "第7節定義的窗口內成長對齊算子正向支付量 $P_j$ 與反向（over-dissipation）支付量 $N_j$。",
      "definition_en": "The within-window positive payment $P_j$ and opposing/over-dissipative payment $N_j$ of the growth-aligned operator, defined in Section 7."
    },
    {
      "id": "ns.c4.c4j.thm_growth_compensation_identity",
      "latex": "P_j=\\Delta E_{1,j}+N_j",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "正負成長補償恆等式（C4-J.2）",
      "label_en": "Exact Growth Compensation Identity (C4-J.2)",
      "definition_zh": "第8節定理8.1證明 $P_j-N_j=\\Delta E_{1,j}=E_1(\\tau_{j+1})-E_1(\\tau_j)>0$，說明任何growth-opposing pulse都不是免費的，必增加須支付的正成長變化量。",
      "definition_en": "Theorem 8.1 of Section 8 proves $P_j-N_j=\\Delta E_{1,j}=E_1(\\tau_{j+1})-E_1(\\tau_j)>0$, showing that any growth-opposing pulse is never free and forces additional positive growth-aligned variation.",
      "defining_relation": "P_j-N_j=\\Delta E_{1,j}>0,\\qquad P_j=\\Delta E_{1,j}+N_j",
      "notes": "Section 10 notes no standard theorem yet gives $\\sum_j(P_j+N_j)<\\infty$, so this exact ledger does not (yet) produce a contradiction."
    },
    {
      "id": "ns.c4.c4j.growth_aligned_scalar",
      "latex": "g(t)=\\zeta r_\\nu",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "成長對齊投影純量",
      "label_en": "Growth-aligned projection scalar",
      "definition_zh": "第9、11節沿用之純量：強反向支路 $g<-1$ 使 $[-h(t)]_+>2\\nu\\|\\Delta S(t)\\|_2^2$（第9節），並為算子分解 $\\widehat Q=-ge_D+Q_\\perp$ 的對齊分量（第11節）。",
      "definition_en": "The scalar used in Sections 9 and 11: the strong-opposing branch $g<-1$ forces $[-h(t)]_+>2\\nu\\|\\Delta S(t)\\|_2^2$ (Section 9), and $g$ is the aligned component of the decomposition $\\widehat Q=-ge_D+Q_\\perp$ (Section 11).",
      "notes": "inherited scalar from the C4-I adjoint-core / operator-angle decomposition."
    },
    {
      "id": "ns.c4.c4j.orthogonal_operator_component",
      "latex": "Q_\\perp,\\quad \\|Q_\\perp\\|_2^2=r_\\nu^2-g^2",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "正交算子分量",
      "label_en": "Orthogonal operator component",
      "definition_zh": "第11節沿用C4-I分解 $\\widehat Q=-ge_D+Q_\\perp$ 中的正交分量，當 $r_\\nu\\gg1$ 且 $|g|\\le1$ 時 $\\|Q_\\perp\\|_2\\sim r_\\nu$。",
      "definition_en": "The orthogonal component in the C4-I decomposition $\\widehat Q=-ge_D+Q_\\perp$ recalled in Section 11; when $r_\\nu\\gg1$ and $|g|\\le1$, $\\|Q_\\perp\\|_2\\sim r_\\nu$.",
      "notes": "decomposition attributed to C4-I; used here to classify the Operator Orthogonal Congestion motif."
    },
    {
      "id": "ns.c4.c4j.motif_operator_orthogonal_congestion",
      "latex": "\\textbf{Operator Orthogonal Congestion}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "算子正交壅塞動機",
      "label_en": "Operator Orthogonal Congestion motif",
      "definition_zh": "第11節命名的補償動機：大算子範數由 $Q_\\perp$（渦度平方或正交平流/應變平方項）承擔而不直接對 $E_1'$ 收費，目前無finite norm budget禁止其反覆出現。",
      "definition_en": "The compensation motif named in Section 11: large operator norm is carried by $Q_\\perp$ (vorticity-quadratic or orthogonal advection/strain-square terms) without directly charging $E_1'$, and no finite norm budget currently forbids its recurrence."
    },
    {
      "id": "ns.c4.c4j.adjoint_core_forcing_identity",
      "latex": "M_\\chi'=-B_\\chi-P_\\chi",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "伴隨核心迫力恆等式",
      "label_en": "Adjoint-core forcing identity",
      "definition_zh": "第13節給出的恆等式，$B_\\chi$ 為局部二次平均迫力、$P_\\chi$ 為局部壓力Hessian平均迫力，兩者共同決定伴隨核心均值 $M_\\chi$ 的變化率。",
      "definition_en": "The identity from Section 13, where $B_\\chi$ is the local quadratic mean forcing and $P_\\chi$ the local pressure-Hessian mean forcing, jointly determining the rate of change of the adjoint-core mean $M_\\chi$."
    },
    {
      "id": "ns.c4.c4j.normalized_integrated_ledger",
      "latex": "\\mathfrak B_I,\\ \\mathfrak V_M(I),\\ \\mathfrak P_I",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "標準化積分記帳量",
      "label_en": "Normalized integrated ledger quantities",
      "definition_zh": "第14節在時間區間 $I$ 與核心尺度 $R$ 上定義的三個標準化量，滿足逐點不等式 $|B_\\chi|\\le|M_\\chi'|+|P_\\chi|$ 導出的 $\\mathfrak B_I\\le\\mathfrak V_M(I)+\\mathfrak P_I$。",
      "definition_en": "The three normalized quantities defined in Section 14 over interval $I$ and core scale $R$, satisfying $\\mathfrak B_I\\le\\mathfrak V_M(I)+\\mathfrak P_I$ derived from the pointwise bound $|B_\\chi|\\le|M_\\chi'|+|P_\\chi|$.",
      "defining_relation": "\\mathfrak B_I=\\frac{1}{\\nu R}\\int_I|B_\\chi|dt,\\quad \\mathfrak V_M(I)=\\frac{1}{\\nu R}\\int_I|M_\\chi'|dt,\\quad \\mathfrak P_I=\\frac{1}{\\nu R}\\int_I|P_\\chi|dt"
    },
    {
      "id": "ns.c4.c4j.thm_integrated_mean_variation_pressure",
      "latex": "\\mathfrak V_M(I)\\ge(1-\\varepsilon)b_0",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "積分平均變化/壓力補償定理（C4-J.3）",
      "label_en": "Integrated Mean-Variation/Pressure Compensation Theorem (C4-J.3)",
      "definition_zh": "第15節定理：若 $\\mathfrak B_I\\ge b_0>0$ 且壓力衝量受壓 $\\mathfrak P_I\\le\\varepsilon b_0$，則 $\\mathfrak V_M(I)\\ge(1-\\varepsilon)b_0$，即壓力若不re-entry，平均應變變化必支付固定debt。",
      "definition_en": "The theorem of Section 15: if $\\mathfrak B_I\\ge b_0>0$ and the pressure impulse is suppressed to $\\mathfrak P_I\\le\\varepsilon b_0$, then $\\mathfrak V_M(I)\\ge(1-\\varepsilon)b_0$ — if pressure does not re-enter, mean-strain variation must pay a fixed debt.",
      "defining_relation": "\\mathfrak B_I\\ge b_0>0,\\ \\mathfrak P_I\\le\\varepsilon b_0\\ (0\\le\\varepsilon<1)\\ \\Longrightarrow\\ \\mathfrak V_M(I)\\ge(1-\\varepsilon)b_0",
      "notes": "Section 16 shows this is not yet a contradiction: C3-V's scale-weighted packing control still allows geometric Zeno-packing of $O(1)$ normalized variation per generation."
    },
    {
      "id": "ns.c4.c4j.local_quadratic_field",
      "latex": "Q(x,t)=S^2+\\frac14\\omega\\otimes\\omega-\\frac14|\\omega|^2I",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "局部二次絕對場",
      "label_en": "Local quadratic absolute field",
      "definition_zh": "第17節定義的張量場，是後續barycenter coherence與seven-point cancellation分析的基本對象。",
      "definition_en": "The tensor field defined in Section 17, the basic object underlying the subsequent barycenter-coherence and seven-point cancellation analysis."
    },
    {
      "id": "ns.c4.c4j.quadratic_barycenter_integrals",
      "latex": "A=\\int\\chi|Q|dx,\\ B=\\int\\chi Qdx",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "二次重心積分",
      "label_en": "Quadratic barycenter integrals",
      "definition_zh": "第17節在固定 $(t,\\chi)$ 下定義的純量 $A$ 與矩陣值積分 $B$，用以構成coherence $\\kappa=|B|/A$。",
      "definition_en": "The scalar $A$ and matrix-valued integral $B$ defined in Section 17 at fixed $(t,\\chi)$, used to form the coherence $\\kappa=|B|/A$.",
      "notes": "reuses the letters A, B in a role unrelated to the record-window thresholds $A_j,B_j$ of Section 2."
    },
    {
      "id": "ns.c4.c4j.coherence_kappa",
      "latex": "\\kappa=\\frac{|B|}{A}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "相干度",
      "label_en": "Coherence",
      "definition_zh": "第17-18節定義的量，等於局部二次方向 $U(x)$ 在機率測度 $\\mu$ 下之重心範數，是本輪判定quadratic cancellation的中心尺度。",
      "definition_en": "The quantity defined in Sections 17-18, equal to the barycenter norm of the local quadratic direction $U(x)$ under the probability measure $\\mu$, the central scalar governing quadratic cancellation in this round.",
      "defining_relation": "\\kappa=\\frac{|B|}{A}=\\left|\\int U\\,d\\mu\\right|,\\qquad d\\mu(x)=\\frac{\\chi(x)|Q(x)|}{A}dx"
    },
    {
      "id": "ns.c4.c4j.orientation_direction_matrix",
      "latex": "U(x)=\\frac{Q(x)}{|Q(x)|}\\in\\operatorname{Sym}(3)",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "二次方向矩陣",
      "label_en": "Quadratic orientation direction matrix",
      "definition_zh": "第18節在 $Q(x)\\ne0$ 之區域定義的正規化對稱矩陣，取值於 $\\operatorname{Sym}(3)\\simeq\\mathbb R^6$，是seven-point witness定理中被表示的對象。",
      "definition_en": "The normalized symmetric matrix defined in Section 18 on the region where $Q(x)\\ne0$, valued in $\\operatorname{Sym}(3)\\simeq\\mathbb R^6$, the object represented by the seven-point witness theorem."
    },
    {
      "id": "ns.c4.c4j.thm_seven_point_witness",
      "latex": "\\sum_{i=1}^{m}\\alpha_iU_i=\\frac{B}{A},\\ m\\le7",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "七點二次消去見證定理（C4-J.5）",
      "label_en": "Seven-Point Quadratic Cancellation Witness Theorem (C4-J.5)",
      "definition_zh": "第20節定理20.1，藉 $\\operatorname{Sym}(3)\\simeq\\mathbb R^6$ 上的Carathéodory定理證明若 $A>0$，則至多7個正規化局部二次矩陣之凸組合即可表示重心 $B/A$，其範數等於 $\\kappa$。",
      "definition_en": "Theorem 20.1 of Section 20, using Carathéodory's theorem on $\\operatorname{Sym}(3)\\simeq\\mathbb R^6$ to show that if $A>0$, a convex combination of at most 7 normalized local quadratic matrices already represents the barycenter $B/A$, with norm equal to $\\kappa$.",
      "defining_relation": "\\sum_{i=1}^{m}\\alpha_iU_i=\\frac{B}{A},\\quad m\\le7,\\quad \\alpha_i\\ge0,\\ \\sum_i\\alpha_i=1\\quad\\Longrightarrow\\quad \\left|\\sum_{i=1}^{m}\\alpha_iU_i\\right|=\\kappa",
      "notes": "the bound 7 = dim(Sym(3))+1 = 6+1; Sections 21-22 extend this to a compactness limit $\\sum_{i=1}^7\\alpha_i^\\ast U_i^\\ast=0$ as $\\kappa_n\\to0$ (the Seven-Point Quadratic Orientation Cancellation Motif) — metadata compactness only, consistent with the C3-H field-compactness no-go (Section 23), not full PDE field compactness."
    },
    {
      "id": "ns.c4.c4j.thm_pressure_avoidance_reduction",
      "latex": "\\mathfrak V_M(I_n)\\ge v_0>0\\ \\vee\\ \\kappa_n^{quad}\\le\\kappa_0<1",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "壓力迴避補償化約定理（C4-J.6）",
      "label_en": "Pressure-Avoidance Compensation Reduction Theorem (C4-J.6)",
      "definition_zh": "第25節定理25.1，證明對收縮伴隨核心事件列，若局部二次迫力非退化且壓力振盪不進入固定下界支路，則沿無窮子列必recurrent出現Mean-Variation Motif（J-P1）或Quadratic Orientation-Cancellation Motif（J-P2）。",
      "definition_en": "Theorem 25.1 of Section 25: for a sequence of shrinking adjoint-core events with nondegenerate local quadratic forcing and pressure oscillation avoiding a fixed lower-bound branch, an infinite subsequence must recurrently exhibit either the Mean-Variation Motif (J-P1) or the Quadratic Orientation-Cancellation Motif (J-P2).",
      "defining_relation": "\\text{J-P1: }\\mathfrak V_M(I_n)\\ge v_0>0\\qquad\\text{or}\\qquad\\text{J-P2: }\\kappa_n^{quad}\\le\\kappa_0<1",
      "notes": "status: PROVED CONDITIONAL ON NONDEGENERATE LOCAL QUADRATIC FORCING; if further $\\kappa_n\\to0$, a Seven-Point zero-barycenter limit witness can be extracted."
    },
    {
      "id": "ns.c4.c4j.local_pressure_concentration",
      "latex": "\\Pi_{R}^{(2)}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "局部壓力集中量",
      "label_en": "Local pressure concentration quantity",
      "definition_zh": "第27節引入的量：當mean variation與quadratic cancellation兩種補償皆失敗時，$\\Pi_{R_n}^{(2)}\\ge\\pi_0>0$ 迫使壓力進入critical concentration branch。",
      "definition_en": "The quantity introduced in Section 27: when both mean-variation and quadratic-cancellation compensation fail, $\\Pi_{R_n}^{(2)}\\ge\\pi_0>0$ forces pressure into the critical concentration branch.",
      "defining_relation": "\\Pi_{R_n}^{(2)}\\ge\\pi_0>0\\ \\Longrightarrow\\ \\int_{B_{CR_n}}|p|^{3/2}dx\\ge c\\pi_0^{3/2}\\nu^3"
    },
    {
      "id": "ns.c4.c4j.compensation_motif_taxonomy",
      "latex": "\\mathcal C=\\{T,O,M,Q,P,D\\}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "補償動機分類",
      "label_en": "Compensation motif taxonomy",
      "definition_zh": "第28節命名、第40節形式化為集合 $\\mathcal C$ 的六種殘餘動機——T時序脈衝、O算子夾角、M平均變化、Q二次方向消去、P壓力集中、D導數閘門缺陷——是C4封階論斷的核心分類。",
      "definition_en": "The six residual motifs named in Section 28 and formalized as the set $\\mathcal C$ in Section 40 — T (temporal pulse), O (operator-angle), M (mean-variation), Q (quadratic orientation-cancellation), P (pressure concentration), D (derivative-gate defect) — the central classification underlying the C4 closure claim.",
      "defining_relation": "\\mathcal C=\\{T,O,M,Q,P,D\\}"
    },
    {
      "id": "ns.c4.c4j.thm_phase_closure",
      "latex": "\\text{C4 branch/synchronization phase is structurally closed}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "C4階段封閉定理（C4-J.7）",
      "label_en": "C4 Phase Closure Theorem (C4-J.7)",
      "definition_zh": "第40節研究方案層級的定理，斷言所有recurrent UV singular events都可路由到synchronized結構或有限補償動機族 $\\mathcal C$，故C4封階，但此僅為research-program phase closure，非Navier–Stokes正則性定理。",
      "definition_en": "The research-program-level theorem of Section 40, asserting every recurrent UV singular event routes to synchronized structures or the finite motif family $\\mathcal C$, so the C4 phase closes — but only as a research-program phase closure, not a Navier-Stokes regularity theorem.",
      "defining_relation": "\\text{every recurrent UV singular event}\\ \\Rightarrow\\ \\text{synchronized structure}\\ \\vee\\ \\mathcal C",
      "notes": "headline result of the document; Section 50 explicitly records Navier-Stokes global regularity as still OPEN."
    },
    {
      "id": "ns.c4.c4j.c5_motif_state_vector",
      "latex": "\\Theta_j^{C5}=\\left\\langle\\mu_j^{mid},\\mu_j^{op,+},\\mu_j^{op,-},\\mu_j^{press},\\mu_j^{work},\\mathcal U_j^{(7)},\\mathsf D_j\\right\\rangle",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "C5記錄窗口動機狀態向量",
      "label_en": "C5-A record-window motif state vector",
      "definition_zh": "第44節為下一階段C5-A提出的狀態向量，將每個record window的中間應變、算子正/負成長、壓力、work-variation測度與seven-point witness、導數閘門缺陷打包為單一metadata物件。",
      "definition_en": "The state vector proposed in Section 44 for the forthcoming C5-A paper, packaging each record window's middle-strain, operator growth/opposing, pressure, and work-variation measures together with the seven-point witness and derivative-gate defect into one metadata object.",
      "notes": "forward-looking setup for C5-A (Record-Window Renormalization and Compensation-Motif State Space), the paper proposed to open the next phase, C5."
    },
    {
      "id": "ns.c4.c4j.six_channel_audit_table",
      "latex": "\\{UV,\\ Strain,\\ \\lambda_2^+,\\ Operator,\\ Helicity,\\ Pressure,\\ Derivative\\}",
      "series": "NS",
      "first_appearance": "C4-J",
      "label_zh": "六通道審計表",
      "label_en": "Six-channel audit table",
      "definition_zh": "第30-37節逐一評定UV、Strain、Middle $\\lambda_2^+$、Operator、Helicity、Pressure、Derivative geometry之最終同步狀態，並在第37節彙整成表，是本輪三大任務之一。",
      "definition_en": "Sections 30-37 assess the final synchronization status of UV, Strain, Middle $\\lambda_2^+$, Operator, Helicity, Pressure, and Derivative geometry individually, summarized in the Section 37 table — one of the round's three main tasks.",
      "notes": "the document's own header calls this the \"six-channel\" table though the Section 37 table lists seven rows (Middle $\\lambda_2^+$ counted separately from Operator/Strain)."
    },
    {
      "id": "ns.c5.c5a.unit_time_variable",
      "latex": "s",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "單位時間變數",
      "label_en": "Unit-time variable",
      "definition_zh": "在第2節中定義，將實體收縮時間窗線性映射至(0,1)區間的無因次化變數。",
      "definition_en": "Defined in Section 2 as the dimensionless variable linearly mapping the physical shrinking time window to the (0,1) interval.",
      "defining_relation": "s=\\frac{t-\\tau_j}{L_j}\\in(0,1)"
    },
    {
      "id": "ns.c5.c5a.compactified_viscous_time",
      "latex": "\\widehat\\Theta_j^{time}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化黏滯時間",
      "label_en": "Compactified viscous time",
      "definition_zh": "在第2節中引入，記錄單位時間重整化後保留的相對黏滯尺度，並映射至[0,1]區間。",
      "definition_en": "Introduced in Section 2, mapping the preserved relative viscous scale under unit-time renormalization to the [0,1] interval.",
      "defining_relation": "\\widehat\\Theta_j^{time} = \\frac{\\Theta_j^{time}}{1+\\Theta_j^{time}}"
    },
    {
      "id": "ns.c5.c5a.ancestry_compact_space",
      "latex": "\\mathcal K_{\\rm anc}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "繼承幾何緊緻空間",
      "label_en": "Ancestry geometry compact space",
      "definition_zh": "在第3節中定義，包含尺度比例與位移距離之緊緻化座標以及方向向量的空間。",
      "definition_en": "Defined in Section 3 as the space containing the compactified coordinates of scale ratios, displacement distances, and direction vectors."
    },
    {
      "id": "ns.c5.c5a.middle_strain_prob_measure",
      "latex": "\\mu_j^{mid}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "中度應變機率測度",
      "label_en": "Middle-strain probability measure",
      "definition_zh": "在第4節中定義，以中度應變特徵值與應變張量範數平方之積分作為權重的單位時間機率測度。",
      "definition_en": "Defined in Section 4 as a unit-time probability measure weighted by the integral of the middle-strain eigenvalue and squared strain tensor norm.",
      "defining_relation": "d\\mu_j^{mid}(s) = \\frac{L_jm_j(t_j(s))}{\\mathcal M_j}ds"
    },
    {
      "id": "ns.c5.c5a.energy_drop_fraction",
      "latex": "\\alpha_j^{mid}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "中度應變能量下降比",
      "label_en": "Middle-strain energy drop fraction",
      "definition_zh": "在第4節中定義，為能量下降量與中度應變總增量之比值。",
      "definition_en": "Defined in Section 4 as the ratio of energy drop to the total middle-strain growth on the window.",
      "defining_relation": "\\alpha_j^{mid} = \\frac{\\Delta E_{0,j}}{\\mathcal M_j}"
    },
    {
      "id": "ns.c5.c5a.dissipation_fraction",
      "latex": "\\delta_j^{mid}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "黏滯耗散分率",
      "label_en": "Dissipation fraction",
      "definition_zh": "在第4節中定義，為應變耗散積分與中度應變總增量之比值。",
      "definition_en": "Defined in Section 4 as the ratio of the strain dissipation integral to the total middle-strain growth on the window.",
      "defining_relation": "\\delta_j^{mid} = \\frac{D_{0,j}}{\\mathcal M_j}"
    },
    {
      "id": "ns.c5.c5a.operator_growth_rate",
      "latex": "h_j(t)",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "算子能量成長率",
      "label_en": "Operator energy growth rate",
      "definition_zh": "在第5節中定義，為包含黏滯耗散與渦度交互作用的算子動能一階導數。",
      "definition_en": "Defined in Section 5 as the first derivative of the operator kinetic energy including viscous dissipation and vorticity interaction.",
      "defining_relation": "h_j(t) = \\nu(\\zeta r_\\nu-1)\\|\\Delta S\\|_2^2"
    },
    {
      "id": "ns.c5.c5a.operator_positive_measure",
      "latex": "\\mu_j^{op,+}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "算子正向成長測度",
      "label_en": "Operator positive growth measure",
      "definition_zh": "在第5節中定義，以算子能量成長率正部為權重歸一化的次機率測度。",
      "definition_en": "Defined in Section 5 as the normalized subprobability measure weighted by the positive part of the operator energy growth rate.",
      "defining_relation": "d\\mu_j^{op,+}(s) = \\frac{L_j[h_j(t_j(s))]_+}{V_j^{op}}ds"
    },
    {
      "id": "ns.c5.c5a.operator_negative_measure",
      "latex": "\\mu_j^{op,-}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "算子負向成長測度",
      "label_en": "Operator negative growth measure",
      "definition_zh": "在第5節中定義，以算子能量衰減量為權重歸一化的次機率測度。",
      "definition_en": "Defined in Section 5 as the normalized subprobability measure weighted by the negative part of the operator energy growth rate.",
      "defining_relation": "d\\mu_j^{op,-}(s) = \\frac{L_j[-h_j(t_j(s))]_+}{V_j^{op}}ds"
    },
    {
      "id": "ns.c5.c5a.compensation_bias",
      "latex": "\\beta_j^{op}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "補償偏移量",
      "label_en": "Compensation bias",
      "definition_zh": "在第5節中定義，反映正負算子能量成長差異相對於總變差的比例。",
      "definition_en": "Defined in Section 5 as the ratio of the net operator energy change to the total variation.",
      "defining_relation": "\\beta_j^{op} = \\frac{P_j-N_j}{P_j+N_j}"
    },
    {
      "id": "ns.c5.c5a.compactified_ratio_coordinate",
      "latex": "\\rho",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化比值座標",
      "label_en": "Compactified ratio coordinate",
      "definition_zh": "在第6節中定義，為容許非線性交互項與耗散項之比值趨於無限大而引入的[0,1]區間座標。",
      "definition_en": "Defined in Section 6 as a [0,1] coordinate to accommodate the ratio of nonlinear interactions to dissipation blowing up to infinity.",
      "defining_relation": "\\rho=\\frac{r_\\nu}{1+r_\\nu}"
    },
    {
      "id": "ns.c5.c5a.compactified_alignment_angle",
      "latex": "\\gamma",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化對齊角",
      "label_en": "Compactified alignment angle coordinate",
      "definition_zh": "在第6節中定義，將算子幾何的對齊參數映射至[-1,1]區間的緊緻座標。",
      "definition_en": "Defined in Section 6 as a compact coordinate mapping the alignment parameter of the operator geometry into the [-1,1] interval.",
      "defining_relation": "\\gamma=\\frac2\\pi\\arctan(g)"
    },
    {
      "id": "ns.c5.c5a.operator_angle_compact_space",
      "latex": "\\mathcal K_{\\rm op}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "算子角度緊緻空間",
      "label_en": "Operator-angle compact space",
      "definition_zh": "在第6節中定義，包含所有緊緻化算子比值與角度座標的封閉緊緻狀態空間。",
      "definition_en": "Defined in Section 6 as the closed, compact state space containing all compactified operator ratio and angle coordinates."
    },
    {
      "id": "ns.c5.c5a.operator_angle_variation_measure",
      "latex": "\\eta_j^{op}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "算子角度變差測度",
      "label_en": "Operator-angle variation measure",
      "definition_zh": "在第7節中定義，以總變差為權重分布於單位時間與算子角度緊緻空間上的機率測度。",
      "definition_en": "Defined in Section 7 as a probability measure on the unit time and operator-angle compact space, weighted by absolute operator growth.",
      "defining_relation": "\\eta_j^{op} = \\left( s, \\Phi_{\\rm op}(r_\\nu,\\zeta) \\right)_\\# \\left[ \\frac{|h_j(t)|dt}{V_j^{op}} \\right]"
    },
    {
      "id": "ns.c5.c5a.mean_variation_compact_amplitude",
      "latex": "a_{M,j}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化平均變差振幅",
      "label_en": "Compactified mean variation amplitude",
      "definition_zh": "在第8節中定義，為伴隨核心無因次化變差總量映射至[0,1]區間的緊緻振幅。",
      "definition_en": "Defined in Section 8 as the compactified amplitude mapping the dimensionless total variation of the adjoint core to the [0,1] interval."
    },
    {
      "id": "ns.c5.c5a.mean_variation_vector_measure",
      "latex": "\\mathbf m_j^M",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "平均變差向量測度",
      "label_en": "Mean-variation vector measure",
      "definition_zh": "在第8節中定義，以伴隨核心的時間導數分布所建構的向量測度。",
      "definition_en": "Defined in Section 8 as a vector measure constructed from the time derivative of the adjoint core.",
      "defining_relation": "d\\mathbf m_j^M(s) = \\frac{L_jM_{\\chi_j}'(t_j(s))} {\\int_{J_j}|M_{\\chi_j}'(t)|dt}ds"
    },
    {
      "id": "ns.c5.c5a.quadratic_coherence",
      "latex": "\\kappa_j^Q",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "二次抵消相干性",
      "label_en": "Quadratic cancellation coherence",
      "definition_zh": "在第9節中定義，衡量二次空間分布淨量相對於總強度的比值，趨於零代表強烈抵消。",
      "definition_en": "Defined in Section 9 as the ratio of net quadratic spatial distribution to absolute intensity, where approaching zero indicates strong cancellation.",
      "defining_relation": "\\kappa_j^Q=\\frac{|B_j^Q|}{A_j^Q}"
    },
    {
      "id": "ns.c5.c5a.seven_point_compact_space",
      "latex": "\\mathcal K_Q",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "七點見證緊緻空間",
      "label_en": "Seven-point witness compact space",
      "definition_zh": "在第9節中定義，透過卡拉西奧多里約化記錄二次抵消特徵方向與權重的有限維緊緻空間。",
      "definition_en": "Defined in Section 9 as a finite-dimensional compact space recording the discrete directions and weights of quadratic cancellation via Carathéodory reduction."
    },
    {
      "id": "ns.c5.c5a.pressure_spatial_prob_measure",
      "latex": "\\nu_j^P",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "壓力空間機率測度",
      "label_en": "Pressure spatial probability measure",
      "definition_zh": "在第10節中定義，在特定壓力激發時刻將局部壓力場分佈正規化所得的空間測度。",
      "definition_en": "Defined in Section 10 as the normalized spatial measure of the local pressure field at a specific active pressure time.",
      "defining_relation": "d\\nu_j^P(y) = \\frac{(R_j^P)^3|p(x_j^P+R_j^Py,t_j^P)|^{3/2}} {Z_j^P}dy"
    },
    {
      "id": "ns.c5.c5a.compactified_pressure_mass",
      "latex": "a_j^P",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化壓力質量",
      "label_en": "Compactified pressure mass",
      "definition_zh": "在第10節中定義，將無因次局部壓力總量映射至[0,1]區間的狀態變數。",
      "definition_en": "Defined in Section 10 as the state variable mapping the dimensionless total local pressure mass to the [0,1] interval."
    },
    {
      "id": "ns.c5.c5a.compactified_pressure_oscillation",
      "latex": "\\widehat\\Pi_j^{(2)}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化壓力震盪",
      "label_en": "Compactified pressure oscillation",
      "definition_zh": "在第10節中定義，捕捉對海森矩陣敏感的局部壓力場震盪並映射至[0,1]區間。",
      "definition_en": "Defined in Section 10 as the parameter capturing the Hessian-sensitive local pressure oscillation, mapped to the [0,1] interval."
    },
    {
      "id": "ns.c5.c5a.compactified_derivative_load",
      "latex": "\\widehat{\\mathfrak L}_j",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "緊緻化導數負載",
      "label_en": "Compactified derivative load",
      "definition_zh": "在第11節中定義，將高階導數閉包負載映射至[0,1]區間的狀態變數。",
      "definition_en": "Defined in Section 11 as the state variable mapping the best higher-order derivative closure load to the [0,1] interval."
    },
    {
      "id": "ns.c5.c5a.motif_activation_vector",
      "latex": "a_j^{motif}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "圖案啟動向量",
      "label_en": "Motif activation vector",
      "definition_zh": "在第12節中定義，記錄六種補償圖案活躍狀態的布林向量，用於抽取最終穩定的圖案組合。",
      "definition_en": "Defined in Section 12 as a boolean vector recording the active states of the six compensation motifs, used to extract an eventually constant pattern."
    },
    {
      "id": "ns.c5.c5a.unified_state_vector",
      "latex": "\\Theta_j^{C5}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "C5統一狀態向量",
      "label_en": "Unified C5 state vector",
      "definition_zh": "在第13節中定義，統合所有緊緻化測度、振幅與幾何元資料的補償圖案狀態向量。",
      "definition_en": "Defined in Section 13 as the state vector unifying all compactified measures, amplitudes, and geometric metadata for the compensation motifs."
    },
    {
      "id": "ns.c5.c5a.limit_state_vector",
      "latex": "\\Theta_\\ast^{C5}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "循環補償極限狀態",
      "label_en": "Recurrent compensation-motif limit state",
      "definition_zh": "在第14節與第25節中證明存在，為無窮生存梯隊在緊緻化拓樸下收斂的元資料相容性極限。",
      "definition_en": "Proved to exist in Sections 14 and 25, representing the convergent metadata compatibility limit of an infinite survivor ladder under compact topologies.",
      "notes": "Does not imply full-field critical compactness, serving only as a necessary metadata limit state."
    },
    {
      "id": "ns.c5.c5a.overlap_spectrum",
      "latex": "\\mathfrak O_{j,n}",
      "series": "NS",
      "first_appearance": "C5-A",
      "label_zh": "尺度相依重疊頻譜",
      "label_en": "Scale-dependent overlap spectrum",
      "definition_zh": "在第19節中定義，用於量化不同時間尺度下中度應變與算子正成長測度的重疊程度。",
      "definition_en": "Defined in Section 19 as the integral quantifying the overlap between the middle-strain and operator positive growth measures at varying time scales.",
      "defining_relation": "\\mathfrak O_{j,n} = \\int_{[0,1]^2} K_n(s,t) d\\mu_j^{mid}(s) d\\mu_j^{op,+}(t)"
    },
    {
      "id": "ns.c5.c5b.middle_normalized_load",
      "latex": "f_j^M(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "中間正規化荷載",
      "label_en": "middle normalized load",
      "definition_zh": "第3節將紀錄窗內中間應變荷載密度 m_j 對總 toll \\mathcal{M}_j 正規化，得到 [0,1] 上積分為一的非負密度 f_j^M。",
      "definition_en": "Section 3 normalizes the middle-strain load density m_j against the total middle toll \\mathcal{M}_j to a nonnegative unit-mass density f_j^M on [0,1].",
      "defining_relation": "f_j^M(s)=\\frac{L_j m_j(t_j(s))}{\\mathcal{M}_j},\\qquad f_j^M\\ge0,\\ \\int_0^1 f_j^M(s)\\,ds=1"
    },
    {
      "id": "ns.c5.c5b.positive_operator_normalized_load",
      "latex": "f_j^+(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "正算子正規化荷載",
      "label_en": "positive-operator normalized load",
      "definition_zh": "第4節將算子增長正部 [h_j]_+ 對總正變分 P_j 正規化，得到單位質量密度 f_j^+。",
      "definition_en": "Section 4 normalizes the positive part [h_j]_+ of operator growth against the total positive variation P_j, yielding the unit-mass density f_j^+.",
      "defining_relation": "f_j^+(s)=\\frac{L_j[h_j(t_j(s))]_+}{P_j},\\qquad f_j^+\\ge0,\\ \\int_0^1 f_j^+\\,ds=1",
      "notes": "P_j>0 is inherited from the C4-J record identity P_j-N_j=\\Delta E_{1,j}>0."
    },
    {
      "id": "ns.c5.c5b.opposing_operator_normalized_load",
      "latex": "f_j^-(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "反相算子正規化荷載",
      "label_en": "opposing-operator normalized load",
      "definition_zh": "第5節在 N_j>0 時將 [-h_j]_+ 對總反相變分正規化為單位質量密度 f_j^-，否則令其恆為零。",
      "definition_en": "Section 5 sets f_j^- to the unit-mass normalization of [-h_j]_+ when N_j>0, and to the zero function when N_j=0.",
      "defining_relation": "f_j^-(s)=\\frac{L_j[-h_j(t_j(s))]_+}{N_j}\\ (N_j>0);\\qquad f_j^-\\equiv0\\ (N_j=0)"
    },
    {
      "id": "ns.c5.c5b.exact_operator_sign_exclusion",
      "latex": "f_j^+ f_j^-=0",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "算子符號精確互斥",
      "label_en": "exact operator-sign exclusion",
      "definition_zh": "第6節由點態 [h_j]_+[-h_j]_+=0 推出兩正規化算子密度幾乎處處相乘為零，作為本輪最基本的精確時相約束。",
      "definition_en": "Section 6 records the exact temporal phase constraint that the two operator-sign normalized densities multiply to zero a.e., inherited from the pointwise identity [h_j]_+[-h_j]_+=0.",
      "notes": "C5-B.2 (§13) upgrades this to the closed-support statement Y_*^ϑ([0,1]×F_{+-})=0 on the Young limit."
    },
    {
      "id": "ns.c5.c5b.rational_thresholds",
      "latex": "\\vartheta=(a,b,c)\\in\\mathbb{Q}_{>0}^3",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "有理閾值三元組",
      "label_en": "rational threshold triple",
      "definition_zh": "第7節固定正有理閾值 ϑ=(a,b,c)，分別切割中間、正算子與反相算子正規化荷載的活躍集合。",
      "definition_en": "Section 7 fixes a rational triple ϑ=(a,b,c) of positive thresholds that cut the three normalized loads into active versus inactive phases."
    },
    {
      "id": "ns.c5.c5b.threshold_phase_indicators",
      "latex": "\\chi_{j,M}^\\vartheta,\\ \\chi_{j,+}^\\vartheta,\\ \\chi_{j,-}^\\vartheta",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "閾值相位指示函數",
      "label_en": "threshold phase indicators",
      "definition_zh": "第7節分別以 {f_j^M≥a}、{f_j^+≥b}、{f_j^-≥c} 的特徵函數定義三個二值時相指示。",
      "definition_en": "Section 7 defines the three binary phase indicators as the characteristic functions of the threshold-active sets {f_j^M≥a}, {f_j^+≥b} and {f_j^-≥c}."
    },
    {
      "id": "ns.c5.c5b.phase_vector",
      "latex": "X_j^\\vartheta(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "相位向量",
      "label_en": "phase vector",
      "definition_zh": "第7節把三個閾值指示函數打包成取值於 {0,1}^3 的聯合相位向量，作為著色時間微狀態。",
      "definition_en": "Section 7 packages the three threshold indicators into the joint phase vector X_j^ϑ taking values in {0,1}^3, which is the colored temporal microstate.",
      "defining_relation": "X_j^\\vartheta(s)=(\\chi_{j,M}^\\vartheta(s),\\chi_{j,+}^\\vartheta(s),\\chi_{j,-}^\\vartheta(s))"
    },
    {
      "id": "ns.c5.c5b.finite_phase_alphabet",
      "latex": "\\mathcal{A}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "有限相位字母表",
      "label_en": "finite phase alphabet",
      "definition_zh": "第8節由 χ_+χ_-=0 把允許相位狀態限制為六個離散字元 {000,100,010,001,110,101}，並禁止 011 與 111。",
      "definition_en": "Section 8 restricts admissible phase states, via χ_+χ_-=0, to the six-element discrete alphabet {000,100,010,001,110,101}, forbidding 011 and 111.",
      "defining_relation": "\\mathcal{A}=\\{000,100,010,001,110,101\\}"
    },
    {
      "id": "ns.c5.c5b.colored_temporal_young_measure",
      "latex": "Y_j^\\vartheta",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "著色時間 Young 測度",
      "label_en": "colored temporal Young measure",
      "definition_zh": "第9節把正規化時間 s 與相位向量 X_j^ϑ(s) 對 Lebesgue 測度 ds 的 push-forward 定義為取值於 P([0,1]×A) 且第一邊緣固定為 ds 的著色時間圖測度。",
      "definition_en": "Section 9 defines the colored temporal graph measure as the push-forward of Lebesgue measure ds under s↦(s,X_j^ϑ(s)), landing in P([0,1]×A) with first marginal ds.",
      "defining_relation": "Y_j^\\vartheta=(s,X_j^\\vartheta(s))_{\\#}(ds)\\in\\mathcal{P}([0,1]\\times\\mathcal{A}),\\qquad(\\pi_s)_{\\#}Y_j^\\vartheta=ds"
    },
    {
      "id": "ns.c5.c5b.young_limit_disintegration",
      "latex": "Y_\\ast^\\vartheta=ds\\,\\nu_s^\\vartheta",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "Young 極限與時間纖維",
      "label_en": "Young limit and temporal disintegration",
      "definition_zh": "第10節定理10.1（C5-B.1）抽出弱極限 Y_*^ϑ，其第一邊緣仍為 ds，並纖維化為幾乎處處取值於 P(A) 的未解析時相分佈 ν_s^ϑ。",
      "definition_en": "Section 10 (Theorem 10.1 / C5-B.1) extracts a weak limit Y_*^ϑ with first marginal ds, disintegrating as ds ν_s^ϑ with ν_s^ϑ∈P(A) for a.e. s.",
      "defining_relation": "Y_\\ast^\\vartheta(ds,d\\xi)=ds\\,\\nu_s^\\vartheta(d\\xi),\\qquad\\nu_s^\\vartheta\\in\\mathcal{P}(\\mathcal{A})"
    },
    {
      "id": "ns.c5.c5b.temporal_phase_spectrum",
      "latex": "\\mathfrak{Y}_\\ast",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "時間相位譜",
      "label_en": "temporal phase spectrum",
      "definition_zh": "第11節對可數集 Q_{>0}^3 對角抽取，得到同一子列上對所有有理 ϑ 同時成立的 Young 極限族，並稱之為時間相位譜。",
      "definition_en": "Section 11 diagonalizes over the countable set Q_{>0}^3 to obtain a single subsequence along which Y_j^ϑ⇀Y_*^ϑ for every rational ϑ, called the temporal phase spectrum.",
      "defining_relation": "\\mathfrak{Y}_\\ast=\\{Y_\\ast^\\vartheta\\}_{\\vartheta\\in\\mathbb{Q}_{>0}^3}"
    },
    {
      "id": "ns.c5.c5b.coactive_phase_mass",
      "latex": "C_{\\ast,M+}^\\vartheta",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "共活相位質量",
      "label_en": "coactive phase mass",
      "definition_zh": "第14節以 Young 測度在相位 110 上的質量定義中間／正算子共活佔空比，有限尺度值為 C_{j,M+}^ϑ、極限值為 C_{*,M+}^ϑ。",
      "definition_en": "Section 14 defines the coactive duty as the Y-mass of the phase 110, with finite-scale value C_{j,M+}^ϑ and Young-limit value C_{*,M+}^ϑ.",
      "defining_relation": "C_{j,M+}^\\vartheta=Y_j^\\vartheta([0,1]\\times\\{110\\}),\\qquad C_{\\ast,M+}^\\vartheta=Y_\\ast^\\vartheta([0,1]\\times\\{110\\})",
      "notes": "Theorem 15.1 (C5-B.3) upgrades C_{*,M+}^ϑ>0 into genuine finite-scale same-time overlap for all sufficiently large j along the extracted subsequence."
    },
    {
      "id": "ns.c5.c5b.barycentric_phase_fractions",
      "latex": "\\bar\\chi_M^\\vartheta(s),\\ \\bar\\chi_+^\\vartheta(s),\\ \\bar\\chi_-^\\vartheta(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "重心相位分數",
      "label_en": "barycentric phase fractions",
      "definition_zh": "第18節將纖維測度 ν_s^ϑ 對各相位座標積分，得到正規化時間 s 處的局部時相佔比重心投影。",
      "definition_en": "Section 18 integrates the disintegrated Young fiber ν_s^ϑ against each phase coordinate to obtain the barycentric (mean) phase fractions at normalized time s."
    },
    {
      "id": "ns.c5.c5b.microscopic_coactivation",
      "latex": "c_{M+}^\\vartheta(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "微觀共活密度",
      "label_en": "microscopic coactivation density",
      "definition_zh": "第18節以纖維測度對乘積 ξ_M ξ_+ 的積分定義微觀共活密度，量測同一微狀態上中間與正算子同時活躍的局部質量。",
      "definition_en": "Section 18 defines the microscopic coactivation density as the fiber integral of the product ξ_M ξ_+, measuring local mass of simultaneous middle and positive-operator activity."
    },
    {
      "id": "ns.c5.c5b.temporal_phase_covariance",
      "latex": "\\operatorname{Cov}_{M+}^\\vartheta(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "時間相位協方差",
      "label_en": "temporal phase covariance",
      "definition_zh": "第19節以微觀共活減去重心分數乘積定義 Cov_{M+}，在完全互斥且兩分數皆正時嚴格為負，用以量化反相關時相混合。",
      "definition_en": "Section 19 defines the temporal phase covariance as microscopic coactivation minus the product of barycentric fractions, which is strictly negative under complete exclusion with both fractions positive.",
      "notes": "The alternating-microcell example of §17 realizes Cov_{M+}=-1/4, repairing the C5-A separate-weak-measure no-go that homogenized to a fake coactive overlap."
    },
    {
      "id": "ns.c5.c5b.operator_compact_with_cemetery",
      "latex": "\\mathcal{K}_{\\mathrm{op}}^\\dagger=\\mathcal{K}_{\\mathrm{op}}\\cup\\{\\partial\\}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "帶墓地點的算子緊化",
      "label_en": "operator compact with cemetery point",
      "definition_zh": "第20節在 C5-A 的算子緊集 K_op 上加入墓地點 ∂，使兩算子閾值皆不活躍的時刻可標記為 ∂ 而空間仍緊。",
      "definition_en": "Section 20 adjoins a cemetery point ∂ to the C5-A operator compact K_op so that times inactive for both operator thresholds can be marked while remaining in a compact space.",
      "notes": "K_op and the operator-angle map Φ_op(r_ν,ζ) carry over from C5-A; the cemetery point is new in C5-B."
    },
    {
      "id": "ns.c5.c5b.operator_angle_marking",
      "latex": "\\kappa_j^{op}(s)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "算子角標記",
      "label_en": "operator-angle marking",
      "definition_zh": "第20節在 χ_{j,+}+χ_{j,-}>0 的正規化時刻把 κ_j^{op} 設為 Φ_op(r_ν,ζ)，否則設為墓地點 ∂。",
      "definition_en": "Section 20 attaches κ_j^{op}(s)=Φ_op(r_ν,ζ) on normalized times where at least one operator threshold is active, and sets κ_j^{op}=∂ otherwise."
    },
    {
      "id": "ns.c5.c5b.marked_temporal_young_measure",
      "latex": "\\widetilde{Y}_j^\\vartheta",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "標記時間 Young 測度",
      "label_en": "marked temporal Young measure",
      "definition_zh": "第21節把 (s,X_j^ϑ(s),κ_j^{op}(s)) 對 ds 的 push-forward 放到緊空間 [0,1]×A×K_op^† 上，從而可抽取標記 Young 弱極限。",
      "definition_en": "Section 21 push-forwards (s,X_j^ϑ(s),κ_j^{op}(s)) under ds onto the compact [0,1]×A×K_op^†, producing a marked Young measure with extractable weak limits.",
      "defining_relation": "\\widetilde{Y}_j^\\vartheta=(s,X_j^\\vartheta(s),\\kappa_j^{op}(s))_{\\#}ds\\in\\mathcal{P}([0,1]\\times\\mathcal{A}\\times\\mathcal{K}_{\\mathrm{op}}^\\dagger)"
    },
    {
      "id": "ns.c5.c5b.phase_angle_compatibility",
      "latex": "\\gamma\\ge\\tfrac12\\ \\text{on}\\ \\mathrm{supp}(\\xi_+=1);\\quad\\gamma\\le\\tfrac12\\ \\text{on}\\ \\mathrm{supp}(\\xi_-=1)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "相位–角度相容性",
      "label_en": "phase–angle compatibility",
      "definition_zh": "第23節定理 C5-B.4 要求標記 Young 極限在正算子相位支撐閉包上滿足 γ≥1/2、在反相相位上滿足 γ≤1/2，並保持 ξ_+ξ_-=0。",
      "definition_en": "Section 23 (C5-B.4) forces the marked Young limit to satisfy γ≥1/2 on the support closure of ξ_+=1, γ≤1/2 on that of ξ_-=1, together with ξ_+ξ_-=0.",
      "notes": "The compact operator coordinate γ=(2/π)arctan(ζ r_ν) and the gate g=ζ r_ν carry over from C5-A; C5-B only adds the joint support constraint with the temporal phase color."
    },
    {
      "id": "ns.c5.c5b.asymptotic_concentration_mass",
      "latex": "\\mathfrak{c}_f^\\infty",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "漸近集中質量",
      "label_en": "asymptotic concentration mass",
      "definition_zh": "第25節先取尾質量 limsup_j ∫_{f_j>K} f_j，再令 K→∞ 得到 [0,1] 中的漸近集中質量，用以捕捉 vanishing-duty 高振幅尖峰。",
      "definition_en": "Section 25 defines the tail mass 𝔠_f(K) as the limsup of load above height K, then takes K→∞ to obtain the asymptotic concentration mass 𝔠_f^∞∈[0,1] that captures vanishing-duty high-amplitude spikes.",
      "defining_relation": "\\mathfrak{c}_f^\\infty=\\lim_{K\\to\\infty}\\limsup_{j\\to\\infty}\\int_{\\{f_j>K\\}}f_j(s)\\,ds\\in[0,1]"
    },
    {
      "id": "ns.c5.c5b.load_concentration_coordinates",
      "latex": "\\mathfrak{c}_M,\\ \\mathfrak{c}_+,\\ \\mathfrak{c}_-",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "荷載加權集中座標",
      "label_en": "load-weighted concentration coordinates",
      "definition_zh": "第27節分別對 f^M、f^+ 與（在反相活躍反覆出現時）f^- 取漸近集中質量，作為荷載加權的時間集中缺陷座標。",
      "definition_en": "Section 27 evaluates the asymptotic concentration mass on f^M, f^+ and (when opposing activity is recurrent) f^-, yielding the load-weighted temporal concentration coordinates.",
      "notes": "Section 26 records the equivalence 𝔠_f^∞=0 ⇔ uniform integrability of the unit-mass family {f_j}."
    },
    {
      "id": "ns.c5.c5b.vanishing_duty_full_concentration",
      "latex": "\\mathfrak{c}_f^\\infty=1",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "消逝佔空比強制完全集中",
      "label_en": "vanishing duty forces full concentration",
      "definition_zh": "第30節定理30.1（C5-B.6）證明：若對每個 a>0 皆有 |{f_j≥a}|→0，則漸近集中質量必等於 1，即全部荷載質量轉成集中缺陷。",
      "definition_en": "Section 30 (Theorem 30.1 / C5-B.6) proves that if |{f_j≥a}|→0 for every a>0, then 𝔠_f^∞=1, so the entire unit load mass becomes a concentration defect."
    },
    {
      "id": "ns.c5.c5b.coactivation_oscillation_concentration_trichotomy",
      "latex": "\\mathrm{B\\text{-}COACT}\\vee\\mathrm{B\\text{-}OSC}\\vee\\mathrm{B\\text{-}CONC}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "共活／振盪／集中三分法",
      "label_en": "coactivation–oscillation–concentration trichotomy",
      "definition_zh": "第33節定理33.1（C5-B.7）斷言中間與正算子正規化荷載序列在抽出的 C5-B 子列上必落入真共活、主體 Young 相位分離／振盪、或時間荷載集中三者之一。",
      "definition_en": "Section 33 (Theorem 33.1 / C5-B.7) asserts that along the extracted C5-B subsequence the middle and positive-operator normalized-load sequences fall into genuine coactivation, bulk Young-phase segregation/oscillation, or temporal load concentration.",
      "notes": "Section 34 records the residual after excluding B-COACT as Young-phase oscillation ∨ DiPerna–Majda-type concentration; the objects remain record-window temporal motifs, not measure-valued Navier–Stokes solutions (§1.2, §35)."
    },
    {
      "id": "ns.c5.c5b.load_colored_dominating_measure",
      "latex": "\\Lambda_j",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "荷載著色支配測度",
      "label_en": "load-colored dominating measure",
      "definition_zh": "第36節取中間與正算子荷載測度的平均作為 [0,1] 上的共同機率測度 Λ_j，用以聯合支配兩種顏色。",
      "definition_en": "Section 36 averages the middle and positive-operator load measures into a common probability Λ_j on [0,1] that jointly dominates both colors.",
      "defining_relation": "\\Lambda_j=\\tfrac12(\\mu_j^M+\\mu_j^+)\\in\\mathcal{P}([0,1]),\\qquad\\mu_j^M=f_j^M\\,ds,\\ \\mu_j^+=f_j^+\\,ds"
    },
    {
      "id": "ns.c5.c5b.radon_nikodym_color_fractions",
      "latex": "z_{j,M},\\ z_{j,+}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "荷載顏色分數",
      "label_en": "load-color Radon–Nikodym fractions",
      "definition_zh": "第36節以 Λ_j 為底取 μ_j^M、μ_j^+ 的 Radon–Nikodym 導數之半，得到 Λ_j-a.e. 和為一的顏色分數。",
      "definition_en": "Section 36 takes half the Radon–Nikodym derivatives of μ_j^M and μ_j^+ with respect to Λ_j, yielding color fractions that sum to 1 Λ_j-a.e."
    },
    {
      "id": "ns.c5.c5b.load_colored_young_graph",
      "latex": "\\Upsilon_j",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "荷載著色 Young 圖",
      "label_en": "load-colored Young graph",
      "definition_zh": "第37節把 (s,z_{j,M},z_{j,+}) 對 Λ_j 的 push-forward 放到 [0,1]×Δ_2 上；第38節證明其弱極限 Υ_* 的重心投影恰好恢復一階分色荷載測度。",
      "definition_en": "Section 37 push-forwards (s,z_{j,M},z_{j,+}) under Λ_j onto [0,1]×Δ_2; Section 38 shows that the weak limit Υ_* recovers the separate first-order load measures as barycentric projections.",
      "defining_relation": "\\Upsilon_j=(s,z_{j,M},z_{j,+})_{\\#}\\Lambda_j\\in\\mathcal{P}([0,1]\\times\\Delta_2)",
      "notes": "Section 39 records that exact finite-scale exclusion f_j^M f_j^+=0 forces supp Υ_* ⊂ [0,1]×{(1,0),(0,1)}, so the load-colored graph likewise retains pulse separation."
    },
    {
      "id": "ns.c5.c5b.lag_correlation_spectrum",
      "latex": "C_j^{a\\to b}(\\ell),\\ C_\\ast^{a\\to b}(\\ell)",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "固定延遲相關譜",
      "label_en": "fixed-lag correlation spectrum",
      "definition_zh": "第42節對二值閾值相位以固定延遲 ℓ 積分乘積，並對有理 ℓ 對角抽取極限 C_*^{a→b}(ℓ) 作為粗轉換譜。",
      "definition_en": "Section 42 integrates products of binary threshold phases at a fixed lag ℓ and extracts limits C_*^{a→b}(ℓ) over rational lags as a coarse transition spectrum.",
      "notes": "Section 43 records that any fixed-lag spectrum still misses moving microscopic periods ε_j→0, which is why C5-C must introduce a two-scale / transition defect."
    },
    {
      "id": "ns.c5.c5b.operator_sign_cycle_bias",
      "latex": "\\beta_j^{op}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "算子符號循環偏置",
      "label_en": "operator sign-cycle bias",
      "definition_zh": "第44節以 (P_j-N_j)/(P_j+N_j)>0 記錄正增長對總變分的偏置；極限 β_*>0 表示正增長質量佔優，β_*=0 則為算子補償循環邊界態。",
      "definition_en": "Section 44 records the strictly positive bias (P_j-N_j)/(P_j+N_j) of operator growth; β_*→positive means positive-growth mass dominance, while β_*=0 is an operator compensation-cycle boundary state.",
      "notes": "The identity P_j-N_j=ΔE_{1,j}>0 is inherited from the C4-J record bias."
    },
    {
      "id": "ns.c5.c5b.compatibility_state",
      "latex": "\\Theta_\\ast^{C5B}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "C5-B 相容極限態",
      "label_en": "C5-B compatibility state",
      "definition_zh": "第45節把 C5-A 極限、時間相位譜、標記 Young 譜、三個集中質量、荷載著色 Young 圖與固定延遲相關元資料打包為增強相容態 Θ_*^{C5B}。",
      "definition_en": "Section 45 packages the C5-A limit together with the phase Young spectrum, marked Young spectrum, concentration masses, load-colored Young graph and lag-correlation metadata into the enhanced compatibility state Θ_*^{C5B}.",
      "defining_relation": "\\Theta_\\ast^{C5B}=\\bigl\\langle\\Theta_\\ast^{C5A},\\mathfrak{Y}_\\ast,\\widetilde{\\mathfrak{Y}}_\\ast,\\mathfrak{c}_M,\\mathfrak{c}_+,\\mathfrak{c}_-,\\Upsilon_\\ast,\\mathfrak{C}_\\ast^{\\mathrm{lag}}\\bigr\\rangle",
      "notes": "This strictly enlarges Θ_*^{C5A} from the previous round; the temporal-Young ETN block Θ_*^{TY} of §49 is the same package without the inherited C5-A motif coordinates."
    },
    {
      "id": "ns.c5.c5b.temporal_defect_classes",
      "latex": "\\mathrm{T1},\\mathrm{T2},\\mathrm{T3},\\mathrm{T4}",
      "series": "NS",
      "first_appearance": "C5-B",
      "label_zh": "第一微結構層時間缺陷四類",
      "label_en": "first-microstructure temporal defect classes",
      "definition_zh": "第46節（C5-B.8）將紀錄窗正規化下的中間／算子補償分類為真共活（T1）、主體 Young 相位分離（T2）、荷載集中（T3）與未解析的次 Young 排序／相關缺陷（T4）。",
      "definition_en": "Section 46 (C5-B.8) classifies middle/operator temporal compensation at the first microstructure level into genuine coactivation (T1), bulk Young-phase segregation (T2), load concentration (T3), and unresolved sub-Young ordering/correlation defect (T4).",
      "notes": "T4 is the declared frontier of C5-C (temporal correlation defects, transition measures, and causal pulse ordering)."
    },
    {
      "id": "ns.c5.c5c.strain_energy_e0",
      "latex": "E_0(t)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "應變能量",
      "label_en": "strain energy",
      "definition_zh": "在光滑預奇異 Navier–Stokes 演化上，應變能量定義為 \\(E_0(t)=\frac12\\|S(t)\\|_2^2\\)（§2）。",
      "definition_en": "On smooth pre-singular Navier–Stokes evolution, the strain energy is defined by \\(E_0(t)=\frac12\\|S(t)\\|_2^2\\) (§2).",
      "defining_relation": "E_0(t)=\\frac12\\|S(t)\\|_2^2",
      "notes": "Miller strain identities (§1.1); all C5-C cumulative identities act on smooth pre-singular solutions."
    },
    {
      "id": "ns.c5.c5c.strain_enstrophy_e1",
      "latex": "E_1(t)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "應變渦能",
      "label_en": "strain enstrophy",
      "definition_zh": "應變渦能（\\(\\dot H^1\\) 能量）定義為 \\(E_1(t)=\frac12\\|S(t)\\|_{\\dot H^1}^2=\frac12\\|\nabla S(t)\\|_2^2\\)（§2）。",
      "definition_en": "Strain enstrophy (the \\(\\dot H^1\\) energy) is defined by \\(E_1(t)=\frac12\\|S(t)\\|_{\\dot H^1}^2=\frac12\\|\nabla S(t)\\|_2^2\\) (§2).",
      "defining_relation": "E_1(t)=\\frac12\\|S(t)\\|_{\\dot H^1}^2=\\frac12\\|\\nabla S(t)\\|_2^2"
    },
    {
      "id": "ns.c5.c5c.record_window",
      "latex": "J_j=(\\tau_j,\\tau_{j+1}),\\quad L_j=|J_j|,\\quad s=(t-\\tau_j)/L_j",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "紀錄窗與正規化時間",
      "label_en": "record window and normalized time",
      "definition_zh": "第 \\(j\\) 個紀錄窗為 \\(J_j=(\tau_j,\tau_{j+1})\\)，長度 \\(L_j=|J_j|\\)，並以 \\(s=(t-\tau_j)/L_j\\in[0,1]\\) 作正規化時間（§2）。",
      "definition_en": "The \\(j\\)-th record window is \\(J_j=(\tau_j,\tau_{j+1})\\) of length \\(L_j=|J_j|\\), with normalized time \\(s=(t-\tau_j)/L_j\\in[0,1]\\) (§2)."
    },
    {
      "id": "ns.c5.c5c.enstrophy_amplification",
      "latex": "a(t)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "渦能放大率",
      "label_en": "enstrophy amplification",
      "definition_zh": "應變渦能放大率定義為 \\(a(t)=-2\\int_{\\mathbb R^3}\\det S\\,dx\\)，並由應變渦能恆等給出 \\(E_0'+2\nu E_1=a(t)\\)（§3）。",
      "definition_en": "Strain-enstrophy amplification is \\(a(t)=-2\\int_{\\mathbb R^3}\\det S\\,dx\\), entering the identity \\(E_0'+2\nu E_1=a(t)\\) (§3).",
      "defining_relation": "E_0'+2\\nu E_1=a(t)=-2\\int_{\\mathbb R^3}\\det S\\,dx"
    },
    {
      "id": "ns.c5.c5c.middle_load",
      "latex": "m(t)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間荷載",
      "label_en": "middle load",
      "definition_zh": "中間荷載定義為 \\(m(t)=\\int\\lambda_2^+|S|^2\\,dx\\)，並由 C4-H 點態矩陣不等式控制 \\(a(t)\\le m(t)\\)（§3）。",
      "definition_en": "Middle load is \\(m(t)=\\int\\lambda_2^+|S|^2\\,dx\\), dominating amplification via the C4-H bound \\(a(t)\\le m(t)\\) (§3).",
      "defining_relation": "m(t)=\\int\\lambda_2^+|S|^2\\,dx,\\qquad a(t)\\le m(t)",
      "notes": "The pointwise bound \\(a\\le m\\) is inherited from C4-H."
    },
    {
      "id": "ns.c5.c5c.middle_slack",
      "latex": "q(t)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間鬆弛",
      "label_en": "middle slack",
      "definition_zh": "中間鬆弛定義為 \\(q(t)=m(t)-a(t)\\ge0\\)，從而給出精確點態帳本 \\(m=E_0'+2\nu E_1+q\\)（§4）。",
      "definition_en": "Middle slack is the nonnegative remainder \\(q(t)=m(t)-a(t)\\ge0\\), yielding the exact pointwise ledger \\(m=E_0'+2\nu E_1+q\\) (§4).",
      "defining_relation": "q(t)=m(t)-a(t)\\ge0,\\qquad m=E_0'+2\\nu E_1+q"
    },
    {
      "id": "ns.c5.c5c.total_middle_toll",
      "latex": "\\mathcal{M}_j",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間總荷載",
      "label_en": "total middle toll",
      "definition_zh": "紀錄窗 \\(J_j\\) 上的中間總荷載（total middle toll）定義為 \\(\\mathcal M_j=\\int_{J_j}m(t)\\,dt>0\\)（§5）。",
      "definition_en": "The total middle toll on the record window \\(J_j\\) is \\(\\mathcal M_j=\\int_{J_j}m(t)\\,dt>0\\) (§5).",
      "defining_relation": "\\mathcal{M}_j=\\int_{J_j}m(t)\\,dt>0"
    },
    {
      "id": "ns.c5.c5c.middle_supply",
      "latex": "C_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間供給",
      "label_en": "middle supply",
      "definition_zh": "正規化中間供給路徑定義為 \\(C_j(s)=\\mathcal M_j^{-1}\\int_{\tau_j}^{t_j(s)}m(t)\\,dt\\)，單調非減且 \\(C_j(0)=0\\)、\\(C_j(1)=1\\)（§6、§8）。",
      "definition_en": "The normalized middle-supply path is \\(C_j(s)=\\mathcal M_j^{-1}\\int_{\tau_j}^{t_j(s)}m(t)\\,dt\\), nondecreasing with \\(C_j(0)=0\\) and \\(C_j(1)=1\\) (§6, §8).",
      "defining_relation": "C_j(s)=\\frac1{\\mathcal{M}_j}\\int_{\\tau_j}^{t_j(s)}m(t)\\,dt"
    },
    {
      "id": "ns.c5.c5c.dissipation_demand",
      "latex": "D_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "應變耗散需求",
      "label_en": "strain-dissipation demand",
      "definition_zh": "正規化應變耗散需求路徑定義為 \\(D_j(s)=(2\nu/\\mathcal M_j)\\int_{\tau_j}^{t_j(s)}E_1(t)\\,dt\\)，單調非減且值域落在 \\([0,1]\\)（§6、§8）。",
      "definition_en": "The normalized strain-dissipation demand path is \\(D_j(s)=(2\nu/\\mathcal M_j)\\int_{\tau_j}^{t_j(s)}E_1(t)\\,dt\\), nondecreasing and valued in \\([0,1]\\) (§6, §8).",
      "defining_relation": "D_j(s)=\\frac{2\\nu}{\\mathcal{M}_j}\\int_{\\tau_j}^{t_j(s)}E_1(t)\\,dt"
    },
    {
      "id": "ns.c5.c5c.cumulative_slack",
      "latex": "Q_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "累積中間鬆弛",
      "label_en": "cumulative middle slack",
      "definition_zh": "正規化中間鬆弛累積路徑定義為 \\(Q_j(s)=\\mathcal M_j^{-1}\\int_{\tau_j}^{t_j(s)}q(t)\\,dt\\)，單調非減且 \\(Q_j(0)=0\\)（§6、§8）。",
      "definition_en": "The normalized cumulative middle-slack path is \\(Q_j(s)=\\mathcal M_j^{-1}\\int_{\tau_j}^{t_j(s)}q(t)\\,dt\\), nondecreasing with \\(Q_j(0)=0\\) (§6, §8)."
    },
    {
      "id": "ns.c5.c5c.e0_record_displacement",
      "latex": "R_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "E0 紀錄位移",
      "label_en": "normalized E0 record displacement",
      "definition_zh": "正規化 \\(E_0\\) 紀錄位移定義為 \\(R_j(s)=(E_0(t_j(s))-E_0(\tau_j))/\\mathcal M_j\\)，並滿足 \\(-2\\le R_j(s)\\le1\\)（§6、§8）。",
      "definition_en": "Normalized \\(E_0\\) record displacement is \\(R_j(s)=(E_0(t_j(s))-E_0(\tau_j))/\\mathcal M_j\\), taking values in \\([-2,1]\\) (§6, §8).",
      "defining_relation": "R_j(s)=\\frac{E_0(t_j(s))-E_0(\\tau_j)}{\\mathcal{M}_j}=C_j(s)-D_j(s)-Q_j(s)"
    },
    {
      "id": "ns.c5.c5c.exact_middle_ledger",
      "latex": "C_j=R_j+D_j+Q_j",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "精確中間累積帳本",
      "label_en": "exact middle cumulative ledger",
      "definition_zh": "定理 7.1（C5-C.1）斷言對所有 \\(s\\in[0,1]\\) 有精確中間累積帳本 \\(C_j(s)=R_j(s)+D_j(s)+Q_j(s)\\)（§7）。",
      "definition_en": "Theorem 7.1 (C5-C.1) asserts the exact middle cumulative ledger \\(C_j(s)=R_j(s)+D_j(s)+Q_j(s)\\) for all \\(s\\in[0,1]\\) (§7).",
      "defining_relation": "C_j(s)=R_j(s)+D_j(s)+Q_j(s)",
      "notes": "Helly compactness of the monotone paths \\(C_j,D_j,Q_j\\) is recorded in §9; the X-guard G-CUMLEDGER requires preserving this identity."
    },
    {
      "id": "ns.c5.c5c.middle_record_fraction",
      "latex": "\\alpha_j^{mid}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間紀錄分數",
      "label_en": "middle record fraction",
      "definition_zh": "中間紀錄分數 \\(\\alpha_j^{mid}=(E_0(\tau_{j+1})-E_0(\tau_j))/\\mathcal M_j>0\\) 滿足端點帳本 \\(1=\\alpha_j^{mid}+\\delta_j^{mid}+Q_j(1)\\)（§8）。",
      "definition_en": "The middle record fraction \\(\\alpha_j^{mid}=(E_0(\tau_{j+1})-E_0(\tau_j))/\\mathcal M_j>0\\) obeys the endpoint ledger \\(1=\\alpha_j^{mid}+\\delta_j^{mid}+Q_j(1)\\) (§8).",
      "notes": "Carries over from C5-A; the quantity \\(1-\\alpha_j^{mid}\\) is the inefficiency budget controlling supply deficit (§12–§13)."
    },
    {
      "id": "ns.c5.c5c.dissipation_demand_mass",
      "latex": "\\delta_j^{mid}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "耗散需求質量",
      "label_en": "dissipation-demand mass",
      "definition_zh": "耗散需求總質量定義為 \\(\\delta_j^{mid}=D_j(1)=(2\nu/\\mathcal M_j)\\int_{J_j}E_1\\,dt\\)，並滿足 \\(0\\le\\delta_j^{mid}\\le1\\)（§8、§16）。",
      "definition_en": "Total dissipation-demand mass is \\(\\delta_j^{mid}=D_j(1)=(2\nu/\\mathcal M_j)\\int_{J_j}E_1\\,dt\\), satisfying \\(0\\le\\delta_j^{mid}\\le1\\) (§8, §16)."
    },
    {
      "id": "ns.c5.c5c.middle_supply_rate",
      "latex": "c_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間供給速率",
      "label_en": "middle supply rate",
      "definition_zh": "中間供給速率 \\(c_j(s)=C_j'(s)=L_j m(t_j(s))/\\mathcal M_j\\ge0\\) 滿足 \\(\\int_0^1 c_j\\,ds=1\\)（§10）。",
      "definition_en": "The middle supply rate \\(c_j(s)=C_j'(s)=L_j m(t_j(s))/\\mathcal M_j\\ge0\\) is a probability density on \\([0,1]\\) (§10)."
    },
    {
      "id": "ns.c5.c5c.demand_rate",
      "latex": "d_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "耗散需求速率",
      "label_en": "dissipation demand rate",
      "definition_zh": "應變耗散需求速率 \\(d_j(s)=D_j'(s)=2\nu L_j E_1(t_j(s))/\\mathcal M_j\\ge0\\) 是交叉曲率所作用的非負路徑，且 \\(\\int_0^1 d_j=\\delta_j^{mid}\\le1\\)（§10、§25）。",
      "definition_en": "The strain-dissipation demand rate \\(d_j(s)=D_j'(s)=2\nu L_j E_1(t_j(s))/\\mathcal M_j\\ge0\\) is the nonnegative path whose curvature encodes operator sign, with \\(\\int_0^1 d_j=\\delta_j^{mid}\\le1\\) (§10, §25).",
      "defining_relation": "d_j(s)=D_j'(s)=\\frac{2\\nu L_j E_1(t_j(s))}{\\mathcal{M}_j}",
      "notes": "C5-C.5 identifies \\(O^+\\)/\\(O^-\\) with the convexity/concavity sources of \\(d_j\\) (§27–§28)."
    },
    {
      "id": "ns.c5.c5c.operator_signed_growth",
      "latex": "h(t)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "算子符號增長",
      "label_en": "operator signed growth",
      "definition_zh": "沿用 C5-A/B 的算子符號增長 \\(h(t)=E_1'(t)=\nu(\\zeta r_\nu-1)\\|\\Delta S\\|_2^2\\)，其正負部分積分分別為 \\(P_j\\) 與 \\(N_j\\)（§20）。",
      "definition_en": "Operator signed growth, carried from C5-A/B, is \\(h(t)=E_1'(t)=\nu(\\zeta r_\nu-1)\\|\\Delta S\\|_2^2\\), with positive/negative integrals \\(P_j\\) and \\(N_j\\) (§20).",
      "defining_relation": "h(t)=E_1'(t)=\\nu(\\zeta r_\\nu-1)\\|\\Delta S\\|_2^2",
      "notes": "Carries over from C5-A/B; C5-C reinterprets \\(\\mathrm{sign}(h)\\) as the curvature sign of \\(d_j\\)."
    },
    {
      "id": "ns.c5.c5c.operator_bv_path",
      "latex": "G_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "算子 BV 路徑",
      "label_en": "operator BV record path",
      "definition_zh": "算子 BV 紀錄路徑 \\(G_j(s)=(E_1(t_j(s))-E_1(\tau_j))/V_j^{op}\\) 滿足 \\(G_j(0)=0\\)、\\(G_j(1)=\beta_j^{op}>0\\) 且 \\(\\operatorname{Var}_{[0,1]}G_j=1\\)（§21）。",
      "definition_en": "The operator BV record path \\(G_j(s)=(E_1(t_j(s))-E_1(\tau_j))/V_j^{op}\\) satisfies \\(G_j(0)=0\\), \\(G_j(1)=\beta_j^{op}>0\\), and \\(\\operatorname{Var}_{[0,1]}G_j=1\\) (§21).",
      "defining_relation": "G_j(s)=\\frac{E_1(t_j(s))-E_1(\\tau_j)}{V_j^{op}},\\quad \\operatorname{Var}_{[0,1]}G_j=1",
      "notes": "Theorem 22.1 (C5-C.3) compactifies \\(G_j\\) to some \\(G_\\ast\\in BV([0,1])\\) with \\(\\operatorname{Var} G_\\ast\\le 1\\)."
    },
    {
      "id": "ns.c5.c5c.cross_curvature",
      "latex": "\\kappa_j^{MO}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "交叉曲率數",
      "label_en": "cross-curvature number",
      "definition_zh": "交叉曲率數定義為 \\(\\kappa_j^{MO}=2\nu L_j V_j^{op}/\\mathcal M_j\\)，並精確等於 \\(\\operatorname{Var}_{[0,1]}d_j\\)，從而使 \\(Dd_j=\\kappa_j^{MO}(\\mu_j^{op,+}-\\mu_j^{op,-})\\)（§26–§27）。",
      "definition_en": "The cross-curvature number is \\(\\kappa_j^{MO}=2\nu L_j V_j^{op}/\\mathcal M_j\\), equal to \\(\\operatorname{Var}_{[0,1]}d_j\\), and satisfies \\(Dd_j=\\kappa_j^{MO}(\\mu_j^{op,+}-\\mu_j^{op,-})\\) (§26–§27).",
      "defining_relation": "\\kappa_j^{MO}=\\frac{2\\nu L_j V_j^{op}}{\\mathcal{M}_j}=\\operatorname{Var}_{[0,1]}d_j,\\qquad Dd_j=\\kappa_j^{MO}(\\mu_j^{op,+}-\\mu_j^{op,-})",
      "notes": "The boxed intro form uses \\((D_j^0)''\\) and \\(P_j+N_j\\); the body identifies \\(d_j=D_j'\\) and \\(V_j^{op}=P_j+N_j\\). Guard G-CROSSCURV."
    },
    {
      "id": "ns.c5.c5c.operator_signed_measures",
      "latex": "\\mu_j^{op,\\pm}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "算子符號測度",
      "label_en": "operator signed measures",
      "definition_zh": "算子導數測度由分布意義 \\(DG_j=\\mu_j^{op,+}-\\mu_j^{op,-}\\) 給出，且 \\(|DG_j|=\\mu_j^{op,+}+\\mu_j^{op,-}\\) 為總質量 1 的測度（§23）。",
      "definition_en": "Operator derivative measures satisfy \\(DG_j=\\mu_j^{op,+}-\\mu_j^{op,-}\\) distributionally, with \\(|DG_j|=\\mu_j^{op,+}+\\mu_j^{op,-}\\) of total mass 1 (§23).",
      "notes": "Weak-star limits of \\(|DG_j|\\) and \\(DG_j\\) produce \\(\\Lambda_\\ast^{op}\\) and \\(DG_\\ast\\) used to define the variation-cancellation defect."
    },
    {
      "id": "ns.c5.c5c.operator_variation_cancellation_defect",
      "latex": "\\mathfrak{D}_\\ast^{op}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "算子變分相消缺陷",
      "label_en": "operator variation-cancellation defect",
      "definition_zh": "算子變分相消缺陷定義為 \\(\\mathfrak D_\\ast^{op}=\\Lambda_\\ast^{op}-|DG_\\ast|\\ge0\\)，記錄在符號 BV 極限中互相抵消的有限尺度 \\(O^+/O^-\\) 微觀變分（§24）。",
      "definition_en": "The operator variation-cancellation defect is \\(\\mathfrak D_\\ast^{op}=\\Lambda_\\ast^{op}-|DG_\\ast|\\ge0\\), recording finite-scale \\(O^+/O^-\\) micro-variation cancelled in the signed BV limit (§24).",
      "defining_relation": "\\mathfrak{D}_\\ast^{op}=\\Lambda_\\ast^{op}-|DG_\\ast|\\ge0"
    },
    {
      "id": "ns.c5.c5c.curvature_variation_defect",
      "latex": "\\mathfrak{D}_\\ast^{curv}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "曲率變分缺陷",
      "label_en": "curvature variation defect",
      "definition_zh": "即使 \\(\\kappa_j^{MO}\\) 有界，曲率變分缺陷 \\(\\mathfrak D_\\ast^{curv}=\\Lambda_\\ast^{curv}-|Dd_\\ast|\\ge0\\) 仍可記錄在極限速率 \\(d_\\ast\\) 中被相消的有限尺度曲率開關（§33）。",
      "definition_en": "Even for bounded \\(\\kappa_j^{MO}\\), the curvature variation defect \\(\\mathfrak D_\\ast^{curv}=\\Lambda_\\ast^{curv}-|Dd_\\ast|\\ge0\\) records finite-scale curvature switches cancelled in the limit rate \\(d_\\ast\\) (§33).",
      "notes": "Guard G-CURVDEF: BV limits must separate \\(|Dd_\\ast|\\) from total curvature mass; this is the C-T2 micro-oscillation regime of §50."
    },
    {
      "id": "ns.c5.c5c.supply_fraction",
      "latex": "\\theta_j(s)",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "供給分數",
      "label_en": "supply fraction",
      "definition_zh": "當 \\(c_j+d_j>0\\) 時供給分數為 \\(\theta_j(s)=c_j/(c_j+d_j)\\in[0,1]\\)，否則取 \\(\theta_j=1/2\\)（§17）。",
      "definition_en": "The supply fraction is \\(\theta_j(s)=c_j/(c_j+d_j)\\in[0,1]\\) when \\(c_j+d_j>0\\), and \\(\theta_j=1/2\\) otherwise (§17).",
      "defining_relation": "\\theta_j(s)=\\frac{c_j(s)}{c_j(s)+d_j(s)}\\in[0,1]"
    },
    {
      "id": "ns.c5.c5c.supply_demand_young_measure",
      "latex": "\\mathscr{S}_j",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "供給–需求 Young 測度",
      "label_en": "supply–demand Young measure",
      "definition_zh": "供給–需求 Young 測度 \\(\\mathscr S_j=(s,\theta_j)_{\\#}\\Lambda_j^{SD}\\) 是 \\([0,1]\times[0,1]\\) 上的推前測度，其弱極限滿足封閉相容約束 \\((1+\\delta_\\ast)\\int[1-2\theta]_+\\,d\\mathscr S_\\ast\\le1-\\alpha_\\ast\\)（§16、§19）。",
      "definition_en": "The supply–demand Young measure \\(\\mathscr S_j=(s,\theta_j)_{\\#}\\Lambda_j^{SD}\\) is a pushforward on \\([0,1]\times[0,1]\\) whose weak limit obeys the closed compatibility constraint \\((1+\\delta_\\ast)\\int[1-2\theta]_+\\,d\\mathscr S_\\ast\\le1-\\alpha_\\ast\\) (§16, §19).",
      "defining_relation": "\\mathscr{S}_j=(s,\\theta_j)_\\#\\Lambda_j^{SD},\\qquad (1+\\delta_\\ast)\\int[1-2\\theta]_+\\,d\\mathscr{S}_\\ast\\le 1-\\alpha_\\ast",
      "notes": "Here \\(d\\Lambda_j^{SD}=(c_j+d_j)/(1+\\delta_j)\\,ds\\) is the supply–demand common probability on \\([0,1]\\) (§16)."
    },
    {
      "id": "ns.c5.c5c.marked_transition_state",
      "latex": "\\mathscr{T}_j",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "帶標記轉移態",
      "label_en": "marked transition state",
      "definition_zh": "帶算子符號標記的供給–需求狀態 \\(\\mathscr T_j=(s,\theta_j,\\sigma_j)_{\\#}\\Lambda_j^{SD}\\) 活在緊集 \\([0,1]\times[0,1]\times\\{-1,0,+1\\}\\) 上（§37）。",
      "definition_en": "The operator-sign-marked supply–demand state \\(\\mathscr T_j=(s,\theta_j,\\sigma_j)_{\\#}\\Lambda_j^{SD}\\) lives on the compact \\([0,1]\times[0,1]\times\\{-1,0,+1\\}\\) (§37).",
      "notes": "The integrand of the supply-deficit constraint is independent of \\(\\sigma\\), so marking does not change the inequality (§38); anti-phase mass \\(\\mathfrak A_\\ast^+\\) is the \\(\\sigma=+1\\) slice (§39)."
    },
    {
      "id": "ns.c5.c5c.curvature_congestion",
      "latex": "\\mathrm{C\\text{-}K}\\infty:\\ \\kappa_j^{MO}\\to\\infty",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "曲率壅塞",
      "label_en": "curvature congestion",
      "definition_zh": "交叉曲率體制 C-K∞ 指 \\(\\kappa_j^{MO}\to\\infty\\)，此時 \\(\\operatorname{Var}d_j\to\\infty\\) 而 \\(\\|d_j\\|_{L^1}\\le1\\)，算子轉移無法被普通 BV 緊緻性吸收而形成曲率壅塞（§30、§34）。",
      "definition_en": "The C-K∞ regime is \\(\\kappa_j^{MO}\to\\infty\\), forcing \\(\\operatorname{Var}d_j\to\\infty\\) while \\(\\|d_j\\|_{L^1}\\le1\\), so operator transitions escape ordinary BV compactness as curvature congestion (§30, §34).",
      "notes": "Sibling regimes are C-K0 (vanishing curvature, demand-rate flattening, §31) and C-KF (finite curvature, BV transition closure, §32); C5-B operator phase measure is the normalized curvature profile of C-K∞ (§34)."
    },
    {
      "id": "ns.c5.c5c.temporal_transition_state",
      "latex": "\\Theta_\\ast^{TC}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "時間轉移狀態",
      "label_en": "temporal transition state",
      "definition_zh": "C5-C 時間轉移狀態 \\(\\Theta_\\ast^{TC}\\) 收集極限累積路徑 \\(C_\\ast,D_\\ast,Q_\\ast,R_\\ast\\)、Young 態 \\(\\mathscr S_\\ast,\\mathscr T_\\ast\\)、算子 BV 路徑 \\(G_\\ast\\)、交叉曲率尺度與各缺陷測度（§59）。",
      "definition_en": "The C5-C temporal transition state \\(\\Theta_\\ast^{TC}\\) collects the limit cumulative paths \\(C_\\ast,D_\\ast,Q_\\ast,R_\\ast\\), Young states \\(\\mathscr S_\\ast,\\mathscr T_\\ast\\), the operator BV path \\(G_\\ast\\), the cross-curvature scale, and the defect measures (§59).",
      "defining_relation": "\\Theta_\\ast^{TC}=\\bigl\\langle C_\\ast,D_\\ast,Q_\\ast,R_\\ast,\\mathscr{S}_\\ast,\\mathscr{T}_\\ast,G_\\ast,\\Lambda_\\ast^{op},\\mathfrak{D}_\\ast^{op},\\kappa_\\ast^{MO},\\mathfrak{D}_\\ast^{curv},\\mathfrak{c}_M,\\mathfrak{c}_+\\bigr\\rangle",
      "notes": "The load-concentration defects \\(\\mathfrak c_M,\\mathfrak c_+\\) carry over from C5-B and appear as the C-T4 alternative in §50."
    },
    {
      "id": "ns.c5.c5c.supply_deficit_budget",
      "latex": "\\int_0^1[d_j-c_j]_+\\,ds\\le 1-\\alpha_j^{mid}",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "中間供給赤字預算",
      "label_en": "middle supply-deficit budget",
      "definition_zh": "定理 13.1（C5-C.2）斷言 \\(\\int_0^1[d_j-c_j]_+\\,ds\\le1-\\alpha_j^{mid}\\)，即正規化供給赤字不得超出中間紀錄無效率預算（§13）。",
      "definition_en": "Theorem 13.1 (C5-C.2) asserts \\(\\int_0^1[d_j-c_j]_+\\,ds\\le1-\\alpha_j^{mid}\\), so normalized supply deficit cannot exceed the middle-record inefficiency budget (§13).",
      "defining_relation": "\\int_0^1[d_j(s)-c_j(s)]_+\\,ds\\le 1-\\alpha_j^{mid}",
      "notes": "Guard G-SUPDEM: middle supply \\(c\\) and strain demand \\(d\\) must not be collapsed into a single load."
    },
    {
      "id": "ns.c5.c5c.scalar_ordering_nogo",
      "latex": "O^+\\to M",
      "series": "NS",
      "first_appearance": "C5-C",
      "label_zh": "純量時序禁則",
      "label_en": "scalar temporal ordering no-go",
      "definition_zh": "結論 48.1（C5-C.7）斷言僅憑 \\(E_0/E_1\\) 帳本、中間上界強迫與交叉曲率恆等，仍不足以禁止分離的 \\(O^+\\to M\\)（及反向）補償排序（§45–§48）。",
      "definition_en": "Conclusion 48.1 (C5-C.7) asserts that the \\(E_0/E_1\\) ledgers, middle upper forcing, and cross-curvature identity still do not forbid a separated \\(O^+\\to M\\) (or reverse) compensation ordering (§45–§48).",
      "notes": "The §45–§47 construction is an abstract scalar ledger, not an N–S orbit (guard G-ORDERNO); it marks the logical boundary of pure temporal-scalar closure (G-TEMPEND)."
    },
    {
      "id": "ns.c5.c5d.local_quadratic_tensor",
      "latex": "Q(S,\\omega)",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "局部二次張量",
      "label_en": "Local quadratic tensor",
      "definition_zh": "在第2節中定義為應變平方與渦度張量二次項的組合，用以捕捉N-S方程式局部真實的非線性交互作用。",
      "definition_en": "Defined in Section 2 as the combination of strain squared and vorticity tensor quadratic terms to capture local true nonlinear interactions of the N-S equations.",
      "defining_relation": "Q(S,\\omega) = S^2 + \\frac{1}{4}\\omega\\otimes\\omega - \\frac{1}{4}|\\omega|^2I"
    },
    {
      "id": "ns.c5.c5d.normalized_strain_direction",
      "latex": "K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "正規化正中間應變方向",
      "label_en": "Normalized positive-middle strain direction",
      "definition_zh": "在第3節中定義為跡數為零、弗羅貝尼烏斯範數為1且中間特徵值嚴格大於零的單位應變張量方向。",
      "definition_en": "Defined in Section 3 as a trace-free, unit Frobenius norm strain tensor direction with a strictly positive middle eigenvalue.",
      "defining_relation": "K \\in \\operatorname{Sym}_0(3), \\quad |K|_F=1, \\quad k_2 > 0"
    },
    {
      "id": "ns.c5.c5d.ordered_eigenvalues",
      "latex": "k_1, k_2, k_3",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "排序特徵值",
      "label_en": "Ordered eigenvalues",
      "definition_zh": "在第3節中引入，表示單位張量 $K$ 的三個由小到大排序的特徵值，其中 $k_1 < 0$ 且 $k_2, k_3 > 0$。",
      "definition_en": "Introduced in Section 3 as the three eigenvalues of the unit tensor $K$, ordered from smallest to largest, where $k_1 < 0$ and $k_2, k_3 > 0$."
    },
    {
      "id": "ns.c5.c5d.most_compressive_eigenvector",
      "latex": "e_1",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "最大壓縮特徵向量",
      "label_en": "Most-compressive eigenvector",
      "definition_zh": "在第3節中定義為對應於最小特徵值 $k_1$ 的單位特徵向量，代表局部流體形變中最強的壓縮方向。",
      "definition_en": "Defined in Section 3 as the unit eigenvector corresponding to the smallest eigenvalue $k_1$, representing the direction of strongest compression in local fluid deformation."
    },
    {
      "id": "ns.c5.c5d.strong_middle_shape_parameter",
      "latex": "\\theta_K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "強中間形狀參數",
      "label_en": "Strong-middle shape parameter",
      "definition_zh": "在第4節中定義為 $k_2$ 與 $k_3$ 的乘積，用以量化正規化應變形狀遠離退化邊界 $\\lambda_2=0$ 的程度。",
      "definition_en": "Defined in Section 4 as the product of $k_2$ and $k_3$, quantifying how far the normalized strain shape is from the degenerate boundary $\\lambda_2=0$.",
      "defining_relation": "\\theta_K = k_2 k_3 = k_1^2 - \\frac{1}{2} > 0"
    },
    {
      "id": "ns.c5.c5d.compressive_axis_test_tensor",
      "latex": "H_K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "壓縮軸測試張量",
      "label_en": "Compressive-axis test tensor",
      "definition_zh": "在第5節中定義的半空間測試泛函，能確保任意渦度與應變平方項在其作用下皆嚴格為正。",
      "definition_en": "Defined in Section 5 as a half-space test functional under which both arbitrary vorticity and strain-square contributions are strictly positive.",
      "defining_relation": "H_K = e_1 \\otimes e_1 - \\frac{1+\\theta_K}{2} I"
    },
    {
      "id": "ns.c5.c5d.strong_middle_cone_radius",
      "latex": "\\delta_K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "強中間錐半徑",
      "label_en": "Strong-middle cone radius",
      "definition_zh": "在第9節中定義的容許擾動半徑，確保在該半徑內的應變方向仍保持二次半空間正定性。",
      "definition_en": "Defined in Section 9 as the allowable perturbation radius ensuring that strain directions within it maintain quadratic half-space positivity."
    },
    {
      "id": "ns.c5.c5d.strong_middle_pointwise_strain_cone",
      "latex": "\\mathcal{C}_K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "強中間逐點應變錐",
      "label_en": "Strong-middle pointwise strain cone",
      "definition_zh": "在第9節中定義為以 $K$ 為中心且半徑為 $\\delta_K$ 的單位應變方向集合，保證局部二次張量的一致同調性。",
      "definition_en": "Defined in Section 9 as the set of unit strain directions within radius $\\delta_K$ from $K$, guaranteeing uniform coherence of the local quadratic tensor.",
      "defining_relation": "\\mathcal{C}_K = \\{ V \\in \\operatorname{Sym}_0(3) : |V|_F = 1, |V-K|_F \\le \\delta_K \\}"
    },
    {
      "id": "ns.c5.c5d.unit_half_space_functional",
      "latex": "\\widehat{H}_K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "單位半空間泛函",
      "label_en": "Unit half-space functional",
      "definition_zh": "在第12節中定義為正規化後的 $H_K$，用以測量二次張量落入嚴格半空間的下界餘裕。",
      "definition_en": "Defined in Section 12 as the normalized $H_K$, used to measure the margin by which the quadratic tensor falls into a strict half-space."
    },
    {
      "id": "ns.c5.c5d.half_space_margin",
      "latex": "\\gamma_K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "半空間餘裕",
      "label_en": "Half-space margin",
      "definition_zh": "在第12節中定義為嚴格正的下界常數，代表強中間錐內二次方向的幾何一致性強度。",
      "definition_en": "Defined in Section 12 as the strictly positive lower bound constant representing the geometric coherence strength of quadratic directions within the strong-middle cone."
    },
    {
      "id": "ns.c5.c5d.strict_quadratic_half_space",
      "latex": "\\mathcal{H}_K^+",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "嚴格二次半空間",
      "label_en": "Strict quadratic half-space",
      "definition_zh": "在第13節中定義的五維球面子集，包含所有由強中間應變錐產生的正規化二次張量方向，其凸包不包含原點。",
      "definition_en": "Defined in Section 13 as the subset of the five-dimensional sphere containing all normalized quadratic tensor directions generated by the strong-middle strain cone, whose convex hull strictly excludes the origin."
    },
    {
      "id": "ns.c5.c5d.weighted_local_quadratic_mass",
      "latex": "A_\\chi^Q",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "加權局部二次質量",
      "label_en": "Weighted local quadratic mass",
      "definition_zh": "在第14節中定義為局部二次張量範數的空間積分，代表該區域內的總二次交互作用強度。",
      "definition_en": "Defined in Section 14 as the spatial integral of the local quadratic tensor norm, representing the total quadratic interaction intensity in the region."
    },
    {
      "id": "ns.c5.c5d.weighted_local_quadratic_mean",
      "latex": "B_\\chi^Q",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "加權局部二次均值",
      "label_en": "Weighted local quadratic mean",
      "definition_zh": "在第14節中定義為局部二次張量本身的空間加權積分矩陣。",
      "definition_en": "Defined in Section 14 as the spatially weighted integral matrix of the local quadratic tensor itself."
    },
    {
      "id": "ns.c5.c5d.local_quadratic_mean_coherence",
      "latex": "\\kappa_\\chi^Q",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "局部二次均值同調性",
      "label_en": "Local quadratic mean coherence",
      "definition_zh": "在第14節中定義為均值矩陣大小與總二次質量之比，用以量化七點抵消機制的失效程度。",
      "definition_en": "Defined in Section 14 as the ratio of the mean matrix magnitude to the total quadratic mass, quantifying the failure of the Seven-Point cancellation mechanism.",
      "defining_relation": "\\kappa_\\chi^Q = \\frac{|B_\\chi^Q|}{A_\\chi^Q}"
    },
    {
      "id": "ns.c5.c5d.quadratic_mass_leakage_fraction",
      "latex": "\\varepsilon_\\chi^K",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "二次質量洩漏比例",
      "label_en": "Quadratic-mass leakage fraction",
      "definition_zh": "在第18節中定義為落在強中間錐外部的二次質量比例，是七點抵消存活的必要代價。",
      "definition_en": "Defined in Section 18 as the fraction of quadratic mass falling outside the strong-middle cone, acting as a necessary cost for Seven-Point cancellation survival.",
      "defining_relation": "\\varepsilon_\\chi^K = \\frac{\\int_{\\mathbb{R}^3 \\setminus G_K} \\chi |Q| dx}{A_\\chi^Q}"
    },
    {
      "id": "ns.c5.c5d.local_mean_strain_matrix",
      "latex": "\\bar{S}_R",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "局部平均應變矩陣",
      "label_en": "Local mean strain matrix",
      "definition_zh": "在第24節中定義為尺度 $R$ 上的空間加權平均應變，為平均應變錐分析的基礎對象。",
      "definition_en": "Defined in Section 24 as the spatially weighted mean strain at scale $R$, serving as the base object for mean-strain cone analysis.",
      "notes": "Must be carefully distinguished from pointwise strain; linking mean coherence to pointwise coherence requires derivative/fluctuation bounds (Sec 23-24)."
    },
    {
      "id": "ns.c5.c5d.normalized_mean_direction",
      "latex": "K_R",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "正規化平均方向",
      "label_en": "Normalized mean direction",
      "definition_zh": "在第24節中定義為 $\\bar{S}_R$ 除以其範數所得到的平均應變單位方向矩陣。",
      "definition_en": "Defined in Section 24 as the unit mean strain direction matrix obtained by dividing $\\bar{S}_R$ by its norm."
    },
    {
      "id": "ns.c5.c5d.relative_strain_fluctuation",
      "latex": "\\eta_R^S",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "相對應變漲落",
      "label_en": "Relative strain fluctuation",
      "definition_zh": "在第25節中定義為局部應變相對於平均應變矩陣大小的最大偏差比例。",
      "definition_en": "Defined in Section 25 as the ratio of maximum local strain deviation to the magnitude of the mean strain matrix.",
      "defining_relation": "\\eta_R^S = \\frac{\\|S - \\bar{S}_R\\|_{L^\\infty}}{|\\bar{S}_R|}"
    },
    {
      "id": "ns.c5.c5d.mean_strain_critical_amplitude",
      "latex": "\\mu_R",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "平均應變臨界振幅",
      "label_en": "Mean-strain critical amplitude",
      "definition_zh": "在第31節中定義的無因次量，代表平均應變在給定尺度下的強度，若夠大則自動保證非退化的二次強度。",
      "definition_en": "Defined in Section 31 as a dimensionless quantity representing the mean strain intensity at a given scale, ensuring nondegenerate quadratic intensity when sufficiently large."
    },
    {
      "id": "ns.c5.c5d.weighted_local_pressure_hessian_mean",
      "latex": "P_\\chi",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "加權局部壓力海森均值",
      "label_en": "Weighted local pressure Hessian mean",
      "definition_zh": "在第32節中定義為伴隨壓力張量的積分，當二次抵消失效時被迫重新進入平衡方程並承擔定向補償。",
      "definition_en": "Defined in Section 32 as the integral of the adjoint pressure tensor, which is forced to re-enter the balance equations and assume oriented compensation when quadratic cancellation fails."
    },
    {
      "id": "ns.c5.c5d.critical_pressure_oscillation",
      "latex": "\\Pi_R^{(2)}",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "臨界壓力振盪",
      "label_en": "Critical pressure oscillation",
      "definition_zh": "在第34節中引入，用以量化壓力相對於仿射函數的局部偏差，受到半空間幾何的強制下界約束。",
      "definition_en": "Introduced in Section 34 to quantify the local deviation of pressure from affine functions, forced into a lower bound by the half-space geometry.",
      "defining_relation": "\\Pi_R^{(2)} = \\nu^{-2} \\inf_{\\ell\\in\\mathcal{A}_1} \\|p - \\ell\\|_{L^{3/2}(B_{CR})}"
    },
    {
      "id": "ns.c5.c5d.far_harmonic_pressure_matrix",
      "latex": "F",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "遠場調和壓力矩陣",
      "label_en": "Far harmonic pressure matrix",
      "definition_zh": "在第36節中定義的無跡對稱矩陣，代表從遠方主導局部核心的共同背景壓力海森。",
      "definition_en": "Defined in Section 36 as a trace-free symmetric matrix representing the common background pressure Hessian dominating local cores from afar."
    },
    {
      "id": "ns.c5.c5d.compressive_axis_stf_projector",
      "latex": "G(e)",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "壓縮軸無跡對稱投影子",
      "label_en": "Compressive-axis STF projector",
      "definition_zh": "在第38節中定義為從半空間泛函中提取出與遠場壓力配對的無跡部分，是引發多核心凸包阻礙的關鍵物件。",
      "definition_en": "Defined in Section 38 as the trace-free part extracted from the half-space functional that pairs with far pressure, serving as the key object triggering multi-core convex-hull obstructions.",
      "defining_relation": "G(e) = e \\otimes e - \\frac{1}{3}I \\in \\operatorname{Sym}_0(3)"
    },
    {
      "id": "ns.c5.c5d.normalized_strain_direction_measure",
      "latex": "\\nu_j^S",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "正規化應變方向測度",
      "label_en": "Normalized strain-direction measure",
      "definition_zh": "在第45節中定義的局部極限測度，用於捕捉應變方向在五維球面上的機率分佈與集中現象。",
      "definition_en": "Defined in Section 45 as a local limit measure used to capture the probability distribution and concentration of strain directions on the five-dimensional sphere."
    },
    {
      "id": "ns.c5.c5d.quadratic_direction_measure",
      "latex": "\\nu_j^Q",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "二次方向測度",
      "label_en": "Quadratic-direction measure",
      "definition_zh": "在第45節中定義的測度，追蹤由七點抵消機制等極限程序產生的正規化二次張量分佈。",
      "definition_en": "Defined in Section 45 as a measure tracking the limit distribution of normalized quadratic tensors arising from procedures like the Seven-Point cancellation mechanism."
    },
    {
      "id": "ns.c5.c5d.strong_middle_shape_variable",
      "latex": "\\vartheta(S)",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "強中間形狀變數",
      "label_en": "Strong-middle shape variable",
      "definition_zh": "在第51節中定義為未正規化應變矩陣的中間正特徵值與最大特徵值之積除以範數平方，提供半空間機制的逐點餘裕來源。",
      "definition_en": "Defined in Section 51 as the product of the positive middle and largest eigenvalues divided by the squared norm of the unnormalized strain, providing the pointwise margin source for the half-space mechanism.",
      "defining_relation": "\\vartheta(S) = \\frac{\\lambda_2^+(S) \\lambda_3(S)}{|S|_F^2}"
    },
    {
      "id": "ns.c5.c5d.spatial_matrix_state",
      "latex": "\\Theta_\\ast^{SM}",
      "series": "NS",
      "first_appearance": "C5-D",
      "label_zh": "C5-D 空間-矩陣狀態",
      "label_en": "C5-D spatial-matrix state",
      "definition_zh": "在第58節中定義的真實ETN更新，囊括了極限狀態下的應變、二次測度、強中間餘裕以及壓縮軸投影組態等空間幾何元數據。",
      "definition_en": "Defined in Section 58 as the True ETN update encapsulating spatial geometric metadata such as strain/quadratic measures, strong-middle margin, and compressive-axis configurations in the limit state.",
      "defining_relation": "\\Theta_\\ast^{SM} = \\langle \\nu_\\ast^S, \\nu_\\ast^Q, \\theta_\\ast, \\gamma_\\ast, \\varepsilon_\\ast^{cone}, \\mathcal{U}_\\ast^{(7)}, \\mu_\\ast^R, \\Pi_\\ast^{(2)}, \\mathcal{G}_\\ast^{axis} \\rangle"
    },
    {
      "id": "ns.c5.c5e.vartheta",
      "latex": "\\vartheta(S)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "正規化中間隙變數",
      "label_en": "normalized middle-gap variable",
      "definition_zh": "第 2 節對非零無跡對稱應變引入正規化中間隙形狀變數，並在中間特徵值非正時令其為零。",
      "definition_en": "Section 2 introduces the normalized middle-gap shape variable of a nonzero trace-free symmetric strain, set to zero when the middle eigenvalue is nonpositive.",
      "defining_relation": "\\vartheta(S)=\\frac{\\lambda_2^+(S)\\lambda_3(S)}{|S|_F^2}",
      "notes": "This is the C5-D shape variable; C5-E.1 proves it is quantitatively equivalent to λ₂⁺/|S| in the positive-middle sector."
    },
    {
      "id": "ns.c5.c5e.xi_2",
      "latex": "\\xi_2(S)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "正規化中間特徵值",
      "label_en": "normalized middle eigenvalue",
      "definition_zh": "第 2 節將 Miller 中間特徵值正部除以 Frobenius 範數，作為正中間扇區的尺度不變形狀座標。",
      "definition_en": "Section 2 normalizes Miller's positive middle eigenvalue by the Frobenius norm, yielding the scale-invariant shape coordinate of the positive-middle sector.",
      "defining_relation": "\\xi_2(S)=\\frac{\\lambda_2^+(S)}{|S|_F}",
      "notes": "C5-E.1 gives (1/√6) ξ₂ ≤ ϑ ≤ (1/√2) ξ₂ whenever λ₂>0, so middle-gap degeneration is genuine middle-eigenvalue degeneration."
    },
    {
      "id": "ns.c5.c5e.lambda_2_plus",
      "latex": "\\lambda_2^+(S)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "中間特徵值正部",
      "label_en": "positive part of the middle eigenvalue",
      "definition_zh": "第 1.1 與第 2 節沿用 Miller 的 λ₂⁺(S)=max(λ₂(S),0)，作為應變中間通道的尺度臨界正則幾何。",
      "definition_en": "Sections 1.1 and 2 take Miller's λ₂⁺(S)=max(λ₂(S),0) as the scale-critical regularity geometry of the middle-strain channel.",
      "notes": "Carries over from Miller (ARMA 2020) and from the C5-D strong-middle analysis."
    },
    {
      "id": "ns.c5.c5e.q_motif",
      "latex": "Q",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "二次抵消 motif",
      "label_en": "quadratic cancellation motif",
      "definition_zh": "第 0 節將 C5-D 的七點零重心二次抵消 motif 視為本輪必須轉成場缺陷的殘餘補償器。",
      "definition_en": "Section 0 treats the C5-D seven-point zero-barycenter quadratic cancellation motif as the residual compensator that this round must convert into field defects.",
      "notes": "Inherited from C4-J and C5-D; C5-E eliminates it as a free motif by the trichotomy Q ⇒ gap concentration ∨ derivative fluctuation ∨ vorticity leakage."
    },
    {
      "id": "ns.c5.c5e.quadratic_coercivity",
      "latex": "|Q(S,\\omega)|\\ge c_Q\\delta(|S|^2+|\\omega|^2)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "中間隙外二次強制性",
      "label_en": "pointwise quadratic coercivity away from the middle gap",
      "definition_zh": "第 4 節 C5-E.2 斷言：在 ϑ≥δ>0 的區域，萬有常數 c_Q>0 使 |Q| 與 |S|²+|ω|² 可比較，故 Q 加權集中不是矩陣正規化假象。",
      "definition_en": "Section 4 (C5-E.2) asserts that on {ϑ≥δ>0} a universal c_Q>0 makes |Q| comparable to |S|²+|ω|², so Q-weighted concentration is genuine quadratic-activity concentration.",
      "defining_relation": "|Q(S,\\omega)|\\ge c_Q\\delta\\bigl(|S|^2+|\\omega|^2\\bigr)\\qquad(\\vartheta(S)\\ge\\delta>0)"
    },
    {
      "id": "ns.c5.c5e.q_mass",
      "latex": "A_j^Q",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "Q 加權質量",
      "label_en": "Q-weighted mass",
      "definition_zh": "第 6 節以選定伴隨／局部核截斷 χ_j≥0 定義活躍 Q motif 的總質量 A_j^Q=∫ χ_j |Q_j| dx。",
      "definition_en": "Section 6 defines the total mass A_j^Q=∫ χ_j |Q_j| dx of an active Q motif against a selected adjoint/local-core cutoff χ_j≥0."
    },
    {
      "id": "ns.c5.c5e.nu_q",
      "latex": "\\nu_j^Q",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "Q 加權機率測度",
      "label_en": "Q-weighted spatial probability measure",
      "definition_zh": "第 6 節在 A_j^Q>0 時把 |Q| 截斷質量正規化為空間機率測度，作為後續方向／間隙缺陷的權重。",
      "definition_en": "Section 6 normalizes truncated |Q|-mass to a spatial probability measure whenever A_j^Q>0, which then weights all subsequent direction/gap defects.",
      "defining_relation": "d\\nu_j^Q(x)=\\frac{\\chi_j(x)|Q_j(x)|}{A_j^Q}\\,dx"
    },
    {
      "id": "ns.c5.c5e.strain_direction",
      "latex": "V_j(x)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "應變方向",
      "label_en": "strain direction",
      "definition_zh": "第 7 節在 S_j≠0 時令 V_j=S_j/|S_j| 取值於無跡對稱矩陣單位球面 S⁴，並在 S_j=0 時送入墓地 ∂_S。",
      "definition_en": "Section 7 sets V_j=S_j/|S_j| in the unit sphere S⁴ of trace-free symmetric matrices when S_j≠0, and sends S_j=0 to the cemetery ∂_S."
    },
    {
      "id": "ns.c5.c5e.joint_pushforward",
      "latex": "\\Xi_j^{S\\theta}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "應變方向／間隙聯合測度",
      "label_en": "joint strain-direction/gap measure",
      "definition_zh": "第 7 節將配對 (應變方向 V_j, 間隙狀態 θ_j=ϑ(S_j)) 對 ν_j^Q 的 push-forward 作為緊緻狀態空間上的 Q 加權缺陷測度。",
      "definition_en": "Section 7 takes the push-forward of the pair (strain direction V_j, gap state θ_j=ϑ(S_j)) under ν_j^Q as the Q-weighted joint defect measure on the compact state space.",
      "defining_relation": "\\Xi_j^{S\\theta}=(V_j,\\theta_j)_{\\#}\\nu_j^Q"
    },
    {
      "id": "ns.c5.c5e.gap_distribution",
      "latex": "\\mathfrak{g}_j(\\delta)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "中間隙分布函數",
      "label_en": "middle-gap distribution function",
      "definition_zh": "第 8 節記錄 Q 加權測度落在 {θ_j≤δ} 的質量，並以弱極限 Ξ_*^{Sθ} 定義極限間隙質量 𝔤_*(δ)。",
      "definition_en": "Section 8 records the Q-weighted mass of {θ_j≤δ} and defines the limit gap mass 𝔤_*(δ) from any weak limit Ξ_*^{Sθ}."
    },
    {
      "id": "ns.c5.c5e.middle_gap_defect_mass",
      "latex": "\\mathfrak{G}_*",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "中間隙缺陷質量",
      "label_en": "middle-gap defect mass",
      "definition_zh": "第 9 節取極限測度在 {θ=0} 的質量（等同 δ↓0 時 𝔤_* 的極限），正值時稱中間隙缺陷測度活躍。",
      "definition_en": "Section 9 takes the limit-measure mass of {θ=0} (equivalently the δ↓0 limit of 𝔤_*), and calls the middle-gap defect measure active when this mass is positive.",
      "defining_relation": "\\mathfrak{G}_*=\\Xi_*^{S\\theta}\\{\\theta=0\\}=\\lim_{\\delta\\downarrow 0}\\mathfrak{g}_*(\\delta)"
    },
    {
      "id": "ns.c5.c5e.barycenter_residual",
      "latex": "\\kappa_j^Q",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "二次重心殘差",
      "label_en": "quadratic barycenter residual",
      "definition_zh": "第 10 節以 Q 方向 U_j=Q_j/|Q_j|∈S⁵ 的 ν_j^Q 平均之模長記錄七點抵消極端分支，第 11 節證明其趨於 0 時不能收斂到單一強中間方向。",
      "definition_en": "Section 10 records the modulus of the ν_j^Q-mean of the Q-direction U_j=Q_j/|Q_j|∈S⁵ as the seven-point cancellation extreme branch, and Section 11 proves that κ_j^Q→0 cannot concentrate on a single strong-middle direction.",
      "notes": "This is the Q-weighted form of the C5-D seven-point zero-barycenter cancellation."
    },
    {
      "id": "ns.c5.c5e.strong_middle_subset",
      "latex": "\\mathcal{S}_\\delta",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "均勻強中間子集",
      "label_en": "uniform strong-middle subset",
      "definition_zh": "第 12 節定義 {V∈S⁴: ϑ(V)≥δ}，其上 C5-D 錐半徑與半空間裕度可取均勻常數 r_δ、γ_δ ≳ δ。",
      "definition_en": "Section 12 defines {V∈S⁴: ϑ(V)≥δ}, on which the C5-D cone radius and half-space margin admit uniform constants r_δ, γ_δ ≳ δ."
    },
    {
      "id": "ns.c5.c5e.strain_carrying_set",
      "latex": "E_S(\\eta)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "應變承載集",
      "label_en": "strain-carrying set",
      "definition_zh": "第 15 節對 0<η<1 定義 {|S|²≥η|Q|}，作為把方向洩漏轉成導數庫存前的應變承載分支。",
      "definition_en": "Section 15 defines {|S|²≥η|Q|} for 0<η<1, the strain-carrying branch used before converting directional leakage into derivative stock."
    },
    {
      "id": "ns.c5.c5e.vorticity_dominant_set",
      "latex": "E_\\omega(\\eta)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "渦量主導集",
      "label_en": "vorticity-dominant set",
      "definition_zh": "第 15–16 節定義 {|S|²<η|Q|}，並由 |Q|≤|S|²+c_ω|ω|²（c_ω=√2/4）得到其上 |ω|² ≳ |Q|。",
      "definition_en": "Sections 15–16 define {|S|²<η|Q|} and, from |Q|≤|S|²+c_ω|ω|² with c_ω=√2/4, obtain |ω|² ≳ |Q| there."
    },
    {
      "id": "ns.c5.c5e.strain_derivative_stock",
      "latex": "\\mathfrak{H}_R",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "應變導數庫存",
      "label_en": "strain-derivative fluctuation stock",
      "definition_zh": "第 20 節將應變承載洩漏經加權 Poincaré 得到的 ∫|∇S|² 無量綱化為高階應變波動庫存，並在 E-DER 分支滿足 ℌ_R ≳ a_R^Q。",
      "definition_en": "Section 20 nondimensionalizes the weighted-Poincaré output ∫|∇S|² of strain-carrying leakage into a higher-derivative strain fluctuation stock, which obeys ℌ_R ≳ a_R^Q on the E-DER branch.",
      "defining_relation": "\\mathfrak{H}_R=\\frac{R^3}{\\nu^2}\\int_{B_{CR}}|\\nabla S|^2"
    },
    {
      "id": "ns.c5.c5e.vorticity_stock",
      "latex": "\\mathfrak{W}_R",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "局部臨界渦量庫存",
      "label_en": "local critical vorticity stock",
      "definition_zh": "第 20 節將渦量主導洩漏上的 ∫χ|ω|² 無量綱化為局部臨界渦量／能散庫存，並在 E-VORT 分支滿足 𝔚_R ≳ a_R^Q。",
      "definition_en": "Section 20 nondimensionalizes ∫χ|ω|² on vorticity-dominant leakage into a local critical vorticity/enstrophy stock, which obeys 𝔚_R ≳ a_R^Q on the E-VORT branch.",
      "defining_relation": "\\mathfrak{W}_R=\\frac{R}{\\nu^2}\\int\\chi|\\omega|^2"
    },
    {
      "id": "ns.c5.c5e.normalized_q_intensity",
      "latex": "a_R^Q",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "正規化 Q 強度",
      "label_en": "normalized Q intensity",
      "definition_zh": "第 20 節定義無量綱局部 Q 強度 a_R^Q=R ν^{-2} A_χ^Q，作為第 21 節三分法的非退化強度閾值尺度。",
      "definition_en": "Section 20 defines the dimensionless local Q intensity a_R^Q=R ν^{-2} A_χ^Q, the nondegeneracy scale for the Section 21 trichotomy."
    },
    {
      "id": "ns.c5.c5e.middle_load",
      "latex": "M_\\delta",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "中間負荷",
      "label_en": "middle load",
      "definition_zh": "第 23 節記錄可測集 G_δ={ϑ≤δ} 所承擔的中間源 ∫ λ₂⁺|S|²，C5-E.7 由此迫使該集上的三次應變至少以 δ^{-1} 放大。",
      "definition_en": "Section 23 records the middle source ∫ λ₂⁺|S|² carried by G_δ={ϑ≤δ}, from which C5-E.7 forces cubic strain on that set at least at rate δ^{-1}.",
      "defining_relation": "M_\\delta=\\int_{G_\\delta}\\lambda_2^+|S|^2\\,dx"
    },
    {
      "id": "ns.c5.c5e.effective_amplitude",
      "latex": "A_{\\mathrm{eff}}(f)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "有效三次振幅",
      "label_en": "effective cubic amplitude",
      "definition_zh": "第 25 節對非零 f∈L²∩L³ 定義三次／二次範數比，並得到 A_eff(f)≤‖f‖_∞。",
      "definition_en": "Section 25 defines the cubic-to-quadratic norm ratio of a nonzero f∈L²∩L³, which satisfies A_eff(f)≤‖f‖_∞.",
      "defining_relation": "A_{\\mathrm{eff}}(f)=\\frac{\\|f\\|_3^3}{\\|f\\|_2^2}"
    },
    {
      "id": "ns.c5.c5e.effective_volume",
      "latex": "V_{\\mathrm{eff}}(f)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "有效活躍體積",
      "label_en": "effective active volume",
      "definition_zh": "第 26 節定義具體積量綱的比 V_eff(f)=‖f‖₂⁶/‖f‖₃⁶，在振幅大致常數的集合上與該集合體積同階。",
      "definition_en": "Section 26 defines the volume-dimensional ratio V_eff(f)=‖f‖₂⁶/‖f‖₃⁶, which is comparable to the volume of a set on which f is roughly constant.",
      "defining_relation": "V_{\\mathrm{eff}}(f)=\\frac{\\|f\\|_2^6}{\\|f\\|_3^6}"
    },
    {
      "id": "ns.c5.c5e.effective_superlevel",
      "latex": "E_c(f)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "有效超水平集",
      "label_en": "effective-amplitude superlevel set",
      "definition_zh": "第 27 節 C5-E.8 定義 |f|≥c A_eff(f) 的顯式集合，其體積 ≤ c^{-2} V_eff(f) 且承載至少 1-c 比例的三次活動。",
      "definition_en": "Section 27 (C5-E.8) defines the explicit set {|f|≥c A_eff(f)}, which has volume ≤ c^{-2} V_eff(f) and carries at least a 1-c fraction of the cubic activity.",
      "defining_relation": "E_c(f)=\\bigl\\{x:\\,|f(x)|\\ge c A_{\\mathrm{eff}}(f)\\bigr\\}"
    },
    {
      "id": "ns.c5.c5e.normalized_effective_volume",
      "latex": "\\phi_{S,3}(R)",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "正規化有效體積比",
      "label_en": "normalized effective-volume ratio",
      "definition_zh": "第 29 節在祖先尺度 R 上將 V_eff(S) 除以 R³，C5-E.9 證明非退化中間負荷與有界應變庫存迫使它隨 δ→0 而崩潰。",
      "definition_en": "Section 29 divides V_eff(S) by the ancestry-scale volume R³, and C5-E.9 proves that nondegenerate middle load plus bounded strain stock forces this ratio to collapse as δ→0.",
      "defining_relation": "\\phi_{S,3}(R)=\\frac{V_{\\mathrm{eff}}(S)}{R^3}"
    },
    {
      "id": "ns.c5.c5e.normalized_middle_load",
      "latex": "b_R^{\\mathrm{mid}}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "正規化中間隙負荷",
      "label_en": "normalized middle-gap load",
      "definition_zh": "第 29 節定義無量綱局部中間隙負荷 b_R^{mid}=R³ ν^{-3} M_δ，C5-E.9 在其有正下界時迫使有效體積崩潰。",
      "definition_en": "Section 29 defines the dimensionless local middle-gap load b_R^{mid}=R³ ν^{-3} M_δ, whose positive lower bound in C5-E.9 forces effective-volume collapse."
    },
    {
      "id": "ns.c5.c5e.strain_enstrophy_stock",
      "latex": "e_R^S",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "應變能散庫存",
      "label_en": "strain-enstrophy stock",
      "definition_zh": "第 29–30 節定義 e_R^S=R ν^{-2} ‖S‖₂²，若它無界則中間隙路線改走應變能散逃逸而非有效體積崩潰。",
      "definition_en": "Sections 29–30 define e_R^S=R ν^{-2} ‖S‖₂², whose unboundedness is the strain-enstrophy-escape alternative to effective-volume collapse on the middle-gap route."
    },
    {
      "id": "ns.c5.c5e.sparseness_radius",
      "latex": "r_{\\mathrm{sp}}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "稀疏化半徑",
      "label_en": "sparseness radius",
      "definition_zh": "第 31 節取 r_sp ≍ δ_sp^{-1} c^{-2/3} V_eff(S)^{1/3}，使有效超水平集在任意基點具一線方向的一維 δ_sp-稀疏性。",
      "definition_en": "Section 31 takes r_sp ≍ δ_sp^{-1} c^{-2/3} V_eff(S)^{1/3}, so that the effective superlevel set is 1D δ_sp-sparse along some line through every base point.",
      "notes": "The underlying volume-to-line occupancy lemma is inherited from C3-W."
    },
    {
      "id": "ns.c5.c5e.defect_state",
      "latex": "\\Theta_*^{SDef}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "C5-E 缺陷狀態",
      "label_en": "C5-E defect state",
      "definition_zh": "第 43 節把本輪 Q 加權方向／間隙測度、間隙質量、方向分散、導數庫存、渦量庫存、有效體積比、稀疏半徑與導數定理介面狀態打包為 True-ETN 缺陷狀態。",
      "definition_en": "Section 43 packages this round's Q-weighted direction/gap measure, gap mass, directional dispersion, derivative stock, vorticity stock, effective-volume ratio, sparseness radius, and derivative-theorem interface status into a True-ETN defect state.",
      "defining_relation": "\\Theta_*^{SDef}=\\bigl\\langle\\Xi_*^{S\\theta},\\,\\mathfrak{G}_*,\\,\\mathfrak{D}_*^{\\mathrm{dir}},\\,\\mathfrak{H}_*,\\,\\mathfrak{W}_*,\\,\\phi_{S,3}^*,\\,r_{\\mathrm{sp}}^*,\\,\\mathsf{G}_{\\mathrm{der}}\\bigr\\rangle",
      "notes": "True-ETN update for C5-E; the interface flag 𝖦_der remains OPEN."
    },
    {
      "id": "ns.c5.c5e.derivative_intermittency_pregate",
      "latex": "\\mathbf{Derivative\\text{-}Intermittency\\ Pre\\text{-}Gate}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "導數間歇前置閘",
      "label_en": "Derivative-Intermittency Pre-Gate",
      "definition_zh": "第 0、34 與 36 節將 ℌ_R、𝔚_R、φ_{S,3} 等已證場缺陷命名為導數／間歇前置閘，強調尚未對接已發表的 Grujić–Xu 正則假設。",
      "definition_en": "Sections 0, 34 and 36 name the proved field defects ℌ_R, 𝔚_R, φ_{S,3} a derivative/intermittency pre-gate, stressing that they do not yet match the published Grujić–Xu regularity hypotheses."
    },
    {
      "id": "ns.c5.c5e.spatial_debt_trichotomy",
      "latex": "Q\\Rightarrow\\mathrm{Gap}\\vee\\mathrm{Derivative}\\vee\\mathrm{Vorticity}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "Q 抵消空間債務三分法",
      "label_en": "Q-cancellation spatial-debt trichotomy",
      "definition_zh": "第 21 節 C5-E.6 斷言：在非退化局部 Q 強度下，反覆小二次均值必走中間隙缺陷、應變導數波動庫存 ℌ_R≳1、或渦量主導洩漏 𝔚_R≳1 之一。",
      "definition_en": "Section 21 (C5-E.6) asserts that under nondegenerate local Q intensity, recurrent small quadratic mean must take at least one of middle-gap defect, strain-derivative stock ℌ_R≳1, or vorticity-dominant leakage 𝔚_R≳1.",
      "defining_relation": "Q\\ \\Rightarrow\\ \\text{Gap Concentration}\\ \\vee\\ \\text{Strain-Derivative Fluctuation}\\ \\vee\\ \\text{Vorticity-Dominant Leakage}"
    },
    {
      "id": "ns.c5.c5e.grujic_xu_theorems",
      "latex": "\\text{Grujić--Xu 2024, Thm.\\ 3.5/3.14}",
      "series": "NS",
      "first_appearance": "C5-E",
      "label_zh": "Grujić–Xu 導數稀疏正則準則",
      "label_en": "Grujić–Xu derivative-sparseness regularity criteria",
      "definition_zh": "第 1.2 與第 35 節援引 J. Math. Fluid Mech. 26, Article 53 (2024) 的 Theorem 3.5／3.14，其前件是 D^k u 或 D^k ω 的分量／符號超水平集在後續解析時間的一維稀疏性。",
      "definition_en": "Sections 1.2 and 35 invoke Theorems 3.5 and 3.14 of J. Math. Fluid Mech. 26, Article 53 (2024), whose antecedents are 1D sparseness of component/sign superlevel sets of D^k u or D^k ω at a later analytic time.",
      "notes": "C5-E.10 records an OPEN INTERFACE (E-G1–E-G5): strain-amplitude / ∇S intermittency is not identified with these D^k-component/sign hypotheses."
    },
    {
      "id": "ns.c5.c5f.k_i",
      "latex": "k_1 \\le k_2 \\le k_3",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "歸一化正中間特徵值",
      "label_en": "Normalized positive-middle eigenvalues",
      "definition_zh": "在第2節中，定義為正規化且跡為零之對稱張量 $K$ 的排序特徵值，並假設 $k_2 \\ge 0$。",
      "definition_en": "Introduced in Section 2 as the ordered eigenvalues of a normalized trace-free symmetric tensor $K$ with $k_2 \\ge 0$."
    },
    {
      "id": "ns.c5.c5f.vartheta",
      "latex": "\\vartheta(K)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "中間能隙參數",
      "label_en": "Middle gap parameter",
      "definition_zh": "在第2節中定義的參數，表示第二與第三主特徵值之積，用以量化能隙退化程度。",
      "definition_en": "Defined in Section 2 as the product of the second and third eigenvalues, quantifying the middle gap degeneration.",
      "defining_relation": "\\vartheta(K) = k_2 k_3"
    },
    {
      "id": "ns.c5.c5f.p1_k",
      "latex": "P_1(K)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "最壓縮軸投影子",
      "label_en": "Compressive-axis projector",
      "definition_zh": "在第4節中定義，對應於最壓縮特徵值 $k_1$ 的秩一光譜投影子。",
      "definition_en": "Defined in Section 4 as the rank-one spectral projector corresponding to the most-compressive eigenvalue $k_1$.",
      "defining_relation": "P_1(K) = e_1(K) \\otimes e_1(K)"
    },
    {
      "id": "ns.c5.c5f.k_ast",
      "latex": "K_\\ast",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "能隙退化極限形狀",
      "label_en": "Middle-gap limit shape",
      "definition_zh": "在第6節中指出，當中能隙退化時，應變張量的極限形狀仍保有明確的最壓縮特徵值。",
      "definition_en": "Identified in Section 6 as the limit strain shape when the middle gap degenerates, retaining a distinct most-compressive eigenvalue.",
      "defining_relation": "K_\\ast = R_\\ast \\operatorname{diag}\\left(-\\frac{1}{\\sqrt{2}}, 0, \\frac{1}{\\sqrt{2}}\\right) R_\\ast^T"
    },
    {
      "id": "ns.c5.c5f.g_e1",
      "latex": "G(e_1)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "遠場壓力補償矩陣",
      "label_en": "Far-pressure compensation matrix",
      "definition_zh": "在第0節與第19節中提及，用於計算與遠場壓力主導矩陣交互作用的二次幾何。",
      "definition_en": "Mentioned in Sections 0 and 19 as the quadratic geometry for evaluating interactions with the far-pressure dominant matrix.",
      "defining_relation": "G(e_1) = e_1 \\otimes e_1 - \\frac{1}{3} I",
      "notes": "Carried over from C5-D as a core pressure-geometry operator."
    },
    {
      "id": "ns.c5.c5f.h_e_delta",
      "latex": "H_{e,\\delta}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "精確共軸半空間算子",
      "label_en": "Exact common-axis half-space operator",
      "definition_zh": "在第7節中定義的半空間矩陣，證明固定壓縮軸能將所有局部二次方向推入嚴格半空間。",
      "definition_en": "Defined in Section 7 as the half-space matrix showing that a fixed compressive axis forces all local quadratic directions into a strict half-space.",
      "defining_relation": "H_{e,\\delta} = e \\otimes e - \\frac{1+\\delta}{2} I"
    },
    {
      "id": "ns.c5.c5f.h_e_sigma",
      "latex": "H_{e,\\sigma}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "軸帽半空間算子",
      "label_en": "Axis-cap half-space operator",
      "definition_zh": "在第11節中定義，為放寬至容許壓縮軸在一個窄圓錐帽內變動的半空間條件算子。",
      "definition_en": "Defined in Section 11 as the half-space condition operator relaxed to allow the compressive axis to vary within a narrow cap.",
      "defining_relation": "H_{e,\\sigma} = e \\otimes e - \\frac{1+\\sigma}{2} I"
    },
    {
      "id": "ns.c5.c5f.alpha_delta",
      "latex": "\\alpha_\\delta",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "能隙軸帽半徑",
      "label_en": "Gap axis-cap radius",
      "definition_zh": "在第13節中定義，為使所有 Q 方向能落入同一半空間所允許的最大軸傾角閾值。",
      "definition_en": "Defined in Section 13 as the maximum axis tilt threshold allowing all Q directions to fall into a common half-space.",
      "defining_relation": "\\alpha_\\delta = \\arcsin \\sqrt{\\frac{\\delta}{2+4\\delta}}"
    },
    {
      "id": "ns.c5.c5f.projective_e",
      "latex": "[e] \\in \\mathbb{RP}^2",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "射影壓縮軸",
      "label_en": "Projective compressive axis",
      "definition_zh": "在第16節中指出，因正負方向產生相同投影子，故真實的軸狀態存在於緊緻射影平面中。",
      "definition_en": "Noted in Section 16 that since opposing directions yield the same projector, the true axis state lies in the compact projective plane."
    },
    {
      "id": "ns.c5.c5f.nu_axis",
      "latex": "\\nu_j^{axis}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "Q權重軸測度",
      "label_en": "Q-weighted axis measure",
      "definition_zh": "在第17節中定義，為推廣至活動 Q 核心上的壓縮軸機率分佈測度。",
      "definition_en": "Defined in Section 17 as the probability measure of the compressive axis over the active Q cores.",
      "defining_relation": "\\nu_j^{axis} = P_1(S_j)_\\# \\nu_j^Q"
    },
    {
      "id": "ns.c5.c5f.far_pressure_f",
      "latex": "F",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "遠場壓力前導矩陣",
      "label_en": "Far-pressure leading matrix",
      "definition_zh": "在第19節中引用的無跡對稱矩陣，代表共同諧和遠場壓力的主導部分。",
      "definition_en": "Referenced in Section 19 as the trace-free symmetric matrix representing the leading part of the common harmonic far pressure."
    },
    {
      "id": "ns.c5.c5f.omega_f_minus",
      "latex": "\\Omega_F^-(c)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "壓力軸約束集",
      "label_en": "Pressure axis constraint set",
      "definition_zh": "在第19節中定義，為使得壓縮軸能與遠場壓力產生足夠負向補償的射影方向集合。",
      "definition_en": "Defined in Section 19 as the set of projective directions where the compressive axis yields sufficient negative compensation with the far pressure.",
      "defining_relation": "\\Omega_F^-(c) = \\{ [e] \\in \\mathbb{RP}^2 : e^T F e \\le -c \\}"
    },
    {
      "id": "ns.c5.c5f.signature_1",
      "latex": "(-,+,+)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "壓力標記 I",
      "label_en": "Pressure signature I",
      "definition_zh": "在第20節中分類的一種壓力慣性標記，具備單一負特徵值，會導致軸向鎖定於射影帽內。",
      "definition_en": "Classified in Section 20 as the pressure inertia signature with a single negative eigenvalue, which forces the axis to lock into a projective cap."
    },
    {
      "id": "ns.c5.c5f.signature_2",
      "latex": "(-,-,+)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "壓力標記 II",
      "label_en": "Pressure signature II",
      "definition_zh": "在第20節中分類的另一種壓力慣性標記，具備雙負特徵值，其負二次區域形成一個帶狀而非單一帽區。",
      "definition_en": "Classified in Section 20 as the pressure inertia signature with two negative eigenvalues, yielding a projective belt rather than a single cap."
    },
    {
      "id": "ns.c5.c5f.alpha_f_c",
      "latex": "\\alpha_F(c)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "壓力帽半徑",
      "label_en": "Pressure cap radius",
      "definition_zh": "在第22節中定義的射影帽半徑，由強負壓力補償邊界 $c$ 與特徵值差距所決定。",
      "definition_en": "Defined in Section 22 as the projective cap radius dictated by the strong negative pressure margin $c$ and the eigenvalue gaps.",
      "defining_relation": "\\alpha_F(c) = \\arcsin \\sqrt{\\frac{|f_1|-c}{|f_1|+f_2}}"
    },
    {
      "id": "ns.c5.c5f.vorticity_stock",
      "latex": "\\mathfrak W_R",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "臨界渦度儲備",
      "label_en": "Critical vorticity stock",
      "definition_zh": "在第29節中重申，為從 C5-E 繼承而來、在局部核心尺度上的渦度 $L^2$ 物理量。",
      "definition_en": "Reiterated in Section 29 as the $L^2$ vorticity quantity over the local core scale, inherited from C5-E."
    },
    {
      "id": "ns.c5.c5f.p_st_omega_sq",
      "latex": "P_{st}(\\omega\\otimes\\omega)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "應變空間渦度二次源",
      "label_en": "Strain-space projected vorticity-quadratic source",
      "definition_zh": "在第31節與32節中討論，為映射至 Miller 正交算子架構中的擁擠項。",
      "definition_en": "Discussed in Sections 31 and 32 as the congestion term mapping to the Miller orthogonal operator architecture."
    },
    {
      "id": "ns.c5.c5f.p_st_perp_omega_sq",
      "latex": "P_{st}^{\\perp}(\\omega\\otimes\\omega)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "渦度約束補集擁擠",
      "label_en": "Vorticity constraint-complement congestion",
      "definition_zh": "在第31節與33節中出現，代表無法直接成為實際壓力黑塞矩陣、而需列入正交約束互補空間的剩餘分量。",
      "definition_en": "Appeared in Sections 31 and 33 representing the residual component forced into the orthogonal constraint complement space, strictly not the actual pressure Hessian."
    },
    {
      "id": "ns.c5.c5f.strain_derivative_stock",
      "latex": "\\mathfrak H_R",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "應變導數儲備",
      "label_en": "Strain-derivative stock",
      "definition_zh": "在第34節中引入，導致必定存在達成尺度臨界振幅的二階速度導數點。",
      "definition_en": "Introduced in Section 34, which forces the existence of points where the second velocity derivative reaches a scale-critical amplitude."
    },
    {
      "id": "ns.c5.c5f.effective_amplitude_as",
      "latex": "\\widehat A_S",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "歸一化有效振幅",
      "label_en": "Normalized effective amplitude",
      "definition_zh": "在第37節中定義，用以衡量 NS 縮放座標下的中間能隙三次方有效振幅。",
      "definition_en": "Defined in Section 37 as the effective cubic amplitude of the middle gap in NS-scaled coordinates.",
      "defining_relation": "\\widehat A_S = \\frac{R^2}{\\nu} \\frac{\\|S\\|_3^3}{\\|S\\|_2^2}"
    },
    {
      "id": "ns.c5.c5f.rho_dir",
      "latex": "\\rho_{\\rm dir}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "直接導數目標尺度",
      "label_en": "Direct derivative target scale",
      "definition_zh": "在第39節中分析，對應固定 $k=1$ 的情況，其在空間指數上比中能隙間歇性路徑來得差。",
      "definition_en": "Analyzed in Section 39 for fixed $k=1$, possessing a spatial exponent structurally less favorable than the middle-gap intermittency route."
    },
    {
      "id": "ns.c5.c5f.rho_chain_k",
      "latex": "\\rho_{\\rm chain}^{(k)}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "導數鏈輔助尺度",
      "label_en": "Chain-assisted derivative scale",
      "definition_zh": "在第43節中引用自 Grujić–Xu 定理 3.14 的輔助尺度，具有形式上更佳的空間指數，但受限於高階導數階層假設。",
      "definition_en": "Referenced in Section 43 from Grujić–Xu Theorem 3.14, having a formally better spatial exponent but constrained by high-order derivative hierarchy hypotheses."
    },
    {
      "id": "ns.c5.c5f.k_best_j",
      "latex": "k_j^{best}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "最佳導數階數",
      "label_en": "Best derivative order",
      "definition_zh": "在第44節中定義，為在每次遞迴事件中綜合各項閥值與定理條件所選出的最有利導數階數。",
      "definition_en": "Defined in Section 44 as the most favorable derivative order selected for each recurrent event based on amplitude, sparsity, and theorem thresholds."
    },
    {
      "id": "ns.c5.c5f.theta_axis",
      "latex": "\\Theta_\\ast^{Axis}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "C5軸向共享狀態",
      "label_en": "C5 shared axis state",
      "definition_zh": "在第52節中定義的新狀態向量，封裝了壓縮軸分佈、能隙缺陷與遠場壓力標記。",
      "definition_en": "Defined in Section 52 as the new state vector encapsulating compressive-axis distribution, gap defect, and far-pressure metadata.",
      "defining_relation": "\\Theta_\\ast^{Axis} = \\left\\langle \\nu_\\ast^{axis}, \\mathfrak G_\\ast, F_\\ast, \\operatorname{sig}F_\\ast, c_\\ast^P, \\mathfrak C_\\ast^{axis} \\right\\rangle"
    },
    {
      "id": "ns.c5.c5f.c_axis_alpha",
      "latex": "\\mathfrak C_{\\rm axis}(\\alpha)",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "軸向集中統計量",
      "label_en": "Axis concentration statistic",
      "definition_zh": "在第53節中定義，用以測量壓縮軸機率測度在任意半徑為 $\\alpha$ 的射影球內的最大質量。",
      "definition_en": "Defined in Section 53 to measure the maximum mass of the compressive-axis probability measure within any projective ball of radius \\alpha.",
      "defining_relation": "\\mathfrak C_{\\rm axis}(\\alpha) = \\sup_{[e]\\in\\mathbb{RP}^2} \\nu_\\ast^{axis} ( B_\\alpha([e]) )"
    },
    {
      "id": "ns.c5.c5f.theta_f",
      "latex": "\\Theta_\\ast^{F}",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "C5-F全域狀態向量",
      "label_en": "C5-F global state vector",
      "definition_zh": "在第56節中定義，為包含軸、壓力與最佳導數階數等多維屬性的更新版 C5-F 真實 ETN 狀態向量。",
      "definition_en": "Defined in Section 56 as the updated C5-F True ETN state vector encompassing axes, pressures, and optimal derivative orders.",
      "defining_relation": "\\Theta_\\ast^{F} = \\left\\langle \\nu_\\ast^{axis}, \\mathfrak G_\\ast, F_\\ast, \\operatorname{sig}F_\\ast, c_\\ast^P, \\mathfrak V_\\ast^{op,\\omega}, \\mathfrak V_\\ast^{\\perp,\\omega}, k_\\ast^{best}, d_\\ast^{der} \\right\\rangle"
    },
    {
      "id": "ns.c5.c5f.c_ast_p",
      "latex": "c_\\ast^P",
      "series": "NS",
      "first_appearance": "C5-F",
      "label_zh": "壓力對齊裕度",
      "label_en": "Pressure alignment margin",
      "definition_zh": "在第52節中作為 C5 共享狀態的一部分引入，代表壓力能有效鎖定軸的負向補償強度。",
      "definition_en": "Introduced in Section 52 as part of the C5 shared state, representing the negative compensation strength by which pressure effectively locks the axis."
    },
    {
      "id": "ns.c5.c5g.theorem_3_5",
      "latex": "\\text{Grujić--Xu Theorem 3.5}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "Grujić–Xu 定理 3.5",
      "label_en": "Grujić–Xu Theorem 3.5",
      "definition_zh": "第1.1節將此定理作為不修改假設的外部閘：若在可容許稍後時刻上選定的 D^ku 分量/符號超水平集在尺度 ρ 上對每個 x_0 呈一維 δ-稀疏，則 T^* 不是爆破時刻。",
      "definition_en": "Section 1.1 takes this theorem as an unmodified external gate: 1D δ-sparseness of a selected D^ku component/sign superlevel set at scale ρ about every x_0 on an admissible later time implies T^* is not a blow-up time.",
      "notes": "Published external gate (J. Math. Fluid Mech. 26, Article 53, 2024); C5-G does not alter its hypotheses and does not mix in Theorem 3.14."
    },
    {
      "id": "ns.c5.c5g.T_star",
      "latex": "T^\\ast",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "假設首次爆破時刻",
      "label_en": "putative first blow-up time",
      "definition_zh": "第1.1節將 T^* 作為假設的首次爆破時刻，而定理 3.5 的結論是在空間稀疏條件成立時它不是爆破時刻。",
      "definition_en": "Section 1.1 treats T^* as a putative first blow-up time, which Theorem 3.5 concludes is not a blow-up time once the spatial sparseness condition holds."
    },
    {
      "id": "ns.c5.c5g.s_t",
      "latex": "s=s(t)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "定理可容許稍後時刻",
      "label_en": "theorem-admissible later time",
      "definition_zh": "第1.1節要求存在稍後時刻 s=s(t) 落在 D^ku 逃逸時刻 t 之後長度約 \\|D^ku(t)\\|_\\infty^{-6/(2k+3)} 的明確區間內，閘閉合只能在此窗口上宣告。",
      "definition_en": "Section 1.1 requires a later time s=s(t) in an explicit interval of length ~\\|D^ku(t)\\|_\\infty^{-6/(2k+3)} after the D^ku escape time t; gate closure may be declared only on this window."
    },
    {
      "id": "ns.c5.c5g.A_k",
      "latex": "A_k(s)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "k 階導數振幅",
      "label_en": "k-th derivative L^∞ amplitude",
      "definition_zh": "第4節在預奇異光滑時刻 s<T^* 上將固定階 k≥1 的導數振幅定義為 A_k(s)=\\|D^ku(s)\\|_\\infty。",
      "definition_en": "Section 4 defines the fixed-order amplitude A_k(s)=\\|D^ku(s)\\|_\\infty at a pre-singular smooth time s<T^*.",
      "defining_relation": "A_k(s)=\\|D^ku(s)\\|_\\infty"
    },
    {
      "id": "ns.c5.c5g.L_k",
      "latex": "L_k(s)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "k 階導數 L^2 質量",
      "label_en": "k-th derivative L^2 mass",
      "definition_zh": "第4節在同一預奇異光滑時刻上將固定階 k 的導數 L^2 質量定義為 L_k(s)=\\|D^ku(s)\\|_2。",
      "definition_en": "Section 4 defines the fixed-order L^2 mass L_k(s)=\\|D^ku(s)\\|_2 at the same pre-singular smooth time.",
      "defining_relation": "L_k(s)=\\|D^ku(s)\\|_2"
    },
    {
      "id": "ns.c5.c5g.V_lambda_k",
      "latex": "V_{\\lambda,k}^{\\zeta,i,\\pm}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "分量/符號超水平集",
      "label_en": "component/sign superlevel set",
      "definition_zh": "第5節對任意多重指標 |ζ|=k、分量 i 與符號 ± 定義超水平集 V_{λ,k}^{ζ,i,±}={x:(D^ζ u_i)^±(x)>λ A_k}，且本輪體積界對所有分量/符號一致成立。",
      "definition_en": "Section 5 defines the component/sign superlevel set V_{λ,k}^{ζ,i,±}={x:(D^ζ u_i)^±(x)>λ A_k} for any multi-index |ζ|=k, component i and sign ±, with the volume bound holding uniformly over all such choices.",
      "defining_relation": "V_{\\lambda,k}^{\\zeta,i,\\pm}=\\{x:(D^\\zeta u_i)^{\\pm}(x)>\\lambda A_k\\}"
    },
    {
      "id": "ns.c5.c5g.V_k_eff",
      "latex": "V_k^{\\mathrm{eff}}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "固定階有效體積",
      "label_en": "fixed-order effective volume",
      "definition_zh": "第8節將固定階有效體積定義為具有體積量綱的比 V_k^{eff}=L_k^2/A_k^2。",
      "definition_en": "Section 8 defines the fixed-order effective volume as the volume-dimension ratio V_k^{eff}=L_k^2/A_k^2.",
      "defining_relation": "V_k^{\\mathrm{eff}}=L_k^2/A_k^2"
    },
    {
      "id": "ns.c5.c5g.r_vol_k",
      "latex": "r_{\\mathrm{vol},k}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "有效體積一維稀疏尺度",
      "label_en": "effective-volume 1D sparseness scale",
      "definition_zh": "第8節在固定定理對 (λ,δ) 後把有效體積轉成一維稀疏尺度 r_{vol,k}=C_{λ,δ} L_k^{2/3} A_k^{-2/3}。",
      "definition_en": "Section 8 converts effective volume into the 1D sparseness scale r_{vol,k}=C_{λ,δ} L_k^{2/3} A_k^{-2/3} after a theorem pair (λ,δ) is fixed.",
      "defining_relation": "r_{\\mathrm{vol},k}=C_{\\lambda,\\delta}\\,L_k^{2/3}A_k^{-2/3}"
    },
    {
      "id": "ns.c5.c5g.r_GX_k",
      "latex": "r_{\\mathrm{GX},k}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "定理 3.5 直接目標尺度",
      "label_en": "Theorem 3.5 direct target scale",
      "definition_zh": "第10節將已發表的定理 3.5 直接目標尺度定義為 r_{GX,k}=1/(2^k c_{GX,k} A_k^{3/(2k+3)})，其中 c_{GX,k}=c(M,\\|u_0\\|_2)。",
      "definition_en": "Section 10 defines the published Theorem 3.5 direct target scale as r_{GX,k}=1/(2^k c_{GX,k} A_k^{3/(2k+3)}), with c_{GX,k}=c(M,\\|u_0\\|_2).",
      "defining_relation": "r_{\\mathrm{GX},k}=\\frac{1}{2^k c_{\\mathrm{GX},k} A_k^{3/(2k+3)}}"
    },
    {
      "id": "ns.c5.c5g.G_k_dir",
      "latex": "\\mathfrak{G}_k^{\\mathrm{dir}}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "固定階直接閘比",
      "label_en": "fixed-order direct gate ratio",
      "definition_zh": "第11節將定理就緒的固定階直接閘比定義為 \\mathfrak{G}_k^{dir}=r_{vol,k}/r_{GX,k}，並寫成 L_k^{2/3} A_k^{-(4k-3)/(3(2k+3))} 的顯式式。",
      "definition_en": "Section 11 defines the theorem-ready fixed-order direct gate ratio \\mathfrak{G}_k^{dir}=r_{vol,k}/r_{GX,k}, equivalently C_{λ,δ} 2^k c_{GX,k} L_k^{2/3} A_k^{-(4k-3)/(3(2k+3))}.",
      "defining_relation": "\\mathfrak{G}_k^{\\mathrm{dir}}=\\frac{r_{\\mathrm{vol},k}}{r_{\\mathrm{GX},k}}=C_{\\lambda,\\delta}2^k c_{\\mathrm{GX},k} L_k^{2/3} A_k^{-(4k-3)/(3(2k+3))}",
      "notes": "C5-G's central theorem-ready interface: \\mathfrak{G}_k^{dir}(s)\\le1 at an admissible later time closes Theorem 3.5 (Theorem 12.1)."
    },
    {
      "id": "ns.c5.c5g.C_k_eff",
      "latex": "\\mathfrak{C}_k^{\\mathrm{eff}}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "固定階有效濃度指標",
      "label_en": "fixed-order effective concentration index",
      "definition_zh": "第20節將固定階有效濃度指標定義為 \\mathfrak{C}_k^{eff}=L_k^2/A_k^{(4k-3)/(2k+3)}，在固定定理歸一化下閘條件即此量不大於 C_{GX,k}。",
      "definition_en": "Section 20 defines the fixed-order effective concentration index \\mathfrak{C}_k^{eff}=L_k^2/A_k^{(4k-3)/(2k+3)}, so that the gate is \\mathfrak{C}_k^{eff}\\le C_{GX,k} under the theorem normalization.",
      "defining_relation": "\\mathfrak{C}_k^{\\mathrm{eff}}=\\frac{L_k^2}{A_k^{(4k-3)/(2k+3)}}"
    },
    {
      "id": "ns.c5.c5g.r_apr_k",
      "latex": "r_{\\mathrm{apr},k}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "定理 3.7 先驗稀疏尺度",
      "label_en": "Theorem 3.7 a-priori sparseness scale",
      "definition_zh": "第2節記錄定理 3.7 僅由 Leray 能量界得到的固定階先驗體積稀疏尺度 r_{apr,k}∼c(\\|u_0\\|_2) A_k^{-2/(2k+3)}，其指數大於定理 3.5 的 3/(2k+3)，即固定階尺度缺口。",
      "definition_en": "Section 2 records the Theorem 3.7 a-priori volumetric sparseness scale r_{apr,k}∼c(\\|u_0\\|_2) A_k^{-2/(2k+3)} coming from the Leray energy bound, whose exponent is larger than the Theorem 3.5 exponent 3/(2k+3) and thus encodes the fixed-order scaling gap."
    },
    {
      "id": "ns.c5.c5g.volume_to_line",
      "latex": "\\text{volume-to-line lemma}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "體積到線段稀疏引理",
      "label_en": "volume-to-line sparseness lemma",
      "definition_zh": "第7節沿用 C3-W 純幾何引理：若可測集 E⊂R^3 滿足 |E|<c_3 δ^3 r^3，則對任意基點 x_0 存在方向使 E 在長度 2r 的線段上佔有率 ≤δ。",
      "definition_en": "Section 7 recalls the C3-W geometric lemma: if a measurable E⊂R^3 satisfies |E|<c_3 δ^3 r^3, then for any base point x_0 there is a line direction along which the one-dimensional occupancy of E on the segment of length 2r is ≤δ.",
      "notes": "Carries over from C3-W; used in C5-G.2 to convert the global volume bound into 1D δ-sparseness at scale r_{vol,k}."
    },
    {
      "id": "ns.c5.c5g.COMPSIGN",
      "latex": "\\mathrm{COMPSIGN}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "分量/符號缺陷（已旁路）",
      "label_en": "component/sign defect (bypassed)",
      "definition_zh": "第16節說明原 C5-A 的 COMPSIGN 缺陷——僅有模長幾何而定理要分量/符號幾何——在 C5-G 直接體積閘中不再是獨立缺陷。",
      "definition_en": "Section 16 records that the C5-A COMPSIGN defect (magnitude geometry versus the theorem's required component/sign geometry) is no longer an independent defect on the C5-G direct-volume gate.",
      "notes": "Introduced in C5-A; bypassed this round by Chebyshev on every (D^ζ u_i)^± superlevel set."
    },
    {
      "id": "ns.c5.c5g.SHELLFULL",
      "latex": "\\mathrm{SHELLFULL}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "殼層/全場轉換缺陷（已旁路）",
      "label_en": "shell/full conversion defect (bypassed)",
      "definition_zh": "第17節說明因本輪直接對完整場 D^ku 估計量而不使用殼層場 u_q，故 SHELLFULL 也不再是固定階直接路線的獨立缺陷。",
      "definition_en": "Section 17 records that SHELLFULL is no longer an independent fixed-k direct-route defect because the argument estimates the full field D^ku and never uses a shell field u_q.",
      "notes": "Earlier C5 shell-to-full conversion debt; bypassed on the C5-G direct-volume route."
    },
    {
      "id": "ns.c5.c5g.D_k_dir",
      "latex": "\\mathfrak{D}_k^{\\mathrm{dir}}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "固定階直接殘餘缺陷族",
      "label_en": "fixed-order direct residual defect family",
      "definition_zh": "第18節將固定階直接路線真正殘餘壓成缺陷族 \\mathfrak{D}_k^{dir}={MULT, TIME}，即有效體積/重數缺陷與稍後時刻缺陷。",
      "definition_en": "Section 18 compresses the true residual of the fixed-order direct route to the defect family \\mathfrak{D}_k^{dir}={MULT, TIME}, namely the effective-volume/multiplicity defect and the later-time defect.",
      "notes": "TIMECHAIN is retained only on the separate Theorem 3.14 chain-assisted route (Section 47)."
    },
    {
      "id": "ns.c5.c5g.T_k",
      "latex": "\\mathsf{T}_k",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "稍後時刻對齊旗標",
      "label_en": "later-time alignment flag",
      "definition_zh": "第46節以 \\mathsf{T}_k∈{0,1} 標記定理 3.5 稍後時刻窗口是否與有利幾何/振幅窗口對齊。",
      "definition_en": "Section 46 lets \\mathsf{T}_k∈{0,1} indicate whether the Theorem 3.5 later-time window aligns with a favorable geometry/amplitude window."
    },
    {
      "id": "ns.c5.c5g.Theta_k_dir",
      "latex": "\\Theta_k^{\\mathrm{dir}}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "固定階直接缺陷狀態",
      "label_en": "fixed-order direct defect state",
      "definition_zh": "第46節將固定階直接缺陷狀態定義為對 Θ_k^{dir}=⟨\\mathfrak{G}_k^{dir}, \\mathsf{T}_k⟩，假設生存者在每個相關逃逸時刻必須維持閘比>1 或時刻旗標為 0。",
      "definition_en": "Section 46 defines the fixed-order direct defect state as the pair Θ_k^{dir}=⟨\\mathfrak{G}_k^{dir}, \\mathsf{T}_k⟩; a hypothetical survivor must recurrently keep the gate ratio >1 or the time flag equal to 0.",
      "defining_relation": "\\Theta_k^{\\mathrm{dir}}=\\langle\\mathfrak{G}_k^{\\mathrm{dir}},\\mathsf{T}_k\\rangle"
    },
    {
      "id": "ns.c5.c5g.F",
      "latex": "F\\in\\operatorname{Sym}_0(3)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "共同遠場壓力矩陣",
      "label_en": "common far-pressure matrix",
      "definition_zh": "第23節沿用 C5-F 的無跡對稱共同遠場壓力矩陣 F∈Sym_0(3) 作為壓力簽名狀態的承載物件。",
      "definition_en": "Section 23 reuses the C5-F traceless symmetric common far-pressure matrix F∈Sym_0(3) as the carrier of pressure-signature state.",
      "notes": "Carries over from C5-F; C3-U far-pressure heredity controls F_{j+1}-F_j."
    },
    {
      "id": "ns.c5.c5g.hat_F",
      "latex": "\\widehat{F}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "正規化遠場壓力矩陣",
      "label_en": "normalized far-pressure matrix",
      "definition_zh": "第23節在 F≠0 時將其正規化為 \\widehat{F}=F/|F|_F，落在無跡對稱矩陣的單位球面 S^4∩Sym_0(3) 上。",
      "definition_en": "Section 23 normalizes a nonzero far-pressure matrix to \\widehat{F}=F/|F|_F on the unit sphere S^4∩Sym_0(3) of traceless symmetric matrices.",
      "defining_relation": "\\widehat{F}=F/|F|_F\\in S^4\\cap\\operatorname{Sym}_0(3)"
    },
    {
      "id": "ns.c5.c5g.S_1_minus",
      "latex": "\\mathcal{S}_{1-}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "單壓縮簽名區域",
      "label_en": "single-compression signature region",
      "definition_zh": "第24節將單壓縮簽名區域定義為開集 \\mathcal{S}_{1-}={\\widehat{F}: sig F=(-,+,+)}。",
      "definition_en": "Section 24 defines the single-compression signature region as the open set \\mathcal{S}_{1-}={\\widehat{F}: sig F=(-,+,+)}."
    },
    {
      "id": "ns.c5.c5g.S_2_minus",
      "latex": "\\mathcal{S}_{2-}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "雙壓縮簽名區域",
      "label_en": "double-compression signature region",
      "definition_zh": "第24節將雙壓縮簽名區域定義為開集 \\mathcal{S}_{2-}={\\widehat{F}: sig F=(-,-,+)}。",
      "definition_en": "Section 24 defines the double-compression signature region as the open set \\mathcal{S}_{2-}={\\widehat{F}: sig F=(-,-,+)}."
    },
    {
      "id": "ns.c5.c5g.Sigma_P",
      "latex": "\\Sigma_P",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "壓力簽名邊界",
      "label_en": "pressure-signature boundary",
      "definition_zh": "第24節將兩簽名開區域的共同邊界定義為行列式零集 Σ_P={\\widehat{F}: det F=0}。",
      "definition_en": "Section 24 defines the common boundary of the two signature regions as the determinant-zero set Σ_P={\\widehat{F}: det F=0}.",
      "defining_relation": "\\Sigma_P=\\{\\widehat{F}:\\det F=0\\}"
    },
    {
      "id": "ns.c5.c5g.d_sig",
      "latex": "d_{\\mathrm{sig}}(F)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "壓力簽名缺陷距離",
      "label_en": "pressure-signature defect distance",
      "definition_zh": "第25節將壓力簽名缺陷距離定義為 d_{sig}(F)=dist(\\widehat{F},Σ_P)，並在 F=0 時令其為 0，從而使 d_{sig}∈[0,1] 成為緊緻簽名元資料。",
      "definition_en": "Section 25 defines the pressure-signature defect distance d_{sig}(F)=dist(\\widehat{F},Σ_P), set to 0 when F=0, so that d_{sig}∈[0,1] is compact signature metadata.",
      "defining_relation": "d_{\\mathrm{sig}}(F)=\\operatorname{dist}(\\widehat{F},\\Sigma_P)\\in[0,1]",
      "notes": "Under strong far-matrix heredity, recurrent opposite-signature switching forces d_{sig}(F_j)\\to0 (C5-G.5); otherwise pressure turnover/fragmentation remains legal."
    },
    {
      "id": "ns.c5.c5g.Delta_p",
      "latex": "\\Delta p=-|S|^2+\\tfrac12|\\omega|^2",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "壓力 Poisson 恆等式",
      "label_en": "pressure-Poisson identity",
      "definition_zh": "第32節由不可壓縮公式寫出精確壓力 Poisson 恆等式 Δp=-|S|^2+(1/2)|ω|^2，用以把渦量主導洩漏同步成正壓力曲率。",
      "definition_en": "Section 32 records the exact incompressible pressure-Poisson identity Δp=-|S|^2+(1/2)|ω|^2, used to synchronize vorticity-dominant leakage with positive pressure curvature.",
      "defining_relation": "\\Delta p=-|S|^2+\\tfrac12|\\omega|^2"
    },
    {
      "id": "ns.c5.c5g.E_omega",
      "latex": "E_\\omega(\\eta)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "渦量主導集",
      "label_en": "vorticity-dominant set",
      "definition_zh": "第33節回顧 C5-E 的渦量主導集 E_ω(η)={|S|^2<η|Q|}，作為把洩漏同步到 (Δp)_+ 的空間承載集。",
      "definition_en": "Section 33 recalls the C5-E vorticity-dominant set E_ω(η)={|S|^2<η|Q|} as the spatial carrier that synchronizes leakage onto (Δp)_+.",
      "defining_relation": "E_\\omega(\\eta)=\\{|S|^2<\\eta|Q|\\}",
      "notes": "Carries over from C5-E; C5-G.6 shows that r_η<1/2 on this set forces Δp>0 pointwise."
    },
    {
      "id": "ns.c5.c5g.r_eta",
      "latex": "r_\\eta",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "渦量主導應變–渦量比",
      "label_en": "vorticity-dominant strain-to-vorticity ratio",
      "definition_zh": "第33節在 E_ω(η) 上得到點態比 |S|^2<r_η|ω|^2，其中 r_η=η c_ω/(1-η) 且 c_ω=√2/4。",
      "definition_en": "Section 33 obtains the pointwise ratio |S|^2<r_η|ω|^2 on E_ω(η), with r_η=η c_ω/(1-η) and c_ω=√2/4.",
      "defining_relation": "r_\\eta=\\frac{\\eta c_\\omega}{1-\\eta},\\qquad c_\\omega=\\frac{\\sqrt{2}}{4}"
    },
    {
      "id": "ns.c5.c5g.P_st_perp",
      "latex": "P_{st}^{\\perp}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "應變約束正交補投影",
      "label_en": "strain-constraint orthogonal complement",
      "definition_zh": "第37節令 P_{st} 為對稱矩陣場到應變約束空間的 L^2 正交投影，並定義補投影 P_{st}^⊥=I-P_{st}。",
      "definition_en": "Section 37 lets P_{st} be the L^2 orthogonal projection of symmetric-matrix fields onto the strain constraint space and defines the complement projection P_{st}^⊥=I-P_{st}.",
      "notes": "Miller–Sawyer Helmholtz-type decomposition for symmetric-matrix fields."
    },
    {
      "id": "ns.c5.c5g.constraint_complement_ledger",
      "latex": "\\nabla^2p=-(C_A+C_S+C_\\omega)",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "約束補壓力帳本",
      "label_en": "constraint-complement pressure ledger",
      "definition_zh": "第39節對應變方程施加 P_{st}^⊥ 得到精確帳本 ∇^2p=-(C_A+C_S+C_ω)，三項分別為平流、應變平方與渦量非線性的約束補。",
      "definition_en": "Section 39 applies P_{st}^⊥ to the strain equation to obtain the exact ledger ∇^2p=-(C_A+C_S+C_ω), the three terms being the advection, strain-square, and vorticity-nonlinear constraint complements.",
      "defining_relation": "\\nabla^2p=-(C_A+C_S+C_\\omega),\\quad C_A=P_{st}^{\\perp}\\mathcal{A},\\; C_S=P_{st}^{\\perp}\\mathcal{S},\\; C_\\omega=P_{st}^{\\perp}\\mathcal{W}",
      "notes": "C5-G.7 trichotomy: large \\|C_ω\\|_2 cannot stand alone and forces actual pressure Hessian or advection complement or strain-square complement."
    },
    {
      "id": "ns.c5.c5g.Theta_DG",
      "latex": "\\Theta_\\ast^{\\mathrm{DG}}",
      "series": "NS",
      "first_appearance": "C5-G",
      "label_zh": "導數閘 True-ETN 狀態",
      "label_en": "derivative-gate True-ETN state",
      "definition_zh": "第60節將 C5-G 導數真 ETN 狀態打包為 Θ_*^{DG}=⟨k, A_k, L_k, V_k^{eff}, r_{vol,k}, r_{GX,k}, \\mathfrak{G}_k^{dir}, \\mathsf{T}_k⟩。",
      "definition_en": "Section 60 packages the C5-G derivative True-ETN state as Θ_*^{DG}=⟨k, A_k, L_k, V_k^{eff}, r_{vol,k}, r_{GX,k}, \\mathfrak{G}_k^{dir}, \\mathsf{T}_k⟩.",
      "notes": "Companion pressure and vorticity-pressure states Θ_*^{PS} and Θ_*^{VP} are defined in the same section."
    },
    {
      "id": "ns.c5.c5h.a_k",
      "latex": "A_k(s)",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "空間導數最大振幅",
      "label_en": "Maximum Derivative Amplitude",
      "definition_zh": "速度場第 k 階空間導數的 L-infinity 範數（定義於第 0 節）。",
      "definition_en": "The L-infinity norm of the k-th order spatial derivative of the velocity field (Sec 0).",
      "defining_relation": "\\|D^ku(s)\\|_\\infty",
      "notes": "Used as the primary norm reference for evaluating structural direct and chain closure scales."
    },
    {
      "id": "ns.c5.c5h.l_k",
      "latex": "L_k(s)",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "空間導數 L2 範數",
      "label_en": "L2 Norm of Derivative",
      "definition_zh": "速度場第 k 階空間導數的 L2 範數（定義於第 0 節）。",
      "definition_en": "The L2 norm of the k-th order spatial derivative of the velocity field (Sec 0)."
    },
    {
      "id": "ns.c5.c5h.r_vol_k",
      "latex": "r_{vol,k}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "有效體積半徑",
      "label_en": "Effective Volume Radius",
      "definition_zh": "由 L2 與 L-infinity 範數組合給出的 global volume superlevel set 空間尺度（定義於第 0 節）。",
      "definition_en": "The spatial scale of the global volume superlevel set derived from L2 and L-infinity norms (Sec 0).",
      "defining_relation": "\\lesssim L_k^{2/3} A_k^{-2/3}"
    },
    {
      "id": "ns.c5.c5h.r_dir_k",
      "latex": "r_{dir,k}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "直接目標半徑",
      "label_en": "Direct Target Radius",
      "definition_zh": "Grujić–Xu Theorem 3.5 所要求的 d=3 fixed-order direct 空間稀疏尺度（定義於第 0 節）。",
      "definition_en": "The d=3 fixed-order direct spatial sparseness target scale required by Grujić–Xu Theorem 3.5 (Sec 0).",
      "defining_relation": "\\frac{1}{2^k c_{dir,k} A_k^{3/(2k+3)}}"
    },
    {
      "id": "ns.c5.c5h.g_k_dir",
      "latex": "\\mathfrak{G}_k^{dir}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "直接門檻比值",
      "label_en": "Direct Gate Ratio",
      "definition_zh": "體積半徑與直接目標半徑的比值，若小於等於 1 則觸發 published regularity theorem（定義於第 0 節）。",
      "definition_en": "The ratio of effective volume radius to the direct target radius, which triggers the regularity theorem if smaller than or equal to 1 (Sec 0).",
      "defining_relation": "r_{vol,k}/r_{dir,k}",
      "notes": "C5-H establishes that this ratio intrinsically tends to infinity for smooth single-scale profiles, serving as a no-go for fixed-order automatic closure."
    },
    {
      "id": "ns.c5.c5h.r_apr_k",
      "latex": "r_{apr,k}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "先驗稀疏尺度",
      "label_en": "A-Priori Sparseness Scale",
      "definition_zh": "Theorem 3.7 基於能量層級所給出的先驗 1D 稀疏空間尺度（定義於第 2 節）。",
      "definition_en": "The a-priori 1D sparseness spatial scale based on energy level given by Theorem 3.7 (Sec 2).",
      "defining_relation": "c(\\|u_0\\|_2) A_k^{-2/(2k+3)}"
    },
    {
      "id": "ns.c5.c5h.r_chain_k",
      "latex": "r_{chain,k}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "鏈尺度",
      "label_en": "Chain Scale",
      "definition_zh": "Theorem 3.14 的 asymptotic critical velocity chain 空間目標尺度（定義於第 3 節）。",
      "definition_en": "The asymptotic critical velocity chain spatial target scale from Theorem 3.14 (Sec 3).",
      "defining_relation": "\\frac{1}{2\\widetilde{\\mathcal C}_k A_k^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5h.r_k_c_t",
      "latex": "\\mathcal R(k,c,t)",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "鏈正規化導數振幅",
      "label_en": "Chain-Normalized Derivative Amplitude",
      "definition_zh": "Grujić–Xu 用於 ascending / descending chains 比較並進行動態插值的精確正規化振幅（定義於第 4 節）。",
      "definition_en": "The exact normalized amplitude used by Grujić–Xu to compare ascending/descending chains for dynamic interpolation (Sec 4).",
      "defining_relation": "\\frac{A_k(t)^{1/(k+1)}}{c^{k/(k+1)} (k!)^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5h.tau_dir_k",
      "latex": "\\tau_{dir,k}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "直接時間窗延遲",
      "label_en": "Direct Time-Window Delay",
      "definition_zh": "Theorem 3.5 直接空間門檻要求的 admissible time window 延遲，帶有指數衰減因子（定義於第 8 節）。",
      "definition_en": "The admissible time window delay for Theorem 3.5 direct spatial gate, bearing an exponential decay factor (Sec 8)."
    },
    {
      "id": "ns.c5.c5h.tau_chain_k",
      "latex": "\\tau_{chain,k}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "鏈時間窗延遲",
      "label_en": "Chain Time-Window Delay",
      "definition_zh": "Theorem 3.14 動態鏈機制的時間窗延遲，不帶直接門檻的指數衰減因子（定義於第 9 節）。",
      "definition_en": "The time window delay for Theorem 3.14 dynamic chain mechanism, which is free of direct-gate exponential decay factors (Sec 9)."
    },
    {
      "id": "ns.c5.c5h.m_k",
      "latex": "M_k(t)",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "L2 旋轉不變矩",
      "label_en": "L2 Rotationally Invariant Moment",
      "definition_zh": "速度場分佈的第 k 階 L2 旋轉不變頻譜矩（定義於第 10 節）。",
      "definition_en": "The k-th order L2 rotationally invariant spectral moment of the velocity field (Sec 10)."
    },
    {
      "id": "ns.c5.c5h.l_k_sharp",
      "latex": "L_k^\\sharp",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "L2 矩平方根",
      "label_en": "Root of L2 Moment",
      "definition_zh": "L2 頻譜矩的平方根，用於控制 fixed-order component 的 L2 norm（定義於第 10 節）。",
      "definition_en": "The square root of the L2 spectral moment, bounding the L2 norms of fixed-order components (Sec 10)."
    },
    {
      "id": "ns.c5.c5h.lambda_k",
      "latex": "\\Lambda_k",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "頻譜階梯頻率",
      "label_en": "Spectral Frequency Ladder",
      "definition_zh": "兩步 L2 頻譜頻率，由 Fourier moment log-convexity 強迫為單調遞增（定義於第 12 節）。",
      "definition_en": "The two-step L2 spectral frequency, forced to be monotone by Fourier moment log-convexity (Sec 12).",
      "defining_relation": "\\left( \\frac{M_{k+2}}{M_k} \\right)^{1/4}"
    },
    {
      "id": "ns.c5.c5h.v_k_eff",
      "latex": "V_k^{eff}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "無因次有效體積",
      "label_en": "Dimensionless Effective Volume",
      "definition_zh": "結合頻譜矩平方根與無窮大範數定義的無因次有效體積（定義於第 14 節）。",
      "definition_en": "The dimensionless effective volume defined by the root L2 moment and infinity norm (Sec 14)."
    },
    {
      "id": "ns.c5.c5h.n_k",
      "latex": "\\mathfrak{N}_k",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "頻譜胞元重數",
      "label_en": "Spectral-Cell Multiplicity",
      "definition_zh": "量化有效體積內含有多少個頻譜尺寸胞元的物理重數，完全不受 log-convexity 控制（定義於第 14 節）。",
      "definition_en": "The physical multiplicity quantifying how many spectral-sized cells fit in the effective volume, which is completely uncontrolled by log-convexity (Sec 14).",
      "defining_relation": "\\Lambda_k^3 V_k^{eff}",
      "notes": "Central motif in establishing that spectral cascade does not force physical geometric concentration."
    },
    {
      "id": "ns.c5.c5h.g_k_chain_vol",
      "latex": "\\mathfrak{G}_k^{chain,vol}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "鏈尺度空間比值",
      "label_en": "Chain-Scale Spatial Ratio",
      "definition_zh": "以全局體積半徑除以 Theorem 3.14 鏈尺度目標定義的體積驗證鏈比值（定義於第 19 節）。",
      "definition_en": "The volume-certified chain ratio dividing the global volume radius by the Theorem 3.14 chain scale (Sec 19)."
    },
    {
      "id": "ns.c5.c5h.x_k",
      "latex": "\\mathfrak{X}_k",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "頻譜振幅根比",
      "label_en": "Spectral-to-Root Ratio",
      "definition_zh": "頻譜頻率與振幅鏈根的比值，將空間重數階梯與動態鏈振幅階梯連接（定義於第 20 節）。",
      "definition_en": "The ratio of spectral frequency to amplitude chain root, connecting the spatial multiplicity ladder with the dynamic chain amplitude ladder (Sec 20)."
    },
    {
      "id": "ns.c5.c5h.theta_k_h",
      "latex": "\\Theta_k^{H}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "高階空間狀態",
      "label_en": "High-Order Spatial State",
      "definition_zh": "包含鏈振幅、頻譜頻率與胞元重數，顯示 all-order 狀態本質上至少是高維度的（定義於第 22 節）。",
      "definition_en": "State vector including chain amplitude, spectral frequency, and cell multiplicity, demonstrating the all-order state is intrinsically multidimensional (Sec 22)."
    },
    {
      "id": "ns.c5.c5h.epsilon_k_eff",
      "latex": "\\varepsilon_k^{eff}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "實際體積集中增益",
      "label_en": "Effective Concentration-Gain Exponent",
      "definition_zh": "相對先驗能量尺度的幾何指數增益，必須大於理論指數缺陷方能跨越門檻（定義於第 29 節）。",
      "definition_en": "The actual concentration gain exponent relative to the a-priori scale, which must strictly beat the theoretical exponent gap (Sec 29).",
      "defining_relation": "\\frac{\\log(r_{apr,k}/r_{eff,k})}{\\log A_k}"
    },
    {
      "id": "ns.c5.c5h.h_spec",
      "latex": "\\text{H-SPEC}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "頻譜根不匹配缺陷",
      "label_en": "Spectral-Root Mismatch",
      "definition_zh": "頻譜頻率相較振幅鏈根過小的高階缺陷型態（定義於第 37 節）。",
      "definition_en": "High-order defect where the spectral frequency is insufficient relative to the amplitude chain root (Sec 37)."
    },
    {
      "id": "ns.c5.c5h.h_mult",
      "latex": "\\text{H-MULT}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "頻譜胞元重數缺陷",
      "label_en": "Spectral-Cell Multiplicity Defect",
      "definition_zh": "物理重數過大導致全局空間體積無法達到 1D sparseness 尺度的高階缺陷（定義於第 37 節）。",
      "definition_en": "High-order defect where the physical multiplicity is too large to allow global volume to reach the 1D sparseness scale (Sec 37)."
    },
    {
      "id": "ns.c5.c5h.h_sat",
      "latex": "\\text{H-SAT}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "漸近先驗飽和缺陷",
      "label_en": "Asymptotic A-Priori Saturation Defect",
      "definition_zh": "高階空間集中度無法比先驗能量稀疏尺度好上任何固定指數冪次的現象（定義於第 32 節）。",
      "definition_en": "The phenomenon where high-order actual spatial concentration does not improve upon the a-priori sparseness by any fixed exponential power (Sec 32)."
    },
    {
      "id": "ns.c5.c5h.h_time",
      "latex": "\\text{H-TIME}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "鏈時間窗缺陷",
      "label_en": "Chain Time Mismatch",
      "definition_zh": "有利的空間集中發生時間未能對齊 Theorem 3.14 動態容許時間窗的缺陷（定義於第 37 節）。",
      "definition_en": "Defect where the favorable spatial geometry fails to align with the dynamic theorem-admissible later time window (Sec 37)."
    },
    {
      "id": "ns.c5.c5h.h_chain",
      "latex": "\\text{H-CHAIN}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "鏈結構缺陷",
      "label_en": "Derivative-Chain Structural Defect",
      "definition_zh": "未能滿足遞增/遞減鏈或 Type-A/B 截面分佈的高階結構缺陷（定義於第 37 節）。",
      "definition_en": "Defect where ascending/descending chains or Type-A/B section topological conditions are not structurally met (Sec 37)."
    },
    {
      "id": "ns.c5.c5h.theta_i_block",
      "latex": "\\Theta_i^{block}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "C5-H 區塊狀態",
      "label_en": "C5-H Block State",
      "definition_zh": "在指數分離的導數區塊上追蹤最大值與各項幾何時間元數據的壓縮狀態向量（定義於第 39 節）。",
      "definition_en": "Compressed state vector tracking maxima and metadata on exponentially separated derivative blocks (Sec 39)."
    },
    {
      "id": "ns.c5.c5h.z_j_i",
      "latex": "Z_{j,i}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "區塊缺陷向量",
      "label_en": "Block Defect Vector",
      "definition_zh": "映射至 [0,1] 以供全無限階序空間抽取 sectionwise 缺陷測度之正規化向量（定義於第 50 節）。",
      "definition_en": "Normalized vector compactified into [0,1] for extracting sectionwise defect measures over the infinite order space (Sec 50)."
    },
    {
      "id": "ns.c5.c5h.theta_h_j_i",
      "latex": "\\Theta^{H}_{j,i}",
      "series": "NS",
      "first_appearance": "C5-H",
      "label_zh": "C5-H 最終全階狀態",
      "label_en": "True ETN C5-H All-Order State",
      "definition_zh": "更新後的 C5 全階狀態張量，正式包含動態鏈狀態與 sign geometry 缺陷追蹤（定義於第 56 節）。",
      "definition_en": "The updated C5 all-order state tensor formally including dynamic chain states and sign geometry defect tracking (Sec 56).",
      "defining_relation": "\\langle \\mathcal R_{j,m_i}, \\Lambda_{j,m_i}, \\mathfrak N_{j,m_i}, \\varepsilon_{j,m_i}^{eff}, \\mathsf{Sign}_{j,i}, \\mathsf{Time}_{j,i}, \\mathsf{ChainType}_{j,i} \\rangle",
      "notes": "This entirely supersedes the scalar static-volume approach and sets the agenda for C5-I."
    },
    {
      "id": "ns.c5.c5i.normalized_chain_amplitude",
      "latex": "\\mathcal{R}(k,c,t)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "正規化鏈振幅",
      "label_en": "normalized chain amplitude",
      "definition_zh": "第1.1節引入的Grujić–Xu正規化導數根振幅，用以在固定區段正規化常數c下比較各階導數的相對大小。",
      "definition_en": "The Grujić–Xu normalized derivative-root amplitude introduced in §1.1, used to compare derivative orders under a fixed section normalization constant c.",
      "defining_relation": "\\mathcal{R}(k,c,t)=\\frac{\\|D^ku(t)\\|_\\infty^{1/(k+1)}}{c^{k/(k+1)}(k!)^{1/(k+1)}}",
      "notes": "Faithful encoding of Grujić–Xu 2024 Definition 3.15; reused as the chain-root coordinate of the compact C5-I state."
    },
    {
      "id": "ns.c5.c5i.derivative_sections",
      "latex": "\\ell_{i+1}=\\phi(\\ell_i),\\quad\\phi(x)\\ge 2x",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "導數區段",
      "label_en": "derivative sections",
      "definition_zh": "第1.1節以指數分離的導數階區間[ℓ_i,ℓ_{i+1}]與增長函數φ(x)≥2x將導數階切成sections。",
      "definition_en": "Exponentially separated derivative-order intervals [ℓ_i,ℓ_{i+1}] with growth φ(x)≥2x, introduced in §1.1 to partition derivative orders into sections.",
      "notes": "Carries over from Grujić–Xu Definition 3.15; later hosts the sectionwise sign-defect measures μ_i^{SG}."
    },
    {
      "id": "ns.c5.c5i.section_maximizer",
      "latex": "m_i",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "區段極大子",
      "label_en": "section maximizer",
      "definition_zh": "第1.1節在每個區段[ℓ_i,ℓ_{i+1}]上選取使R(j,c(ℓ_i),t)達到最大的階m_i。",
      "definition_en": "The order m_i in each section [ℓ_i,ℓ_{i+1}] that maximises R(j,c(ℓ_i),t), selected in §1.1.",
      "defining_relation": "\\mathcal{R}(m_i,c(\\ell_i),t)=\\max_{\\ell_i\\le j\\le\\ell_{i+1}}\\mathcal{R}(j,c(\\ell_i),t)"
    },
    {
      "id": "ns.c5.c5i.type_A",
      "latex": "\\text{Type-\\mathcal{A}}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "A型區段",
      "label_en": "Type-A section",
      "definition_zh": "第2節稱區段為Type-A，若存在更高階k_i>ℓ_{i+1}使其正規化振幅追上或超越當前區段極大值。",
      "definition_en": "A section is Type-A, as defined in §2, if some higher order k_i>ℓ_{i+1} eventually catches or overtakes the current section maximum.",
      "notes": "C5-I does not prove Type-A implies all levels spatially good; only a same-time ascent-gain puncture is obtained in §30–31."
    },
    {
      "id": "ns.c5.c5i.type_B",
      "latex": "\\text{Type-\\mathcal{B}}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "B型區段",
      "label_en": "Type-B section",
      "definition_zh": "第2節稱區段為Type-B，若當前區段極大值嚴格支配整個更高階尾部。",
      "definition_en": "A section is Type-B, as defined in §2, if the current section maximum strictly dominates the entire higher-order tail.",
      "notes": "Sign-thick descending-root toll is geometrically compatible with Type-B, but published Theorem 3.9 / Corollary 3.12 already stabilize descending chains."
    },
    {
      "id": "ns.c5.c5i.chain_scale_radius",
      "latex": "r_k(s)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "鏈尺度半徑",
      "label_en": "chain-scale radius",
      "definition_zh": "第3節定義Theorem 3.14可容許的最大空間檢驗半徑，作為符號幾何與下降引理的鏈尺度。",
      "definition_en": "The maximal spatially admissible inspection radius of Theorem 3.14, defined in §3 and used as the chain scale for sign geometry and the descent lemma.",
      "defining_relation": "r_k(s)=\\frac{1}{2\\widetilde{\\mathcal{C}}(\\|u_0\\|,\\ell,k)\\|D^ku(s)\\|_\\infty^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5i.theorem_constant",
      "latex": "\\widetilde{\\mathcal{C}}_k",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "定理常數",
      "label_en": "theorem constant",
      "definition_zh": "第3節與第24節中依賴初值、區段與階的published常數，同時控制鏈半徑與定理時間窗。",
      "definition_en": "The published constant depending on initial data, section, and order, appearing in §3 and §24, that controls both the chain radius and the theorem time window.",
      "notes": "C5-I does not assume a growth bound on \\widetilde{\\mathcal{C}}_k sufficient to force d_k\\to 1."
    },
    {
      "id": "ns.c5.c5i.selected_sign_high_set",
      "latex": "V_{\\lambda,k}^{j,\\pm}(s)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "選定符號高位集",
      "label_en": "selected sign high set",
      "definition_zh": "第3節定義選定分量與符號的超位集，作為1D弦佔有率與調和測度檢驗的載體。",
      "definition_en": "The selected-component/sign superlevel set defined in §3, serving as the carrier of 1D chord occupancy and harmonic-measure tests.",
      "defining_relation": "V_{\\lambda,k}^{j,\\pm}(s)=\\{x:(D^ku)_j^{\\pm}(x,s)>\\lambda A_k(s)\\}"
    },
    {
      "id": "ns.c5.c5i.derivative_amplitude",
      "latex": "A_k(s)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "導數振幅",
      "label_en": "derivative amplitude",
      "definition_zh": "第3節以A_k(s)=\\|D^ku(s)\\|_\\infty記第k階導數的空間上確界振幅。",
      "definition_en": "The spatial supremum amplitude of the k-th derivative, defined in §3 by A_k(s)=\\|D^ku(s)\\|_\\infty.",
      "defining_relation": "A_k(s)=\\|D^ku(s)\\|_\\infty"
    },
    {
      "id": "ns.c5.c5i.theorem_parameters",
      "latex": "(\\lambda,\\delta)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "定理參數對",
      "label_en": "theorem parameter pair",
      "definition_zh": "第4節給出與調和測度條件一致選取的published參數對，並要求1/(1+λ)<δ<1。",
      "definition_en": "The published parameter pair of §4, chosen consistently with the harmonic-measure condition and required to satisfy 1/(1+λ)<δ<1."
    },
    {
      "id": "ns.c5.c5i.occupancy_margin",
      "latex": "\\kappa_{\\lambda,\\delta}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "佔有裕度",
      "label_en": "occupancy margin",
      "definition_zh": "第4節與第22節定義由參數對誘導的正裕度，作為符號厚弦下降引理的係數。",
      "definition_en": "The positive margin induced by the theorem parameter pair, defined in §4 and §22, which appears as the coefficient in the sign-thick chord descent lemma.",
      "defining_relation": "\\kappa_{\\lambda,\\delta}=(1+\\lambda)\\delta-1>0"
    },
    {
      "id": "ns.c5.c5i.chord_occupancy",
      "latex": "b_E(x_0,r,[\\nu])",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "弦佔有率",
      "label_en": "chord occupancy",
      "definition_zh": "第5節定義可測集E沿過x_0、方向[ν]、半徑r的單位化1D弦佔有率，取值於[0,1]。",
      "definition_en": "The normalized 1D chord occupancy of a measurable set E through x_0 in projective direction [ν] at radius r, defined in §5 and valued in [0,1].",
      "defining_relation": "b_E(x_0,r,[\\nu])=\\frac{1}{2r}\\mathcal{H}^1\\bigl(E\\cap(x_0-r\\nu,x_0+r\\nu)\\bigr)"
    },
    {
      "id": "ns.c5.c5i.best_directional_occupancy",
      "latex": "\\beta_E(x_0,r)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "最佳方向佔有率",
      "label_en": "best directional occupancy",
      "definition_zh": "第6節取所有射影方向上弦佔有率的下確界；β_E≤δ等價於存在定理可用的稀疏方向。",
      "definition_en": "The infimum of chord occupancies over all projective directions, defined in §6; β_E≤δ is equivalent to existence of a theorem-usable sparse direction.",
      "defining_relation": "\\beta_E(x_0,r)=\\inf_{[\\nu]\\in\\mathbb{RP}^2}b_E(x_0,r,[\\nu])"
    },
    {
      "id": "ns.c5.c5i.spatial_geometry_pass",
      "latex": "\\mathsf{SG}_k(s)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "空間幾何通過指標",
      "label_en": "spatial geometry pass",
      "definition_zh": "第7節以二值指標SG_k(s)=1表示level k在時間s對所有基點皆存在半徑≤r_k的δ-稀疏方向，此即Theorem 3.14的真正空間通過。",
      "definition_en": "The binary indicator of §7, with SG_k(s)=1 iff every basepoint at time s admits a δ-sparse direction at some radius ≤r_k; this is the genuine spatial pass of Theorem 3.14.",
      "defining_relation": "\\mathsf{SG}_k(s)=1\\iff\\forall x_0\\,\\exists\\,0<\\rho\\le r_k(s):\\ \\beta_{V_{\\lambda,k}^{j(x_0),\\pm(x_0)}}(x_0,\\rho)\\le\\delta"
    },
    {
      "id": "ns.c5.c5i.angular_sign_profile",
      "latex": "b_k([\\nu])",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "角向符號剖面",
      "label_en": "angular sign profile",
      "definition_zh": "第9節在壞見證點x_k與最大鏈半徑r_k上記錄各方向的弦佔有率，得到取值於(δ,1]的L^∞(RP^2)函數。",
      "definition_en": "The L^∞(RP^2) function of §9 recording chord occupancy in every direction at the bad witness x_k and maximal chain radius r_k, with values strictly above δ.",
      "notes": "Recurrent profiles compactify by C5-I.1 as a weak-* limit in L^∞(RP^2), yielding an isotropically sign-thick derivative core."
    },
    {
      "id": "ns.c5.c5i.harmonic_measure_map",
      "latex": "h(\\beta)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "調和測度映射",
      "label_en": "harmonic-measure map",
      "definition_zh": "第12節以Solynin極值估計把線上主動集佔有率β映成調和測度下界，且h嚴格遞減。",
      "definition_en": "The strictly decreasing Solynin extremal lower bound of §12 converting line active-set occupancy β into a harmonic-measure lower bound.",
      "defining_relation": "h(\\beta)=\\frac{2}{\\pi}\\arcsin\\frac{1-\\beta^2}{1+\\beta^2}"
    },
    {
      "id": "ns.c5.c5i.descent_coefficient",
      "latex": "d_k(c)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "下降係數",
      "label_en": "descent coefficient",
      "definition_zh": "第25節將鏈尺度振幅下降改寫成相鄰正規化根振幅的乘子，並含階乘與定理常數修正。",
      "definition_en": "The adjacent-order multiplier of §25 converting chain-scale amplitude descent into a comparison of normalized roots, including factorial and theorem-constant corrections.",
      "defining_relation": "d_k(c)=\\left(\\frac{\\kappa_{\\lambda,\\delta}}{2\\widetilde{\\mathcal{C}}_k}\\right)^{1/k}c^{1/[k(k+1)]}\\frac{(k!)^{1/(k+1)}}{((k-1)!)^{1/k}}",
      "notes": "C5-I does not assume d_k→1; that would require extra control on the growth of \\widetilde{\\mathcal{C}}_k."
    },
    {
      "id": "ns.c5.c5i.bad_core_mass_fraction",
      "latex": "\\Phi_k^{bad}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "壞核局部質量分數",
      "label_en": "bad-core mass fraction",
      "definition_zh": "第17節定義壞核球B_{r_k}(x_k)內第k階導數L^2質量佔全局L_k^2的分數。",
      "definition_en": "The fraction of global k-th derivative L^2 mass sitting in the bad-core ball B_{r_k}(x_k), defined in §17.",
      "defining_relation": "\\Phi_k^{bad}=\\frac{\\int_{B_{r_k}(x_k)}|D^ku|^2\\,dx}{L_k^2},\\qquad L_k=\\|D^ku\\|_2"
    },
    {
      "id": "ns.c5.c5i.effective_volume",
      "latex": "V_k^{eff}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "有效體積",
      "label_en": "effective volume",
      "definition_zh": "第17–18節沿用C5-H有效體積V_k^{eff}=L_k^2/A_k^2，並用以把壞核分數改寫成r_k^3/V_k^{eff}。",
      "definition_en": "The C5-H effective volume V_k^{eff}=L_k^2/A_k^2, reused in §17–18 to rewrite the bad-core fraction as a multiple of r_k^3/V_k^{eff}.",
      "defining_relation": "V_k^{eff}=L_k^2/A_k^2=\\mathfrak{N}_k\\Lambda_k^{-3}",
      "notes": "Carries over from C5-H; couples sign-thick cores to spectral-cell multiplicity."
    },
    {
      "id": "ns.c5.c5i.spectral_cell_multiplicity",
      "latex": "\\mathfrak{N}_k",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "譜胞重數",
      "label_en": "spectral-cell multiplicity",
      "definition_zh": "第18節沿用C5-H譜胞重數，說明巨大重數可使單個符號厚核只承擔很小的全局導數質量分數。",
      "definition_en": "The C5-H spectral-cell multiplicity reused in §18, showing that a large multiplicity can make a single sign-thick core carry only a small global derivative-mass fraction.",
      "notes": "Carries over from C5-H; sign-thick cores and spectral-cell multiplicity are compatible but coupled motifs."
    },
    {
      "id": "ns.c5.c5i.bad_core_multiplicity",
      "latex": "N_k",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "壞核重數",
      "label_en": "bad-core multiplicity",
      "definition_zh": "第19節以同一level/time兩兩不交壞球的個數為壞核重數，並由有效體積對r_k^3的比上界控制。",
      "definition_en": "The number of pairwise disjoint bad balls at a fixed level and time, introduced in §19 and bounded by a multiple of V_k^{eff}/r_k^3."
    },
    {
      "id": "ns.c5.c5i.section_defect_measure",
      "latex": "\\mu_i^{SG}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "區段符號缺陷測度",
      "label_en": "section sign-defect measure",
      "definition_zh": "第37節把區段內各階的二值幾何失敗指標平均成[0,1]上的子概率測度，用以緊緻化符號缺陷在階空間的分佈。",
      "definition_en": "The subprobability measure on [0,1] of §37 averaging binary geometry-failure indicators along a section, used to compactify the order-space distribution of sign defects.",
      "defining_relation": "\\mu_i^{SG}=\\frac{1}{\\ell_{i+1}-\\ell_i+1}\\sum_{k=\\ell_i}^{\\ell_{i+1}}g_{i,k}\\,\\delta_{\\theta_{i,k}},\\quad g_{i,k}=1-\\mathsf{SG}_k,\\quad\\theta_{i,k}=\\frac{k-\\ell_i}{\\ell_{i+1}-\\ell_i}",
      "notes": "Vanishing μ_*^{SG}=0 does not imply Theorem 3.14 closure, since a single bad level per ever-larger section still blocks an all-order hypothesis."
    },
    {
      "id": "ns.c5.c5i.normalized_chain_time",
      "latex": "\\tau_k",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "正規化鏈時間",
      "label_en": "normalized chain time",
      "definition_zh": "第43節把Theorem 3.14的後期時間窗正規化到[1/4,1]，無對齊定理時間時則取值於墳場點∂_T。",
      "definition_en": "The normalized Theorem 3.14 later-time coordinate of §43, valued in [1/4,1] when an admissible evaluation exists and at the cemetery point ∂_T otherwise.",
      "defining_relation": "\\tau_k=\\widetilde{\\mathcal{C}}_k A_k(t)^{2/(k+1)}(s-t)\\in[1/4,1]",
      "notes": "Different derivative orders generally use different theorem-admissible times, so same-time descent inequalities cannot be blindly multiplied across k (G-TIMESTITCH)."
    },
    {
      "id": "ns.c5.c5i.compact_state",
      "latex": "\\Theta^{I}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "C5-I緊緻狀態",
      "label_en": "C5-I compact state",
      "definition_zh": "第57節把鏈根、型態、空間通過、正規化時間、佔有率、角向剖面、壞核質量、譜胞重數與壞階位置打包成可子列緊緻化的區塊狀態。",
      "definition_en": "The block/level state of §57 packing chain roots, type, spatial pass, normalized time, occupancy, angular profile, bad-core mass, multiplicity, and bad-order location into a subsequentially compactifiable object.",
      "defining_relation": "\\Theta^{I}=\\langle\\mathcal{R}_k,\\mathsf{T}_i,\\mathsf{SG}_k,\\tau_k,\\beta_k,b_k(\\cdot),\\Phi_k^{bad},\\mathfrak{N}_k,\\theta_i^{bad}\\rangle"
    },
    {
      "id": "ns.c5.c5i.sign_thick_chord_descent",
      "latex": "A_{k-1}\\ge\\kappa_{\\lambda,\\delta} r_k A_k",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "符號厚弦下降引理",
      "label_en": "sign-thick chord descent lemma",
      "definition_zh": "第22節的C5-I.3：在壞見證點沿選定座標線積分同號厚弦，強迫低一階振幅至少為裕度乘上r_k A_k。",
      "definition_en": "C5-I.3 of §22: integrating a same-sign thick chord through a bad witness along a selected coordinate line forces the lower-order amplitude to be at least the occupancy margin times r_k A_k.",
      "defining_relation": "A_{k-1}\\ge\\kappa_{\\lambda,\\delta} r_k A_k"
    },
    {
      "id": "ns.c5.c5i.harmonic_or_descent_dichotomy",
      "latex": "\\mathsf{SG}_k(s)=1\\ \\vee\\ \\mathcal{R}(k-1,c,s)\\ge d_k(c)\\mathcal{R}(k,c,s)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "調和或下降二分法",
      "label_en": "harmonic-or-descent dichotomy",
      "definition_zh": "第27節的C5-I.4：在定理可容許時間，每個k≥1要嘛空間幾何通過，要嘛正規化根振幅付出下降係數d_k的toll。",
      "definition_en": "C5-I.4 of §27: at a theorem-admissible time, every k≥1 either spatially passes or pays a descending-root toll with coefficient d_k.",
      "defining_relation": "\\mathsf{SG}_k(s)=1\\quad\\text{or}\\quad\\mathcal{R}(k-1,c,s)\\ge d_k(c)\\mathcal{R}(k,c,s)",
      "notes": "Main new bridge of this round; the contrapositive in §28 says a sufficiently steep adjacent root ascent forces chain-scale sign sparseness."
    },
    {
      "id": "ns.c5.c5i.type_a_puncture_criterion",
      "latex": "\\frac{\\mathcal{R}(K,c,s)}{\\mathcal{R}(J,c,s)}>\\prod_{n=J+1}^{K}d_n(c)^{-1}\\ \\Rightarrow\\ \\exists n\\in(J,K]:\\mathsf{SG}_n(s)=1",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "A型穿刺準則",
      "label_en": "Type-A puncture criterion",
      "definition_zh": "第30節的C5-I.5：同一時間上若區間[J,K]的上升增益超過下降係數倒數之積，則該區間不能全部空間失敗，至少一階必須調和通過。",
      "definition_en": "C5-I.5 of §30: a same-time ascent gain on [J,K] larger than the product of inverse descent coefficients cannot have every intermediate level spatially bad, so at least one harmonic-pass level must puncture the interval."
    },
    {
      "id": "ns.c5.c5i.harmonic_critical_saturation",
      "latex": "\\beta_\\ast=\\delta",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "調和臨界飽和缺陷",
      "label_en": "harmonic critical-saturation defect",
      "definition_zh": "第11節與第34節稱反覆壞核的最佳佔有率弱極限恰等於定理閾值δ的情形為調和臨界飽和，對應I-SGCRIT而非有限階通過。",
      "definition_en": "The I-SGCRIT regime of §11 and §34 in which recurrent bad-core best occupancies weak-limit exactly to the theorem threshold δ; this is a boundary saturation of finite-level failures, not a finite-level pass.",
      "notes": "Analogous to C5-H a-priori saturation: the survivor lives increasingly close to the sufficient regularity boundary rather than violating it by a fixed margin (G-HSAT)."
    },
    {
      "id": "ns.c5.c5i.joint_section_alphabet",
      "latex": "\\mathsf{Z}_\\ast\\in\\{A_G,A_B,B_G,B_B,\\mathrm{TIME}\\}",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "型態–幾何聯合字母",
      "label_en": "joint type-geometry alphabet",
      "definition_zh": "第40–41節把每個區段標成Type-A/B與GOOD/BAD的有限字母，另加無對齊定理時間的TIME墳場，從而使無窮區段列具有最終常值聯合型態。",
      "definition_en": "The finite alphabet of §40–41 labelling each section by Type-A/B and GOOD/BAD, plus a TIME cemetery when no theorem-admissible evaluation can be aligned, so that infinite section sequences admit eventually constant joint types."
    },
    {
      "id": "ns.c5.c5i.line_section_sign_process",
      "latex": "\\chi_k([\\nu],s)",
      "series": "NS",
      "first_appearance": "C5-I",
      "label_zh": "線截符號過程",
      "label_en": "line-section sign process",
      "definition_zh": "第62節在正規化弦s∈[-1,1]上追蹤選定高位集的指示函數，作為下一輪C5-J的主要對象。",
      "definition_en": "The indicator of the selected high set along the normalized chord s∈[-1,1], introduced in §62 as the primary object of the next round C5-J.",
      "defining_relation": "\\chi_k([\\nu],s)=1_{V_{\\lambda,k}^{j,\\pm}}(x_k+r_k s\\nu)",
      "notes": "Scheduled as the main object of C5-J, living on RP^2×[-1,1] and recording occupancy, interval fragmentation, and neighboring-order correlations beyond total chord occupancy."
    },
    {
      "id": "ns.c5.c5j.A_k",
      "latex": "A_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "連鎖尺度導數上限",
      "label_en": "Chain-Scale Derivative Bound",
      "definition_zh": "第0節回顧 C5-I 引入的選定高階導數在空間失效點附近的連鎖尺度上限。",
      "definition_en": "Bounding supremum of the selected high-order derivative at the chain scale introduced in C5-I, reviewed in Section 0.",
      "notes": "Carries over from C5-I."
    },
    {
      "id": "ns.c5.c5j.kappa_lambda_delta",
      "latex": "\\kappa_{\\lambda,\\delta}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "降階積分懲罰乘子",
      "label_en": "Descent Toll Multiplier",
      "definition_zh": "第8節定義的降階積分懲罰乘子，僅依賴於符號一致高值集的總長度。",
      "definition_en": "The lower-order descent toll multiplier defined in Section 8, depending only on the total length of the same-sign high set.",
      "defining_relation": "\\kappa_{\\lambda,\\delta} = (1+\\lambda)\\delta-1 > 0"
    },
    {
      "id": "ns.c5.c5j.grujic_xu_root",
      "latex": "\\mathcal{R}_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "正規化導數根",
      "label_en": "Normalized Derivative Root",
      "definition_zh": "第18節引用的 Grujić–Xu 正規化導數根，用於建立跨階空間相容性條件。",
      "definition_en": "The Grujić–Xu normalized derivative root referenced in Section 18 to establish cross-order spatial compatibility conditions.",
      "defining_relation": "\\mathcal R_k = \\frac{A_k(s)^{1/(k+1)}}{c^{k/(k+1)}(k!)^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5j.active_occupancy",
      "latex": "\\beta",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "活躍佔有率",
      "label_en": "Active Occupancy",
      "definition_zh": "第3節定義的選定符號高值集在正規化弦上的長度佔有率。",
      "definition_en": "The occupancy of the selected sign-high set on the normalized chord, defined in Section 3.",
      "defining_relation": "\\beta = \\frac{1}{2}|E_{\\rm act}|"
    },
    {
      "id": "ns.c5.c5j.solynin_harmonic_bound",
      "latex": "h(\\beta)",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "Solynin 調和下界",
      "label_en": "Solynin Harmonic Bound",
      "definition_zh": "第3節由 Solynin 極值定理導出的、僅依賴於佔有率的通用調和測度下界。",
      "definition_en": "The universal harmonic measure lower bound derived from Solynin's extremal theorem, depending only on occupancy, defined in Section 3.",
      "defining_relation": "h(\\beta) := \\frac{2}{\\pi} \\arcsin \\frac{1-\\beta^2}{1+\\beta^2}"
    },
    {
      "id": "ns.c5.c5j.selected_scalar_derivative",
      "latex": "f_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "選定純量導數",
      "label_en": "Selected Scalar Derivative",
      "definition_zh": "第7節定義的選定純量導數函數，其取向由最大符號表示決定。",
      "definition_en": "The selected scalar derivative function defined in Section 7, oriented by the positive representation sign.",
      "defining_relation": "f_k = \\sigma D^\\zeta u_a"
    },
    {
      "id": "ns.c5.c5j.chain_scale_radius",
      "latex": "r_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "連鎖半徑",
      "label_en": "Chain Radius",
      "definition_zh": "第24節引用的 Grujić–Xu 連鎖半徑，在跨階推導的三明治不等式中會被完全對消。",
      "definition_en": "The Grujić–Xu chain radius referenced in Section 24, which cancels out completely in the cross-order sandwich inequality.",
      "defining_relation": "r_k = \\frac{1}{2\\widetilde{\\mathcal C}_k A_k^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5j.hysteresis_thresholds",
      "latex": "\\lambda_0, \\lambda_1",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "遲滯雙閾值",
      "label_en": "Hysteresis Thresholds",
      "definition_zh": "第11節固定的雙閾值遲滯參數，用於穩健計算線截面的碎裂次數。",
      "definition_en": "The two-threshold hysteresis parameters fixed in Section 11, used to robustly count fragmentation on line sections."
    },
    {
      "id": "ns.c5.c5j.hysteretic_fragment_count",
      "latex": "N_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "遲滯碎裂數",
      "label_en": "Hysteretic Fragment Count",
      "definition_zh": "第12節與第15節定義的遲滯碎裂數，代表由真實低谷分隔的穩健高值島嶼數量。",
      "definition_en": "The hysteretic fragmentation count defined in Sections 12 and 15, representing the number of robust high islands separated by genuine low excursions.",
      "defining_relation": "N_k = N_{\\lambda_0,\\lambda_1}(f_k;[-r_k,r_k])"
    },
    {
      "id": "ns.c5.c5j.derivative_conversion_constant",
      "latex": "C_D",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "導數轉換常數",
      "label_en": "Derivative Conversion Constant",
      "definition_zh": "第14節引入的導數範數轉換常數，用於計算上限導數的總變差。",
      "definition_en": "The derivative norm conversion constant introduced in Section 14 to compute the total variation of the upper derivative."
    },
    {
      "id": "ns.c5.c5j.order_sandwich_ratio",
      "latex": "\\mathfrak{C}_k^{ord}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "三階三明治比值",
      "label_en": "Three-Order Sandwich Ratio",
      "definition_zh": "第17節定義的三階三明治比值，用於量化跨階導數的相對增長。",
      "definition_en": "The three-order sandwich ratio defined in Section 17, used to quantify the relative growth across adjacent derivative orders.",
      "defining_relation": "\\mathfrak{C}_k^{ord} = \\frac{A_{k-1}A_{k+1}}{A_k^2}"
    },
    {
      "id": "ns.c5.c5j.sandwich_constant",
      "latex": "c_{\\lambda_0,\\lambda_1,\\delta}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "三明治曲率常數",
      "label_en": "Sandwich Curvature Constant",
      "definition_zh": "第17節給出的三階曲率不等式的常數因子，結合了降階懲罰與遲滯間距。",
      "definition_en": "The constant factor for the three-order curvature inequality given in Section 17, combining the descent toll and hysteresis gap.",
      "defining_relation": "c_{\\lambda_0,\\lambda_1,\\delta} = \\frac{((1+\\lambda_1)\\delta-1)(\\lambda_1-\\lambda_0)}{C_D}"
    },
    {
      "id": "ns.c5.c5j.log_chain_root",
      "latex": "Y_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "對數連鎖根",
      "label_en": "Log-Chain Root",
      "definition_zh": "第20節定義的對數連鎖根，用以分析跨階導數輪廓的凸性。",
      "definition_en": "The log-chain root defined in Section 20, used to analyze the convexity of the cross-order derivative profile.",
      "defining_relation": "Y_k = (k+1)\\log\\mathcal{R}_k"
    },
    {
      "id": "ns.c5.c5j.log_chain_curvature",
      "latex": "\\Delta^2Y_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "對數連鎖曲率算子",
      "label_en": "Log-Chain Curvature Operator",
      "definition_zh": "第20節定義的對數連鎖曲率算子，大量碎裂會迫使其呈現嚴格的局部凸性。",
      "definition_en": "The log-chain curvature operator defined in Section 20, which is forced into strict local convexity by high fragmentation.",
      "defining_relation": "\\Delta^2Y_k = Y_{k-1} + Y_{k+1} - 2Y_k"
    },
    {
      "id": "ns.c5.c5j.chain_scale_line_roughness",
      "latex": "\\mathfrak{U}_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "線粗糙度",
      "label_en": "Line Roughness",
      "definition_zh": "第23節定義的無因次連鎖尺度線粗糙度，為遲滯碎裂數提供上界。",
      "definition_en": "The dimensionless chain-scale line roughness defined in Section 23, bounding the hysteretic fragmentation count from above.",
      "defining_relation": "\\mathfrak{U}_k = r_k \\frac{A_{k+1}}{A_k}"
    },
    {
      "id": "ns.c5.c5j.normalized_line_profile",
      "latex": "\\psi_k(s)",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "正規化線輪廓",
      "label_en": "Normalized Line Profile",
      "definition_zh": "第26節定義的正規化線輪廓函數，將選定的壞核心沿弦映射至單位區間。",
      "definition_en": "The normalized line profile function defined in Section 26, mapping the selected bad core along a chord to the unit interval.",
      "defining_relation": "\\psi_k(s) = \\frac{\\sigma D^\\zeta u_a(x_k+r_kse_q)}{A_k}"
    },
    {
      "id": "ns.c5.c5j.continuous_recurrent_profile",
      "latex": "\\psi_\\ast",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "遞歸線輪廓極限",
      "label_en": "Recurrent Line-Profile Limit",
      "definition_zh": "第27節中，若粗糙度有界，透過 Arzelà–Ascoli 定理取得的連續遞歸線輪廓極限。",
      "definition_en": "The continuous recurrent line-profile limit obtained via the Arzelà–Ascoli theorem under bounded roughness, as shown in Section 27."
    },
    {
      "id": "ns.c5.c5j.full_angular_line_process",
      "latex": "\\Psi_k(\\nu,s)",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "全角度線過程",
      "label_en": "Full Angular Line Process",
      "definition_zh": "第30節定義的全角度線過程，描述指定純量導數在各方向上的正規化分佈。",
      "definition_en": "The full angular line process defined in Section 30, describing the normalized distribution of the specified scalar derivative across all directions.",
      "defining_relation": "\\Psi_k(\\nu,s) = \\frac{f_k(x_k+r_ks\\nu)}{A_k}"
    },
    {
      "id": "ns.c5.c5j.harmonic_critical_saturation",
      "latex": "\\beta_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "調和臨界佔有率",
      "label_en": "Harmonic Critical Saturation",
      "definition_zh": "第35節回顧的調和臨界飽和佔有率，探討其向下逼近臨界值 $\\delta$ 的動態。",
      "definition_en": "The harmonic critical saturation occupancy reviewed in Section 35, examining its dynamics as it approaches the critical threshold $\\delta$ from above.",
      "defining_relation": "\\beta_k = \\inf_\\nu \\frac{1}{2} |\\{s: \\Psi_k(\\nu,s) > \\lambda_1\\}|"
    },
    {
      "id": "ns.c5.c5j.root_order_curvature",
      "latex": "\\mathfrak{K}_k^{root}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "根階數曲率",
      "label_en": "Root Order Curvature",
      "definition_zh": "第44節定義的正規化根的局部三階曲率狀態，其直接給出碎裂次數的上界。",
      "definition_en": "The local three-order normalized-root curvature state defined in Section 44, which directly bounds the fragmentation number.",
      "defining_relation": "\\mathfrak{K}_k^{root} = \\frac{k+1}{k} \\frac{\\mathcal{R}_{k-1}^{k} \\mathcal{R}_{k+1}^{k+2}}{\\mathcal{R}_k^{2k+2}}"
    },
    {
      "id": "ns.c5.c5j.compactified_order_curvature",
      "latex": "\\widehat{\\mathfrak{K}}_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "緊緻化階數曲率",
      "label_en": "Compactified Order Curvature",
      "definition_zh": "第45節定義的緊緻化階數曲率，當其趨近於 1 時即代表階數曲率擁塞。",
      "definition_en": "The compactified order curvature defined in Section 45, which indicates order-curvature congestion as it approaches 1.",
      "defining_relation": "\\widehat{\\mathfrak{K}}_k = \\frac{\\mathfrak{K}_k^{root}}{1+\\mathfrak{K}_k^{root}}"
    },
    {
      "id": "ns.c5.c5j.chain_time_state",
      "latex": "\\Theta_k^{time-line}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "連鎖時間狀態",
      "label_en": "Chain-Time State",
      "definition_zh": "第52節提出的連鎖時間狀態元組，將空間微幾何與時間動態屬性結合儲存。",
      "definition_en": "The chain-time state tuple proposed in Section 52, storing combined spatial microgeometry and temporal dynamic properties.",
      "defining_relation": "\\Theta_k^{time-line} = \\langle t_k, s_k, \\tau_k, \\beta_k, \\mathfrak{U}_k, \\mathfrak{K}_k^{root}, \\dots \\rangle"
    },
    {
      "id": "ns.c5.c5j.normalized_theorem_time",
      "latex": "\\tau_k",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "正規化定理時間",
      "label_en": "Normalized Theorem Time",
      "definition_zh": "第52節定義的正規化定理時間差，用於評估不同階層間時間的對齊情況。",
      "definition_en": "The normalized theorem time difference defined in Section 52, used to evaluate the temporal alignment across different derivative orders.",
      "defining_relation": "\\tau_k = \\widetilde{\\mathcal{C}}_k A_k(t_k)^{2/(k+1)} (s_k-t_k)"
    },
    {
      "id": "ns.c5.c5j.root_transfer_factor",
      "latex": "\\mathfrak{T}_{k \\to k+1}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "根轉移因子",
      "label_en": "Root Transfer Factor",
      "definition_zh": "第53節定義的根轉移因子，用以量化同一階導數在不同評估時間間的乘性變化。",
      "definition_en": "The root transfer factor defined in Section 53, quantifying the multiplicative change of the same derivative root between different evaluation times.",
      "defining_relation": "\\mathfrak{T}_{k \\to k+1} = \\max\\left\\{ \\frac{\\mathcal{R}_k(s_{k+1})}{\\mathcal{R}_k(s_k)}, \\frac{\\mathcal{R}_k(s_k)}{\\mathcal{R}_k(s_{k+1})} \\right\\}",
      "notes": "Sets up the temporal turnover defect problem for the next round, C5-K."
    },
    {
      "id": "ns.c5.c5j.j_comp",
      "latex": "\\text{J-COMP}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "緊緻遲滯核心條件",
      "label_en": "Compact Hysteretic Core Condition",
      "definition_zh": "第34節界定的壞核心線過程分支之一，表示存在粗糙度有界的緊緻遲滯符號核心。",
      "definition_en": "One branch of the bad-core line-process dichotomy defined in Section 34, representing a compact hysteretic sign-core with bounded roughness."
    },
    {
      "id": "ns.c5.c5j.j_up",
      "latex": "\\text{J-UP}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "高階導數粗糙化條件",
      "label_en": "Upper-Order Roughness Condition",
      "definition_zh": "第34節界定的另一個分支，表示上限階導數粗糙度趨於無限大。",
      "definition_en": "The alternative branch defined in Section 34, indicating that the upper-order derivative roughness blows up."
    },
    {
      "id": "ns.c5.c5j.j_hc",
      "latex": "\\text{J-HC}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "緊緻臨界核心",
      "label_en": "Compact Critical Core",
      "definition_zh": "第36節界定的調和臨界飽和模式之一，對應線過程緊緻化為一連續的遲滯臨界輪廓。",
      "definition_en": "One mode of harmonic critical saturation defined in Section 36, corresponding to a line process compactifying into a continuous hysteretic critical profile."
    },
    {
      "id": "ns.c5.c5j.j_hr",
      "latex": "\\text{J-HR}",
      "series": "NS",
      "first_appearance": "C5-J",
      "label_zh": "粗糙臨界核心",
      "label_en": "Rough Critical Core",
      "definition_zh": "第36節界定的另一調和臨界飽和模式，對應臨界佔有率伴隨高階導數的粗糙化。",
      "definition_en": "The alternative harmonic critical saturation mode defined in Section 36, where critical occupancy is accompanied by upper-order derivative roughness."
    },
    {
      "id": "ns.c5.c5k.theorem_3_14",
      "latex": "\\mathrm{Theorem\\ 3.14}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "Grujić–Xu 定理 3.14",
      "label_en": "Grujić–Xu Theorem 3.14",
      "definition_zh": "第 1.1 節重審 Grujić–Xu 2024 定理 3.14：對每個滿足設置與剩餘時間的 (k,t)，若可容許窗 I_k(t) 內存在達到 1D 稀疏性的時刻，則 T* 不是爆破時刻。",
      "definition_en": "Section 1.1 re-audits Grujić–Xu 2024 Theorem 3.14: if every required pair (k,t) admits a 1D-sparseness time inside I_k(t), then T* is not a blow-up time.",
      "notes": "External-closure object of this round; C5-K does not reprove it, but records that order-dependent times and Type-A/B switching are already handled once all window passes hold."
    },
    {
      "id": "ns.c5.c5k.type_a_string",
      "latex": "\\mathcal{A}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "A 型字串",
      "label_en": "Type-A string",
      "definition_zh": "第 2 節依公開定義 3.15 與條件 (3.43) 將導數區段字串標為 A 型，表示該區段呈由時間相關最大化元主導的上升型行為。",
      "definition_en": "Section 2, following published Definition 3.15 and condition (3.43), labels a derivative-section string as Type-A when it exhibits maximizer-dominated ascending behaviour.",
      "notes": "Once Theorem 3.14 hypotheses hold, C5-K.1 removes generic TYPE-SWITCH as an independent C5 residual."
    },
    {
      "id": "ns.c5.c5k.type_b_string",
      "latex": "\\mathcal{B}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "B 型字串",
      "label_en": "Type-B string",
      "definition_zh": "第 2 節依公開定義 3.15 與條件 (3.44) 將導數區段字串標為 B 型，表示該區段呈下降或尾部主導的行為。",
      "definition_en": "Section 2, following published Definition 3.15 and condition (3.44), labels a derivative-section string as Type-B when it exhibits descending or tail-dominating behaviour.",
      "notes": "Paired with Type-A under published dynamic interpolation; not retained as a standalone C5 survivor motif."
    },
    {
      "id": "ns.c5.c5k.lemma_3_16",
      "latex": "\\mathrm{Lemma\\ 3.16}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "A 型控制引理",
      "label_en": "Type-A control lemma",
      "definition_zh": "第 3 節核對公開 Lemma 3.16：若字串以 A 型起頭且空間假設可用，則直至首次 A-to-B 切換，導數根被控制在初始最大值的 (1+ε̃)^{1/ℓ_{i+q}} 倍之內。",
      "definition_en": "Section 3 audits published Lemma 3.16: if a string starts Type-A and the spatial hypothesis is available, derivative roots stay bounded by a (1+ε̃)^{1/ℓ_{i+q}} factor times the initial string maximum until the first A-to-B switch.",
      "notes": "Treated as EXTERNAL/VERIFIED; C5-K does not replace the published Type-A dynamic proof."
    },
    {
      "id": "ns.c5.c5k.lemma_3_17",
      "latex": "\\mathrm{Lemma\\ 3.17}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "B 型控制引理",
      "label_en": "Type-B control lemma",
      "definition_zh": "第 4 節核對公開 Lemma 3.17：若字串以 B 型起頭，則直至首次 B-to-A 切換，導數根的時間上確界不超過該區段的初始最大值。",
      "definition_en": "Section 4 audits published Lemma 3.17: if a string starts Type-B, the time-supremum of derivative roots stays at most the initial section/string maximum until the first B-to-A switch.",
      "notes": "Treated as EXTERNAL/VERIFIED; together with Lemma 3.16 it supplies the published switch-interval stitching of Theorem 3.14."
    },
    {
      "id": "ns.c5.c5k.possible_blowup_time",
      "latex": "T^\\ast",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "可能爆破時刻",
      "label_en": "possible blow-up time",
      "definition_zh": "第 1.1 節將 T* 取為可能爆破時刻：若所有所需 (k,t) 的窗內空間通過皆成立，則定理 3.14 斷言它不是爆破時刻。",
      "definition_en": "Section 1.1 takes T* as the candidate blow-up time, which Theorem 3.14 concludes is not a blow-up time once every required window pass holds.",
      "notes": "Inherited from Grujić–Xu 2024; global regularity remains OPEN at the end of C5-K."
    },
    {
      "id": "ns.c5.c5k.amplitude_A_k",
      "latex": "A_k(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "階振幅",
      "label_en": "order-k amplitude",
      "definition_zh": "第 1.1 節沿用鏈振幅 A_k(t) 作為定理 3.14 剩餘時間條件與可容許窗長度的輸入。",
      "definition_en": "Section 1.1 uses the inherited chain amplitude A_k(t) as the input that sets both the remaining-time gate of Theorem 3.14 and the length of the admissible window.",
      "notes": "Carries over from C5-I/J and the Grujić–Xu derivative chain; not redefined in this round."
    },
    {
      "id": "ns.c5.c5k.theorem_constant_C_tilde",
      "latex": "\\widetilde{\\mathcal{C}}_k",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "階相關定理常數",
      "label_en": "order-dependent theorem constant",
      "definition_zh": "第 1.1 節沿用 Grujić–Xu 定理 3.14 的階相關常數 𝒞̃_k，同時決定可容許窗長度 τ_k 與空間尺度 r_k。",
      "definition_en": "Section 1.1 inherits the Grujić–Xu Theorem 3.14 order-dependent constant 𝒞̃_k, which simultaneously sets the admissible-window length τ_k and the spatial scale r_k.",
      "notes": "Published theorem constant; appears in every C5-K clock, window, and scale formula."
    },
    {
      "id": "ns.c5.c5k.chain_clock_tau",
      "latex": "\\tau_k(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "鏈時鐘",
      "label_en": "chain clock",
      "definition_zh": "第 9 節將定理鏈時鐘定義為 τ_k(t)=[𝒞̃_k A_k(t)^{2/(k+1)}]^{-1}，作為階 k 在基時 t 的可容許時間長度。",
      "definition_en": "Section 9 defines the theorem chain clock τ_k(t)=[𝒞̃_k A_k(t)^{2/(k+1)}]^{-1} as the admissible time length of order k at base time t.",
      "defining_relation": "\\tau_k(t)=\\bigl[\\widetilde{\\mathcal{C}}_k A_k(t)^{2/(k+1)}\\bigr]^{-1}"
    },
    {
      "id": "ns.c5.c5k.admissible_window",
      "latex": "I_k(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "可容許定理窗",
      "label_en": "admissible theorem window",
      "definition_zh": "第 9 節（並於第 1.1 節依定理 3.14 核對）將可容許窗定義為 I_k(t)=[t+τ_k/4, t+τ_k]，即定理要求存在空間通過的整段區間。",
      "definition_en": "Section 9 (matching the Section 1.1 audit of Theorem 3.14) defines the admissible window I_k(t)=[t+τ_k/4, t+τ_k] as the full interval in which a spatial pass is required to exist.",
      "defining_relation": "I_k(t)=[t+\\tau_k(t)/4,\\,t+\\tau_k(t)]"
    },
    {
      "id": "ns.c5.c5k.chain_clock_frequency",
      "latex": "\\Omega_k(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "鏈時鐘頻率",
      "label_en": "chain-clock frequency",
      "definition_zh": "第 9 節將鏈時鐘頻率定義為 Ω_k(t)=τ_k(t)^{-1}=𝒞̃_k A_k(t)^{2/(k+1)}。",
      "definition_en": "Section 9 defines the chain-clock frequency by Ω_k(t)=τ_k(t)^{-1}=𝒞̃_k A_k(t)^{2/(k+1)}.",
      "defining_relation": "\\Omega_k(t)=\\tau_k(t)^{-1}=\\widetilde{\\mathcal{C}}_k A_k(t)^{2/(k+1)}"
    },
    {
      "id": "ns.c5.c5k.spatial_scale_r",
      "latex": "r_k(s)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "定理空間尺度",
      "label_en": "theorem spatial scale",
      "definition_zh": "第 10 節將定理選取的 1D 稀疏性尺度定為 r_k(s)=(2𝒞̃_k A_k(s)^{1/(k+1)})^{-1}。",
      "definition_en": "Section 10 sets the theorem-selected 1D-sparseness scale to r_k(s)=(2𝒞̃_k A_k(s)^{1/(k+1)})^{-1}.",
      "defining_relation": "r_k(s)=\\frac{1}{2\\widetilde{\\mathcal{C}}_k A_k(s)^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5k.spatial_defect_score",
      "latex": "\\beta_k(s)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "空間缺陷分數",
      "label_en": "spatial defect score",
      "definition_zh": "第 10 節將 β_k(s) 定義為在尺度 r_k(s) 內、對所有基點與方向，符號超水平集 1D 厚度的上確界–下確界。",
      "definition_en": "Section 10 defines β_k(s) as the sup-inf 1D thickness of the selected sign superlevel set over basepoints, scales up to r_k(s), and directions.",
      "defining_relation": "\\beta_k(s)=\\sup_{x_0}\\inf_{0<\\rho\\le r_k(s)}\\inf_{[\\nu]\\in\\mathbb{RP}^2}b_{V_{\\lambda,k}(x_0,s)}(x_0,\\rho,[\\nu])"
    },
    {
      "id": "ns.c5.c5k.window_spatial_score",
      "latex": "\\beta_k^{win}(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "窗空間分數",
      "label_en": "window spatial score",
      "definition_zh": "第 11 節將窗空間分數定義為 β_k^{win}(t)=inf_{s∈I_k(t)} β_k(s)，並以 β_k^{win}(t)>δ 判定嚴格整窗失敗。",
      "definition_en": "Section 11 defines the window spatial score by β_k^{win}(t)=inf_{s∈I_k(t)} β_k(s), with strict whole-window failure when β_k^{win}(t)>δ.",
      "defining_relation": "\\beta_k^{win}(t)=\\inf_{s\\in I_k(t)}\\beta_k(s)"
    },
    {
      "id": "ns.c5.c5k.window_persistent_sign_defect",
      "latex": "\\mathrm{Window\\text{-}Persistent\\ Sign\\ Defect}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "窗持久符號缺陷",
      "label_en": "window-persistent sign defect",
      "definition_zh": "第 12 節將窗持久符號缺陷定義為整個可容許窗 I_k(t) 內沒有任何時刻通過定理 1D 稀疏性條件，從而使每個 s∈I_k(t) 都帶有過厚的符號核。",
      "definition_en": "Section 12 defines a window-persistent sign defect as the failure of the 1D-sparseness condition at every time in the admissible window I_k(t), so every s∈I_k(t) carries a too-thick sign core.",
      "notes": "This is the true spatial survivor of C5-K, replacing generic Type-switch and order-dependent-time mismatch as independent residuals."
    },
    {
      "id": "ns.c5.c5k.descent_margin_kappa",
      "latex": "\\kappa_{\\lambda,\\delta}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "下降邊際常數",
      "label_en": "descent-margin constant",
      "definition_zh": "第 13 節定義 κ_{λ,δ}=(1+λ)δ−1>0，作為整窗符號失敗所強制的下階振幅下降係數。",
      "definition_en": "Section 13 defines κ_{λ,δ}=(1+λ)δ−1>0 as the lower-order amplitude descent coefficient forced by whole-window sign failure.",
      "defining_relation": "\\kappa_{\\lambda,\\delta}=(1+\\lambda)\\delta-1>0"
    },
    {
      "id": "ns.c5.c5k.descent_factor_d_k",
      "latex": "d_k(c)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "根下降因子",
      "label_en": "root-descent factor",
      "definition_zh": "第 14 節沿用 C5-I 在固定區段正規化 c 下的下降因子 d_k(c)，使窗內每一時刻皆有 ℛ(k−1,c,s)≥d_k(c)ℛ(k,c,s)。",
      "definition_en": "Section 14 inherits the C5-I descent factor d_k(c) in a fixed section normalization c, so that ℛ(k−1,c,s)≥d_k(c)ℛ(k,c,s) holds at every time in the window.",
      "notes": "Carries over from C5-I; C5-K upgrades the one-time witness to a window-persistent inequality."
    },
    {
      "id": "ns.c5.c5k.derivative_root_R",
      "latex": "\\mathcal{R}(j,c,t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "導數根",
      "label_en": "derivative root",
      "definition_zh": "第 2 節沿用公開導數根 ℛ(j,c(ℓ_i),t) 作為區段最大化、Type-A/B 控制、以及窗內下降帶的對象。",
      "definition_en": "Section 2 inherits the published derivative root ℛ(j,c(ℓ_i),t) as the object of section maximizers, Type-A/B control, and the window-persistent descent strip.",
      "notes": "Carries over from Grujić–Xu and C5-I/J; C5-K evaluates it throughout entire admissible windows rather than at isolated times."
    },
    {
      "id": "ns.c5.c5k.window_persistent_descent_strip",
      "latex": "\\mathrm{Window\\text{-}Persistent\\ Descent\\ Strip}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "窗持久下降帶",
      "label_en": "window-persistent descent strip",
      "definition_zh": "第 13 節證明：若整窗空間條件失敗，則對所有 s∈I_k(t) 皆有 A_{k−1}(s)≥κ_{λ,δ} r_k(s) A_k(s)，從而使 C5-I 下降升為整條時間帶。",
      "definition_en": "Section 13 proves that whole-window spatial failure yields A_{k−1}(s)≥κ_{λ,δ} r_k(s) A_k(s) for every s∈I_k(t), promoting the C5-I descent toll to a full temporal strip.",
      "defining_relation": "A_{k-1}(s)\\ge\\frac{\\kappa_{\\lambda,\\delta}}{2\\widetilde{\\mathcal{C}}_k}A_k(s)^{k/(k+1)}\\qquad\\forall s\\in I_k(t)"
    },
    {
      "id": "ns.c5.c5k.harmonic_temporal_critical_saturation",
      "latex": "\\mathrm{Harmonic\\text{--}Temporal\\ Critical\\ Saturation}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "諧和–時間臨界飽和",
      "label_en": "harmonic–temporal critical saturation",
      "definition_zh": "第 16 節將諧和–時間臨界飽和定義為一連串失敗窗滿足 β_{k_j}^{win}(t_j)↓δ，使整窗最小值逼近但未跨越諧和閾值。",
      "definition_en": "Section 16 defines harmonic–temporal critical saturation as a sequence of failing windows with β_{k_j}^{win}(t_j)↓δ, so the whole-window minimum approaches but never crosses the harmonic threshold.",
      "notes": "Stronger than C5-I pointwise saturation: the closest approach to a harmonic pass is measured across the full admissible window."
    },
    {
      "id": "ns.c5.c5k.block_clock_spread",
      "latex": "\\mathfrak{S}_{J,K}^{clock}(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "區塊時鐘展度",
      "label_en": "block clock spread",
      "definition_zh": "第 21 節將區塊時鐘展度定義為導數區塊 [J,K] 上最大與最小鏈時鐘之比。",
      "definition_en": "Section 21 defines the block clock spread as the ratio of the maximal to the minimal chain clock over the derivative block [J,K].",
      "defining_relation": "\\mathfrak{S}_{J,K}^{clock}(t)=\\frac{\\max_{J\\le k\\le K}\\tau_k(t)}{\\min_{J\\le k\\le K}\\tau_k(t)}"
    },
    {
      "id": "ns.c5.c5k.adjacent_window_overlap_lemma",
      "latex": "I_k(t)\\cap I_{k+1}(t)\\ne\\varnothing",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "相鄰鏈窗重疊引理",
      "label_en": "Adjacent Chain-Window Overlap Lemma",
      "definition_zh": "第 19 節的 C5-K.3 斷言：同一基時 t 的相鄰窗相交若且唯若 1/4≤τ_{k+1}(t)/τ_k(t)≤4。",
      "definition_en": "Section 19 (C5-K.3) asserts that adjacent windows at the same base time t overlap if and only if 1/4≤τ_{k+1}(t)/τ_k(t)≤4.",
      "defining_relation": "I_k(t)\\cap I_{k+1}(t)\\ne\\varnothing\\iff\\frac14\\le\\frac{\\tau_{k+1}(t)}{\\tau_k(t)}\\le 4"
    },
    {
      "id": "ns.c5.c5k.chain_window_helly_lemma",
      "latex": "\\mathfrak{S}_{J,K}^{clock}(t)\\le 4",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "鏈窗 Helly 引理",
      "label_en": "Chain-Window Helly Lemma",
      "definition_zh": "第 21 節的 C5-K.4 斷言：區塊時鐘展度 ≤4 等價於存在對所有階 J,…,K 同時可容許的共同時刻 s。",
      "definition_en": "Section 21 (C5-K.4) asserts that block clock spread ≤4 is equivalent to existence of one time s admissible for every order J,…,K.",
      "defining_relation": "\\mathfrak{S}_{J,K}^{clock}(t)\\le 4\\iff\\bigcap_{k=J}^{K}I_k(t)\\ne\\varnothing"
    },
    {
      "id": "ns.c5.c5k.common_time_block_descent",
      "latex": "\\mathcal{R}(J,c,s_\\ast)\\ge\\Bigl(\\prod_{k=J+1}^{K}d_k(c)\\Bigr)\\mathcal{R}(K,c,s_\\ast)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "共同時間區塊下降",
      "label_en": "common-time block descent",
      "definition_zh": "第 23 節的 C5-K.5 證明：在共同可容許時刻 s_* 上，若 J+1,…,K 皆有窗持久符號缺陷，則低階根至少是高階根乘上下降因子的連乘積。",
      "definition_en": "Section 23 (C5-K.5) proves that at a common admissible time s_*, if every level J+1,…,K has a window-persistent sign defect, then the lower-order root is at least the product of the descent factors times the top-order root.",
      "defining_relation": "\\mathcal{R}(J,c,s_\\ast)\\ge\\Bigl(\\prod_{k=J+1}^{K}d_k(c)\\Bigr)\\mathcal{R}(K,c,s_\\ast)"
    },
    {
      "id": "ns.c5.c5k.chain_clock_separation",
      "latex": "\\mathrm{Chain\\text{-}Clock\\ Separation}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "鏈時鐘分離",
      "label_en": "chain-clock separation",
      "definition_zh": "第 25 節將鏈時鐘分離定義為區塊時鐘展度 𝔖_{J,K}^{clock}>4，從而使整塊導數階無法共享同一定理時刻。",
      "definition_en": "Section 25 defines chain-clock separation as block clock spread 𝔖_{J,K}^{clock}>4, so that no common theorem time exists across the full derivative block.",
      "notes": "C5-K.7 shows this is not free temporal noise: it is equivalent to a large adjacent derivative-root clock jump."
    },
    {
      "id": "ns.c5.c5k.adjacent_clock_ratio",
      "latex": "\\frac{\\tau_{k+1}(t)}{\\tau_k(t)}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "相鄰時鐘比",
      "label_en": "adjacent clock ratio",
      "definition_zh": "第 26–27 節（C5-K.7）將相鄰鏈時鐘比寫成振幅幾何，並證明比落出 [1/4,4] 等價於強向上或向下的根–時鐘跳躍。",
      "definition_en": "Sections 26–27 (C5-K.7) express the adjacent chain-clock ratio in amplitude geometry and prove that leaving [1/4,4] is a strong upward or downward root-clock jump.",
      "defining_relation": "\\frac{\\tau_{k+1}}{\\tau_k}=\\frac{\\widetilde{\\mathcal{C}}_k}{\\widetilde{\\mathcal{C}}_{k+1}}\\left(\\frac{A_k^{1/(k+1)}}{A_{k+1}^{1/(k+2)}}\\right)^2"
    },
    {
      "id": "ns.c5.c5k.theorem_time_pass_flag",
      "latex": "\\mathsf{W}_k(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "定理時通過旗標",
      "label_en": "theorem-time pass flag",
      "definition_zh": "第 30 節將 𝖶_k(t) 定義為指示函數：若 I_k(t) 內存在滿足空間條件 (3.41) 的時刻則為 1，否則為 0。",
      "definition_en": "Section 30 defines 𝖶_k(t) as the indicator that equals 1 iff some s∈I_k(t) satisfies the spatial condition (3.41), and 0 otherwise.",
      "defining_relation": "\\mathsf{W}_k(t)=\\begin{cases}1,&\\exists s\\in I_k(t)\\text{ satisfying (3.41)},\\\\0,&\\text{otherwise.}\\end{cases}"
    },
    {
      "id": "ns.c5.c5k.theorem_setup_defect",
      "latex": "\\mathrm{Theorem\\text{-}Setup\\ Defect}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "定理設置缺陷",
      "label_en": "theorem-setup defect",
      "definition_zh": "第 33 節將定理設置缺陷定義為在到達空間問題之前，定理 3.14 的鏈條件、參數相容、剩餘時間或正則性假設已不可用。",
      "definition_en": "Section 33 defines a theorem-setup defect as failure of the chain, parameter-compatibility, remaining-time, or regularity hypotheses of Theorem 3.14 before the spatial question is reached.",
      "notes": "Kept strictly distinct from window-persistent sign defect; C5-K does not treat setup hypotheses as automatic."
    },
    {
      "id": "ns.c5.c5k.window_root_turnover",
      "latex": "\\mathfrak{T}_k^{win}(t)",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "窗內根週轉因子",
      "label_en": "window root-turnover factor",
      "definition_zh": "第 34 節將失敗定理窗內導數根的上確界與下確界之比定義為 𝔗_k^{win}(t)∈[1,∞]，其發散即為同階窗內根週轉缺陷。",
      "definition_en": "Section 34 defines the window root-turnover factor 𝔗_k^{win}(t)∈[1,∞] as the ratio of sup to inf of ℛ(k,c,·) on a failing theorem window, with divergence recording same-order temporal turnover.",
      "defining_relation": "\\mathfrak{T}_k^{win}(t)=\\frac{\\sup_{s\\in I_k(t)}\\mathcal{R}(k,c,s)}{\\inf_{s\\in I_k(t)}\\mathcal{R}(k,c,s)}\\in[1,\\infty]"
    },
    {
      "id": "ns.c5.c5k.ascent_clock_harmonic_trichotomy",
      "latex": "\\mathrm{K\\text{-}HARM}\\vee\\mathrm{K\\text{-}CLOCK}\\vee\\mathrm{K\\text{-}SETUP}",
      "series": "NS",
      "first_appearance": "C5-K",
      "label_zh": "強上升–時鐘–諧和三分法",
      "label_en": "strong ascent–clock–harmonic trichotomy",
      "definition_zh": "第 45 節的 C5-K.9 斷言：若區塊根上升強到超過下降因子連乘的倒數，則至少發生窗內諧和通過、鏈時鐘分離、或共同定理設置不可用三者之一。",
      "definition_en": "Section 45 (C5-K.9) asserts that if block root ascent exceeds the reciprocal product of descent factors, then at least one of window-harmonic pass, chain-clock separation, or unavailable common theorem setup must hold.",
      "notes": "Upgrades C5-K.6 (clock-synchronized Type-A puncture) by including the setup gate as a third alternative."
    },
    {
      "id": "ns.c5.c5l.a_k",
      "latex": "A_k(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "高階振幅",
      "label_en": "derivative amplitude",
      "definition_zh": "速度場第 k 階空間導數在時刻 t 的 L^∞ 振幅，於 §2 引入。",
      "definition_en": "The L^∞ amplitude of the k-th spatial derivative of the velocity at time t, introduced in §2.",
      "defining_relation": "A_k(t)=\\|D^ku(t)\\|_\\infty",
      "notes": "Carries over from C5-I/K and Grujić–Xu; logarithmic turnover of this quantity is the PDE object compressed in C5-L.4."
    },
    {
      "id": "ns.c5.c5l.tau_k",
      "latex": "\\tau_k(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "鏈時鐘",
      "label_en": "chain clock",
      "definition_zh": "附著於定理對 (k,t) 的鏈時鐘時間尺度，於 §2 定義為 \\widetilde{\\mathcal C}_k A_k(t)^{2/(k+1)} 之倒數。",
      "definition_en": "The chain-clock time scale attached to the theorem pair (k,t), defined in §2 as the reciprocal of \\widetilde{\\mathcal C}_k A_k(t)^{2/(k+1)}.",
      "defining_relation": "\\tau_k(t)=\\frac{1}{\\widetilde{\\mathcal C}_k A_k(t)^{2/(k+1)}}",
      "notes": "Published Grujić–Xu Theorem 3.14 clock, already audited as order-dependent in C5-K; C5-L compresses mismatches of these clocks into order-space total variation."
    },
    {
      "id": "ns.c5.c5l.i_k",
      "latex": "I_k(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "定理視窗",
      "label_en": "theorem window",
      "definition_zh": "定理對 (k,t) 的 Grujić–Xu 可容許時間區間 [t+\\tau_k/4,t+\\tau_k]，於 §2 定義。",
      "definition_en": "The Grujić–Xu admissible time interval [t+\\tau_k/4, t+\\tau_k] for the pair (k,t), defined in §2.",
      "defining_relation": "I_k(t)=[t+\\tau_k/4,\\,t+\\tau_k]",
      "notes": "Existential quantifier of Theorem 3.14; C5-L treats failure as persistent over the whole window rather than as a timing mismatch."
    },
    {
      "id": "ns.c5.c5l.w_k",
      "latex": "\\mathsf W_k(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "視窗失敗指示",
      "label_en": "window-failure indicator",
      "definition_zh": "視窗成敗指示量，\\mathsf W_k(t)=0 表示 I_k(t) 內無一刻滿足定理所需的一維符號稀疏性，於 §2 引入。",
      "definition_en": "The window-success indicator, with \\mathsf W_k(t)=0 meaning no instant in I_k(t) meets the theorem-required 1D sign sparseness, introduced in §2.",
      "notes": "The C5-K Window-Persistent Sign Defect, rewritten in this round as a carrier-free descent/load debt (L-WINDOW, as opposed to L-SETUP)."
    },
    {
      "id": "ns.c5.c5l.bad_carrier_set",
      "latex": "\\mathcal B_k(s)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "壞載體集",
      "label_en": "bad-carrier set",
      "definition_zh": "時刻 s 上所選分量/符號對所有可許容直線與尺度測試皆失敗的空間點集，於 §3 定義。",
      "definition_en": "The set of points at time s where the selected component/sign fails every admissible line/scale test, defined in §3.",
      "notes": "No continuous selection s\\mapsto x_k(s) is assumed; C5-L.1 quotients the carrier identity out of the amplitude-chain descent strip."
    },
    {
      "id": "ns.c5.c5l.r_k",
      "latex": "r_k(s)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "鏈尺度",
      "label_en": "chain scale",
      "definition_zh": "由當下振幅決定的瞬時鏈空間尺度，於 §4 定義。",
      "definition_en": "The instantaneous chain spatial scale determined by the current amplitude, defined in §4.",
      "defining_relation": "r_k(s)=\\frac{1}{2\\widetilde{\\mathcal C}_k A_k(s)^{1/(k+1)}}"
    },
    {
      "id": "ns.c5.c5l.kappa_lambda_delta",
      "latex": "\\kappa_{\\lambda,\\delta}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "下降係數",
      "label_en": "descent coefficient",
      "definition_zh": "調和閾值 \\delta 處均勻嚴格正的振幅下降係數，於 §4 定義並於 §10 證明臨界飽和時仍不消失。",
      "definition_en": "The uniformly strictly positive amplitude-descent coefficient at the harmonic threshold \\delta, defined in §4 and shown in §10 to remain nonvanishing under critical saturation.",
      "defining_relation": "\\kappa_{\\lambda,\\delta}=(1+\\lambda)\\delta-1>0",
      "notes": "Inherited from the C5-I sign-descent lemmas; C5-L.3 uses positivity of this constant to rule out harmonic-temporal critical saturation as a zero-cost boundary."
    },
    {
      "id": "ns.c5.c5l.l_k",
      "latex": "L_k(s)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "導數L2負荷",
      "label_en": "L2 derivative load",
      "definition_zh": "第 k 階導數的全域 L^2 範數，於 §6 由壞核局部 L^2 下界推出並於 §7 積成載體無關的視窗負荷帶。",
      "definition_en": "The global L^2 norm of the k-th derivative, introduced in §6 from the bad-core local L^2 lower bound and integrated in §7 into a carrier-independent window-load strip.",
      "defining_relation": "L_k(s)^2=\\|D^ku(s)\\|_2^2\\ge c_{\\lambda,\\delta}A_k(s)^2 r_k(s)^3"
    },
    {
      "id": "ns.c5.c5l.mu_k_r",
      "latex": "\\mu_k^{R}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "根負荷測度",
      "label_en": "root-load measure",
      "definition_zh": "定理視窗 I_k(t) 上以歸一根量為密度的正測度 \\mathcal R(k,c,s)\\,ds，於 §8 定義。",
      "definition_en": "The positive measure \\mathcal R(k,c,s)\\,ds on the theorem window I_k(t), defined in §8.",
      "defining_relation": "d\\mu_k^{R}(s)=\\mathcal R(k,c,s)\\,ds",
      "notes": "Window-persistent failure upgrades C5-I pointwise descent to the measure domination \\mu_{k-1}^{R}\\ge d_k(c)\\mu_k^{R}, which survives carrier relay and weak time limits (G-WROOTMEAS)."
    },
    {
      "id": "ns.c5.c5l.cal_r_k",
      "latex": "\\mathcal R_k(s)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "歸一根量",
      "label_en": "normalized root",
      "definition_zh": "第 k 階導數振幅經截面常數 c 與階乘正規化後的根量，於 §8 引入並於 §12 寫成對數形式。",
      "definition_en": "The section-and-factorial normalized root of the k-th derivative amplitude, introduced in §8 and rewritten in logarithmic form in §12.",
      "defining_relation": "\\log\\mathcal R_k(s)=\\frac{1}{k+1}\\log A_k(s)+\\mathrm{const}",
      "notes": "C5 chain root; because c and k! are time-independent, within-window log-root variation is exactly a multiple of log-amplitude variation."
    },
    {
      "id": "ns.c5.c5l.d_k",
      "latex": "d_k(c)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "根下降因子",
      "label_en": "root-descent factor",
      "definition_zh": "迫使失敗視窗內相鄰階根量逐點（因而作為測度）下降的正因子，於 §8 引入。",
      "definition_en": "The positive factor forcing adjacent-order roots to descend pointwise, hence as measures, throughout a failing window, introduced in §8.",
      "defining_relation": "\\mu_{k-1}^{R}\\ge d_k(c)\\,\\mu_k^{R}"
    },
    {
      "id": "ns.c5.c5l.beta_k_win",
      "latex": "\\beta_k^{win}(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "視窗調和佔有",
      "label_en": "window harmonic occupancy",
      "definition_zh": "調和佔有量 \\beta_k 在定理視窗 I_k(t) 上的下確界，於 §9 定義。",
      "definition_en": "The infimum of the harmonic occupancy \\beta_k over the theorem window I_k(t), defined in §9.",
      "defining_relation": "\\beta_k^{win}(t)=\\inf_{s\\in I_k(t)}\\beta_k(s)",
      "notes": "C5-J harmonic occupancy restricted to the theorem window; the regime \\beta_k^{win}\\downarrow\\delta is Harmonic-Temporal Critical Saturation, shown in C5-L.3 not to kill the descent coefficient."
    },
    {
      "id": "ns.c5.c5l.turnover_factor",
      "latex": "\\mathfrak T_k^{win}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "根翻轉因子",
      "label_en": "root-turnover factor",
      "definition_zh": "定理視窗內歸一根量的上確界對下確界之比，於 §13 定義，其對數由 \\operatorname{Var}_{I_k}\\log\\mathcal R_k 控制。",
      "definition_en": "The ratio of sup to inf of the normalized root over the theorem window, defined in §13, with logarithm controlled by \\operatorname{Var}_{I_k}\\log\\mathcal R_k.",
      "defining_relation": "\\mathfrak T_k^{win}=\\frac{\\sup_{s\\in I_k(t)}\\mathcal R_k(s)}{\\inf_{s\\in I_k(t)}\\mathcal R_k(s)}\\ge 1",
      "notes": "Generic within-window root turnover is removed as an independent survivor motif once C5-L.4 routes it into viscous or projected-nonlinear tolls."
    },
    {
      "id": "ns.c5.c5l.n_k_proj",
      "latex": "\\mathcal N_k^{proj}(s)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "投影非線性源",
      "label_en": "projected nonlinear source",
      "definition_zh": "Leray 投影非線性項第 k 階空間導數的真實 L^∞ 範數，於 §15 定義，不作 Calderón–Zygmund 替換。",
      "definition_en": "The genuine L^∞ norm of the k-th spatial derivative of the Leray-projected nonlinearity, defined in §15 and not replaced by a Calderón–Zygmund bound.",
      "defining_relation": "\\mathcal N_k^{proj}(s)=\\bigl\\|D^k\\mathbb P\\bigl((u\\cdot\\nabla)u\\bigr)(s)\\bigr\\|_\\infty",
      "notes": "Guard G-PROJNL: keep the actual projected source; C5-L does not claim Grujić–Xu Theorem 3.8 automatically bounds this instantaneous quantity."
    },
    {
      "id": "ns.c5.c5l.v_k_visc",
      "latex": "\\mathfrak V_k^{visc}(I)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "黏性翻轉通行費",
      "label_en": "viscous turnover toll",
      "definition_zh": "在區間 I 上積分 A_{k+2}/A_k 所得的黏性兩階翻轉通行費，於 §17 定義。",
      "definition_en": "The viscous two-order turnover toll obtained by integrating A_{k+2}/A_k over an interval I, defined in §17.",
      "defining_relation": "\\mathfrak V_k^{visc}(I)=\\frac{C_\\Delta\\nu}{k+1}\\int_I\\frac{A_{k+2}(s)}{A_k(s)}\\,ds",
      "notes": "Unboundedness of this toll is the L-VISC alternative D^{k+2}u viscous-order congestion in C5-L.4/L.5."
    },
    {
      "id": "ns.c5.c5l.v_k_nl",
      "latex": "\\mathfrak V_k^{NL}(I)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "非線性翻轉通行費",
      "label_en": "nonlinear turnover toll",
      "definition_zh": "在區間 I 上積分 \\mathcal N_k^{proj}/A_k 所得的投影非線性翻轉通行費，於 §17 定義。",
      "definition_en": "The projected-nonlinear turnover toll obtained by integrating \\mathcal N_k^{proj}/A_k over an interval I, defined in §17.",
      "defining_relation": "\\mathfrak V_k^{NL}(I)=\\frac{1}{k+1}\\int_I\\frac{\\mathcal N_k^{proj}(s)}{A_k(s)}\\,ds",
      "notes": "The L-NL alternative of C5-L.4/L.5; together with \\mathfrak V_k^{visc} it exhausts within-window root turnover."
    },
    {
      "id": "ns.c5.c5l.root_turnover_pde_compression",
      "latex": "\\operatorname{Var}_I\\log\\mathcal R_k\\le\\mathfrak V_k^{visc}(I)+\\mathfrak V_k^{NL}(I)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "根翻轉PDE壓縮",
      "label_en": "Root-Turnover PDE Compression",
      "definition_zh": "C5-L.4 根翻轉 PDE 壓縮定理：任意活躍預奇異區間上對數根的全變差由黏性與投影非線性通行費之和控制，於 §18 陳述。",
      "definition_en": "C5-L.4 Root-Turnover PDE Compression Theorem: on any active pre-singular interval the total variation of the log-root is bounded by the sum of the viscous and projected-nonlinear tolls, stated in §18.",
      "defining_relation": "\\operatorname{Var}_I\\log\\mathcal R_k\\le\\mathfrak V_k^{visc}(I)+\\mathfrak V_k^{NL}(I)"
    },
    {
      "id": "ns.c5.c5l.y_root",
      "latex": "y_n(\\theta)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "中心化根路徑",
      "label_en": "centered root path",
      "definition_zh": "將定理區間正規化到 [0,1] 後、以視窗起點為原點的對數根路徑，於 §20 定義，在有界通行費下 BV 緊緻。",
      "definition_en": "The log-root path on a theorem interval time-normalized to [0,1] and centered at the window start, defined in §20 and BV-compact under bounded turnover tolls.",
      "notes": "C5-L.5 alternative L-RCOMP: bounded tolls yield BV-compact normalized root paths in L^1([0,1]); otherwise L-RFORCE forcing congestion."
    },
    {
      "id": "ns.c5.c5l.c_k_clock",
      "latex": "c_k^{clock}(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "對數鏈時鐘",
      "label_en": "logarithmic chain clock",
      "definition_zh": "固定基底時刻 t 處鏈時鐘的對數，於 §23 定義，並於 §26 精確分解為定理正規化減兩倍對數根。",
      "definition_en": "The logarithm of the chain clock at a fixed base time t, defined in §23 and decomposed in §26 as theorem normalization minus twice the log-root.",
      "defining_relation": "c_k^{clock}=g_k^{th}-2\\log\\mathcal R_k",
      "notes": "Exact clock/root decomposition of §26–§28: chain-clock separation is order-root variation plus theorem/factorial normalization drift, not an independent temporal coordinate."
    },
    {
      "id": "ns.c5.c5l.v_clock",
      "latex": "\\mathfrak V_{J,K}^{clock}(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "時鐘全變差",
      "label_en": "clock total variation",
      "definition_zh": "導數區塊 [J,K] 上相鄰對數鏈時鐘增量絕對值之和，於 §23 定義為階空間變差預算。",
      "definition_en": "The sum of absolute adjacent increments of the logarithmic chain clocks over the derivative block [J,K], defined in §23 as an order-space variation budget.",
      "defining_relation": "\\mathfrak V_{J,K}^{clock}(t)=\\sum_{k=J}^{K-1}|\\log\\tau_{k+1}(t)-\\log\\tau_k(t)|",
      "notes": "C5-K chain-clock separation compressed into a finite order-space variation; the threshold \\log 4 is the factor-4 common-time criterion of C5-K."
    },
    {
      "id": "ns.c5.c5l.g_k_th",
      "latex": "g_k^{th}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "定理時鐘正規化",
      "label_en": "theorem-clock normalization",
      "definition_zh": "由定理常數 \\widetilde{\\mathcal C}_k、截面 c 與階乘 k! 組成的決定性時鐘漂移項，於 §26 定義。",
      "definition_en": "The deterministic clock-drift term assembled from the theorem constants \\widetilde{\\mathcal C}_k, the section c, and the factorial k!, defined in §26.",
      "defining_relation": "g_k^{th}=-\\log\\widetilde{\\mathcal C}_k-\\frac{2k}{k+1}\\log c-\\frac{2}{k+1}\\log(k!)",
      "notes": "Guard G-CLOCKTH: theorem/factorial drift \\mathfrak V_{J,K}^{th}=\\sum|\\Delta g_k^{th}| must be separated from genuine root-order variation \\mathfrak V_{J,K}^{root}."
    },
    {
      "id": "ns.c5.c5l.mu_clock",
      "latex": "\\mu_{J,K}^{clock}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "時鐘缺陷測度",
      "label_en": "clock defect measure",
      "definition_zh": "將時鐘全變差正規化為正規化階坐標 [0,1] 上的機率測度，於 §29 定義。",
      "definition_en": "The probability measure on the normalized order coordinate [0,1] obtained by normalizing the clock total variation, defined in §29.",
      "defining_relation": "\\mu_{J,K}^{clock}=\\frac{1}{\\mathfrak V_{J,K}^{clock}}\\sum_{k=J}^{K-1}|\\Delta c_k^{clock}|\\,\\delta_{\\theta_k}",
      "notes": "C5-L.7: recurrent blocks yield subsequential compactness of (a^{clock},\\mu^{clock}) in [0,1]\\times\\mathcal P([0,1]), so clock separation is an order-space defect measure."
    },
    {
      "id": "ns.c5.c5l.a_clock",
      "latex": "a_{J,K}^{clock}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "時鐘振幅緊化",
      "label_en": "compactified clock amplitude",
      "definition_zh": "將時鐘全變差映入 [0,1] 的緊化振幅，於 §29 定義，零變差時改用固定墓地測度。",
      "definition_en": "The compactification of clock total variation into [0,1], defined in §29, with a fixed cemetery measure used when the variation vanishes.",
      "defining_relation": "a_{J,K}^{clock}=\\frac{\\mathfrak V_{J,K}^{clock}}{1+\\mathfrak V_{J,K}^{clock}}\\in[0,1]"
    },
    {
      "id": "ns.c5.c5l.n_sync",
      "latex": "N_{sync}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "同步簇數",
      "label_en": "synchronized-cluster count",
      "definition_zh": "將導數區塊貪婪切成時鐘極差不超過 \\log 4 的相鄰同步簇後所得的簇數，於 §31–§32 引入。",
      "definition_en": "The number of greedy contiguous order clusters whose clock range stays at most \\log 4, introduced in §31–§32.",
      "defining_relation": "N_{sync}\\le 1+\\frac{\\mathfrak V_{J,K}^{clock}}{\\log 4}",
      "notes": "C5-L.8 Clock-Cluster Packing Lemma: each extra completed cluster consumes more than \\log 4 of clock variation, so bounded clock TV yields only finitely many same-time C5-I/J clusters."
    },
    {
      "id": "ns.c5.c5l.theta_k_l",
      "latex": "\\Theta_k^{L}(t)",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "統一視窗缺陷態",
      "label_en": "unified window defect state",
      "definition_zh": "彙集視窗調和佔有、根負荷測度、黏性/非線性通行費、根路徑與視窗指示量的 C5-L 合法定理對缺陷態，於 §45 定義。",
      "definition_en": "The C5-L defect state of a legal theorem pair, assembling window harmonic occupancy, root-load measures, viscous/nonlinear tolls, the root path, and the window indicator, defined in §45.",
      "defining_relation": "\\Theta_k^{L}(t)=\\bigl\\langle\\beta_k^{win},\\mu_{k-1}^{R},\\mu_k^{R},\\mathfrak V_k^{visc},\\mathfrak V_k^{NL},y_k^{root},\\mathsf W_k\\bigr\\rangle",
      "notes": "For an order block one adjoins (a^{clock},\\mu^{clock},N_{sync}); this finite family of compact defect objects is the intended input to C5-M."
    },
    {
      "id": "ns.c5.c5l.carrier_relay_quotient",
      "latex": "A_{k-1}(s)\\ge\\frac{\\kappa_{\\lambda,\\delta}}{2\\widetilde{\\mathcal C}_k}A_k(s)^{k/(k+1)}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "載體中繼商定理",
      "label_en": "Carrier-Relay Quotient Theorem",
      "definition_zh": "C5-L.1 載體中繼商定理：視窗內任意壞載體選取仍給出不含載體座標的同一振幅下降帶，於 §4 陳述。",
      "definition_en": "C5-L.1 Carrier-Relay Quotient Theorem: an arbitrary choice of bad carrier throughout the window still yields the same carrier-free amplitude descent strip, stated in §4.",
      "defining_relation": "A_{k-1}(s)\\ge\\frac{\\kappa_{\\lambda,\\delta}}{2\\widetilde{\\mathcal C}_k}A_k(s)^{k/(k+1)}\\qquad\\forall s\\in I_k(t)",
      "notes": "Removes Carrier Relay as an independent high-order motif at the amplitude/load level (G-CARRIERQ); carrier motion remains relevant only for local pressure/ancestry geometry."
    },
    {
      "id": "ns.c5.c5l.clock_variation_sync",
      "latex": "\\mathfrak V_{J,K}^{clock}(t)\\le\\log 4\\ \\Rightarrow\\ \\bigcap_{k=J}^{K}I_k(t)\\ne\\varnothing",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "時鐘變差同步準則",
      "label_en": "Clock-Variation Synchronization Criterion",
      "definition_zh": "C5-L.6 時鐘變差同步準則：區塊時鐘全變差不超過 \\log 4 則必有共同定理時刻，於 §25 陳述。",
      "definition_en": "C5-L.6 Clock-Variation Synchronization Criterion: clock total variation at most \\log 4 forces a nonempty common theorem time, stated in §25.",
      "defining_relation": "\\bigcap_{k=J}^{K}I_k(t)=\\varnothing\\ \\Rightarrow\\ \\mathfrak V_{J,K}^{clock}(t)>\\log 4",
      "notes": "Contrapositive of the factor-4 common-time lemma of C5-K: absence of a common theorem time is a strictly positive order-clock variation debt, not an unstructured timing defect."
    },
    {
      "id": "ns.c5.c5l.persistent_window_compression",
      "latex": "\\mathsf W_k(t)=0\\ \\Rightarrow\\ \\text{descent/load strip}+(\\text{BV root}\\vee\\text{viscous}\\vee\\text{projected-NL})",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "持續視窗壓縮",
      "label_en": "Persistent Window Compression",
      "definition_zh": "C5-L.9 持續視窗壓縮定理：合法定理對若在整個 I_k(t) 上空間閘門失敗，則必攜帶非消失下降帶與導數 L^2 負荷帶，並在 BV 緊根路徑、黏性 k+2 擁塞、投影非線性擁塞三者中擇一，於 §47 陳述。",
      "definition_en": "C5-L.9 Persistent Window Compression: legal window-wide spatial-gate failure necessarily carries a nonvanishing descent strip and an L^2 load strip, plus one of BV-compact root path, viscous k+2 congestion, or projected-nonlinear congestion, stated in §47.",
      "notes": "Headline compression of this round; remaining independent high-order interfaces are L-R1–L-R5 of §51, to be audited in C5-M."
    },
    {
      "id": "ns.c5.c5l.clock_descent_block_alternative",
      "latex": "\\text{L-BSYNC}\\vee\\text{L-BCONG}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "時鐘下降區塊擇一",
      "label_en": "Clock/Descent Block Alternative",
      "definition_zh": "C5-L.10 時鐘/下降區塊擇一定理：全程視窗持續失敗的合法同基底導數區塊 [J,K] 要麼可分成至多 1+\\mathfrak V^{clock}/\\log 4 個同步簇（簇內 C5-I/J 同時約束全部適用），要麼時鐘全變差本身很大並由時鐘缺陷測度記錄，於 §48 陳述。",
      "definition_en": "C5-L.10 Clock/Descent Block Alternative: a legal same-base block [J,K] with window-persistent sign defect at every level either partitions into at most 1+\\mathfrak V^{clock}/\\log 4 synchronized clusters (inside which all C5-I/J same-time constraints apply) or else records large clock variation by a clock defect measure, stated in §48."
    },
    {
      "id": "ns.c5.c5l.theorem_setup_defect",
      "latex": "\\textbf{Theorem-Setup Defect}",
      "series": "NS",
      "first_appearance": "C5-L",
      "label_zh": "定理設置缺陷",
      "label_en": "Theorem-Setup Defect",
      "definition_zh": "定理對 (k,t) 並未合法落入 Grujić–Xu Theorem 3.14 前提（升鏈、參數關係、剩餘時間、常數等）的殘差，於 §40–§41 與 L-SETUP 重新劃定，且不得與空間符號失敗 L-WINDOW 合併。",
      "definition_en": "The residual that the pair (k,t) is not legally inside the antecedents of Grujić–Xu Theorem 3.14 (ascending-chain, parameter relation, remaining time, constants), restated in §40–§41 as L-SETUP and forbidden from being merged with spatial-sign failure L-WINDOW.",
      "notes": "Retained from C5-K; C5-L explicitly does not prove that every C5 record event satisfies every Theorem 3.14 antecedent, so setup remains an external gate (G-SETUP2) among the five true remaining interfaces of §51."
    },
    {
      "id": "ns.c5.c5m.residual_class_a",
      "latex": "\\mathsf{A}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "殘差類 A（合法性／譜系）",
      "label_en": "Residual class A",
      "definition_zh": "第 3 節將殘差類 A 定義為涵蓋 UV 譜系合法性、最終局部源主導、合法父路由與 Grujić–Xu 定理入場／參數／剩餘時間前置條件的證明／定理入場合法性缺陷，而非物理奇異機制。",
      "definition_en": "Section 3 defines residual class A as the proof/theorem-entry legality defect covering UV ancestry legality, eventual local-source dominance, legal parent routing, and Grujić–Xu theorem-setup antecedents, rather than a physical singularity mechanism.",
      "notes": "Carries C3-G/C4 UV-ancestry legality into the C5 residual alphabet; Section 23 and the X-integration guards forbid treating A as a physical SCC."
    },
    {
      "id": "ns.c5.c5m.residual_class_t",
      "latex": "\\mathsf{T}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "殘差類 T（時間相位）",
      "label_en": "Residual class T",
      "definition_zh": "第 4 節將殘差類 T 定義為由 Young 相位振盪、時間荷載集中與分離純量補償循環組成的時間相位缺陷類。",
      "definition_en": "Section 4 defines residual class T as the temporal-phase defect comprising Young phase oscillation, temporal load concentration, and separated scalar compensation cycles.",
      "notes": "Compactified from C5-B/C; Section 26 records it as compact but not eliminated, and it is C6 likely target I as a possibly isolated SCC."
    },
    {
      "id": "ns.c5.c5m.residual_class_g",
      "latex": "\\mathsf{G}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "殘差類 G（場幾何）",
      "label_en": "Residual class G",
      "definition_zh": "第 5 節將殘差類 G 定義為由中間間隙缺陷、應變方向／壓縮軸散布、導數應變漲落、渦度主導洩漏與三次應變間歇／活躍體積塌縮組成的場幾何退化類。",
      "definition_en": "Section 5 defines residual class G as the field-geometry degeneration comprising middle-gap defects, strain-direction/compressive-axis dispersion, derivative-strain fluctuation, vorticity-dominant leakage, and cubic-strain intermittency/active-volume collapse.",
      "notes": "Absorbs free Seven-Point Q-cancellation from C5-D/E; Section 27 records G as highly compressed but not eliminated."
    },
    {
      "id": "ns.c5.c5m.residual_class_p",
      "latex": "\\mathsf{P}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "殘差類 P（壓力補償）",
      "label_en": "Residual class P",
      "definition_zh": "第 6 節將殘差類 P 定義為由平均旋轉補償、壓力集中、遠場壓力符號 (-,+,+)／(-,-,+)、行列式零符號邊界、壓力源碎裂／週轉與壓縮軸壓力鎖定組成的壓力補償／來源類。",
      "definition_en": "Section 6 defines residual class P as the pressure compensation/provenance class comprising mean-rotation compensation, pressure concentration, far-pressure signatures (-,+,+) / (-,-,+), det-zero signature boundary, pressure-source fragmentation/turnover, and compressive-axis pressure locking.",
      "notes": "C5-D/F supply finite-dimensional incompatibilities but do not kill all provenance routes; Section 28 records P as geometrically constrained but not eliminated."
    },
    {
      "id": "ns.c5.c5m.residual_class_h",
      "latex": "\\mathsf{H}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "殘差類 H（調和／定理窗）",
      "label_en": "Residual class H",
      "definition_zh": "第 7 節將殘差類 H 定義為由固定 k 階直接閘門失敗、窗持續符號缺陷、調和－時間臨界飽和、持續壞導數簇、以及定理設置合法但調和窗永不通過所組成的高階調和／定理窗缺陷類。",
      "definition_en": "Section 7 defines residual class H as the high-order harmonic/theorem-window defect comprising fixed-k direct-gate failure, Window-Persistent Sign Defect, harmonic-temporal critical saturation, persistent bad derivative clusters, and legal theorem setups whose harmonic window never passes.",
      "notes": "Absorbs C5-G/I/J/K/L theorem-window motifs after deleting SHELLFULL, COMPSIGN, generic Type switching, and line fragmentation as independent nodes."
    },
    {
      "id": "ns.c5.c5m.residual_class_f",
      "latex": "\\mathsf{F}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "殘差類 F（強迫／階債務）",
      "label_en": "Residual class F",
      "definition_zh": "第 8 節將殘差類 F 定義為由黏性 D^{k+2}u 週轉通行費、投影非線性週轉、階曲率、鏈時鐘變動、根階變動與定理／階乘正規化漂移組成的強迫／階變動債務類。",
      "definition_en": "Section 8 defines residual class F as the forcing/order-variation debt comprising viscous D^{k+2}u turnover toll, projected nonlinear turnover, order curvature, chain-clock variation, root-order variation, and theorem/factorial normalization drift.",
      "notes": "Generic root turnover and generic clock mismatch from C5-L are deleted as independent motifs and routed into F; pairs with H as C6 likely target III."
    },
    {
      "id": "ns.c5.c5m.residual_alphabet",
      "latex": "\\mathfrak{D}_{C5}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "C5 六類殘差字母表",
      "label_en": "C5 residual alphabet",
      "definition_zh": "第 9 節將 C5 最終殘差字母表定義為六個殘差類組成的有限集合 {A, T, G, P, H, F}。",
      "definition_en": "Section 9 defines the C5 residual alphabet as the finite six-class set {A, T, G, P, H, F}.",
      "defining_relation": "\\mathfrak{D}_{C5}=\\{\\mathsf{A},\\mathsf{T},\\mathsf{G},\\mathsf{P},\\mathsf{H},\\mathsf{F}\\}",
      "notes": "Re-certified as structurally closed by the Six-Class Closure Theorem (Section 31); C6 works on cycles in this alphabet rather than adding a seventh class."
    },
    {
      "id": "ns.c5.c5m.compatibility_graph",
      "latex": "\\mathcal{G}_{C5}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "C5 認證相容圖",
      "label_en": "C5 compatibility graph",
      "definition_zh": "第 11 節將認證相容圖定義為以六類殘差加上正則頂點 REG 為頂、並以 U/C/E 標記邊的有向圖。",
      "definition_en": "Section 11 defines the certified compatibility graph as the directed graph on the six residual classes plus the regularity vertex REG, with edges marked U/C/E.",
      "defining_relation": "\\mathcal{G}_{C5}=(V,E)",
      "notes": "C6-A's first obligation is an edge-completeness audit of this graph; SCC extraction is only certified when the representation is complete for the survivor path."
    },
    {
      "id": "ns.c5.c5m.graph_vertex_set",
      "latex": "V",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "相容圖頂點集",
      "label_en": "Graph vertex set",
      "definition_zh": "第 11 節將相容圖頂點集定義為六個殘差類與正則頂點 REG 的聯集。",
      "definition_en": "Section 11 defines the compatibility-graph vertex set as the six residual classes together with the regularity vertex REG.",
      "defining_relation": "V=\\{\\mathsf{A},\\mathsf{T},\\mathsf{G},\\mathsf{P},\\mathsf{H},\\mathsf{F},\\mathrm{REG}\\}"
    },
    {
      "id": "ns.c5.c5m.reg_vertex",
      "latex": "\\mathrm{REG}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "正則頂點 REG",
      "label_en": "Regularity vertex",
      "definition_zh": "第 11 節將 REG 定義為相容圖中表示被外部正則性閘門殺死的正則頂點，因而不列為殘差節點。",
      "definition_en": "Section 11 introduces REG as the compatibility-graph vertex representing an external-theorem kill, and therefore not a residual node.",
      "notes": "X-integration guard in Section 42 forbids listing external theorem kill states as residual nodes; H→REG and P→REG are the principal E-type exits."
    },
    {
      "id": "ns.c5.c5m.edge_status",
      "latex": "\\mathrm{U}/\\mathrm{C}/\\mathrm{E}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "邊狀態標記 U/C/E",
      "label_en": "Edge-status labels",
      "definition_zh": "第 11 節將相容圖的每條邊標記為 U（無條件恆等／路由）、C（依賴額外／局部化／譜系閘門的條件邊）或 E（外部定理閉合）。",
      "definition_en": "Section 11 marks each compatibility-graph edge as U (unconditional identity/routing), C (conditional on an extra/localization/ancestry gate), or E (external theorem closure).",
      "notes": "Section 42 requires preserving U/C/E proof status and forbids declaring an SCC from an unproved reverse edge."
    },
    {
      "id": "ns.c5.c5m.miller_qsv",
      "latex": "\\mathcal{Q}_{SV}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "Miller 應變－渦度算子",
      "label_en": "Miller strain-vorticity operator",
      "definition_zh": "第 2 節將 Miller 應變－渦度算子定義為對 (u·∇)S + S² + (3/4) ω⊗ω 的無跡對稱投影 P_st。",
      "definition_en": "Section 2 defines the Miller strain-vorticity operator as the traceless-symmetric projection P_st of (u·∇)S + S² + (3/4) ω⊗ω.",
      "defining_relation": "\\mathcal{Q}_{SV}=P_{st}\\bigl((u\\cdot\\nabla)S+S^{2}+\\tfrac{3}{4}\\omega\\otimes\\omega\\bigr)",
      "notes": "Imported as an external theorem gate from Miller's strain-vorticity work; used alongside the middle-eigenvalue gate rather than as a residual node."
    },
    {
      "id": "ns.c5.c5m.miller_sv_identity",
      "latex": "\\langle-\\Delta S,\\omega\\otimes\\omega\\rangle=0",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "Miller 應變－渦度恆等式",
      "label_en": "Miller strain-vorticity identity",
      "definition_zh": "第 2 節將 Miller 應變－渦度閘門的核心恆等式表述為 ⟨−ΔS, ω⊗ω⟩ = 0。",
      "definition_en": "Section 2 records the core identity of the Miller strain-vorticity operator gate as ⟨−ΔS, ω⊗ω⟩ = 0.",
      "defining_relation": "\\langle-\\Delta S,\\omega\\otimes\\omega\\rangle=0"
    },
    {
      "id": "ns.c5.c5m.miller_middle_eigenvalue_gate",
      "latex": "\\text{Miller middle-eigenvalue gate}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "Miller 中間特徵值閘門",
      "label_en": "Miller middle-eigenvalue gate",
      "definition_zh": "第 2 節將 Miller 中間特徵值閘門表述為：有限時間爆破必須逃出中間應變尺度臨界正則性體制。",
      "definition_en": "Section 2 states the Miller middle-eigenvalue gate as the requirement that a finite-time blow-up must escape the middle-strain scale-critical regularity regime.",
      "notes": "External E-type gate from Miller's middle-eigenvalue criterion; finite-time blow-up must leave this regime before any residual cycle can be physical."
    },
    {
      "id": "ns.c5.c5m.grujic_xu_thm_3_5",
      "latex": "\\text{Gruji\\'{c}--Xu Theorem 3.5}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "Grujić–Xu 固定階閘門",
      "label_en": "Grujić–Xu fixed-order gate",
      "definition_zh": "第 2 節將 Grujić–Xu 定理 3.5 表述為：在定理可接受的後期時刻，若選定的 D^k u 或 D^k ω 分量／符號超水平集在定理尺度上為一維稀疏，則 T* 不是爆破時刻。",
      "definition_en": "Section 2 records Grujić–Xu Theorem 3.5: on a theorem-admissible later time, if a selected D^k u or D^k ω component/sign superlevel set is 1D-sparse at theorem scale, then T* is not a blow-up time.",
      "notes": "Principal H→REG edge of type E; prepared as a theorem-ready interface in C5-G, with antecedents still tracked by class A."
    },
    {
      "id": "ns.c5.c5m.grujic_xu_thm_3_14",
      "latex": "\\text{Gruji\\'{c}--Xu Theorem 3.14}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "Grujić–Xu 鏈閘門",
      "label_en": "Grujić–Xu chain gate",
      "definition_zh": "第 2 節將 Grujić–Xu 定理 3.14 表述為：在導數鏈設置、時間窗與分量／符號幾何等假設成立時，動態插值／調和測度機制排除爆破。",
      "definition_en": "Section 2 records Grujić–Xu Theorem 3.14: under the derivative-chain, time-window, and component/sign-geometry hypotheses, dynamic interpolation/harmonic-measure machinery rules out blow-up.",
      "notes": "Second H→REG edge of type E; C5-K audited the dynamic interpolation/switch interface, while remaining-time antecedents stay in class A."
    },
    {
      "id": "ns.c5.c5m.q_cancellation_routing",
      "latex": "Q\\text{-cancellation}\\Rightarrow\\mathsf{G}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "Q 相消路由至 G",
      "label_en": "Q-cancellation routing",
      "definition_zh": "第 5 節將自由七點 Q 相消從獨立節點刪除，並強制路由為 Q-cancellation ⇒ G。",
      "definition_en": "Section 5 deletes free Seven-Point Q-cancellation as an independent node and routes it by the implication Q-cancellation ⇒ G.",
      "defining_relation": "Q\\text{-cancellation}\\Rightarrow\\mathsf{G}",
      "notes": "Pseudo-defect deletion item 1 in Section 10; the C4/C5-D free Q-cancellation motif is no longer an independent residual class."
    },
    {
      "id": "ns.c5.c5m.turnover_clock_routing",
      "latex": "\\text{TURNOVER/CLOCK}\\Rightarrow\\mathsf{F}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "週轉／時鐘路由至 F",
      "label_en": "Turnover/clock routing",
      "definition_zh": "第 8 節將泛型根週轉與泛型時鐘失配從獨立節點刪除，並強制路由為 TURNOVER/CLOCK ⇒ F。",
      "definition_en": "Section 8 deletes generic root turnover and generic clock mismatch as independent nodes and routes them by the implication TURNOVER/CLOCK ⇒ F.",
      "defining_relation": "\\text{TURNOVER/CLOCK}\\Rightarrow\\mathsf{F}",
      "notes": "Pseudo-defect deletion items 4–5 in Section 10, compressing C5-L turnover/clock motifs into residual class F."
    },
    {
      "id": "ns.c5.c5m.finite_recurrence_principle",
      "latex": "D_1,D_2,\\ldots\\in\\mathfrak{D}_{C5}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "有限復現原理",
      "label_en": "Finite recurrence principle",
      "definition_zh": "第 18 節證明：任意無限假設倖存標籤序列因殘差字母表有限，必有某個殘差類出現無限多次，故每個無限倖存者都有復現殘差類。",
      "definition_en": "Section 18 proves that any infinite hypothetical survivor label sequence in the finite residual alphabet has some class occurring infinitely often, hence every infinite survivor has a recurrent residual class.",
      "defining_relation": "D_1,D_2,\\ldots,\\qquad D_n\\in\\mathfrak{D}_{C5}",
      "notes": "Graph-theoretic, not PDE-theoretic; Section 25 stresses that a finite defect graph need not imply global regularity."
    },
    {
      "id": "ns.c5.c5m.sink_scc",
      "latex": "\\text{sink SCC}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "匯點強連通分量",
      "label_en": "Sink SCC",
      "definition_zh": "第 19 節在認證路由圖對該倖存路徑完備時，證明其 SCC 縮合圖有限且無環，故無限路徑經暫態後必在某匯點 SCC 中復現。",
      "definition_en": "Section 19 proves that if the certified routing graph is complete for a survivor path, then its SCC condensation is finite and acyclic, so an infinite path is eventually recurrent in some sink SCC.",
      "notes": "Section 45 records the sink-SCC principle as proved only conditionally on graph completeness; uniqueness of a sink SCC is not proved, and C6's true object is the recurrent sink SCC rather than a new isolated defect node."
    },
    {
      "id": "ns.c5.c5m.finite_defect_recurrence_theorem",
      "latex": "\\text{Finite Defect Recurrence Theorem}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "有限缺陷復現定理",
      "label_en": "Finite Defect Recurrence Theorem",
      "definition_zh": "第 24 節將有限缺陷復現定理陳述為：在當前認證 C5 圖內，任何避開 REG 的無限假設倖存路徑都有復現殘差子序列，且在圖表示完備時最終復現支撐於某匯點 SCC。",
      "definition_en": "Section 24 states the Finite Defect Recurrence Theorem: every infinite hypothetical survivor path avoiding REG in the certified C5 graph has a recurrent residual subsequence, and if the graph representation is complete then eventual recurrence is supported on a sink SCC.",
      "notes": "Boxed status is PROVED AS FINITE GRAPH THEORY; graph completeness itself remains a C6 edge-audit problem."
    },
    {
      "id": "ns.c5.c5m.six_class_closure_theorem",
      "latex": "\\text{Six-Class Closure Theorem}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "六類閉包定理",
      "label_en": "Six-Class Closure Theorem",
      "definition_zh": "第 31 節將六類閉包定理陳述為：在當前 C3/C4/C5 守衛與條件譜系框架下，所有 C5-A–L 遭遇的復現倖存狀態都可編碼為六類殘差字母表加各類緊緻元數據，無需第七個獨立殘差類。",
      "definition_en": "Section 31 states the Six-Class Closure Theorem: under current C3/C4/C5 guards and the conditional ancestry framework, every C5-A–L recurrent survivor state encodes into the six-class residual alphabet plus compact metadata, with no seventh independent residual class required.",
      "notes": "This is research-program closure of C5 defect-state classification, not PDE proof closure of Navier–Stokes regularity."
    },
    {
      "id": "ns.c5.c5m.c5_phase_closure_theorem",
      "latex": "\\text{C5 Phase Closure Theorem}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "C5 相位閉包定理",
      "label_en": "C5 Phase Closure Theorem",
      "definition_zh": "第 33 節將 C5 相位閉包定理陳述為：C5-A–L 已為所有復現母題族提供緊緻狀態表示並把主要偽缺陷路由到有限殘差字母表或外部正則閘門，故研究相位關閉而整體正則性仍開放。",
      "definition_en": "Section 33 states the C5 Phase Closure Theorem: C5-A–L supplied compact state representations for all recurrent motif families and routed the main pseudo-defects into the finite residual alphabet or external regularity gates, so the research phase is closed while global regularity remains open.",
      "notes": "Formal status PHASE CLOSED versus PDE status GLOBAL REGULARITY OPEN; NG-M5 forbids equating C5 phase closure with Millennium-problem closure."
    },
    {
      "id": "ns.c5.c5m.hf_forcing_loop",
      "latex": "\\mathsf{H}\\longrightarrow\\mathsf{F}\\longrightarrow\\mathsf{H}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "高階強迫迴路 H→F→H",
      "label_en": "High-order forcing loop",
      "definition_zh": "第 20 節將候選復現迴路 I 定義為持續壞定理窗經導數下降／荷載支付高階黏性、投影非線性與時鐘債務後把活動推到更高導數層並再次產生壞窗的 H→F→H 迴路。",
      "definition_en": "Section 20 defines candidate recurrent cycle I as the loop H→F→H in which a persistent bad theorem window pays high-order viscous/projected-nonlinear/clock debt, pushes activity to a higher derivative level, and regenerates a bad window.",
      "defining_relation": "\\mathsf{H}\\longrightarrow\\mathsf{F}\\longrightarrow\\mathsf{H}",
      "notes": "C5 has no finite all-order budget excluding this loop; Section 39 flags H↔F as the hardest likely sink-SCC candidate for C6."
    },
    {
      "id": "ns.c5.c5m.gp_loop",
      "latex": "\\mathsf{G}\\leftrightarrow\\mathsf{P}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "幾何－壓力迴路 G↔P",
      "label_en": "Geometry–pressure loop",
      "definition_zh": "第 21 節將候選復現迴路 II 定義為由強中間相干逼出壓力／平均旋轉、壓力軸／符號回饋應變幾何、以及間隙塌縮／雙負壓力／源碎裂避開有限維障礙所形成的 G↔P 補償迴路。",
      "definition_en": "Section 21 defines candidate recurrent cycle II as the compensation loop G↔P in which strong-middle coherence forces pressure/mean rotation, pressure axis/signature feeds back into strain geometry, and gap collapse/two-negative pressure/source fragmentation can evade finite-dimensional obstructions.",
      "defining_relation": "\\mathsf{G}\\leftrightarrow\\mathsf{P}",
      "notes": "C5 already obtained two finite-dimensional incompatibilities; C6 likely target II is whether the remaining routes must accumulate derivative/high-order debt."
    },
    {
      "id": "ns.c5.c5m.isolated_temporal_scc",
      "latex": "\\mathsf{T}_{\\mathrm{SCC}}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "孤立時間 SCC",
      "label_en": "Isolated temporal SCC",
      "definition_zh": "第 22 節將候選復現類 III 定義為可在純量時間層以振盪、集中與分離補償自我復現的孤立 T 強連通分量。",
      "definition_en": "Section 22 defines candidate recurrent class III as the isolated temporal SCC in which T can self-recur at the scalar temporal layer via oscillation, concentration, and separated compensation.",
      "notes": "Removing this candidate SCC requires a universal T→G/P/H shared-source theorem, which Section 12 records as CONDITIONAL and Section 37 lists as C6 likely target I."
    },
    {
      "id": "ns.c5.c5m.temporal_shared_source_coupling",
      "latex": "\\mathsf{T}\\to\\mathsf{G}/\\mathsf{P}/\\mathsf{H}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "時間共用源耦合",
      "label_en": "Temporal shared-source coupling",
      "definition_zh": "第 22 節將移除孤立時間 SCC 所需的萬有橋接表述為 T→G/P/H 共用源定理，目前仍開放。",
      "definition_en": "Section 22 formulates the universal bridge needed to kill the isolated temporal SCC as a shared-source theorem T→G/P/H, which remains open.",
      "defining_relation": "\\mathsf{T}\\to\\mathsf{G}/\\mathsf{P}/\\mathsf{H}",
      "notes": "Section 12 marks T→G/P as CONDITIONAL on a shared-source/localization bridge; this is the cross-domain coupling C6-A must audit."
    },
    {
      "id": "ns.c5.c5m.window_persistent_sign_defect",
      "latex": "\\text{Window-Persistent Sign Defect}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "窗持續符號缺陷",
      "label_en": "Window-Persistent Sign Defect",
      "definition_zh": "第 7 節將窗持續符號缺陷列為殘差類 H 的組成部分，表示定理設置合法但符號／調和窗在定理尺度上持續不通過。",
      "definition_en": "Section 7 lists the Window-Persistent Sign Defect as a constituent of residual class H: the theorem setup is legal, yet the sign/harmonic window never passes at theorem scale.",
      "notes": "Carried from C5-K/L persistent-window compression into class H after generic Type switching and line fragmentation were deleted as independent motifs."
    },
    {
      "id": "ns.c5.c5m.etn_c5_final",
      "latex": "\\mathfrak{T}^{C5}_{final}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "C5 最終 ETN 狀態",
      "label_en": "C5 final ETN state",
      "definition_zh": "第 43 節將 C5 最終 ETN 狀態定義為殘差類、緊緻元數據、邊狀態、債務向量與外部殺死閘門的五元組。",
      "definition_en": "Section 43 defines the C5 final ETN state as the 5-tuple of residual class, compact metadata, edge status, debt vector, and external kill gates.",
      "defining_relation": "\\mathfrak{T}^{C5}_{final}=(\\text{residual class},\\text{compact metadata},\\text{edge status},\\text{debt vector},\\text{external kill gates})",
      "notes": "True-ETN transition object of this round; compact metadata inventory for each of the six classes is listed in Section 32."
    },
    {
      "id": "ns.c5.c5m.etn_c6",
      "latex": "\\mathfrak{T}^{C6}",
      "series": "NS",
      "first_appearance": "C5-M",
      "label_zh": "C6 ETN 狀態",
      "label_en": "C6 ETN state",
      "definition_zh": "第 43 節將 C6 狀態定義為 SCC、迴路、迴路債務、復現頻率與迴路不相容性的五元組。",
      "definition_en": "Section 43 defines the C6 ETN state as the 5-tuple of SCC, cycle, cycle debt, recurrence frequency, and cycle incompatibility.",
      "defining_relation": "\\mathfrak{T}^{C6}=(\\text{SCC},\\text{cycle},\\text{cycle debt},\\text{recurrence frequency},\\text{cycle incompatibility})",
      "notes": "Target state of proposed C6 / C6-A: cycle extraction, sink-SCC identification, and cross-domain closure rather than construction of a new isolated defect node."
    }
  ]
}