--- title: "Navier–Stokes Forest Coercive Budget Program 05：Long-Age Observability、Sharp Half-Exponent Window Threshold、Combined Anti-Phantom Detection 與 Paid-Side Recurrence" short_title: "NS-FCBP 05" series: "Navier–Stokes Forest Coercive Budget Program" cycle: "VI" version: "v0.1" date: "2026-08-16" author: "Neo.K / EveMissLab" language: "zh-TW" status: "Temporal criticality / combined-observability compiler / paid-side audit" epistemic_status: "Proves that a direct universal bridge from fresh vorticity source to one signed kinetic-energy-work channel is impossible: the Navier-Stokes nonlinearity is globally energy-orthogonal while its curl can be nonzero. Replaces the single-channel CRW target by combined pressure/flux/positive-energy/adjoint-trace/model-cone observability. Proves a sharp sequence theorem for finite-time horizon schedules: if r_k decreases and sum r_k^2 is finite, with normalized horizon thickness delta_k=1-(r_{k+1}/r_k)^2, then sum r_k delta_k^a is finite for every a>1/2; the exponent a=1/2 is sharp, with explicit logarithmic schedules for which the series diverges. Consequently, for moving-window observability growth M_k less than or comparable to delta_k^{-gamma} and depletion exponent q, gamma q=1/2 is the sharp temporal threshold for schedule-based depletion effectiveness: gamma q<1/2 admits power schedules, gamma q=1/2 admits logarithmic borderline schedules, while gamma q>1/2 cannot be forced effective from this growth law alone on any finite-time monotone schedule. Integrates Tao's quantitative L^3 back-propagation as an external strong-critical Long-Age State Propagation module with parabolic scale migration, and recent finite-window anti-phantom/recursive-audit results as conditional combined detector modules. Proves an abstract signed-telescope total-variation no-go showing endpoint-energy control alone cannot bound weighted backscatter. The remaining Horizon Critical Lift is reduced to causal-to-audit realization, combined-window growth at or below the half-exponent threshold, invisible-cascade exclusion, and paid-side sign/leakage control. No Forest Coercive Budget, Finite Forest Obstruction, atomic CN3, or Navier-Stokes regularity is proved." canonical_source: "UTF-8 Markdown" --- # Navier–Stokes Forest Coercive Budget Program 05 # Long-Age Observability、Sharp Half-Exponent Window Threshold、Combined Anti-Phantom Detection 與 Paid-Side Recurrence ## 0. 本文定位 FCBP-04 separated the horizon temporal problem into: $$ \boxed{ \text{LAO} \vee \text{Thin-Window Amplification}. } $$ It also showed that the filter endpoint problem is no longer the principal obstruction. The current questions are: 1. can dangerous information remain observable over many local parabolic ages?; 2. on horizon-aligned thin windows, how fast may observability constants deteriorate while a non-summable depletion schedule remains possible?; 3. can fresh vorticity renewal be forced into one signed pressure--flux work channel?; 4. can the paid-side backscatter/leakage be controlled by endpoint energy alone? The answers produce a sharper temporal phase diagram. --- # 1. Velocity nonlinearity and global energy orthogonality Let: $$ \boxed{ \mathcal N = -\mathbb P\nabla\cdot(u\otimes u). } $$ For smooth divergence-free finite-energy velocity: $$ \boxed{ \langle \mathcal N,u \rangle_{L^2} = 0. } $$ ### Proof Since the Leray projector is self-adjoint and: $$ \mathbb Pu=u, $$ $$ \langle\mathcal N,u\rangle = - \langle \nabla\cdot(u\otimes u), u \rangle. $$ Integrate by parts: $$ = \int u_i u_j \partial_j u_i dx = \frac12 \int u_j \partial_j |u|^2 dx = 0. $$ $\square$ --- # 2. Vorticity forcing may remain nonzero Define: $$ \boxed{ \mathcal G = -\nabla\times\mathcal N. } $$ Consider the explicit Schwartz divergence-free field: $$ \boxed{ u(x,y,z) = \left( -2y e^{-R^2}, 2x e^{-R^2}, 0 \right), \qquad R^2=x^2+y^2+z^2. } $$ A direct computation gives: $$ \boxed{ \mathcal G = \left( 16yz e^{-2R^2}, -16xz e^{-2R^2}, 0 \right), } $$ up to the harmless sign convention used in: $$ \mathcal N. $$ Thus: $$ \boxed{ \mathcal G\not\equiv0 } $$ while the global nonlinear kinetic-energy work is zero. --- # 3. CIV/VI-5.1 — Single Energy-Work Bridge No-Go ## Theorem 3.1 There is no universal coercive estimate of the form: $$ \boxed{ | \langle \mathcal N,u \rangle | \ge c \mathfrak S(\mathcal G) } $$ for all smooth divergence-free finite-energy fields, with: $$ c>0 $$ and any nonnegative source functional: $$ \mathfrak S $$ that is strictly positive on every nonzero: $$ \mathcal G. $$ ### Proof The left-hand side vanishes identically by Section 1. Section 2 gives a field with: $$ \mathcal G\neq0. $$ $\square$ --- # 4. Meaning for CRW The FCBP-04 bridge: $$ \text{fresh causal renewal} \Longrightarrow \text{work} $$ cannot be interpreted as a universal one-channel global kinetic-energy-work inequality. The correct target is a **combined observation hierarchy**. Define: $$ \boxed{ \textbf{CCRW — Combined Causal-Renewal Observability} } $$ to mean that a dangerous fresh-renewal package must become visible in at least one of: - active pressure; - resolved flux; - positive energy/dissipation; - selected adjoint trace; - model-cone excess; - or an explicitly paid residual channel. --- # 5. External combined observability framework Recent finite-window Navier--Stokes work defines a combined observation map: $$ \boxed{ O_W^{comb} = ( O_W^P, O_W^F, O_W^E, O_W^T ), } $$ with active pressure, flux, positive energy, and adjoint-trace channels. The corresponding moving-window branch observability constant is: $$ \boxed{ M_n = \sup_{ 0\neq d\in Y_n^{NS} } \frac{ \|d\|_{Y_n} }{ \mathsf O_n(d) }. } $$ When: $$ M_n<\infty, $$ every normalized NS-realizable defect obeys: $$ \boxed{ \mathsf O_n(\widehat d) \ge M_n^{-1}. } $$ ### Status $$ \boxed{ \mathrm{EXTERNAL/CONDITIONAL\ FRAMEWORK}. } $$ --- # 6. External observable depletion On an extraction set: $$ I=\{n_k\}, $$ the moving-window framework assumes/proves in controlled modules a depletion inequality of the form: $$ \boxed{ \mathscr B_k - \mathscr B_{k+1} \ge c \lambda_{n_k} \mathsf O_{n_k}(\widehat d_{n_k})^q - e_k, } $$ with: $$ \boxed{ \sum_ke_k<\infty. } $$ The moving windows are depletion-effective when: $$ \boxed{ \sum_k \lambda_{n_k} M_{n_k}^{-q} = \infty. } $$ Under the paper's extraction/depletion/finite-window hypotheses, failure of regularity is then reduced to non-effective moving-window observability or an NS-realizable combined-invisible defect cascade. ### Status $$ \boxed{ \mathrm{EXTERNAL/CONDITIONAL}. } $$ --- # 7. Horizon-aligned radius sequence Let: $$ r_k>0 $$ be decreasing with: $$ \boxed{ \sum_k r_k^2 < \infty. } $$ This is the finite-time requirement for a parabolic horizon schedule. Define: $$ \boxed{ d_k = 1- \left( \frac{ r_{k+1} }{ r_k } \right)^2 \in[0,1]. } $$ The horizon-aligned normalized slab thickness in FCBP-04 is: $$ \delta_k = c\,d_k. $$ Since constants do not affect summability, the analysis below uses: $$ d_k. $$ --- # 8. CIV/VI-5.2 — Sharp Half-Exponent Schedule Theorem ## Theorem 8.1 Let: $$ r_k $$ be positive, nonincreasing, and: $$ \sum_kr_k^2<\infty. $$ Then for every: $$ \boxed{ a>\frac12, } $$ $$ \boxed{ \sum_k r_k d_k^a < \infty. } $$ Moreover the exponent: $$ a=\frac12 $$ is sharp: there exist decreasing sequences with: $$ \sum_kr_k^2<\infty $$ but: $$ \boxed{ \sum_k r_k d_k^{1/2} = \infty. } $$ --- # 9. Proof for $a\ge1$ Since: $$ 0\le d_k\le1, $$ $$ d_k^a\le d_k. $$ Write: $$ q_k = r_{k+1}/r_k. $$ Then: $$ d_k = 1-q_k^2 = (1-q_k)(1+q_k) \le 2(1-q_k). $$ Therefore: $$ r_kd_k \le 2(r_k-r_{k+1}). $$ Hence: $$ \boxed{ \sum_kr_kd_k^a \le 2r_0. } $$ --- # 10. Proof for $1/20, $$ the remaining exponential factor is summable. There is at most one block-exit index per level, whose contribution is: $$ O(2^{-m/2}), $$ also summable. Thus: $$ \sum_kr_kd_k^a<\infty. $$ $\square$ --- # 11. Sharpness at $a=1/2$ Choose: $$ \boxed{ r_k = \frac1{ \sqrt{k+2} [ \log(k+2) ]^b }, \qquad \frac12\frac12, } $$ Theorem 8.1 shows that the polynomial growth control: $$ M_k\le C d_k^{-\gamma} $$ alone cannot force the lower-bound series: $$ \sum_k r_kd_k^{a} $$ to diverge on any monotone finite-time schedule. ### Meaning $$ \boxed{ \gamma q=\frac12 } $$ is the sharp schedule-level temporal threshold for this depletion mechanism. $\square$ --- # 14. Comparison with the single-work thin-slab requirement FCBP-04's single pressure--flux work compiler required approximately: $$ \boxed{ \mathcal W_k^+ \gtrsim1 } $$ on thin slabs, or equivalently an inverse-thickness normalized work rate. The combined-observability compiler is weaker. It permits the detector to deteriorate polynomially with window thickness as long as: $$ \boxed{ \gamma q\le\frac12. } $$ Thus: $$ \boxed{ \text{combined observability changes the temporal critical exponent}. } $$ --- # 15. CIV/VI-5.4 — Combined Horizon Depletion Compiler ## Theorem 15.1 Assume a dangerous horizon branch produces NS-realizable normalized defect packages: $$ \widehat d_k $$ with: 1. depletion: $$ \mathscr B_k-\mathscr B_{k+1} \ge c w_k \mathsf O_k(\widehat d_k)^q - e_k; $$ 2.: $$ \sum_ke_k<\infty; $$ 3. observability: $$ \mathsf O_k(\widehat d_k) \ge M_k^{-1}; $$ 4.: $$ M_k \le C d_k^{-\gamma}; $$ 5.: $$ \gamma q \le \frac12; $$ 6. the horizon schedule is chosen from Theorem 13.1. Then the branch cannot persist indefinitely. ### Proof Theorem 13.1 gives: $$ \sum_k w_k M_k^{-q} = \infty. $$ Hence: $$ \sum_k ( \mathscr B_k-\mathscr B_{k+1} ) $$ would diverge after subtracting a summable error, contradicting nonnegativity and finiteness of the initial budget. $\square$ --- # 16. Causal-to-audit realization Theorem 15.1 starts with an NS-realizable normalized defect package. ANP/CFOP starts with a dangerous causal node or a fresh-renewal packet. Define: $$ \boxed{ \textbf{CAR — Causal-to-Audit Realization}. } $$ CAR is a theorem which maps an ANP/CFOP dangerous causal package into the finite-window quotient geometry used by the combined detector while preserving a uniform nontrivial baseline distance modulo explicitly controlled residuals. Current status: $$ \boxed{ \mathrm{OPEN}. } $$ --- # 17. External finite-window anti-phantom module Recent local-to-clean work proves, under synchronized representatives, quotient lifting, component comparisons, residual-ledger closure, detector comparison, chart visibility, and a clean quotient gap: $$ \boxed{ \text{baseline-visible package} \Longrightarrow \text{localized detector} \vee \text{explicit residual ledger}. } $$ A later recursive-audit paper proves finite-chain anti-phantom lower bounds under explicit synchronization/localization/projection/harmonic/gate/detector/chart ledgers. ### FCBP interpretation If CAR supplies a nontrivial baseline defect and the anti-phantom residuals are controlled, then the combined detector cannot disappear for free. ### Status $$ \boxed{ \mathrm{EXTERNAL/CONDITIONAL}. } $$ --- # 18. Anti-phantom moving-window route The combined moving-window paper gives the structural alternative: $$ \boxed{ \text{non-effective moving-window observability} \vee \text{NS-realizable combined-invisible defect cascade} } $$ under its extraction/depletion/hierarchy hypotheses. FCBP-05 refines the first branch: a polynomial window-growth model: $$ M_k \lesssim d_k^{-\gamma} $$ is schedule-compatible with depletion whenever: $$ \boxed{ \gamma q\le1/2. } $$ Thus a surviving non-effective branch must either: - grow faster than this threshold in the relevant schedule class; - be extracted too sparsely; - fail CAR/residual transfer; - or enter the combined-invisible cascade. --- # 19. Long-Age State Propagation in the strong $L^3$ subbranch Tao's quantitative critical: $$ L_t^\infty L_x^3 $$ theory provides an external iterated back-propagation theorem. Assume: $$ \boxed{ \|u\|_{L_t^\infty L_x^3} \le A. } $$ If a terminal point has a critical frequency bubble: $$ |P_{N_0}u(t_0,x_0)| \ge A_1^{-1}N_0, $$ then for every admissible longer time scale: $$ T_1, $$ there exists an earlier bubble: $$ (t_1,x_1,N_1), $$ with: $$ \boxed{ N_1 = A^{O(1)} T_1^{-1/2}, } $$ and: $$ \boxed{ |x_1-x_0| \le A^{O(1)} T_1^{1/2}, } $$ while the critical amplitude threshold is preserved up to the fixed quantitative constants. ### Status $$ \boxed{ \mathrm{EXTERNAL/PROVED} } $$ in the strong: $$ L_t^\infty L_x^3 $$ branch. --- # 20. LAO scale migration Theorem 19 shows that long-age state observability is naturally accompanied by parabolic scale migration: $$ \boxed{ R_{\rm obs} \sim T_1^{1/2}. } $$ If the terminal local scale is: $$ r, $$ and one propagates over: $$ K $$ local parabolic ages: $$ T_1 \sim Kr^2, $$ then: $$ \boxed{ R_{\rm obs} \sim \sqrt K\,r. } $$ Thus LAO should not be formulated as "the same tiny scale remains visible far into the past." The natural state observable coarsens with elapsed parabolic age. --- # 21. LAO safety Tao's theorem assumes a uniform strong: $$ L^3 $$ critical bound. The ANP Type-I branch was formulated at the weak endpoint: $$ L^{3,\infty}. $$ Therefore: $$ \boxed{ \text{strong }L^3\text{ LAO} } $$ cannot be imported as a theorem for the general weak: $$ L^{3,\infty} $$ Type-I branch. It is an external strong-subbranch calibration. --- # 22. State propagation is not work observability Even in the strong: $$ L^3 $$ subbranch, Tao's theorem produces: - a velocity/frequency bubble; - spatial displacement control; - scale migration. It does not prove: $$ \boxed{ \text{active pressure--flux work} } $$ or the FCBP combined finite-window detector is bounded below. Thus LAO-state and combined-work observability remain distinct layers. --- # 23. Signed endpoint budgets do not control backscatter variation Let: $$ B_k\ge0 $$ be bounded endpoint budgets and suppose: $$ \boxed{ B_k-B_{k+1}=a_k. } $$ A bound on: $$ B_k $$ controls the signed partial sums of: $$ a_k. $$ It does not control: $$ \sum_ka_k^-. $$ --- # 24. CIV/VI-5.5 — Signed-Telescope Total-Variation No-Go ## Theorem 24.1 There exist bounded nonnegative: $$ B_k $$ for which: $$ a_k=B_k-B_{k+1} $$ has: $$ \boxed{ \sum_ka_k^-=\infty } $$ and: $$ \boxed{ \sum_ka_k^+=\infty. } $$ ### Proof Take: $$ B_{2m}=1, \qquad B_{2m+1}=0. $$ Then: $$ a_{2m}=1, $$ and: $$ a_{2m+1}=-1. $$ Both positive and negative total variations diverge while: $$ 0\le B_k\le1. $$ $\square$ --- # 25. Meaning for pressure--flux backscatter The pressure--flux endpoint telescope is a signed budget. Endpoint kinetic-energy boundedness alone does not imply finite weighted backscatter variation. Therefore: $$ \boxed{ \text{weighted backscatter closure} } $$ requires an additional: - sign-coherence; - total-variation; - recurrence; - or observable-depletion theorem. It cannot be obtained solely from the endpoint energy telescope. --- # 26. Paid-side recurrence If a horizon branch avoids forward depletion by repeatedly generating large negative combined work: $$ \mathcal W_k^-, $$ this is not a bookkeeping error. It is a recurrent backscatter mechanism. Define: $$ \boxed{ \textbf{BSR — Backscatter Recurrence}. } $$ Likewise persistent localization payment defines: $$ \boxed{ \textbf{LKR — Leakage Recurrence}. } $$ These are paid-side recurrence objects. Neither is universally excluded. --- # 27. Temporal criticality trichotomy After FCBP-05, a persistent dangerous horizon branch must enter at least one of: ### TC-OBS depletion-effective combined observability; ### TC-INVIS an NS-realizable combined-invisible defect cascade; ### TC-PAID backscatter/leakage recurrence or another explicit residual ledger; ### TC-REAL failure of causal-to-audit realization; ### TC-GROW moving-window observability deteriorates beyond the schedule-effective threshold. The first branch is depleting. The remaining branches are the true temporal obstruction classes. --- # 28. Sharp temporal threshold For polynomial thin-window growth: $$ M_k \lesssim d_k^{-\gamma}, $$ and depletion exponent: $$ q, $$ the schedule threshold is: $$ \boxed{ \gamma q = \frac12. } $$ This exponent comes from only two requirements: 1. finite total parabolic horizon time: $$ \sum r_k^2<\infty; $$ 2. divergent moving-window depletion: $$ \sum r_kd_k^{\gamma q} = \infty. $$ Thus it is a genuine temporal geometry threshold, not a detector-specific constant. --- # 29. Relation to logarithmic admissibility The external moving-window framework allows observability constants with more complicated growth: $$ M_n \le C N_n^a \exp(CN_n^b), $$ and defines logarithmically admissible windows by divergence of the corresponding weighted reciprocal series. FCBP-05's half-exponent theorem is complementary: it treats deterioration explicitly as a power of normalized physical window thickness and identifies the exact finite-time horizon threshold. Both approaches reduce temporal observability to a weighted divergent-series problem. --- # 30. Combined Horizon Critical-Lift criterion ## Theorem 30.1 Suppose an ANP/CFOP horizon branch satisfies: 1. CAR; 2. finite-window anti-phantom transfer with summable residuals; 3. an observable-depletion ledger with: $$ \lambda_k\asymp r_k/r_0; $$ 4. moving-window combined observability: $$ M_k\le C d_k^{-\gamma}; $$ 5.: $$ \gamma q\le1/2; $$ 6. no combined-invisible cascade; 7. no non-summable paid-side backscatter/leakage recurrence. Then the dangerous horizon branch cannot persist indefinitely. ### Status $$ \boxed{ \mathrm{PROVED\ CONDITIONAL}. } $$ ### Proof Use CAR and anti-phantom transfer to obtain an NS-realizable observed defect. Use Theorem 13.1 to choose a finite-time horizon schedule with: $$ \sum w_kM_k^{-q}=\infty. $$ Observable depletion then exhausts the finite selected budget. The remaining assumptions exclude the alternate invisible/paid residual branches. $\square$ --- # 31. What remains open The scale and filter compatibility problems are no longer the primary barriers. The unresolved theorems are now: ### CAR causal renewal/danger to finite-window NS-defect realization; ### MWG moving-window growth at or below the sharp temporal threshold, or another depletion-effective growth law; ### INV exclusion of NS-realizable combined-invisible defect cascades; ### PAID weighted backscatter/leakage recurrence control; ### weak-endpoint LAO long-age propagation in the general weak: $$ L^{3,\infty} $$ Type-I branch. --- # 32. Next paper The next paper should attack the last combined-observability objects directly: $$ \boxed{ \textbf{ NS-FCBP 06 — Causal-to-Audit Transfer、 Combined-Invisible Cascades、 Backscatter Sign Coherence 與 Cycle-VI Closure Audit }. } $$ Primary tasks: 1. map ANP/CFOP causal renewal packets into the finite-window defect quotient; 2. compare source/provenance coordinates with pressure--flux--energy--trace packages; 3. test clean-gap/anti-phantom hypotheses on the canonical Footprint/Dual Node class; 4. derive or refute a weighted backscatter sign-coherence theorem; 5. classify combined-invisible recurrent objects against model-cone and filtered commutator recurrence; 6. decide whether Cycle VI yields a genuine Forest Coercive Budget or only a final combined-invisible reduction. --- # 33. Formal status ledger $$ \boxed{ \begin{aligned} \text{Single global energy-work CRW} &:\ \mathrm{NO\mbox{-}GO},\\ \text{combined CRW semantics} &:\ \mathrm{DEFINED},\\ \text{Sharp Half-Exponent Schedule Theorem} &:\ \mathrm{PROVED},\\ \text{Thin-Window Observability Threshold} &:\ \mathrm{PROVED},\\ \text{Combined Horizon Depletion Compiler} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{strong-}L^3\text{ Long-Age State Propagation} &:\ \mathrm{EXTERNAL/PROVED},\\ \text{general weak-}L^{3,\infty}\text{ LAO} &:\ \mathrm{OPEN},\\ \text{finite-window anti-phantom detection} &:\ \mathrm{EXTERNAL/CONDITIONAL},\\ CAR &:\ \mathrm{OPEN},\\ \text{Signed-Telescope Total-Variation No-Go} &:\ \mathrm{PROVED},\\ \text{weighted backscatter closure} &:\ \mathrm{OPEN},\\ \text{combined-invisible cascade exclusion} &:\ \mathrm{OPEN},\\ \text{Critical Lift} &:\ \mathrm{OPEN},\\ \text{Forest Coercive Budget} &:\ \mathrm{OPEN},\\ \text{Finite Forest Obstruction} &:\ \mathrm{OPEN},\\ CN3_{\rm Atomic} &:\ \mathrm{OPEN},\\ \text{Navier--Stokes regularity} &:\ \mathrm{NOT\ PROVED}. \end{aligned} } $$ --- # 34. Conclusion FCBP-05 resolves the temporal scaling question more sharply than the previous single-work route. Fresh vorticity source cannot be universally coerced into one signed kinetic-energy-work observable. The correct target is combined finite-window observability. For a finite-time horizon schedule: $$ \sum_kr_k^2<\infty, $$ define the relative thin-window decrement: $$ d_k = 1- (r_{k+1}/r_k)^2. $$ Then the exact sequence threshold is: $$ \boxed{ \sum_kr_kd_k^a<\infty \quad \text{for every } a>1/2, } $$ while: $$ a=1/2 $$ admits explicit logarithmic divergent examples. Therefore a combined observability constant with polynomial deterioration: $$ M_k \lesssim d_k^{-\gamma} $$ is compatible with depletion-effective horizon scheduling precisely up to the sharp borderline: $$ \boxed{ \gamma q\le1/2. } $$ This is substantially weaker than requiring order-one signed pressure--flux work on every vanishing window. In the strong critical: $$ L_t^\infty L_x^3 $$ subbranch, Tao's quantitative back-propagation shows that long-age state observability can persist, but only with natural parabolic scale migration. The remaining problem is not whether there exist detectors in principle. Finite-window anti-phantom and recursive-audit frameworks already show that under explicit structural hypotheses, a baseline-visible NS-generated defect cannot be both detector-silent and residual-cheap. The missing step is to place the ANP/CFOP causal renewal package inside that quotient/detector architecture with horizon-uniform control. At the same time, signed endpoint energy alone cannot control total backscatter variation. Thus the final temporal critical-lift frontier is: $$ \boxed{ \textbf{ Causal-to-Audit Realization + sharp moving-window observability + invisible-cascade exclusion + paid-side sign control. } } $$ That is FCBP-06. --- # References 1. T. Tao, *Quantitative bounds for critically bounded solutions to the Navier--Stokes equations*, arXiv:1908.04958. 2. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756. 3. R. Yu, *Finite-Window Local-to-Clean Transfer and Anti-Phantom Detection for Sharp Navier--Stokes Packages*, arXiv:2606.18476. 4. R. Yu, *Finite-Window Recursive Audit Chains for Navier--Stokes Generated Packages*, arXiv:2606.20899. 5. R. Yu, *Finite-Window Singularity Audits and Local-to-Clean Defect Transfer for Navier--Stokes*, arXiv:2606.15086. 6. `NS_FCBP_04_MovingFilter_HorizonAlignment_v0.1.md`.