--- title: "Navier–Stokes Impulsive Defect Recurrence Program 02: Source-Impulse Visibility, Filtered Trace Variation, PFET Burst Coupling, Logarithmic Atom Thickening and Moving-Window Depletion" short_title: "NS-IDRP 02" series: "Navier–Stokes Impulsive Defect Recurrence Program" cycle: "IX" version: "v0.1" date: "2026-08-16" author: "Neo.K / EveMissLab" language: "en" status: "Burst-visibility compiler / logarithmic packet-or-burst thickening / PFET realization frontier" epistemic_status: "Advances the Burst Visibility Problem. Starting from the exact localized filtered-enstrophy identity, proves a filtered mechanism-burst theorem: if a selected filtered-enstrophy atom loses a fixed amount on a window, then the loss is paid by diffusion, negative near-field stretching, far-field strain, commutator forcing, or localization. Inserting the external derivative-compatible commutator estimate converts the commutator branch into a scale-invariant critical increment defect modulo diffusion/localization. Thus rapid filtered-trace loss is not mechanism-invisible. Combines this with the DCRP logarithmic far-field atom floor to prove a full-parabolic-window packet-or-burst theorem: every selected logarithmic FAR atom either persists as a scale-critical filtered-enstrophy spacetime packet of the same logarithmic amplitude or generates an equally large mechanism burst on that same full parabolic window. Proves an abstract dual-source burst amplification theorem and applies it to velocity/vorticity nonlinear forcing: under uniform dual-profile cost, fixed causal fresh-source pairing on shrinking disjoint windows forces an L_t^{4/3} weak-topology source-action burst, with a global energy-class packing law. Defines Burst-to-Defect Realization (BDR) as the missing non-tautological bridge from source/mechanism bursts into the native finite-window NS defect quotient. Once BDR holds, the external finite-window combined observability constant gives a quantitative PFET lower bound; if the primitive channels are removed, the external trace-cost theorem yields controlled trace exactification or a relative left-singular invisible burst residual. Proves a conditional amplitude-weighted moving-window depletion compiler and a logarithmic-series threshold: if burst amplitude b_n is comparable to n^{-2/3}, depletion weights behave like n^{-s}, observability loss like n^{gamma}, and an unnormalized burst-depletion law with exponent q holds, then the burst series diverges when s+q(2/3+gamma)<=1 (with the expected logarithmic caveat at equality). Under the external exponential/logarithmically-admissible window growth model, the corresponding strict condition is s+2q/3+q C alpha<1. This shows logarithmic FAR bursts can only close through sufficiently linear/low-loss depletion; the general BDR/PFET bridge and invisible-burst exclusion remain open. No Impulsive Diffuse Recurrence exclusion, Forest Coercive Budget, Finite Forest Obstruction, atomic CN3, or Navier-Stokes regularity is proved." canonical_source: "UTF-8 Markdown" --- # Navier–Stokes Impulsive Defect Recurrence Program 02 # Source-Impulse Visibility, Filtered Trace Variation, PFET Burst Coupling, Logarithmic Atom Thickening and Moving-Window Depletion ## 0. Positioning of this paper IDRP-01 proved two sharply different temporal facts. First, in the strong Type-I critical branch, external concentration theory already gives full parabolic-thickness state packets. Second, for a generic energy-class spectral atom, the universal: $$ L_t^{4/3}\dot H_x^{-1} $$ temporal action gives only a very subparabolic persistence floor. The missing problem was: $$ \boxed{\textbf{BVP — Burst Visibility Problem}.} $$ The present paper shows that for **filtered vorticity traces**, rapid loss is not invisible at the mechanism-ledger level. It also formulates the remaining non-tautological bridge to PFET moving-window observability. --- # 1. Exact filtered enstrophy identity Let: $$ U_{\ell}=\varphi_{\ell}*u, \qquad \Omega_{\ell}=\nabla\times U_{\ell}. $$ For a nonnegative localized cutoff: $$ \chi, $$ the exact external filtered-enstrophy identity is: $$ \boxed{ E_{\chi}(s_1) - E_{\chi}(s_0) + P_{\chi} = V_{\chi}^{near} + V_{\chi}^{rem} + R_{\chi} + L_{\chi}. } $$ Here: $$ E_{\chi}(t) = \frac r2 \int \chi |\Omega_{\ell}|^2dx, $$ $$ P_{\chi} = r \iint \chi |\nabla\Omega_{\ell}|^2, $$ and the remaining terms are: - near-field stretching; - far/remainder strain; - commutator forcing; - cutoff/transport localization. ### Status $$ \boxed{\mathrm{EXTERNAL/EXACT}.} $$ --- # 2. Trace drop Define the filtered trace loss: $$ \boxed{ \Delta E_{\chi} = E_{\chi}(s_0)-E_{\chi}(s_1). } $$ Suppose: $$ \boxed{\Delta E_{\chi}\ge a>0.} $$ Write: $$ V_{\chi}^{near} = (V_{\chi}^{near})_+ - (V_{\chi}^{near})_-. $$ --- # 3. CIV/IX-2.1 — Filtered Mechanism-Burst Visibility ## Theorem 3.1 Under Sections 1--2: $$ \boxed{ a \le P_{\chi} + (V_{\chi}^{near})_- + |V_{\chi}^{rem}| + |R_{\chi}| + |L_{\chi}|. } $$ ### Proof The exact identity gives: $$ \Delta E_{\chi} = P_{\chi} - V_{\chi}^{near} - V_{\chi}^{rem} - R_{\chi} - L_{\chi}. $$ Use: $$ - V_{\chi}^{near} \le (V_{\chi}^{near})_-, $$ and absolute values for the remaining signed terms. $\square$ --- # 4. Interpretation A rapidly disappearing filtered atom is not dynamically invisible. Its loss is paid by at least one of: $$ \boxed{\text{DIFFUSION}} $$ $$ \boxed{\text{NEGATIVE NEAR-FIELD STRETCHING}} $$ $$ \boxed{\text{FAR-FIELD}} $$ $$ \boxed{\text{COMMUTATOR}} $$ or: $$ \boxed{\text{LOCALIZATION}.} $$ The first two are dynamically favorable depletion channels. The last three are explicit residual/mechanism bursts. --- # 5. Good-depletion versus residual-burst split Fix: $$ 0<\theta<1. $$ If: $$ \boxed{ P_{\chi} + (V_{\chi}^{near})_- \ge \theta a, } $$ then a fixed fraction of the trace loss is paid by good depletion. Otherwise: $$ \boxed{ |V_{\chi}^{rem}| + |R_{\chi}| + |L_{\chi}| \ge (1-\theta)a. } $$ Therefore one residual channel has magnitude at least: $$ \boxed{(1-\theta)a/3.} $$ --- # 6. External commutator insertion At dyadic scale: $$ r_k=2^{-k}, \qquad \ell_k=\sigma_f r_k, $$ the external filtered theory gives: $$ \boxed{ F_k^{com} \le \eta P_k + C_{\eta,\varphi} \widetilde{\mathcal S}_k^{(p)} + L_{k,inc}^{com}, } $$ for: $$ p\in[2,4]. $$ The quantity: $$ \widetilde{\mathcal S}_k^{(p)} $$ is the derivative-compatible scale-invariant increment defect. ### Status $$ \boxed{\mathrm{EXTERNAL/PROVED}.} $$ --- # 7. CIV/IX-2.2 — Filtered Burst Reduction to Critical Increment Defect ## Theorem 7.1 Suppose: $$ \Delta E_k\ge a $$ on one admissible dyadic filtered slab. Then: $$ \boxed{ a \le (1+\eta)P_k + (V_k^{near})_- + |V_k^{far}| + C_{\eta,\varphi} \widetilde{\mathcal S}_k^{(p)} + L_k + L_{k,inc}^{com}. } $$ Consequently, if diffusion, negative near-field stretching, far-field work, and localization together contribute at most: $$ (1-\vartheta)a $$ for some: $$ \vartheta>0, $$ then: $$ \boxed{ \widetilde{\mathcal S}_k^{(p)} \ge c_{\eta,\varphi} \vartheta a. } $$ ### Meaning A filtered trace burst which is not already paid by good depletion, FAR, or localization forces a scale-critical increment defect. $\square$ --- # 8. Partial closure of filtered BVP For filtered vorticity traces: $$ \boxed{\text{rapid atom loss}} $$ has now been compiled into explicit mechanism channels. Thus: $$ \boxed{\textbf{filtered BVP is closed at the mechanism-ledger layer}.} $$ It is **not** yet closed at the PFET moving-window layer. --- # 9. Source impulse Let: $$ X $$ be a Banach space and: $$ X^\ast $$ its dual. Let: $$ F\in L^{p}(I;X), \qquad \Phi\in L^{p'}(I;X^\ast), $$ where: $$ 10. $$ Then: $$ \boxed{ \sum_n \frac{A_n^{4/3}}{K_n^{4/3}|I_n|^{1/3}} < \infty } $$ up to the fixed factor: $$ \sigma^{-4/3}. $$ Thus the IDR-S branch has a true weak-source action packing law. --- # 16. Source-action visibility is not PFET visibility The nonlinear source may be large while one signed kinetic-energy-work pairing vanishes. Therefore: $$ \boxed{ \text{large source action} \not\Rightarrow \text{single signed work detector}. } $$ This is consistent with the FCBP single-energy-work no-go. The correct target remains combined finite-window observability. --- # 17. Burst-to-Defect Realization Define: $$ \boxed{\textbf{BDR — Burst-to-Defect Realization}.} $$ For a burst package: $$ \mathfrak B_W $$ on a finite window: $$ W, $$ BDR asks for a native NS-realizable cleaned defect: $$ d_W\in Y_W^{NS} $$ such that: $$ \boxed{ \|d_W\|_{Y_W} \ge c_B \mathfrak b_W - \varepsilon_W, } $$ where: $$ \mathfrak b_W $$ is the burst amplitude and: $$ \varepsilon_W $$ is an explicit residual. ### Safety The burst amplitude may not be copied into an artificial detector coordinate. BDR must be non-tautological. --- # 18. External combined observability For: $$ d\in Y_W^{NS}, $$ the external finite-window observability constant is: $$ \boxed{ M_W = \sup_{0\neq d\in Y_W^{NS}} \frac{\|d\|_{Y_W}}{\mathsf O_W(d)}, } $$ where: $$ \mathsf O_W $$ is the combined pressure/flux/energy/trace observed strength. Whenever: $$ M_W<\infty, $$ $$ \boxed{ \mathsf O_W(d) \ge \frac{\|d\|_{Y_W}}{M_W}. } $$ ### Status $$ \boxed{\mathrm{EXTERNAL/FINITE\mbox{-}WINDOW}.} $$ --- # 19. CIV/IX-2.5 — BDR-to-PFET Visibility Compiler ## Theorem 19.1 Assume BDR on: $$ W. $$ If: $$ M_W<\infty, $$ then: $$ \boxed{ \mathsf O_W(d_W) \ge \frac{c_B\mathfrak b_W-\varepsilon_W}{M_W}. } $$ If: $$ \varepsilon_W \le \frac12c_B\mathfrak b_W, $$ then: $$ \boxed{ \mathsf O_W(d_W) \ge \frac{c_B}{2M_W}\mathfrak b_W. } $$ $\square$ --- # 20. Residual trace exactification Suppose pressure, flux, and energy reductions leave a residual: $$ g_W $$ with native scale: $$ \rho_W. $$ The external trace-cost theorem gives: $$ \boxed{\text{controlled combined trace cost}} $$ or: $$ \boxed{\text{relative left-singular invisible failure}.} $$ In particular, if: $$ \|g_W\|\lesssim\rho_W, $$ failure of a uniform combined estimate produces normalized dual directions whose pressure/flux/energy/trace observations vanish relative to the residual pairing. ### Status $$ \boxed{\mathrm{EXTERNAL/PROVED\ FINITE\mbox{-}WINDOW\ ALTERNATIVE}.} $$ --- # 21. Burst amplitude can be the relative residual scale For a decaying burst sequence: $$ \mathfrak b_n\to0, $$ one may set the branch-native relative scale: $$ \boxed{\rho_n\asymp\mathfrak b_n} $$ **only if** the BDR/residual construction proves: $$ \|g_n\|\lesssim\mathfrak b_n. $$ Then amplitude decay does not make the residual meaningless. It becomes a **relative PFET trace-obstruction problem**. --- # 22. Logarithmic FAR atom DCRP-04 proved a selected-time filtered-vorticity atom: $$ \boxed{ a_k \gtrsim \left[ 1+\log(C/r_k) \right]^{-2/3} } $$ whenever the annular FAR output is order one. For dyadic: $$ r_k=2^{-k}, $$ this is: $$ \boxed{a_k\gtrsim(k+1)^{-2/3}} $$ up to harmless constants. --- # 23. Full parabolic window Let: $$ I_k $$ be the full: $$ r_k^2 $$ filtered slab. At the selected atom time: $$ t_k^\ast, $$ let the localized filtered-enstrophy trace satisfy: $$ \boxed{E_k(t_k^\ast)\ge c a_k.} $$ Apply the trace persistence theorem over the remaining admissible full-scale window, using a symmetric or forward/backward version compatible with the location of: $$ t_k^\ast. $$ --- # 24. CIV/IX-2.6 — Logarithmic FAR Packet-or-Burst Thickening ## Theorem 24.1 Relative to the DCRP-04 FAR atom and an admissible full parabolic filtered slab, one of the following occurs. ### filtered packet There exists a subwindow of normalized thickness bounded below by a fixed geometric constant on which: $$ E_k(t)\ge c a_k. $$ Consequently: $$ \boxed{ r_k^{-1} \iint \chi_k |\Omega_k|^2 \gtrsim a_k. } $$ ### mechanism burst The integrated mechanism ledger on the full scale window satisfies: $$ \boxed{ P_k + (V_k^{near})_- + |V_k^{far}| + C \widetilde{\mathcal S}_k^{(p)} + L_k + L_{k,inc}^{com} \gtrsim a_k. } $$ ### Meaning The selected logarithmic FAR atom cannot remain a purely instantaneous object. It becomes a full-window packet or a full-window mechanism burst of the same logarithmic order. $\square$ --- # 25. Improvement over generic energy persistence IDRP-01 energy-class spectral regularity gave only an: $$ r^6 $$ scale persistence floor. Theorem 24.1 uses the exact filtered Navier--Stokes balance and works on the full: $$ r_k^2 $$ window. Thus branch-specific structure recovers the four powers of: $$ r $$ which generic temporal regularity could not. ### Safety The output amplitude: $$ a_k $$ still decays logarithmically. Full time thickness does not imply fixed native separation. --- # 26. Amplitude-aware depletion model Suppose a burst sequence is BDR-realized as: $$ d_n $$ and a selected budget obeys an **unnormalized** depletion law: $$ \boxed{ \mathscr B_n-\mathscr B_{n+1} \ge c \lambda_n \mathsf O_n(d_n)^q - e_n, } $$ with: $$ \sum_ne_n<\infty. $$ ### Safety This is a new conditional burst-depletion hypothesis. The external moving-window theorem is formulated for normalized extracted defects with a fixed extraction lower bound. The unnormalized law is not silently imported from that theorem. --- # 27. CIV/IX-2.7 — Amplitude-Weighted Burst Depletion Compiler ## Theorem 27.1 Assume: $$ \|d_n\| \ge c_B b_n, $$ $$ M_n<\infty, $$ and Section 26. Then: $$ \boxed{ \mathscr B_n-\mathscr B_{n+1} \ge c' \lambda_n b_n^q M_n^{-q} - e_n. } $$ Therefore, if: $$ \boxed{ \sum_n \lambda_n b_n^q M_n^{-q} = \infty, } $$ the burst branch cannot persist indefinitely. $\square$ --- # 28. Polynomial logarithmic threshold Assume: $$ \boxed{b_n\gtrsim(n+2)^{-\beta},} $$ $$ \boxed{\lambda_n\gtrsim(n+2)^{-s},} $$ and: $$ \boxed{M_n\lesssim(n+2)^\gamma.} $$ Then: $$ \lambda_n b_n^q M_n^{-q} \gtrsim (n+2)^{-s-q(\beta+\gamma)}. $$ --- # 29. CIV/IX-2.8 — Logarithmic Burst Series Threshold ## Theorem 29.1 Under Section 28, the amplitude-weighted depletion series diverges if: $$ \boxed{s+q(\beta+\gamma)\le1.} $$ For the DCRP logarithmic FAR amplitude: $$ \beta=\frac23, $$ the threshold is: $$ \boxed{ s+q\left(\frac23+\gamma\right)\le1. } $$ ### Consequences For: $$ q=1, $$ one needs: $$ \boxed{s+\gamma\le\frac13.} $$ For: $$ q\ge\frac32, $$ there is no room with: $$ s,\gamma\ge0 $$ except degenerate borderline cases. $\square$ --- # 30. External exponential/logarithmic window growth The external moving-window framework allows: $$ \boxed{ M_n \le C N_n^a \exp(CN_n^b), } $$ and logarithmically admissible windows with: $$ \boxed{N_n^b\le\alpha\log(n+2).} $$ Thus: $$ M_n^{-q} \gtrsim (n+2)^{-qC\alpha} (\log(n+2))^{-aq/b}. $$ --- # 31. CIV/IX-2.9 — Amplitude-Weighted Logarithmic Admissibility ## Theorem 31.1 Assume the amplitude-aware depletion model of Section 26, the DCRP burst amplitude: $$ b_n\gtrsim(n+2)^{-2/3}, $$ the external growth form of Section 30, and: $$ \lambda_n\gtrsim(n+2)^{-s}. $$ Then the depletion series diverges whenever: $$ \boxed{ s+\frac{2q}{3}+qC\alpha<1. } $$ At equality, divergence additionally depends on the logarithmic power: $$ aq/b. $$ ### Meaning The logarithmic FAR packet-or-burst is compatible with moving-window depletion only when the depletion law is sufficiently close to linear and the observation/window loss is mild enough. $\square$ --- # 32. Comparison with external normalized moving-window theory The external theorem assumes a fixed extracted defect norm: $$ \|d_{n_k}\|\ge c_0>0 $$ and applies depletion to: $$ \widehat d_{n_k}. $$ The logarithmic FAR amplitude: $$ b_n\to0 $$ does not automatically satisfy that extraction hypothesis. Therefore Theorems 27.1--31.1 do not close the external moving-window theorem by themselves. They identify the exact amplitude-aware extension needed for IDRP. --- # 33. Relative invisible burst normal form If BDR produces a residual of size: $$ \rho_n\asymp b_n $$ but combined observability degenerates relative to that scale, the external trace-cost theorem permits a relative left-singular invisible residual. Define: $$ \boxed{\textbf{RIB — Relative Invisible Burst}.} $$ A RIB sequence has: - burst-native residual scale: $$ \rho_n\to0; $$ - nontrivial residual pairing relative to: $$ \rho_n; $$ - pressure/flux/energy/trace dual observations vanishing relative to that pairing. This is the natural surviving IDRP obstruction after burst amplitude is allowed to decay. --- # 34. Filtered BVP status At the mechanism level: $$ \boxed{ \text{filtered BVP} : \mathrm{SUBSTANTIALLY\ CLOSED}. } $$ Rapid filtered trace loss forces explicit mechanism activity. At the PFET level: $$ \boxed{ \text{BDR/PFET BVP} : \mathrm{OPEN}. } $$ The remaining task is native burst-to-defect realization and relative invisible-burst exclusion. --- # 35. Source impulse status Source bursts have a genuine: $$ L_t^{4/3} $$ weak-topology packing law. But large weak source action does not, by itself, imply pressure-flux-energy-trace visibility. The missing theorem is again BDR or an equivalent source-to-audit transfer theorem. The 2026 forced Navier--Stokes quantitative theory of Barker--Popkin shows that critical quantitative propagation can incorporate forcing under additional critical-bound/forcing hypotheses, but the forcing is itself a nontrivial source of large-scale loss. This is useful calibration, not a universal BDR theorem. --- # 36. IDRP-02 surviving temporal obstruction After this paper, a recurrent impulsive branch must enter at least one of: $$ \boxed{\textbf{RIB}} $$ — relative invisible burst; $$ \boxed{\textbf{AMP}} $$ — burst amplitude decays too fast for the available depletion weights; $$ \boxed{\textbf{OBS}} $$ — moving-window observability constants grow too fast; $$ \boxed{\textbf{BDR}} $$ — burst-to-native-defect realization fails; $$ \boxed{\textbf{RES}} $$ — localization/exterior/representation residual remains unabsorbed. This is a narrower temporal normal form than the original IDR class. --- # 37. Strongest positive result The main positive result is: $$ \boxed{ \textbf{ rapid filtered trace loss is mechanism-visible, and logarithmic FAR atoms are full-window packet-or-burst objects. } } $$ Thus the temporal obstruction can no longer hide solely in instantaneous filtered trace loss. --- # 38. Strongest remaining gap The missing bridge is not temporal regularity. It is: $$ \boxed{ \textbf{ mechanism/source burst} \Longrightarrow \textbf{ native PFET defect with effective relative observability}. } } $$ That is BDR plus relative moving-window rigidity. --- # 39. Next paper The next paper should attack the relative invisible burst directly: $$ \boxed{ \textbf{ NS-IDRP 03 — Relative Invisible Burst Kernels, Burst-to-Defect Realization, Source/Trace Dual Compatibility, Amplitude-Normalized Audit and Temporal Rigidity }. } $$ Primary tasks: 1. construct a non-tautological BDR map for filtered/source bursts; 2. compare ANP source duals with PFET adjoint-trace duals; 3. normalize decaying burst residuals without copying the burst amplitude; 4. classify the relative left-singular kernel; 5. test whether model-cone/increment channels eliminate relative invisibility; 6. derive an amplitude-normalized depletion law or prove a no-go; 7. decide whether Cycle IX can exclude recurrent impulsive bursts. --- # 40. Formal status ledger $$ \boxed{ \begin{aligned} \text{Filtered Mechanism-Burst Visibility} &:\ \mathrm{PROVED},\\ \text{Filtered Burst to Critical Increment} &:\ \mathrm{PROVED\ USING\ EXTERNAL\ COMMUTATOR\ ESTIMATE},\\ \text{Dual Source-Burst Amplification} &:\ \mathrm{PROVED},\\ \text{Disjoint Source-Burst Packing} &:\ \mathrm{PROVED},\\ \text{BDR-to-PFET Visibility} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Logarithmic FAR Packet-or-Burst Thickening} &:\ \mathrm{PROVED\ RELATIVE\ TO\ DCRP\ ATOM/EXTERNAL\ BALANCE},\\ \text{Amplitude-Weighted Burst Depletion} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Logarithmic Burst Series Threshold} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Amplitude-Weighted Logarithmic Admissibility} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Burst-to-Defect Realization} &:\ \mathrm{OPEN},\\ \text{Relative Invisible Burst exclusion} &:\ \mathrm{OPEN},\\ \text{Impulsive Diffuse Recurrence exclusion} &:\ \mathrm{OPEN},\\ \text{Forest Coercive Budget} &:\ \mathrm{OPEN},\\ \text{Finite Forest Obstruction} &:\ \mathrm{OPEN},\\ CN3_{\rm Atomic} &:\ \mathrm{OPEN},\\ \text{Navier--Stokes regularity} &:\ \mathrm{NOT\ PROVED}. \end{aligned} } $$ --- # 41. Conclusion IDRP-02 partially solves the Burst Visibility Problem. The exact filtered-vorticity balance shows that rapid loss of a selected filtered atom cannot occur without explicit mechanism activity. After derivative-compatible commutator insertion, a disappearing atom is paid by good depletion, far-field activity, a scale-critical increment defect, or localization. The logarithmic FAR atoms from DCRP therefore admit a full-parabolic-window dichotomy: either they persist as filtered-enstrophy packets or they generate mechanism bursts of the same logarithmic order. Fresh causal-source impulses have an analogous quantitative structure. Duality converts fixed source pairing on short windows into an: $$ L_t^{4/3} $$ weak-source action burst, and disjoint bursts satisfy a global energy-class packing law. The remaining difficulty is no longer showing that a burst exists. It is showing that the burst has a **native PFET realization**. Once a non-tautological burst-to-defect map exists, finite-window combined observability gives a direct observed-strength lower bound. After pressure/flux/energy removal, the external trace-cost machinery further reduces failure to a relative left-singular invisible burst. The logarithmic amplitude makes this distinction essential. A full-window burst of size: $$ k^{-2/3} $$ is not a fixed defect. An amplitude-aware depletion law would close it only under a sharp series condition, for example: $$ s+q(2/3+\gamma)\le1 $$ in the polynomial-loss model. Thus the new frontier is precise: $$ \boxed{ \textbf{ relative burst realization + relative PFET rigidity. } } $$ That is IDRP-03. --- # References 1. R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier--Stokes Equations*, arXiv:2606.27560. 2. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756. 3. R. 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