--- title: "Navier–Stokes Impulsive Defect Recurrence Program 01: Impulse Persistence, Trace Thickening, Temporal Action Packing, Moving-Window Visibility and Recurrent Defect Normal Forms" short_title: "NS-IDRP 01" series: "Navier–Stokes Impulsive Defect Recurrence Program" cycle: "IX" version: "v0.1" date: "2026-08-16" author: "Neo.K / EveMissLab" language: "en" status: "Temporal persistence foundation / burst-packing audit / Type-I thickening module" epistemic_status: "Launches the Impulsive Defect Recurrence Program from the Cycle-VIII normal form K_IDR. Imports Barker-Prange's Type-I concentration theorem and derives a parabolic-thick critical L3 spacetime packet on every late horizon slab in the strong L3 Type-I branch. Proves an abstract trace persistence-or-variation theorem and applies it to localized filtered-enstrophy balances: a selected-time filtered atom either persists or pays a fixed total-variation/mechanism debt. Proves the universal energy-class temporal regularity u_t in L_t^{4/3} H_x^{-1}. For dyadic velocity-shell atoms, proves a spectral impulse-action lower bound: if a local scale-r critical kinetic atom of strength a drops by a fixed factor within physical time tau=delta r^2, then the H^{-1} temporal action costs at least c a^{2/3} r^{4/3} delta^{-1/3}. Pairwise-disjoint impulses therefore satisfy a global burst-packing inequality. This yields an energy-class persistence floor: infinitely many disjoint atoms cannot all disappear on times tau <= c a^2 r^6. It also proves a no-go: the generic energy-class modulus is far too weak to force parabolic r^2 thickening; geometric scale sequences can have much shorter normalized persistence while keeping the temporal-action budget summable. Applying the result to the logarithmic FAR atoms from DCRP-04 shows that energy-class regularity alone does not upgrade the selected-time logarithmic atom floor to a fixed moving-window packet. The remaining bridge is therefore structural: causal/source impulses, filtered ledger variation, or combined moving-window observability must convert rapid trace loss into a scale-critical visible burst. 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 01 # Impulse Persistence, Trace Thickening, Temporal Action Packing, Moving-Window Visibility and Recurrent Defect Normal Forms ## 0. Program objective DCRP Cycle VIII ended with the surviving normal form: $$ \boxed{ \mathcal K_{\rm IDR} } $$ — Impulsive Diffuse Recurrence. Its branches were: - asymptotically propagated high-shell impulses; - super-height fresh-source bursts; - logarithmically weak FAR atoms; - recurrent commutator defect work/microstructure; - combined-invisible WORK/RES recurrence. The central new question is temporal: > can every positive carrier exist only at isolated times, so that every spacetime budget remains summable? IDRP begins by separating: $$ \boxed{ \text{PERSISTENCE} \vee \text{BURST DEBT}. } $$ --- # 1. External Type-I concentration module Barker--Prange prove a concentration theorem near a possible Type-I blow-up. Under their Type-I hypotheses, if: $$ (0,T_\ast) $$ is singular, then for late times: $$ \boxed{ \|u(\cdot,t)\|_{L^3(B_{R(t)}(0))} \ge \gamma_{\rm univ}, } $$ where: $$ \boxed{ R(t) \le C \sqrt{ T_\ast-t }. } $$ Their framework also treats critical Lorentz/Besov variants. ### Status $$ \boxed{ \mathrm{EXTERNAL/PROVED}. } $$ --- # 2. Horizon slab Fix small: $$ r>0. $$ Define: $$ \boxed{ J_r = [ T_\ast-2r^2, T_\ast-r^2 ]. } $$ For: $$ t\in J_r, $$ $$ R(t) \le C\sqrt2\,r. $$ Therefore the external concentration theorem gives: $$ \boxed{ \int_{ B_{C\sqrt2\,r} } |u(x,t)|^3dx \ge \gamma_{\rm univ}^3. } $$ for every admissible late: $$ t\in J_r. $$ --- # 3. CIV/IX-1.1 — Type-I Parabolic Thickening Corollary ## Theorem 3.1 Under the external Barker--Prange Type-I concentration hypotheses: $$ \boxed{ r^{-2} \int_{J_r} \int_{ B_{C\sqrt2\,r} } |u(x,t)|^3dxdt \ge \gamma_{\rm univ}^3. } $$ Thus every sufficiently small horizon scale carries a full normalized parabolic-thickness critical: $$ L^3 $$ spacetime packet. ### Proof The pointwise-in-time lower bound holds for every time in: $$ J_r, $$ whose length is: $$ r^2. $$ Integrate in time and divide by: $$ r^2. $$ $\square$ --- # 4. Meaning for IDRP In this strong Type-I critical branch: $$ \boxed{ \text{selected-time state impulse} } $$ is not the correct normal form. The critical state concentration is already persistent on a full: $$ O(r^2) $$ horizon window. Thus the hardest impulse-persistence problem lies in: - non-Type-I branches; - weak trace carriers not covered by the strong: $$ L^3 $$ packet; - filtered/commutator/WORK residual carriers. --- # 5. Abstract trace observable Let: $$ A: [t_0,t_0+\tau] \to [0,\infty) $$ be absolutely continuous. Assume: $$ \boxed{ A(t_0)\ge a>0. } $$ Define total variation: $$ \boxed{ \operatorname{Var}_I(A) = \int_I |A'(t)|dt. } $$ --- # 6. CIV/IX-1.2 — Trace Persistence-or-Variation ## Theorem 6.1 Exactly one of the following broad alternatives holds. ### persistence $$ \boxed{ A(t)\ge a/2 \qquad \forall t\in I; } $$ ### variation debt $$ \boxed{ \operatorname{Var}_I(A) \ge a/2. } $$ ### Proof If persistence fails, choose: $$ t_1 $$ with: $$ A(t_1) a/2. $$ $\square$ --- # 7. Persistent trace gives a spacetime packet Suppose: $$ A_r(t) = r \int_Q |\Omega_r(x,t)|^2dx, $$ where: $$ Q $$ is an: $$ r $$ scale cell. If: $$ A_r(t)\ge a/2 $$ for an interval of length: $$ \tau=\delta r^2, $$ then: $$ \boxed{ r^{-1} \int_I \int_Q |\Omega_r|^2dxdt \ge \frac{ a\delta }{ 2 }. } $$ This is a scale-invariant spacetime enstrophy packet. --- # 8. Filtered enstrophy ledger The external filtered-vorticity theory supplies a localized filtered-enstrophy balance of schematic exact form: $$ \boxed{ A_r'(t) + D_r(t) = V_r^{near}(t) + V_r^{far}(t) + C_r^{com}(t) + L_r(t). } $$ The terms denote: - filtered diffusion; - near-field stretching; - far-field strain; - commutator forcing; - localization residual. ### Status $$ \boxed{ \mathrm{EXTERNAL\ IDENTITY/FRAMEWORK}. } $$ --- # 9. Mechanism variation action Define: $$ \boxed{ \mathfrak B_r(I) = \int_I \left[ D_r + |V_r^{near}| + |V_r^{far}| + |C_r^{com}| + |L_r| \right]dt. } $$ Then: $$ \boxed{ \operatorname{Var}_I(A_r) \le \mathfrak B_r(I). } $$ --- # 10. CIV/IX-1.3 — Filtered Trace Thickening-or-Burst ## Theorem 10.1 If: $$ A_r(t_0)\ge a, $$ then on every chosen forward interval: $$ I=[t_0,t_0+\tau] $$ either: $$ \boxed{ A_r(t)\ge a/2 \quad \forall t\in I, } $$ or: $$ \boxed{ \mathfrak B_r(I) \ge a/2. } $$ ### Meaning A filtered selected-time atom can disappear rapidly only by paying a fixed absolute mechanism-variation debt. $\square$ --- # 11. Variation-budget barrier Theorem 10.1 does not close the impulse problem. The global energy inequality controls signed/net budgets and selected positive actions. It does not give a universal finite total-variation bound for: $$ \mathfrak B_r $$ across infinitely many scales. This is the temporal analogue of the earlier backscatter total-variation no-go. Therefore: $$ \boxed{ \text{trace variation debt} } $$ must be routed into separately controlled mechanism channels before it becomes coercive. --- # 12. Universal temporal regularity of velocity For a Leray--Hopf energy-class solution on: $$ [0,T], $$ $$ \partial_tu = \nu\Delta u - \mathbb P \nabla\cdot ( u\otimes u ). $$ The viscous term obeys: $$ \boxed{ \|\Delta u\|_{\dot H^{-1}} \le C \|\nabla u\|_2. } $$ The nonlinear term obeys: $$ \boxed{ \| \mathbb P \nabla\cdot ( u\otimes u ) \|_{\dot H^{-1}} \le C \|u\|_4^2. } $$ --- # 13. Energy interpolation In three dimensions: $$ \boxed{ \|u\|_4 \le C \|u\|_2^{1/4} \|\nabla u\|_2^{3/4}. } $$ Hence: $$ \boxed{ \|u\|_4^{8/3} \le C \|u\|_2^{2/3} \|\nabla u\|_2^2. } $$ The right-hand side is integrable in time by finite energy. --- # 14. CIV/IX-1.4 — Universal Temporal Action ## Theorem 14.1 Every finite-energy Leray--Hopf solution satisfies: $$ \boxed{ \int_0^T \| \partial_tu(t) \|_{\dot H^{-1}}^{4/3} dt < \infty. } $$ ### Proof Use: $$ (a+b)^{4/3} \le C ( a^{4/3}+b^{4/3} ). $$ For viscosity: $$ \int_0^T \|\nabla u\|_2^{4/3}dt < \infty $$ by Hölder and: $$ \nabla u\in L_t^2L_x^2. $$ For the nonlinearity, use Section 13 and: $$ \sup_t \|u(t)\|_2<\infty. $$ $\square$ --- # 15. Dyadic shell Let: $$ u_k=P_ku $$ be supported in an annulus: $$ |\xi| \asymp r^{-1}, \qquad r=2^{-k}. $$ Let: $$ Q $$ be a fixed spatial cell. Define the local critical kinetic atom: $$ \boxed{ \mathcal A_{k,Q}(t) = r^{-1} \int_Q |u_k(x,t)|^2dx. } $$ --- # 16. Spectral norm equivalence For every band-limited: $$ g $$ with frequency: $$ |\xi|\asymp r^{-1}, $$ $$ \boxed{ \|g\|_{\dot H^{-1}} \ge c r \|g\|_2. } $$ --- # 17. Atom disappearance Assume: $$ \boxed{ \mathcal A_{k,Q}(t_0) \ge a, } $$ while at: $$ t_1=t_0+\tau $$ $$ \boxed{ \mathcal A_{k,Q}(t_1) \le a/4. } $$ Then: $$ \|u_k(t_0)\|_{L^2(Q)} \ge \sqrt{ ar }, $$ and: $$ \|u_k(t_1)\|_{L^2(Q)} \le \frac12 \sqrt{ ar }. $$ Thus: $$ \boxed{ \| u_k(t_1)-u_k(t_0) \|_2 \ge \frac12 \sqrt{ ar }. } $$ --- # 18. CIV/IX-1.5 — Spectral Impulse Action Lower Bound ## Theorem 18.1 Under Sections 15--17: $$ \boxed{ \int_{t_0}^{t_1} \| \partial_tu_k \|_{\dot H^{-1}}^{4/3} dt \ge c a^{2/3} r^2 \tau^{-1/3}. } $$ Writing: $$ \boxed{ \tau = \delta r^2, } $$ this becomes: $$ \boxed{ r^{-4/3} \int_{t_0}^{t_1} \| \partial_tu_k \|_{\dot H^{-1}}^{4/3} dt \ge c a^{2/3} \delta^{-1/3}. } $$ ### Proof By Section 16: $$ \| u_k(t_1)-u_k(t_0) \|_{\dot H^{-1}} \ge c r \sqrt{ ar } = c a^{1/2} r^{3/2}. $$ The fundamental theorem in: $$ \dot H^{-1} $$ gives: $$ c a^{1/2} r^{3/2} \le \int_{t_0}^{t_1} \| \partial_tu_k \|_{\dot H^{-1}}dt. $$ By Hölder: $$ \int_I b(t)dt \le \tau^{1/4} \left( \int_I b(t)^{4/3}dt \right)^{3/4}. $$ Rearrange. $\square$ --- # 19. Interpretation Rapid disappearance of a scale-critical spectral atom has a quantitative temporal-action cost. The cost grows like: $$ \boxed{ \delta^{-1/3} } $$ in normalized time. This is the first universal burst-packing mechanism in IDRP. --- # 20. Pairwise-disjoint impulses Let: $$ I_n $$ be pairwise-disjoint atom-disappearance intervals. Let: $$ r_n, \quad a_n, \quad \delta_n $$ be their scales, atom strengths, and normalized durations. Since: $$ P_{k_n} $$ is bounded on: $$ \dot H^{-1}, $$ and the time intervals are disjoint: $$ \sum_n \int_{I_n} \| \partial_tu_{k_n} \|_{\dot H^{-1}}^{4/3}dt \le C \int_0^T \| \partial_tu \|_{\dot H^{-1}}^{4/3}dt. $$ --- # 21. CIV/IX-1.6 — Temporal Burst Packing ## Theorem 21.1 For pairwise-disjoint spectral atom disappearance intervals: $$ \boxed{ \sum_n a_n^{2/3} r_n^{4/3} \delta_n^{-1/3} < \infty. } $$ More quantitatively: $$ \boxed{ \sum_n a_n^{2/3} r_n^{4/3} \delta_n^{-1/3} \le C \int_0^T \| \partial_tu \|_{\dot H^{-1}}^{4/3}dt. } $$ $\square$ --- # 22. Energy-class persistence floor Suppose infinitely many disjoint atoms all disappear within: $$ \boxed{ \delta_n \le c_0 a_n^2 r_n^4. } $$ Then every summand in Theorem 21.1 is bounded below by a positive constant depending only on: $$ c_0. $$ Hence this is impossible. --- # 23. CIV/IX-1.7 — Minimal Spectral Persistence Floor ## Corollary 23.1 For infinitely many pairwise-disjoint critical spectral atoms, it is impossible that every atom disappears by a fixed factor on a time scale: $$ \boxed{ \tau_n \lesssim a_n^2 r_n^6. } $$ Equivalently, along an infinite subsequence one must have: $$ \boxed{ \tau_n \gtrsim a_n^2 r_n^6 } $$ unless the selected impulse intervals cease to be disjoint. ### Safety This is an extremely subparabolic persistence floor. It is not: $$ O(r_n^2). $$ --- # 24. CIV/IX-1.8 — Energy-Class Parabolic Thickening No-Go ## Theorem 24.1 The universal temporal action of Theorem 14.1 does not force parabolic normalized persistence: $$ \delta_n\gtrsim1. $$ ### Reason For a geometric scale sequence: $$ r_n=2^{-n}, $$ choose, for example: $$ \delta_n=r_n^\beta, \qquad 0<\beta<4, $$ and: $$ a_n\asymp1. $$ Then the burst lower-bound series is: $$ \sum_n r_n^{(4-\beta)/3}, $$ which converges. Thus a finite: $$ L_t^{4/3}\dot H^{-1} $$ temporal-action budget is compatible with normalized persistence: $$ \delta_n\to0. $$ $\square$ --- # 25. Balanced impulse scale Set: $$ \boxed{ \delta=a^2. } $$ Then Theorem 18.1 gives the scale-critical normalized burst: $$ \boxed{ r^{-4/3} \int_I \| \partial_tu_k \|_{\dot H^{-1}}^{4/3}dt \ge c. } $$ Thus: $$ \boxed{ \text{drop before }a^2r^2 } $$ forces order-one **normalized** temporal action. But the corresponding physical action carries: $$ r^{4/3} $$ and is summable over geometric scales. Therefore: $$ \boxed{ \text{normalized burst coercivity} \neq \text{global Critical Lift}. } $$ --- # 26. Persistent branch at the balanced scale If instead: $$ \mathcal A_{k,Q}(t) \ge a/2 $$ for: $$ \tau=a^2r^2, $$ then: $$ \boxed{ r^{-3} \int_I \int_Q |u_k|^2dxdt \ge c a^3. } $$ Thus the balanced impulse scale gives a dichotomy: $$ \boxed{ \text{critical spacetime packet of size }a^3 } $$ or: $$ \boxed{ \text{order-one normalized temporal-action burst}. } $$ --- # 27. Application to logarithmic FAR atoms DCRP-04 produced selected-time FAR atoms of scale-dependent strength: $$ \boxed{ a_k \gtrsim [ 1+\log(C/r_k) ]^{-2/3}. } $$ If a spectrally compatible atom of comparable strength is available, Corollary 23.1 yields only the persistence floor: $$ \boxed{ \tau_k \gtrsim r_k^6 [ 1+\log(C/r_k) ]^{-4/3} } $$ along an infinite disjoint subsequence. This is far below the parabolic scale: $$ r_k^2. $$ ### Meaning Energy-class temporal regularity alone cannot convert the DCRP logarithmic FAR atom into a full moving-window carrier. --- # 28. Filtered versus spectral atom safety The DCRP-04 FAR atom is a filtered-vorticity cell. Theorem 18.1 is a dyadic spectral velocity-shell theorem. The two are not identified automatically. A spectralization/reprofiling theorem is required to transfer one carrier into the other with comparable native strength. This remains open. --- # 29. Type-I versus generic impulses The program now has a sharp contrast. ### Type-I strong critical state External localized concentration gives: $$ \boxed{ \text{full parabolic thickening}. } $$ ### generic energy-class spectral atom Universal temporal regularity gives only: $$ \boxed{ \tau \gtrsim a^2r^6 } $$ in the packing sense. Thus the missing four powers of: $$ r $$ must come from singular-branch structure, not generic energy regularity. --- # 30. Moving-window observability interface The external moving-window framework reduces persistent badness to: $$ \boxed{ \text{effective combined observability} } $$ or: $$ \boxed{ \text{combined-invisible defect recurrence}. } $$ IDRP supplies the temporal input: - persistent packet; - mechanism variation burst; - spectral temporal-action burst; - source impulse. The missing theorem is to map at least one of these burst classes into a depletion-effective combined detector with controlled moving-window growth. --- # 31. Burst visibility problem Define: $$ \boxed{ \textbf{BVP — Burst Visibility Problem}. } $$ BVP asks for a theorem of the form: $$ \boxed{ \text{rapid loss of native atom} \Longrightarrow \text{PFET/model-cone/commutator/source observable} } $$ with a lower bound strong enough to enter the moving-window depletion series. Current status: $$ \boxed{ \mathrm{OPEN}. } $$ --- # 32. Trace thickening problem Define: $$ \boxed{ \textbf{TTP — Trace Thickening Problem}. } $$ Given a selected-time native atom: $$ A_r(t_0)\ge a_r, $$ prove either: $$ \boxed{ A_r(t)\ge c a_r } $$ on a window of normalized thickness: $$ \delta_r $$ large enough for effective observability, or prove a BVP-visible burst. Type-I strong: $$ L^3 $$ concentration solves TTP externally on that subbranch with: $$ \delta_r\asymp1. $$ General TTP remains open. --- # 33. Recurrent impulse normal forms After IDRP-01 a selected-time recurrence falls into: ### IR-P persistent critical packet; ### IR-V large filtered ledger variation; ### IR-T large: $$ L_t^{4/3}\dot H^{-1} $$ temporal-action burst; ### IR-S fresh-source impulse; ### IR-I combined-invisible moving-window recurrence. These overlap. They are not declared independent physical mechanisms. --- # 34. CIV/IX-1.9 — Impulse Temporalization Compiler ## Theorem 34.1 Consider a selected critical carrier on scale: $$ r. $$ Relative to the branch-specific hypotheses developed in this paper, at least one of the following occurs: 1. a critical spacetime packet persists; 2. the filtered mechanism ledger pays fixed variation; 3. a spectralized carrier pays the temporal-action lower bound; 4. the causal ledger pays a fresh-source impulse; 5. the carrier remains in a combined-invisible moving-window residual class. ### Safety The theorem is a compiler of the prior exact alternatives. It does not prove that alternatives 2--5 have a globally non-summable budget. $\square$ --- # 35. Strongest positive result The strongest full-thickness result is external and branch-specific: $$ \boxed{ \text{Type-I critical state} \Longrightarrow \text{parabolic-thick }L^3\text{ packet}. } $$ The strongest universal energy-class result is weaker but unconditional: $$ \boxed{ \text{rapid spectral atom loss} \Longrightarrow \text{quantitative }H^{-1}\text{ temporal-action burst}. } $$ --- # 36. Strongest no-go Generic energy-class temporal regularity cannot bridge the entire impulse gap. The scaling: $$ \boxed{ \tau_{\min} \sim a^2r^6 } $$ is much shorter than: $$ r^2. $$ Hence an IDRP closure must exploit one of: - Type-I/localized smoothing structure; - causal source renewal; - filtered exact mechanism ledgers; - moving-window combined observability; - another singular-branch persistence theorem. --- # 37. Next paper The next paper should attack BVP directly: $$ \boxed{ \textbf{ NS-IDRP 02 — Source-Impulse Visibility, Filtered Trace Variation, PFET Burst Coupling, Logarithmic Atom Thickening and Moving-Window Depletion }. } $$ Primary tasks: 1. convert super-height source impulses into pressure/flux/energy/trace or model-cone visibility; 2. separate fast filtered-atom loss into diffusion, far-field, commutator, and localization burst channels; 3. couple logarithmic FAR atom floors to the filtered balance; 4. seek a better-than-energy-class persistence exponent from branch-specific structure; 5. connect burst lower bounds to the FCBP half-exponent moving-window threshold; 6. determine whether an impulsive burst can remain combined-invisible; 7. build a temporal critical-lift compiler or isolate the exact invisible burst normal form. --- # 38. Formal status ledger $$ \boxed{ \begin{aligned} \text{Type-I Parabolic Thickening} &:\ \mathrm{EXTERNAL/PROVED\ COROLLARY},\\ \text{Trace Persistence-or-Variation} &:\ \mathrm{PROVED},\\ \text{Filtered Trace Thickening-or-Burst} &:\ \mathrm{PROVED\ FROM\ EXTERNAL\ BALANCE},\\ \text{Universal Temporal Action} &:\ \mathrm{PROVED},\\ \text{Spectral Impulse Action Lower Bound} &:\ \mathrm{PROVED},\\ \text{Temporal Burst Packing} &:\ \mathrm{PROVED},\\ \text{Minimal Spectral Persistence Floor} &:\ \mathrm{PROVED},\\ \text{generic parabolic thickening} &:\ \mathrm{NO\mbox{-}GO\ FROM\ ENERGY\ ACTION\ ALONE},\\ \text{FAR filtered-to-spectral transfer} &:\ \mathrm{OPEN},\\ \text{Burst Visibility Problem} &:\ \mathrm{OPEN},\\ \text{general Trace Thickening Problem} &:\ \mathrm{OPEN},\\ \text{Impulsive Diffuse Recurrence exclusion} &:\ \mathrm{OPEN},\\ \text{Forest Coercive Budget} &:\ \mathrm{OPEN},\\ \text{Finite Forest Obstruction} &:\ \mathrm{OPEN},\\ CN3_{\rm Atomic} &:\ \mathrm{OPEN},\\ \text{Navier--Stokes regularity} &:\ \mathrm{NOT\ PROVED}. \end{aligned} } $$ --- # 39. Conclusion IDRP-01 makes the temporal impulse problem quantitative. In the strong Type-I critical branch, localized concentration already persists on every late parabolic horizon slab. This gives a genuine scale-critical spacetime packet and shows that "instantaneous only" is not the correct Type-I state normal form. For a generic energy-class spectral atom, however, the universal temporal modulus is much weaker. The finite-energy equation gives: $$ \partial_tu \in L_t^{4/3} \dot H_x^{-1}. $$ Band localization converts rapid loss of a critical scale-$r$ atom into the lower bound: $$ r^{-4/3} \int_I \|\partial_tu_k\|_{\dot H^{-1}}^{4/3} \gtrsim a^{2/3}\delta^{-1/3}. $$ Disjoint impulses therefore satisfy a true global packing law. But the induced persistence floor is only: $$ \tau \gtrsim a^2r^6, $$ not the parabolic: $$ r^2 $$ scale required by the strongest moving-window arguments. This four-power gap is the central result of the paper. It proves that temporal regularity alone cannot solve the impulsive recurrence problem. The next theorem must use singular-branch structure to convert rapid trace loss into visible source/work/mechanism activity. Thus IDRP has a precise target: $$ \boxed{ \textbf{ impulse disappearance} \Longrightarrow \textbf{ depletion-effective visible burst}. } } $$ That is the Burst Visibility Problem. --- # References 1. T. Barker, C. Prange, *Localized smoothing for the Navier--Stokes equations and concentration of critical norms near singularities*, arXiv:1812.09115. 2. T. Tao, *Quantitative bounds for critically bounded solutions to the Navier--Stokes equations*, arXiv:1908.04958. 3. R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier--Stokes Equations*, arXiv:2606.27560. 4. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756. 5. `NS_DCRP_CYCLE_VIII_HANDOFF_v1.0.md`. 6. `NS_DCRP_04_TemporalRecurrence_FinalAudit_v0.1.md`.