--- title: "Navier–Stokes Impulsive Defect Recurrence Program 03：Relative Invisible Burst Kernels、Dual-Compatible Burst-to-Defect Realization、Amplitude-Normalized Audit 與 Temporal Rigidity" short_title: "NS-IDRP 03" series: "Navier–Stokes Impulsive Defect Recurrence Program" cycle: "IX" version: "v0.1" date: "2026-08-16" author: "Neo.K / EveMissLab" language: "zh-TW" status: "Relative-burst kernel compactification / source-native BDR compiler / moving-window singular-limit reduction" epistemic_status: "Partially closes the Burst-to-Defect Realization problem by identifying an exact native route through the finite-window source quotient. In a finite-dimensional Hilbert source-coordinate model, a causal forcing burst whose induced source dual has a fixed transverse component to the clean/model source subspace forces a quantitative source-quotient distance; if the sharp package norm dominates that native source coordinate, this yields a non-tautological BDR lower bound. Proves an approximate causal-audit dual compatibility compiler which transfers causal source pairing into a native audit residual pairing modulo a quantified dual mismatch and residual-ledger error. Defines intrinsic residual normalization rho=||g|| and proves that this normalization is legitimate for kernel analysis because it occurs only after a native NS residual has been constructed. Proves a compact-family moving-window theorem: after transporting finite-window combined dual observability maps to one fixed finite-dimensional model, operator-norm precompactness plus kernel-freeness of every limit map gives a uniform positive minimum singular value and excludes every relative invisible burst. Conversely, a strong relative invisible burst with compact normalized residuals and precompact observability operators converges to a genuine nonzero limit kernel/phantom direction. Proves a no-go showing that a merely relative left-singular failure with normalized residual pairing tending to zero need not compactify to a true defect phantom. Finally proves an amplitude-normalization no-free-lunch theorem: normalizing a native residual removes amplitude from kernel geometry but not from any physical depletion law; a budget driven by the actual observed amplitude still carries the factor rho^q. The surviving IDRP obstruction is therefore reduced to source-transversality failure, causal/audit dual mismatch, operator-family noncompactness or singular limit kernels, relative weak-pairing degeneration, and physical amplitude/depletion loss. No universal BDR, relative invisible burst exclusion, 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 03 # Relative Invisible Burst Kernels、Dual-Compatible Burst-to-Defect Realization、Amplitude-Normalized Audit 與 Temporal Rigidity ## 0. 本文定位 IDRP-02 established: $$ \boxed{ \text{rapid filtered trace loss} \Longrightarrow \text{explicit mechanism burst}, } $$ and: $$ \boxed{ \text{fresh causal source pairing} \Longrightarrow L_t^{4/3} \text{ weak-source action burst}. } $$ The remaining bridge was: $$ \boxed{ \textbf{BDR — Burst-to-Defect Realization}. } $$ The surviving finite-window obstruction was: $$ \boxed{ \textbf{RIB — Relative Invisible Burst}. } $$ The present paper separates these problems into native quotient geometry, dual compatibility, and moving-window operator compactness. --- # 1. Native source coordinate Let: $$ X_{\rm src} $$ be a finite-dimensional Hilbert source-coordinate space on one normalized finite window. Let: $$ S_{\rm cl} \subset X_{\rm src} $$ be the clean/model source subspace. Let: $$ F\in X_{\rm src} $$ be the physical active/source-residual coordinate produced from the Navier--Stokes package. Define the native source quotient distance: $$ \boxed{ \delta_{\rm src}(F) = \operatorname{dist}_{X_{\rm src}} ( F,S_{\rm cl} ). } $$ This coordinate is generated from the PDE package. No burst amplitude coordinate is inserted. --- # 2. Forcing map Let: $$ \boxed{ \mathcal B: X_{\rm src} \to X_{\rm frc} } $$ be the native forcing map. For the velocity nonlinear source tensor one may schematically take: $$ \boxed{ \mathcal B F = - \mathbb P \nabla\cdot F. } $$ Let: $$ \Phi \in X_{\rm frc}^{\ast} $$ be a causal/source dual. Its induced source dual is: $$ \boxed{ \psi = \mathcal B^{\ast}\Phi \in X_{\rm src}^{\ast}. } $$ --- # 3. Clean-source transverse dual Since: $$ X_{\rm src} $$ is Hilbert in the reduced finite-window model, write: $$ \boxed{ \psi_\perp = P_{S_{\rm cl}^{\perp}} \psi. } $$ Define the source-transverse burst pairing: $$ \boxed{ J_\perp = | \langle F,\psi_\perp \rangle |. } $$ The total forcing pairing is: $$ \boxed{ J = | \langle \mathcal BF,\Phi \rangle | = | \langle F,\psi \rangle |. } $$ --- # 4. CIV/IX-3.1 — Source-Quotient BDR Theorem ## Theorem 4.1 If: $$ \psi_\perp\neq0, $$ then: $$ \boxed{ \delta_{\rm src}(F) \ge \frac{ J_\perp }{ \|\psi_\perp\|_{X_{\rm src}^{\ast}} }. } $$ In particular, if: $$ \boxed{ J_\perp \ge \theta J } $$ for: $$ \theta>0, $$ and: $$ \boxed{ \|\psi_\perp\| \le K, } $$ then: $$ \boxed{ \delta_{\rm src}(F) \ge \frac{ \theta J }{ K }. } $$ ### Proof For every: $$ s\in S_{\rm cl}, $$ $$ \langle s,\psi_\perp\rangle=0. $$ Hence: $$ J_\perp = | \langle F-s,\psi_\perp \rangle | \le \|F-s\| \|\psi_\perp\|. $$ Take the infimum over: $$ s. $$ $\square$ --- # 5. Meaning A causal source burst becomes a native source defect whenever a fixed fraction of its dual pairing is transverse to the clean/model source class. This is a genuine BDR mechanism. The burst amplitude is not copied into the package. The PDE input is the transversality: $$ \boxed{ J_\perp/J \ge \theta. } $$ --- # 6. Source-tangency alternative If: $$ \boxed{ J_\perp/J \to0, } $$ the causal source pairing becomes asymptotically tangent to the selected clean/model source space. Define: $$ \boxed{ \textbf{STG — Source Tangency}. } $$ STG is not automatically harmless. It says only that the selected source quotient cannot distinguish the burst. A different native channel or a larger clean-source model may be required. --- # 7. Sharp package geometry Recent finite-window sharp package frameworks include native coordinates for: - active source; - model source/covariance; - source residual; - active pressure; - harmonic pressure; - flux; - energy; - selected trace; - localization/slack residuals. Let: $$ \boxed{ \delta_{\rm pkg}(D) } $$ be a sharp quotient distance. Assume the sharp package geometry dominates the source quotient: $$ \boxed{ \delta_{\rm pkg}(D) \ge w_{\rm src} \delta_{\rm src}(F) } $$ for one fixed positive source weight: $$ w_{\rm src}>0. $$ --- # 8. CIV/IX-3.2 — Source-Native BDR Compiler ## Theorem 8.1 Under Theorem 4.1 and Section 7: $$ \boxed{ \delta_{\rm pkg}(D) \ge w_{\rm src} \frac{ J_\perp }{ \|\psi_\perp\|} . } $$ If: $$ J_\perp\ge\theta J, \qquad \|\psi_\perp\|\le K, $$ then: $$ \boxed{ \delta_{\rm pkg}(D) \ge c_{\rm BDR} J, \qquad c_{\rm BDR} = w_{\rm src}\theta/K. } $$ ### Status $$ \boxed{ \mathrm{PROVED\ CONDITIONAL}. } $$ The conditional inputs are native source transversality and source-coordinate visibility in the chosen sharp geometry. --- # 9. Audit source insertion Let: $$ \boxed{ \mathcal J_{W}^{src}: X_{\rm src}/S_{\rm cl} \to Y_W } $$ be the source-residual insertion into the finite-window active defect quotient. Let: $$ y\in Y_W^{\ast} $$ be an audit dual. Then: $$ \boxed{ (\mathcal J_W^{src})^{\ast}y } $$ is the source-side dual seen by the audit residual. --- # 10. Causal/audit dual mismatch Define: $$ \boxed{ \varepsilon_{\rm dual} = \left\| (\mathcal J_W^{src})^{\ast}y - \psi_\perp \right\|_{X_{\rm src}^{\ast}}. } $$ Let: $$ R_W\in Y_W $$ collect the remaining native residual-ledger terms. Write: $$ \boxed{ d_W = \mathcal J_W^{src}[F] + R_W. } $$ --- # 11. CIV/IX-3.3 — Causal-to-Audit Dual Compatibility ## Theorem 11.1 Assume: $$ \|y\|_{Y_W^{\ast}} \le K_y. $$ Then: $$ \boxed{ | \langle d_W,y \rangle | \ge J_\perp - \varepsilon_{\rm dual} \, \delta_{\rm src}(F) - | \langle R_W,y \rangle |. } $$ Consequently: $$ \boxed{ \|d_W\|_{Y_W} \ge \frac{ J_\perp - \varepsilon_{\rm dual}\delta_{\rm src}(F) - |\langle R_W,y\rangle| }{ K_y } } $$ whenever the numerator is positive. ### Proof Use: $$ \langle \mathcal J_W^{src}[F],y \rangle = \langle [F], (\mathcal J_W^{src})^\ast y \rangle. $$ Insert: $$ \psi_\perp + \left[ (\mathcal J_W^{src})^\ast y-\psi_\perp \right] $$ and estimate the error by quotient duality. Add the residual pairing. $\square$ --- # 12. Dual-Compatible BDR condition If: $$ \boxed{ \varepsilon_{\rm dual}\delta_{\rm src}(F) + | \langle R_W,y \rangle | \le \eta J_\perp, \qquad 0\le\eta<1, } $$ then: $$ \boxed{ \|d_W\| \ge \frac{ (1-\eta)J_\perp }{ K_y }. } $$ Thus generic BDR has been reduced to: $$ \boxed{ \textbf{SOURCE TRANSVERSALITY} + \textbf{DUAL COMPATIBILITY} + \textbf{RESIDUAL CONTROL}. } $$ --- # 13. External status of the package layer Finite-window package-realizability and pressure/source residual coordinates exist on explicit smooth/reduced classes. The literature also explicitly states that baseline/source visibility and broad kernel-freeness are not automatic for arbitrary NS-generated packages. Therefore Theorems 8.1 and 11.1 are native compilers, not universal PDE closure theorems. --- # 14. Intrinsic residual amplitude Let: $$ g_n\in Y_{W_n} $$ be an NS-realizable residual surviving the pressure--flux--energy reductions. Define its **intrinsic** scale: $$ \boxed{ \rho_n = \|g_n\|_{Y_{W_n}}. } $$ Assume: $$ \rho_n>0. $$ Define: $$ \boxed{ \widehat g_n = g_n/\rho_n. } $$ Then: $$ \boxed{ \|\widehat g_n\|=1. } $$ --- # 15. Why intrinsic normalization is non-tautological The residual: $$ g_n $$ is first generated by the NS finite-window package. Only after that realization do we divide by its native norm. No additional detector coordinate is introduced. Thus: $$ \boxed{ \text{intrinsic residual normalization} } $$ is legitimate for compactness/kernel analysis. It is not a copied-gate BDR. --- # 16. Normalized combined dual map For each window: $$ W_n, $$ let: $$ \boxed{ \mathfrak O_n^{\ast} : Y_{W_n}^{\ast} \to Z_{W_n} } $$ be the dual combined observation map containing: - adjoint trace; - energy; - active pressure; - flux. Fixed-window finite-dimensional theory gives: $$ \boxed{ \ker \mathfrak O_n^\ast = \{0\} \Longleftrightarrow M_n^\ast<\infty. } $$ --- # 17. Common normalized chart Assume finite-dimensional isomorphisms: $$ \boxed{ U_n: Y_{W_n}\to Y, } $$ and: $$ \boxed{ V_n: Z_{W_n}\to Z, } $$ to fixed finite-dimensional Hilbert spaces: $$ Y,Z. $$ Transport the dual operators to: $$ \boxed{ \widetilde{\mathfrak O}_n^\ast: Y^\ast\to Z. } $$ Assume the family: $$ \{ \widetilde{\mathfrak O}_n^\ast \} $$ is precompact in operator norm. --- # 18. CIV/IX-3.4 — Compact Operator-Family RIB No-Go ## Theorem 18.1 Assume every operator-norm cluster limit: $$ \widetilde{\mathfrak O}_\ast^\ast $$ is injective. Then: $$ \boxed{ \inf_n \inf_{\|y\|=1} \| \widetilde{\mathfrak O}_n^\ast y \| > 0. } $$ Consequently no relative invisible burst can occur on this normalized operator family. ### Proof Suppose the conclusion fails. Then there exist: $$ n_j $$ and unit vectors: $$ y_j $$ such that: $$ \| \widetilde{\mathfrak O}_{n_j}^\ast y_j \| \to0. $$ By finite-dimensional compactness and operator precompactness: $$ y_j\to y_\ast, \qquad \widetilde{\mathfrak O}_{n_j}^\ast \to \widetilde{\mathfrak O}_\ast^\ast $$ along a subsequence. Then: $$ \|y_\ast\|=1, $$ and: $$ \widetilde{\mathfrak O}_\ast^\ast y_\ast=0, $$ contradicting injectivity. $\square$ --- # 19. Meaning Moving-window observability can degenerate in a compact normalized finite-dimensional class only by approaching a genuinely singular combined operator. Thus: $$ \boxed{ \text{RIB} \Longrightarrow \text{operator-family noncompactness} \vee \text{singular limit operator}. } $$ This makes moving-window degeneration a normal-form problem. --- # 20. Strong relative invisible burst Assume unit audit duals: $$ \boxed{ \|y_n\|=1, } $$ and strong relative residual pairing: $$ \boxed{ | \langle \widehat g_n,y_n \rangle | \ge c_0>0. } $$ Assume: $$ \boxed{ \| \widetilde{\mathfrak O}_n^\ast y_n \| \to0. } $$ --- # 21. Residual compactness Assume the normalized residual directions: $$ U_n\widehat g_n $$ are precompact in: $$ Y. $$ This is a genuine compactness assumption on the NS-realizable residual class. It is not automatic from finite-dimensionality if the window charts themselves degenerate. --- # 22. CIV/IX-3.5 — Strong RIB Compactification ## Theorem 22.1 Under Sections 17, 20, and 21, after passing to a subsequence there exist: $$ \boxed{ g_\ast\in Y, \qquad y_\ast\in Y^\ast } $$ such that: $$ \boxed{ \|g_\ast\|=1, \qquad \|y_\ast\|=1, } $$ $$ \boxed{ | \langle g_\ast,y_\ast \rangle | \ge c_0, } $$ and: $$ \boxed{ \widetilde{\mathfrak O}_\ast^\ast y_\ast = 0. } $$ If the normalized NS-realizable residual class is closed under this limit, then: $$ \boxed{ (g_\ast,y_\ast) } $$ is a genuine normalized true-phantom limit pair. $\square$ --- # 23. Strong RIB consequence A strong RIB cannot remain an indefinitely weak moving-window artifact inside a compact normalized family. It converges to an actual kernel object. Therefore its exclusion reduces to: $$ \boxed{ \text{limit-kernel rigidity}. } $$ --- # 24. Relative failure need not give a true phantom The external trace-cost theorem explicitly distinguishes strong residual pairing from merely relative left-singular failure. The distinction is necessary. --- # 25. CIV/IX-3.6 — Relative-Failure-to-Phantom No-Go ## Theorem 25.1 A relative left-singular ratio can diverge while the normalized residual pairing tends to zero, so that the limit kernel has no nontrivial residual pairing. ### Example Let: $$ Y=\mathbb R^2. $$ Set: $$ \boxed{ \widetilde{\mathfrak O}_n^\ast = \begin{pmatrix} 1&0\\ 0&n^{-2} \end{pmatrix}, } $$ $$ \boxed{ y_n=e_2, } $$ and: $$ \boxed{ \widehat g_n = \frac{ e_1+n^{-1}e_2 }{ \sqrt{1+n^{-2}} }. } $$ Then: $$ \boxed{ | \langle \widehat g_n,y_n \rangle | \asymp n^{-1} \to0, } $$ while: $$ \boxed{ \| \widetilde{\mathfrak O}_n^\ast y_n \| = n^{-2}. } $$ Hence: $$ \boxed{ \frac{ |\langle \widehat g_n,y_n \rangle| }{ \| \widetilde{\mathfrak O}_n^\ast y_n \| } \asymp n \to\infty. } $$ But: $$ \widehat g_n\to e_1, \qquad y_n\to e_2, $$ and: $$ \boxed{ \langle e_1,e_2\rangle=0. } $$ The limit operator has a kernel, but the limit residual does not pair with that kernel direction. $\square$ --- # 26. Meaning A weak RIB may certify: $$ \boxed{ \text{singular-value degeneration} } $$ without producing a true NS defect phantom. Therefore strong normalized residual pairing is a separate rigidity input. --- # 27. Compact-family trichotomy Relative moving-window failure now has three canonical causes: $$ \boxed{ \textbf{OP-NC} } $$ operator-family noncompactness; $$ \boxed{ \textbf{OP-KER} } $$ a singular limit combined operator; $$ \boxed{ \textbf{PAIR-DEG} } $$ normalized residual pairing degenerates even faster than the operator singular value. This replaces the undifferentiated label "observability constant grows." --- # 28. Amplitude normalization Suppose a native residual is: $$ \boxed{ d_n = \rho_n \widehat d_n, \qquad \|\widehat d_n\|=1. } $$ Let: $$ \mathcal O_n $$ be a linear observed-strength map. Then: $$ \boxed{ \mathcal O_n(d_n) = \rho_n \mathcal O_n(\widehat d_n). } $$ --- # 29. CIV/IX-3.7 — Amplitude-Normalization No-Free-Lunch ## Theorem 29.1 Suppose a physical selected budget obeys: $$ \boxed{ \mathscr B_n-\mathscr B_{n+1} \ge c \lambda_n \| \mathcal O_n(d_n) \|^q - e_n. } $$ Then: $$ \boxed{ \mathscr B_n-\mathscr B_{n+1} \ge c \lambda_n \rho_n^q \| \mathcal O_n(\widehat d_n) \|^q - e_n. } $$ Thus intrinsic normalization can remove amplitude from **kernel geometry**, but it cannot remove: $$ \boxed{ \rho_n^q } $$ from the physical depletion law. $\square$ --- # 30. Meaning Amplitude normalization is mathematically legitimate for: - compactness; - operator limits; - invisible-kernel classification. It is not a free Critical Lift. The physical amplitude problem from IDRP-02 remains. --- # 31. Source burst normal forms Combining the source-native BDR theorem and dual compatibility gives the following alternatives. ### SB-TR Source burst has uniform transverse pairing to the clean source quotient and bounded dual mismatch. Then: $$ \boxed{ \text{native BDR} } $$ holds. ### SB-TAN The source burst becomes asymptotically tangent to the clean/model source space. ### SB-DUAL Causal/audit dual mismatch is comparable to the burst pairing. ### SB-RES The residual ledger pairing is comparable to the burst pairing. Only SB-TR is closed by the present theorem. --- # 32. RIB normal forms Once a native residual exists, relative invisibility reduces to: ### RIB-K compact family with a true singular limit kernel; ### RIB-NC operator/residual family noncompactness; ### RIB-P weak normalized residual pairing degeneration. If every normalized limit operator is kernel-free and the normalized residual/operator families are compact, RIB is excluded. --- # 33. External finite-window alignment The finite-window literature already proves: - NS-generated source/residual coordinate realizability on explicit classes; - fixed-window anti-phantom detection under clean-gap/visibility hypotheses; - finite-dimensional equivalence between kernel-freeness and finite observability; - conditional finite-chain CKN-compatible audit defect extraction; - relative left-singular failure as the final trace obstruction. The present paper does not replace those hypotheses. It turns the moving-window burst problem into their exact native inputs. --- # 34. IDRP-03 strongest positive result The generic BDR problem is no longer one black box. A causal source burst is natively realized whenever: $$ \boxed{ \text{source transversality} + \text{dual compatibility} + \text{residual smallness} } $$ hold. And once a native residual is realized, a compact normalized kernel-free operator family cannot support RIB. --- # 35. IDRP-03 strongest remaining gap The unresolved PDE content is now: $$ \boxed{ \textbf{TRAN} } $$ source burst transversality to the clean model class; $$ \boxed{ \textbf{DUAL} } $$ compatibility between ANP/source dual propagation and PFET finite-window adjoint geometry; $$ \boxed{ \textbf{KERN} } $$ exclusion of singular limit combined operators on NS-realizable normalized classes; $$ \boxed{ \textbf{AMP} } $$ physical amplitude/depletion packing. --- # 36. Conditional relative-burst closure ## Theorem 36.1 Assume along a recurrent source-burst branch: 1. uniform source transversality: $$ J_{\perp,n}\ge\theta_0J_n; $$ 2. bounded source-dual cost; 3. causal/audit dual mismatch and residual-ledger pairings are: $$ o(J_n); $$ 4. normalized source/audit residuals are precompact; 5. normalized combined operator family is precompact; 6. every limit combined operator is kernel-free; 7. the physical amplitude/depletion series is non-summable. Then the recurrent source-burst branch is impossible. ### Proof Items 1--3 give native BDR. Items 4--6 exclude RIB by Theorem 18.1. The resulting uniformly observed native residual enters the physical depletion law. Item 7 exhausts the selected finite budget. $\square$ ### Safety This is a conditional compiler. Items 1--7 are not universally proved. --- # 37. Next paper The next paper should attack the remaining PDE content rather than the finite-dimensional operator algebra: $$ \boxed{ \textbf{ NS-IDRP 04 — Source Transversality、 Adjoint Compatibility、 Singular Limit Kernels、 Physical Burst Amplitude 與 Cycle-IX Closure Audit }. } $$ Primary tasks: 1. derive source-transversality from the actual NS quadratic source geometry; 2. compare ANP dual propagators with the finite-window linearized PFET adjoint; 3. identify whether source-tangent bursts are already absorbed by active pressure/model covariance; 4. study singular limit combined operators on the smooth/reduced NS-generated package class; 5. combine model-cone and increment channels with the limit kernel; 6. decide whether physical amplitude decay can remain summable after relative kernel exclusion; 7. close Cycle IX or isolate a canonical singular-limit impulsive phantom. --- # 38. Formal status ledger $$ \boxed{ \begin{aligned} \text{Source-Quotient BDR} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Source-Native BDR Compiler} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Causal-to-Audit Dual Compatibility} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{intrinsic residual normalization} &:\ \mathrm{DEFINED/VALID},\\ \text{Compact Operator-Family RIB No-Go} &:\ \mathrm{PROVED},\\ \text{Strong RIB Compactification} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Relative-Failure-to-Phantom No-Go} &:\ \mathrm{PROVED},\\ \text{Amplitude-Normalization No-Free-Lunch} &:\ \mathrm{PROVED},\\ \text{generic BDR} &:\ \mathrm{OPEN},\\ \text{source transversality} &:\ \mathrm{OPEN},\\ \text{causal/audit dual compatibility as a PDE theorem} &:\ \mathrm{OPEN},\\ \text{singular limit kernel exclusion} &:\ \mathrm{OPEN},\\ \text{physical amplitude Critical Lift} &:\ \mathrm{OPEN},\\ \text{Relative Invisible Burst exclusion} &:\ \mathrm{OPEN/PARTIAL},\\ \text{Impulsive Diffuse Recurrence exclusion} &:\ \mathrm{OPEN},\\ \text{Forest Coercive Budget} &:\ \mathrm{OPEN},\\ \text{Finite Forest Obstruction} &:\ \mathrm{OPEN},\\ CN3_{\rm Atomic} &:\ \mathrm{OPEN},\\ \text{Navier--Stokes regularity} &:\ \mathrm{NOT\ PROVED}. \end{aligned} } $$ --- # 39. Conclusion IDRP-03 converts the relative invisible burst problem into a native quotient and operator-limit problem. A causal source burst does not automatically imply a finite-window audit defect. But if the induced source dual has a fixed transverse component to the clean/model source space, elementary quotient duality gives a quantitative native source defect. If the causal source dual also matches the audit source dual up to controlled error, this native source defect becomes a genuine finite-window active residual. Thus BDR can be closed on a precise source-transverse/dual-compatible branch without copying the burst amplitude into the detector. After a native residual exists, intrinsic amplitude normalization is legitimate. The moving-window invisibility problem then becomes finite-dimensional operator geometry. A precompact normalized family of combined observability operators whose every cluster limit is kernel-free has a uniform positive minimum singular value. Such a family cannot support relative invisible bursts. Conversely, a strong relative invisible burst in a compact family converges to a genuine nonzero limit kernel/phantom pair. A weak relative failure need not do so: the singular value may collapse faster than the normalized residual pairing, leaving a limit kernel with no true residual coupling. Finally, amplitude normalization does not create physical depletion. The actual budget retains the residual factor: $$ \rho_n^q. $$ The remaining problem is therefore sharply separated into: $$ \boxed{ \textbf{ source transversality + dual compatibility + limit-kernel rigidity + physical amplitude packing. } } $$ That is the target of IDRP-04. --- # References 1. R. Yu, *Finite-Window Local-to-Clean Transfer and Anti-Phantom Detection for Sharp Navier--Stokes Packages*, arXiv:2606.18476. 2. R. Yu, *Finite-Window Recursive Audit Chains for Navier--Stokes Generated Packages*, arXiv:2606.20899. 3. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756. 4. T. Barker, H. Popkin, *Quantitative estimates for the forced Navier--Stokes equations and applications*, arXiv:2602.09951. 5. `NS_IDRP_02_BurstVisibility_MovingWindow_v0.1.md`. 6. `NS_DCRP_CYCLE_VIII_HANDOFF_v1.0.md`.