# NS-DCRP-21 — Far-Field Annular Escape, Core-Profile Collapse, and Harmonic-Jet Reduction to Spatial Infinity - date: 2026-08-17 - status: research proof checkpoint - canonical source: UTF-8 Markdown - canonical math delimiters: `$...$` and `$$...$$` - objective: 1. attack the DCRP-20 far-field-only survivor without assuming an unproved affine-jet cancellation; 2. combine the exact annular reassignment formula with the already-proved collapse of the core filtered-enstrophy reservoir; 3. prove that a persistent far-field stretching surplus forces the source annulus to escape to infinite relative spatial radius with diverging normalized annular vorticity amplitude; 4. show that bounded-relative harmonic affine jets cannot be the final zero-cost survivor. - no full Navier--Stokes regularity claim is made. - principal external primary source: - Runlong Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier--Stokes Equations*, arXiv:2606.27560v1. - internal dependencies: - DCRP-18 two-sided scale/spatial completion; - DCRP-20 filtered-enstrophy diffusion/IR dichotomy and far-field-only survivor reduction. - no novelty/priority claim is made without independent audit. --- # 1. Executive result DCRP-20 reduced the zero-cost/no-IR filtered-vorticity branch to a far-field-only survivor: $$ \boxed{ O_k\to0, \qquad V_k^{+,\mathrm{far}} \ge b_0>0. } \tag{1.1} $$ The external far-field paper gives the annular reassignment bound $$ \boxed{ \mu_k^{\mathrm{far,ann}} \le C_0 \sum_{j=0}^{k} 2^{-(k-j)} \mathfrak A_{j,k} \mathcal Q_k, } \tag{1.2} $$ where $$ \boxed{ \mathfrak A_{j,k} = \left( r_j^{-1} \iint_{I_k\times\widetilde A_j} |\Omega_k|^2 \right)^{1/2}, } \tag{1.3} $$ and $$ \boxed{ \mathcal Q_k = \left[ \int_{I_k} \left( \int_{B_{2r_k}} \chi_k|\Omega_k|^2dx \right)^2dt \right]^{1/2}. } \tag{1.4} $$ The first new theorem of this round is the core time-profile estimate $$ \boxed{ \mathcal Q_k \le C_{\sigma} M_k^{1/2} O_k^{1/2}, } \tag{1.5} $$ where $$ M_k $$ is the fixed-relative local kinetic-energy bound used in the filtered-vorticity theorem. Therefore $$ \boxed{ O_k\to0 \Longrightarrow \mathcal Q_k\to0. } \tag{1.6} $$ This is stronger than the observation in DCRP-20 that only the time-integrated core reservoir vanishes. The second new theorem is the far-field annular amplification theorem. Assume: $$ V_k^{+,\mathrm{far}} \ge b_0>0, $$ the exterior tail beyond a fixed physical base radius is separated as in the external far-field decomposition, and $$ O_k\to0. $$ The exterior tail satisfies $$ \boxed{ V_k^{+,\mathrm{ext}} \le C r_k O_k \to0. } \tag{1.7} $$ Hence for sufficiently large: $$ k, $$ $$ \boxed{ \mu_k^{\mathrm{far,ann}} \ge \frac{b_0}{2}. } \tag{1.8} $$ Using (1.2) and (1.5), $$ \boxed{ \sum_{j=0}^{k} 2^{-(k-j)} \mathfrak A_{j,k} \ge \frac{ c\,b_0 }{ M_k^{1/2} O_k^{1/2} }. } \tag{1.9} $$ Since: $$ \sum_{m=0}^{\infty}2^{-m}=2, $$ there exists: $$ j_k\le k $$ such that: $$ \boxed{ \mathfrak A_{j_k,k} \ge \frac{ c\,b_0 }{ M_k^{1/2} O_k^{1/2} }. } \tag{1.10} $$ Thus if: $$ \sup_k M_k<\infty, $$ $$ \boxed{ \mathfrak A_{j_k,k} \to\infty. } \tag{1.11} $$ The third new theorem shows that this amplified annulus cannot remain at bounded relative spatial distance from the core. Let: $$ m=k-j. $$ For every fixed: $$ M<\infty, $$ assume the local energy on the fixed enlarged normalized ball is uniformly bounded: $$ \boxed{ \sup_k M_k^{(M)} < \infty. } \tag{1.12} $$ Then for: $$ 0\le m\le M, $$ the local filter smoothing bound gives: $$ \boxed{ \mathfrak A_{k-m,k} \le C_{\sigma,M} \left( M_k^{(M)} \right)^{1/2}. } \tag{1.13} $$ Consequently the annuli selected in (1.10) must satisfy: $$ \boxed{ m_k = k-j_k \to\infty. } \tag{1.14} $$ Equivalently: $$ \boxed{ \frac{ r_{j_k} }{ r_k } = 2^{m_k} \to\infty. } \tag{1.15} $$ Hence: $$ \boxed{ \textbf{ persistent far-field work with a collapsing core enstrophy profile forces the source vorticity reservoir to escape to normalized spatial infinity. } } \tag{1.16} $$ Moreover its normalized annular amplitude diverges. This result changes the interpretation of the harmonic-jet frontier. The external paper correctly notes that fixed exterior annular sources generate harmonic strain fields in the core and that low-order affine jets are the modes that can recur across nested scales. DCRP-21 proves: $$ \boxed{ \textbf{ an affine jet sourced at bounded relative spatial radius cannot sustain the DCRP-20 far-field-only survivor. } } \tag{1.17} $$ If a fixed-relative source annulus remains inside: $$ |y-x_0| \lesssim 2^M r_k, $$ its normalized annular reservoir is uniformly bounded by local energy and filter smoothing, while the core profile: $$ \mathcal Q_k $$ tends to zero. Its work therefore tends to zero. Thus any recurrent affine harmonic jet capable of paying: $$ V_k^{+,\mathrm{far}} \ge b_0 $$ must be sourced at: $$ \boxed{ \frac{ |y-x_0| }{ r_k } \to\infty. } \tag{1.18} $$ This is not a mysterious finite-dimensional jet recurrence. It is an exterior-source spatial-escape branch. Therefore the DCRP-20 far-field-only survivor reduces further to: $$ \boxed{ \textbf{ spatial-infinity annular vorticity amplification. } } \tag{1.19} $$ In a transition-complete package that retains: - absolute annular filtered-vorticity amplitude; - normalized spatial source position; - the point at spatial infinity; one has: $$ \boxed{ \textbf{ zero spatial-defect branch} \Longrightarrow V_k^{+,\mathrm{far}}\to0. } \tag{1.20} $$ Combining with DCRP-20: $$ \boxed{ \textbf{ zero diffusion} + \textbf{ zero IR-frequency defect} + \textbf{ zero commutator defect} + \textbf{ zero localization} + \textbf{ zero spatial-source escape} \Longrightarrow \textbf{ no positive filtered-enstrophy surplus}. } } \tag{1.21} $$ Thus the far-field harmonic-jet obstruction is closed **at the level of compactness alternatives**. The remaining major bridge is no longer a stretching decomposition. It is: $$ \boxed{ \textbf{ Singular/CKN Badness} \Longrightarrow \textbf{ Persistent Filtered-Enstrophy Surplus or an Already-Paid Defect}. } } \tag{1.22} $$ Equivalently, the next question is whether every singular local branch must actually activate the filtered-vorticity mechanism strongly enough for the now-closed mechanism decomposition to apply. A useful next target is: $$ \boxed{ \textbf{ Local Supplier / Filtered-Enstrophy Activation Lemma}. } \tag{1.23} $$ The DCRP-16 supplier theorem is a natural starting point. --- # 2. External annular reassignment audited The external paper defines: $$ I_k = (t_0-r_k^2,t_0), $$ and for: $$ j\le k, $$ $$ \boxed{ \widetilde A_j = \left\{ y: (\Gamma-1)r_j < |y-x_0| \le (2\Gamma+1)r_j \right\}. } \tag{2.1} $$ The reassigned annular reservoir is: $$ \boxed{ \mathfrak A_{j,k} = \left( r_j^{-1} \iint_{I_k\times\widetilde A_j} |\Omega_k|^2 \right)^{1/2}. } \tag{2.2} $$ The core time profile is: $$ \boxed{ \mathcal Q_k = \left[ \int_{I_k} \left( \int_{B_{2r_k}} \chi_k|\Omega_k|^2dx \right)^2 dt \right]^{1/2}. } \tag{2.3} $$ The exact moving-shell reassignment estimate is: $$ \boxed{ \mu_k^{\mathrm{far,ann}} \le C_0 \sum_{j=0}^{k} 2^{-(k-j)} \mathfrak A_{j,k} \mathcal Q_k. } \tag{2.4} $$ The dyadic weight is summable: $$ \boxed{ \sum_{j=0}^{k} 2^{-(k-j)} < 2. } \tag{2.5} $$ This summability is the key new leverage once: $$ \mathcal Q_k $$ is shown to vanish. --- # 3. Core filtered-vorticity profile Let: $$ \boxed{ F_k(t) = \int_{B_{2r_k}} \chi_k(x,t) |\Omega_k(x,t)|^2dx. } \tag{3.1} $$ Then: $$ \boxed{ O_k = r_k^{-1} \int_{I_k} F_k(t)dt. } \tag{3.2} $$ Also: $$ \boxed{ \mathcal Q_k = \|F_k\|_{L_t^2(I_k)}. } \tag{3.3} $$ The problem is that in general: $$ L_t^1\to0 $$ does not imply: $$ L_t^2\to0. $$ The filtered-vorticity smoothing bound supplies the missing: $$ L_t^\infty $$ control. --- # 4. NEW THEOREM — Core Time-Profile Collapse ## Theorem 4.1 Assume: $$ \ell_k = \sigma r_k $$ with fixed: $$ \sigma>0. $$ Assume the fixed-relative local kinetic-energy coordinate satisfies: $$ \boxed{ M_k \le M_\ast. } \tag{4.1} $$ Then: $$ \boxed{ \mathcal Q_k \le C_{\sigma} M_\ast^{1/2} O_k^{1/2}. } \tag{4.2} $$ In particular: $$ \boxed{ O_k\to0 \Longrightarrow \mathcal Q_k\to0. } \tag{4.3} $$ ### Proof The local filtered-vorticity bound gives: $$ \boxed{ \|\Omega_k(t)\|_{L^\infty(B_{2r_k})} \le C_\sigma M_\ast^{1/2} r_k^{-2}. } \tag{4.4} $$ Therefore: $$ F_k(t) \le C r_k^3 \|\Omega_k(t)\|_\infty^2 \le C_\sigma M_\ast r_k^{-1}. $$ Hence: $$ \boxed{ \|F_k\|_{L^\infty_t} \le C_\sigma M_\ast r_k^{-1}. } \tag{4.5} $$ Now: $$ \mathcal Q_k^2 = \int F_k^2dt \le \|F_k\|_\infty \int F_kdt. $$ But: $$ \int F_kdt = r_kO_k. $$ Therefore: $$ \mathcal Q_k^2 \le C_\sigma M_\ast O_k. $$ Take square roots. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 5. Exterior tail beyond the fixed base radius The annular reassignment of the external paper treats the source shells: $$ 0\le m\le k. $$ The more distant shells: $$ m>k $$ lie beyond a fixed physical base radius comparable to: $$ r_0. $$ Let: $$ S_k^{\mathrm{ext}} $$ be the filtered strain generated from source points separated from the core by at least: $$ c r_0. $$ The strain-kernel: $$ L^2 $$ tail gives: $$ \boxed{ \| K\mathbf1_{|z|>cr_0} \|_2 \le C r_0^{-3/2}. } \tag{5.1} $$ The global filtered-vorticity bound gives: $$ \boxed{ \|\Omega_k(t)\|_2 \le C \ell_k^{-1} \|u(t)\|_2 \le C_\sigma r_k^{-1} M_E^{1/2}. } \tag{5.2} $$ Therefore: $$ \boxed{ \|S_k^{\mathrm{ext}}(t)\|_\infty \le C_\sigma r_0^{-3/2} r_k^{-1} M_E^{1/2}. } \tag{5.3} $$ The normalized exterior positive work obeys: $$ \begin{aligned} V_k^{+,\mathrm{ext}} &\le r_k \|S_k^{\mathrm{ext}}\|_\infty \iint_{Q_k} \chi_k|\Omega_k|^2 \\ &= r_k \|S_k^{\mathrm{ext}}\|_\infty (r_kO_k). \end{aligned} $$ Hence: $$ \boxed{ V_k^{+,\mathrm{ext}} \le C_\sigma r_0^{-3/2} M_E^{1/2} r_k O_k. } \tag{5.4} $$ Thus if: $$ O_k $$ is bounded, and in particular if: $$ O_k\to0, $$ $$ \boxed{ V_k^{+,\mathrm{ext}}\to0. } \tag{5.5} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 6. NEW THEOREM — Annular Source Amplification ## Theorem 6.1 Assume: $$ \boxed{ V_k^{+,\mathrm{far}} \ge b_0>0, } \tag{6.1} $$ $$ \boxed{ O_k\to0, } \tag{6.2} $$ and: $$ \boxed{ M_k\le M_\ast. } \tag{6.3} $$ Then, after discarding finitely many: $$ k, $$ there exists: $$ j_k\le k $$ such that: $$ \boxed{ \mathfrak A_{j_k,k} \ge \frac{ c\,b_0 }{ M_\ast^{1/2} O_k^{1/2} }. } \tag{6.4} $$ Consequently: $$ \boxed{ \mathfrak A_{j_k,k} \to\infty. } \tag{6.5} $$ ### Proof By Theorem 5.1: $$ V_k^{+,\mathrm{ext}}\to0. $$ The far-field positive work is bounded above by the annular absolute contribution plus the exterior tail budget. Therefore for sufficiently large: $$ k, $$ $$ \mu_k^{\mathrm{far,ann}} \ge \frac{b_0}{2}. $$ Apply the external annular reassignment estimate: $$ \frac{b_0}{2} \le C_0 \mathcal Q_k \sum_{j=0}^{k} 2^{-(k-j)} \mathfrak A_{j,k}. $$ By Theorem 4.1: $$ \mathcal Q_k \le C_\sigma M_\ast^{1/2} O_k^{1/2}. $$ Thus: $$ \sum_{j=0}^{k} 2^{-(k-j)} \mathfrak A_{j,k} \ge \frac{ c\,b_0 }{ M_\ast^{1/2} O_k^{1/2} }. $$ Since the weights sum to less than two: $$ \max_{0\le j\le k} \mathfrak A_{j,k} \ge \frac12 \sum_{j=0}^{k} 2^{-(k-j)} \mathfrak A_{j,k}. $$ Choose: $$ j_k $$ realizing the maximum. This proves (6.4). $$ \square $$ Status: $$ \boxed{ \textbf{PROVED using Proposition 8.6 of arXiv:2606.27560 plus Theorems 4.1 and 5.1 above}. } $$ --- # 7. Fixed-relative annuli cannot amplify without bound Let: $$ m=k-j. $$ Then: $$ r_j = 2^m r_k. $$ Fix: $$ M<\infty. $$ For: $$ 0\le m\le M, $$ the annulus: $$ \widetilde A_{k-m} $$ lies inside a fixed enlarged normalized ball: $$ B_{R_Mr_k}(x_0), $$ where: $$ R_M $$ depends only on: $$ M $$ and: $$ \Gamma. $$ Assume: $$ \boxed{ M_k^{(M)} := r_k^{-1} \operatorname*{ess\,sup}_{t\in I_k} \int_{ B_{R_Mr_k}(x_0) } |u(x,t)|^2dx \le M_M. } \tag{7.1} $$ The local filter smoothing estimate gives: $$ \boxed{ \|\Omega_k(t)\|_{ L^\infty(B_{R_Mr_k}) } \le C_{\sigma,M} M_M^{1/2} r_k^{-2}. } \tag{7.2} $$ --- # 8. NEW THEOREM — Bounded-Relative Annular Reservoir Bound ## Theorem 8.1 Under (7.1), for every: $$ 0\le m\le M, $$ $$ \boxed{ \mathfrak A_{k-m,k} \le C_{\sigma,\Gamma,M} M_M^{1/2} 2^m. } \tag{8.1} $$ In particular: $$ \boxed{ \sup_k \max_{0\le m\le M} \mathfrak A_{k-m,k} < \infty. } \tag{8.2} $$ ### Proof The annulus: $$ \widetilde A_{k-m} $$ has volume: $$ \boxed{ |\widetilde A_{k-m}| \le C_\Gamma r_{k-m}^3 = C_\Gamma 2^{3m} r_k^3. } \tag{8.3} $$ The time interval has length: $$ |I_k| = r_k^2. $$ Therefore: $$ \begin{aligned} \mathfrak A_{k-m,k}^2 &= r_{k-m}^{-1} \iint_{ I_k\times\widetilde A_{k-m} } |\Omega_k|^2 \\ &\le r_{k-m}^{-1} r_k^2 C_\Gamma r_{k-m}^3 \|\Omega_k\|_\infty^2 \\ &\le C r_k^2 r_{k-m}^2 \left[ M_M r_k^{-4} \right] \\ &= C M_M \left( \frac{ r_{k-m} }{ r_k } \right)^2 \\ &= C M_M 2^{2m}. \end{aligned} $$ Take square roots. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 9. NEW THEOREM — Far-Field Source Spatial Escape ## Theorem 9.1 Assume the hypotheses of Theorem 6.1. Assume in addition that for every fixed: $$ M<\infty, $$ the enlarged local-energy bound: $$ \sup_kM_k^{(M)}<\infty $$ holds. Let: $$ j_k $$ be the amplified annulus supplied by Theorem 6.1 and define: $$ \boxed{ m_k = k-j_k. } \tag{9.1} $$ Then: $$ \boxed{ m_k\to\infty. } \tag{9.2} $$ Equivalently: $$ \boxed{ \frac{ r_{j_k} }{ r_k } \to\infty. } \tag{9.3} $$ ### Proof Suppose not. Then after a subsequence: $$ m_k\le M $$ for some fixed: $$ M. $$ Theorem 8.1 gives: $$ \sup_k \mathfrak A_{j_k,k} < \infty. $$ But Theorem 6.1 gives: $$ \mathfrak A_{j_k,k}\to\infty. $$ Contradiction. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 10. Quantitative spatial-escape strength Theorem 6.1 actually gives more than: $$ m_k\to\infty. $$ The selected annular reservoir satisfies: $$ \boxed{ \mathfrak A_{j_k,k} \gtrsim O_k^{-1/2}. } \tag{10.1} $$ up to the fixed local-energy and far-surplus constants. Thus the source does not merely move outward. Its normalized annular filtered-vorticity amplitude diverges while its relative radius diverges. Therefore the survivor is: $$ \boxed{ \textbf{ spatial escape} + \textbf{ annular critical-amplitude blowup}. } \tag{10.2} $$ This is substantially more rigid than a bounded external harmonic background. --- # 11. Harmonic affine-jet interpretation The external paper replaces moving shells by a fixed smooth annular partition: $$ \psi_j(y) $$ supported where: $$ |y-x_0| \simeq r_j, $$ and defines: $$ \boxed{ H_{j,k}(x,t) = \int K(x-y) \psi_j(y) \Omega_k(y,t)dy. } \tag{11.1} $$ For: $$ j If the core filtered-enstrophy time profile collapses, then any affine jet sourced at bounded relative radius has vanishing work. Thus the only affine jets relevant to the DCRP-20 far-field survivor are sourced at unbounded relative distance. Those are already a spatial noncompactness phenomenon. Therefore, on a transition-complete branch satisfying: $$ \boxed{ \text{no spatial-source escape}, } \tag{13.1} $$ one does not need a separate universal affine-cancellation theorem. The far-field source is forced into the near/finite-relative compact sector, where its work vanishes because: $$ \mathcal Q_k\to0. $$ This is an alternative closure route to the paper's proposed affine-jet cancellation module. --- # 14. Spatial carrier completion For the far-field source, a natural native carrier is the family: $$ \boxed{ \left( m, \mathfrak A_{k-m,k} \right), \qquad m\in\mathbb N_0. } \tag{14.1} $$ Compactify relative source radius by: $$ \boxed{ \overline{\mathbb N}_0^{sp} = \mathbb N_0 \cup \{ +\infty_{sp} \}. } \tag{14.2} $$ Retain separately: 1. normalized source-position distribution; 2. absolute annular amplitude. Theorem 9.1 says that a far-field-only survivor produces: $$ \boxed{ +\infty_{sp} } $$ with divergent absolute amplitude. This is a native PDE-generated spatial carrier. It does not copy a singularity label. --- # 15. Transition-complete zero-spatial-defect implication Suppose a normalized filtered mechanism sequence satisfies the DCRP-20 zero-cost conditions: $$ P_k\to0, $$ $$ \widetilde{\mathcal S}^{(3)}_k\to0, $$ $$ L_k+L_k^{\mathrm{com}}+L_k^\omega\to0, $$ no IR-frequency defect, and fixed-relative local-energy bounds. DCRP-20 gives: $$ O_k\to0. $$ If the completed spatial-source carrier also has no defect at: $$ +\infty_{sp}, $$ then Theorem 9.1 rules out: $$ V_k^{+,\mathrm{far}}\ge b_0. $$ Therefore: $$ \boxed{ V_k^{+,\mathrm{far}}\to0. } \tag{15.1} $$ Together with DCRP-20: $$ \boxed{ V_k^{+,\mathrm{near}}\to0, } \tag{15.2} $$ $$ \boxed{ F_k^{\mathrm{com}}\to0, } \tag{15.3} $$ and: $$ \boxed{ L_k\to0. } \tag{15.4} $$ Hence: $$ \boxed{ \textbf{ all positive filtered-vorticity mechanism channels vanish. } } \tag{15.5} $$ Status: $$ \boxed{ \textbf{PROVED at the mechanism-package level under the stated zero-defect compactness assumptions}. } $$ --- # 16. Filtered-surplus consequence Let: $$ \mathfrak B_k $$ be the post-near-field filtered-enstrophy surplus used in DCRP-20 and in the external filtered-vorticity theorem. Under the zero-cost/no-IR/no-spatial-defect hypotheses above: $$ V_k^{+,\mathrm{far}} \to0, $$ $$ \widetilde{\mathcal S}^{(3)}_k \to0, $$ and all localization terms vanish. Therefore: $$ \boxed{ \mathfrak B_k\to0. } \tag{16.1} $$ Thus: $$ \boxed{ \textbf{ a persistent positive filtered-enstrophy surplus cannot be an exact zero-cost compact obstruction. } } \tag{16.2} $$ This substantially closes the mechanism decomposition. --- # 17. What this does not yet prove A singular suitable weak solution is known to remain CKN-bad at every sufficiently small scale around a singular point. But the current chain has not yet proved the implication: $$ \boxed{ \text{persistent CKN badness} \Longrightarrow \inf_k \mathfrak B_k > 0. } \tag{17.1} $$ Nor has it proved that every local supplier event forces a fixed positive: $$ \mathfrak B_k $$ at a comparable filtered scale. Therefore eliminating a hypothetical persistent positive filtered-vorticity surplus does not yet eliminate every possible singular branch. This is now the principal interface gap. --- # 18. Why this is the correct next gap The external structural program already separates: - full CKN badness; - coarse resolved badness; - subfilter residual badness. DCRP-19 reduced full critical supply to: - transition influx; - coarse resolved mechanism; - subfilter residual. DCRP-20/21 now substantially close the **filtered-vorticity mechanism** whenever it is activated. The remaining question is whether singularity must activate that mechanism at a fixed critical strength. This is a detector-to-mechanism lower-bound problem, not another decomposition problem. --- # 19. Supplier route as the activation candidate DCRP-16 proves that every first singular point admits: $$ t_n\uparrow T, $$ $$ x_n\to x_\ast, $$ $$ \lambda_n\to\infty, $$ with: $$ \boxed{ \lambda_n^{-1} | \Delta_{\lambda_n}u(x_n,t_n) | \ge c_{\rm loc}\nu. } \tag{19.1} $$ DCRP-09/14 then produce an actual same-history nonlinear increment at the same scale. A band-limited divergence-free supplier also has the global Fourier identity: $$ \boxed{ \| \nabla\times u_q \|_2^2 \asymp \lambda_q^2 \|u_q\|_2^2. } \tag{19.2} $$ Together with: $$ \lambda_q \|u_q\|_2^2 \gtrsim \nu^2, $$ this gives a critical instantaneous vorticity-shell lower bound. The unresolved part is to convert this instantaneous bandpass vorticity atom into a **fixed spacetime filtered-enstrophy surplus**: $$ \mathfrak B_k\ge b_0. $$ This is where possible ultrashort temporal spikes and low-pass/bandpass cancellation still matter. --- # 20. New exact frontier The next target is: $$ \boxed{ \textbf{ Local Supplier / Filtered-Enstrophy Activation Lemma}. } $$ A useful sufficient statement is: > Let: > > $$ > (x_n,t_n,\lambda_n) > $$ > > be the local supplier sequence of DCRP-16. > > Then after passing to: > > $$ > r_n\asymp\lambda_n^{-1}, > \qquad > \ell_n=\sigma r_n, > $$ > > at least one of: > > 1. a fixed positive post-near-field filtered-enstrophy surplus: > > $$ > \mathfrak B_n\ge b_0; > $$ > > 2. a fixed positive filtered diffusion cost; > 3. a derivative-compatible commutator defect; > 4. a localization/pressure residual; > 5. a temporal concentration defect; > > occurs. If alternative 1 occurs, DCRP-20/21 eliminate the zero-cost compact branch. Alternatives 2--5 are already paid/native defect channels after completion. This would finally connect local singular supplier capture to the now-closed filtered-vorticity mechanism calculus. --- # 21. Possible temporal concentration coordinate The main technical difference between a supplier endpoint and the filtered-enstrophy surplus is time. A supplier may, a priori, be a short spike. Define the normalized bandpass enstrophy profile: $$ \boxed{ e_n(\tau) = \int_{B_R} | \omega_{q_n}^{(n)}(y,\tau) |^2dy. } \tag{21.1} $$ At the supplier endpoint: $$ \boxed{ e_n(0)\ge c\nu^2. } \tag{21.2} $$ There are two possibilities. ### positive normalized residence For some fixed: $$ \tau_0>0, $$ $$ \boxed{ \int_{-\tau_0}^{0} e_n(\tau)d\tau \ge c_0>0. } \tag{21.3} $$ Then a fixed filtered/bandpass enstrophy spacetime reservoir is activated. ### temporal concentration For every fixed: $$ \tau_0>0, $$ the profile mass collapses toward: $$ \tau=0. $$ Then the supplier produces a nontrivial temporal concentration defect. A transition-complete package should retain this concentration rather than silently lose it. Thus even before a quantitative residence-time theorem, the activation problem admits a compactness alternative. --- # 22. Why the harmonic-jet frontier has changed The external paper states that a complete harmonic-rigidity theorem should control affine jets from a fixed annular source decomposition directly. DCRP-21 does not prove that general theorem. Instead, in the specific DCRP zero-core-reservoir regime, it proves: $$ \boxed{ \text{bounded-relative source} \Longrightarrow \text{bounded annular amplitude} \Longrightarrow \text{vanishing far work}. } \tag{22.1} $$ Therefore the only harmonic jets still relevant to the DCRP survivor are those whose **source annuli themselves escape to normalized spatial infinity**. This is a stronger classification in the specific zero-cost branch, but it does not supersede the external paper's general harmonic-jet problem for arbitrary filtered flows. --- # 23. Source ledger ## Filtered Vortex Stretching and Subgrid Defects The following primary results are used: ### far-field moving-shell decomposition $$ \mathbb S_k^{far} = \sum_m \mathbb S_{k,m}. $$ ### bounded-overlap annular reassignment The moving shell at relative separation: $$ m $$ is contained in a fixed annulus at scale: $$ r_{k-m}, $$ and the fixed annuli have uniformly bounded overlap. ### exact reassigned bound $$ \mu_k^{far,ann} \le C \sum_{j=0}^k 2^{-(k-j)} \mathfrak A_{j,k} \mathcal Q_k. $$ ### fixed-annulus harmonic route A fixed exterior annular source generates a smooth harmonic strain in the smaller core, and after subtraction of its affine Taylor jet the higher-order remainder gains powers of scale separation. The paper explicitly does not prove unconditional affine-jet cancellation. DCRP-21 uses the exact annular bound, not an assumed cancellation theorem. --- # 24. End state The far-field-only survivor from DCRP-20 has been reduced to a spatial-escape object. The key new estimates are: $$ \boxed{ \mathcal Q_k \le C_\sigma M_\ast^{1/2} O_k^{1/2}, } $$ and, if: $$ V_k^{+,\mathrm{far}} \ge b_0, \qquad O_k\to0, $$ then there exists: $$ j_k\le k $$ such that: $$ \boxed{ \mathfrak A_{j_k,k} \gtrsim O_k^{-1/2} \to\infty, } $$ and: $$ \boxed{ k-j_k\to\infty. } $$ Thus: $$ \boxed{ \textbf{ far-field survivor} \Longrightarrow \textbf{ spatial-infinity annular vorticity amplification}. } $$ Consequently a zero-spatial-defect, zero-IR, zero-diffusion, zero-commutator, zero-localization branch has: $$ \boxed{ \mathfrak B_k\to0. } $$ The next single frontier is: $$ \boxed{ \textbf{ Local Supplier / Filtered-Enstrophy Activation Lemma}. } $$ The mechanism decomposition is now substantially closed. The remaining question is whether a singular branch must activate it at a fixed critical strength, or else leave a temporal/paid defect.