# NS-DCRP-31 — DSS Radial PFET Rigidity, Far-Tail Pressure Decoupling, and the Core-to-Tail Matching Flux - date: 2026-08-17 - status: research proof checkpoint - canonical source: UTF-8 Markdown - canonical math delimiters: `$...$` and `$$...$$` - objective: 1. attack the DCRP-30 critical-tail DSS Euler survivor; 2. test the proposed idea that the infinite tail must directly force the core through a large far-field pressure multipole; 3. prove the exact period-averaged radial energy-flux identity for Euler DSS profiles; 4. show that a smooth nonzero core cannot connect to the critical DSS tail without a finite-radius inward Euler pressure--kinetic flux; 5. compress the resulting continuum-radius statement to a finite native PFET witness on compact normalized profile classes; 6. identify the remaining issue as recurrent critical PFET taxation/summability rather than tail invisibility. - no full Navier--Stokes regularity claim is made. - principal external calibration: - D. Chae, T.-P. Tsai, *On discretely self-similar solutions of the Euler equations*, arXiv:1304.7414; - L. Xue, *Discretely self-similar singular solutions for the incompressible Euler equations*, arXiv:1408.6619v2; - D. Chae, *Euler's equations and the maximum principle*, arXiv:1308.1051; - D. Chae, J. Wolf, *On the Discretely Self-similar Solutions to the Euler Equations in R^3*, J. Nonlinear Sci. 33 (2023), 115; - P. Constantin, M. Ignatova, V. Vicol, *On putative self-similarity for incompressible 3D Euler*, arXiv:2602.17570v3. - internal dependencies: - DCRP-29 raw-energy atomic/material split; - DCRP-30 same-parent DSS recurrence and exponent window; - MORP/FCBP pressure--flux--energy--trace observation architecture. - no novelty/priority claim is made without independent audit. - several identities below are direct reorganizations of the standard DSS local-energy equation and are likely implicit in the existing DSS Euler literature. --- # 1. Executive result DCRP-30 reduced the strict compact same-parent Type-II branch to a smooth Euler DSS profile in the exponent window $$ \boxed{ 1<\alpha<\frac32, } \tag{1.1} $$ equivalently $$ \boxed{ \frac25<\gamma<\frac12, \qquad \gamma=\frac1{1+\alpha}, } \tag{1.2} $$ with: - vanishing raw physical core energy; - infinite normalized global kinetic energy; - mandatory global-energy escape to normalized spatial infinity; - non-outgoing/trapped similarity dynamics; - no finite-time material crossing; - no anomalous Navier--Stokes viscous residue; - no Reynolds/trace/localization defect in the strong compact branch. The initial DCRP-31 proposal was: > the infinite DSS tail must exert a non-negligible far-field pressure on the core. That proposal is false in the natural Calderon--Zygmund pressure class. Let $$ \boxed{ \kappa = 3-2\alpha. } \tag{1.3} $$ In the strict window: $$ \boxed{ 0<\kappa<1. } \tag{1.4} $$ Assume the DSS profile satisfies the Xue-type critical energy envelope $$ \boxed{ \sup_{s\in[0,S_0]} \int_{B_R} |V(y,s)|^2dy \le C_E R^\kappa \qquad (R\ge1). } \tag{1.5} $$ Assume the pressure is represented by the standard Calderon--Zygmund formula, modulo the declared harmonic/gauge class. For a fixed core $$ |y|\le R_0 $$ and tail cutoff $$ L\gg R_0, $$ the contribution to the pressure from $$ |z|\ge L $$ obeys $$ \boxed{ \|P_{>L}\|_{ L^\infty( B_{R_0}\times[0,S_0] ) } \le C C_E L^{-2\alpha}, } \tag{1.6} $$ and $$ \boxed{ \|\nabla P_{>L}\|_{ L^\infty( B_{R_0}\times[0,S_0] ) } \le C C_E L^{-1-2\alpha}. } \tag{1.7} $$ Hence $$ \boxed{ \textbf{ the arbitrarily remote infinite-energy tail can be instantaneously pressure-decoupled from a fixed core. } } \tag{1.8} $$ The required infinite Euler tail does **not** imply a large direct pressure multipole at the core. This is the main NO-GO of this round. The actual unavoidable coupling occurs in the finite-radius **core-to-tail matching layer**. The Euler DSS similarity variables satisfy $$ \boxed{ \partial_sV + \frac{\alpha}{\alpha+1}V + \frac1{\alpha+1} (y\cdot\nabla)V + (V\cdot\nabla)V + \nabla P = 0, } \tag{1.9} $$ $$ \boxed{ \nabla\cdot V=0, } \tag{1.10} $$ with period $$ S_0. $$ Define the period-averaged local energy $$ \boxed{ \mathcal E(R) = \int_0^{S_0} \int_{B_R} \frac{|V(y,s)|^2}{2} dyds. } \tag{1.11} $$ Define the period-averaged **physical Euler pressure--kinetic flux** across the fixed similarity sphere: $$ \boxed{ \mathcal F(R) = \int_0^{S_0} \int_{\partial B_R} \left( \frac{|V|^2}{2} + P \right) V\cdot n dSds. } \tag{1.12} $$ Then for almost every $$ R>0, $$ one has the exact identity $$ \boxed{ \mathcal F(R) = \frac1{\alpha+1} \left[ \kappa \mathcal E(R) - R\mathcal E'(R) \right]. } \tag{1.13} $$ Equivalently, $$ \boxed{ \mathcal F(R) = - \frac1{\alpha+1} R^{\kappa+1} \frac d{dR} \left[ R^{-\kappa} \mathcal E(R) \right]. } \tag{1.14} $$ This is the central exact identity of DCRP-31. For a smooth core: $$ \boxed{ \mathcal E(R)=O(R^3) \qquad (R\downarrow0). } \tag{1.15} $$ Since: $$ 0<\kappa<1, $$ $$ \boxed{ R^{-\kappa} \mathcal E(R) \to0 \qquad (R\downarrow0). } \tag{1.16} $$ If: $$ V\not\equiv0, $$ there exists a finite: $$ R_1 $$ with: $$ \mathcal E(R_1)>0. $$ Integrating (1.14) from: $$ 0 $$ to: $$ R_1 $$ gives $$ \boxed{ \int_0^{R_1} \left( -\mathcal F(R) \right) R^{-\kappa-1} dR = \frac{ R_1^{-\kappa} \mathcal E(R_1) }{ \alpha+1 }. } \tag{1.17} $$ Consequently $$ \boxed{ \int_0^{R_1} \left( -\mathcal F(R) \right)_+ R^{-\kappa-1} dR \ge \frac{ R_1^{-\kappa} \mathcal E(R_1) }{ \alpha+1 } > 0. } \tag{1.18} $$ Therefore: $$ \boxed{ \textbf{ every nonzero smooth DSS profile with }\alpha<3/2 \textbf{ has a finite-radius inward period-averaged Euler energy flux.} } \tag{1.19} $$ The sign convention is: $$ \mathcal F<0 $$ for inward physical pressure--kinetic energy flux across the outward-oriented sphere. Thus the critical DSS tail cannot be joined to a regular smooth core with all physical PFET channels zero. This result does **not** require the very remote tail pressure to be large. The mandatory flux occurs somewhere in the finite core-to-tail transition region. The exact identity also supplies a rigidity statement. If $$ \boxed{ \mathcal F(R)=0 } \tag{1.20} $$ for every $$ R $$ in an interval: $$ I, $$ then: $$ \boxed{ \mathcal E(R) = C_I R^\kappa \qquad (R\in I). } \tag{1.21} $$ Thus **zero physical radial flux is equivalent to exact critical power-law energy scaling** on each connected zero-flux interval. If: $$ \mathcal F(R)=0 $$ for every: $$ R>0, $$ smoothness at the origin forces: $$ C_I=0, $$ and hence: $$ \boxed{ V\equiv0. } \tag{1.22} $$ This eliminates the exact zero-PFET strict DSS strong profile. A finite compiler version is available. Let: $$ \mathscr C_{\rm DSS} $$ be a sequentially compact normalized class satisfying: -: $$ \alpha\in [1+\delta,3/2-\delta]; $$ -: $$ \mathcal E(R_1)\ge e_0>0; $$ - a uniform local smoothness bound near the core; - fixed translation/pressure gauges. Then there exists: $$ \boxed{ 00 } \tag{1.24} $$ such that every profile in: $$ \mathscr C_{\rm DSS} $$ satisfies $$ \boxed{ \int_{R_0}^{R_1} \left( -\mathcal F(R) \right)_+ R^{-\kappa-1} dR \ge c_{\rm PFET}. } \tag{1.25} $$ Thus one fixed finite annular radial PFET observation detects every nonzero compact strict DSS strong profile. No infinite radius family is needed in the compiler. This gives: $$ \boxed{ \textbf{ strict compact same-parent DSS} + \textbf{ zero PFET} = \varnothing. } \tag{1.26} $$ provided the native PFET package includes the finite annular radial aggregate above, or an equivalent smooth-cutoff realization. The remaining difficulty is **not visibility**. It is taxation/summability. A DSS cascade may repeat a fixed normalized inward flux while the corresponding raw physical energy transfer decreases geometrically with scale. Indeed: $$ \beta_{n+1} \sim q_\ast\beta_n, \qquad 0 can the exact same-parent discrete return convert the mandatory inward PFET matching-layer flux into a strict normalized return tax, rather than merely a geometrically summable raw energy transfer? This is narrower than a tail-pressure or Euler-Liouville problem. --- # 2. External DSS similarity equation For backward Euler DSS with exponent: $$ \alpha>-1, $$ the similarity variables are: $$ y = (-t)^{-1/(\alpha+1)}x, $$ and a logarithmic time: $$ s. $$ The profile is periodic in: $$ s $$ and satisfies: $$ \boxed{ \partial_sV + aV + b(y\cdot\nabla)V + (V\cdot\nabla)V + \nabla P = 0, } \tag{2.1} $$ where: $$ \boxed{ a = \frac{\alpha}{\alpha+1}, } \tag{2.2} $$ and: $$ \boxed{ b = \frac1{\alpha+1}. } \tag{2.3} $$ The profile is divergence free. This is the standard Chae--Tsai/Xue DSS profile equation. --- # 3. Tail energy exponent Define: $$ \boxed{ \kappa = 3-2\alpha. } \tag{3.1} $$ In the DCRP strict Type-II window: $$ 1<\alpha<3/2, $$ so: $$ \boxed{ 0<\kappa<1. } \tag{3.2} $$ Xue's admissible nontrivial DSS profile classes exhibit or are bounded by the critical energy law: $$ \boxed{ \mathcal E(R) \sim R^\kappa } \tag{3.3} $$ under the corresponding global integrability/regularity assumptions. DCRP-31 does not assume the lower asymptotic unless explicitly stated. For the pressure-tail NO-GO, only the upper envelope: $$ \sup_s \int_{B_R}|V|^2 \le C_ER^\kappa $$ is used. --- # 4. Conditional far-tail pressure decomposition Assume the pressure is in the standard Calderon--Zygmund representation class: $$ \boxed{ P(y,s) = -\frac13|V(y,s)|^2 + \operatorname{p.v.} \int K_{ij}(y-z) V_i(z,s)V_j(z,s)dz } \tag{4.1} $$ up to the declared pressure gauge/harmonic class. The kernel satisfies: $$ \boxed{ |K(z)| \le C|z|^{-3}, } \tag{4.2} $$ and: $$ \boxed{ |\nabla K(z)| \le C|z|^{-4}. } \tag{4.3} $$ For a core: $$ |y|\le R_0 $$ and: $$ L\ge4R_0, $$ define the remote tail: $$ \boxed{ P_{>L}(y,s) = \int_{|z|>L} K_{ij}(y-z) V_i(z,s)V_j(z,s)dz. } \tag{4.4} $$ Any harmonic/gauge terms are handled separately in the declared pressure package. --- # 5. NEW THEOREM — Far-Tail Pressure Decoupling ## Theorem 5.1 Assume: $$ \boxed{ \sup_s \int_{B_R} |V(y,s)|^2dy \le C_E R^\kappa } \tag{5.1} $$ for all: $$ R\ge1, $$ with: $$ 0<\kappa<3. $$ Then for every fixed: $$ R_0 $$ and: $$ L\ge4R_0, $$ $$ \boxed{ \sup_s \|P_{>L}(\cdot,s)\|_{L^\infty(B_{R_0})} \le C C_E L^{\kappa-3}. } \tag{5.2} $$ Also: $$ \boxed{ \sup_s \|\nabla P_{>L}(\cdot,s)\|_{L^\infty(B_{R_0})} \le C C_E L^{\kappa-4}. } \tag{5.3} $$ For: $$ \kappa=3-2\alpha, $$ this becomes: $$ \boxed{ P_{>L} = O(L^{-2\alpha}), } \tag{5.4} $$ and: $$ \boxed{ \nabla P_{>L} = O(L^{-1-2\alpha}). } \tag{5.5} $$ ### Proof Decompose the remote region into dyadic shells: $$ A_j = \left\{ 2^jL < |z| \le 2^{j+1}L \right\}. $$ For: $$ |y|\le R_0, $$ and: $$ z\in A_j, $$ $$ |y-z| \ge c2^jL. $$ Therefore: $$ \begin{aligned} |P_{>L}(y,s)| &\le C \sum_{j\ge0} (2^jL)^{-3} \int_{A_j} |V(z,s)|^2dz \\ &\le C C_E \sum_{j\ge0} (2^jL)^{-3} (2^{j+1}L)^\kappa \\ &\le C C_E L^{\kappa-3} \sum_{j\ge0} 2^{-j(3-\kappa)}. \end{aligned} $$ The geometric series converges because: $$ \kappa<3. $$ The gradient bound is identical with: $$ |K| $$ replaced by: $$ |\nabla K|. $$ $$ \square $$ Status: $$ \boxed{ \textbf{PROVED under the stated pressure-representation and energy-envelope assumptions}. } $$ --- # 6. Consequence — remote tail work is not the unavoidable coupling Let: $$ \chi $$ be a fixed core cutoff. If: $$ V $$ is locally bounded in the core, then: $$ \boxed{ \left| \int_0^{S_0} \int P_{>L} V\cdot\nabla\chi dyds \right| \le C_\chi L^{-2\alpha}. } \tag{6.1} $$ Thus: $$ \boxed{ \textbf{ the arbitrarily remote tail cannot be forced to provide a fixed material pressure-work payment. } } \tag{6.2} $$ This invalidates the strongest form of the DCRP-30 tail-pressure proposal. The coupling must be sought at finite relative radius or through scale recurrence. --- # 7. Exact DSS local energy equation Set: $$ \boxed{ e = \frac12 |V|^2. } \tag{7.1} $$ Dot (2.1) with: $$ V. $$ Using: $$ \nabla\cdot V=0, $$ $$ V\cdot (V\cdot\nabla V) = \nabla\cdot(eV), $$ and: $$ V\cdot\nabla P = \nabla\cdot(PV), $$ one gets: $$ \partial_se + 2ae + b y\cdot\nabla e + \nabla\cdot \left[ (e+P)V \right] = 0. $$ Since: $$ y\cdot\nabla e = \nabla\cdot(ye) - 3e, $$ $$ \boxed{ \partial_se + \nabla\cdot \left[ bye + (e+P)V \right] + (2a-3b)e = 0. } \tag{7.2} $$ But: $$ \boxed{ 2a-3b = -\frac{ 3-2\alpha }{ \alpha+1 } = -b\kappa. } \tag{7.3} $$ Thus: $$ \boxed{ \partial_se + \nabla\cdot \left[ bye + (e+P)V \right] - b\kappa e = 0. } \tag{7.4} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 8. Period-averaged local energy Define: $$ \boxed{ \mathcal E(R) = \int_0^{S_0} \int_{B_R} e dyds. } \tag{8.1} $$ For almost every: $$ R, $$ coarea gives: $$ \boxed{ \mathcal E'(R) = \int_0^{S_0} \int_{\partial B_R} e dSds. } \tag{8.2} $$ Define the physical Euler energy current flux: $$ \boxed{ \mathcal F(R) = \int_0^{S_0} \int_{\partial B_R} (e+P) V\cdot n dSds. } \tag{8.3} $$ Because the profile is periodic: $$ e(y,S_0)=e(y,0). $$ Integrate (7.4) over: $$ B_R\times[0,S_0]. $$ The time derivative vanishes. The similarity-drift boundary term is: $$ \boxed{ bR \mathcal E'(R). } \tag{8.4} $$ Therefore: $$ \boxed{ bR \mathcal E'(R) + \mathcal F(R) - b\kappa \mathcal E(R) = 0. } \tag{8.5} $$ --- # 9. NEW THEOREM — DSS Radial PFET Identity ## Theorem 9.1 For almost every: $$ R>0, $$ $$ \boxed{ \mathcal F(R) = b \left[ \kappa \mathcal E(R) - R \mathcal E'(R) \right], } \tag{9.1} $$ where: $$ b = 1/(\alpha+1). $$ Equivalently: $$ \boxed{ \mathcal F(R) = - b R^{\kappa+1} \frac d{dR} \left[ R^{-\kappa} \mathcal E(R) \right]. } \tag{9.2} $$ ### Proof Equation (9.1) is (8.5). For: $$ G(R) = R^{-\kappa} \mathcal E(R), $$ $$ G'(R) = R^{-\kappa-1} \left[ R\mathcal E'(R) - \kappa\mathcal E(R) \right]. $$ Substitute. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 10. Smooth-core behavior Suppose: $$ V $$ is locally bounded near: $$ y=0 $$ uniformly over one period. Then: $$ \boxed{ \mathcal E(R) \le C R^3. } \tag{10.1} $$ For: $$ 0<\kappa<3, $$ $$ \boxed{ R^{-\kappa} \mathcal E(R) \to0 } \tag{10.2} $$ as: $$ R\downarrow0. $$ In the DCRP strict DSS window: $$ 0<\kappa<1, $$ so the conclusion applies. --- # 11. NEW THEOREM — Core-to-Tail Inward PFET Gap ## Theorem 11.1 Let: $$ V $$ be a smooth nonzero: $$ S_0 $$ -periodic DSS Euler profile with: $$ \alpha<3/2. $$ Let: $$ R_1 $$ satisfy: $$ \mathcal E(R_1)>0. $$ Then: $$ \boxed{ \int_0^{R_1} \left( -\mathcal F(R) \right)_+ R^{-\kappa-1} dR \ge \frac{ R_1^{-\kappa} \mathcal E(R_1) }{ \alpha+1 } > 0. } \tag{11.1} $$ Consequently: $$ \boxed{ \mathcal F(R)<0 } \tag{11.2} $$ on a set of positive measure in: $$ (0,R_1). $$ ### Proof By Theorem 9.1: $$ \frac d{dR} \left[ R^{-\kappa} \mathcal E(R) \right] = -(\alpha+1) \mathcal F(R) R^{-\kappa-1}. $$ Integrate from: $$ 0 $$ to: $$ R_1. $$ The lower endpoint vanishes by Section 10: $$ R^{-\kappa}\mathcal E(R)\to0. $$ Hence: $$ R_1^{-\kappa} \mathcal E(R_1) = (\alpha+1) \int_0^{R_1} \left( -\mathcal F(R) \right) R^{-\kappa-1} dR. $$ The positive part of: $$ -\mathcal F $$ dominates the signed integral. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 12. Interpretation of the sign The sphere normal: $$ n $$ points outward. The physical Euler energy current is: $$ (e+P)V. $$ Therefore: $$ \boxed{ \mathcal F(R)<0 } $$ means net period-averaged **inward** pressure--kinetic energy flux through: $$ \partial B_R. $$ Thus a nonzero smooth DSS core in: $$ \alpha<3/2 $$ cannot be sustained by pure similarity drift alone at every radius. Some finite matching layer imports physical Euler energy into the core. --- # 13. Zero-flux rigidity Suppose: $$ \boxed{ \mathcal F(R)=0 } \tag{13.1} $$ for almost every: $$ R $$ in a connected interval: $$ I. $$ Then Theorem 9.1 gives: $$ \boxed{ \frac d{dR} \left[ R^{-\kappa} \mathcal E(R) \right] = 0 } \tag{13.2} $$ on: $$ I. $$ Hence: $$ \boxed{ \mathcal E(R) = C_I R^\kappa } \tag{13.3} $$ on: $$ I. $$ Thus zero radial physical PFET is equivalent to exact critical energy scaling on that interval. If: $$ \mathcal F(R)=0 $$ for all: $$ R>0, $$ smooth-core behavior forces: $$ C_I=0, $$ and: $$ \boxed{ V\equiv0. } \tag{13.4} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 14. Why the critical tail can be asymptotically flux-silent Suppose formally: $$ \mathcal E(R) \sim C R^\kappa $$ at large: $$ R. $$ Then the two terms: $$ \kappa\mathcal E(R) $$ and: $$ R\mathcal E'(R) $$ have the same leading order. Therefore: $$ \mathcal F(R) $$ can be lower order at large radius. This is compatible with: - Xue's critical tail energy law; - the remote pressure-decoupling estimate. Hence the theorem does **not** force a large PFET at arbitrarily large radius. It forces a finite **matching transition** between: $$ O(R^3) $$ smooth-core energy and: $$ O(R^\kappa) $$ critical-tail energy. --- # 15. Uniform annular gap on a compact DSS class Let: $$ \mathscr C_{\rm DSS} $$ be sequentially compact in a topology giving uniform local: $$ C^0 $$ control and continuous local energy/flux functionals. Assume: $$ \boxed{ \alpha \in [1+\delta,3/2-\delta] } \tag{15.1} $$ for a fixed: $$ \delta>0. $$ Assume a normalized nontriviality condition: $$ \boxed{ \mathcal E(R_1) \ge e_0>0 } \tag{15.2} $$ for one fixed: $$ R_1. $$ Uniform local boundedness gives: $$ \mathcal E(R) \le C_0R^3. $$ Because: $$ \kappa \le 1-2\delta $$ and: $$ \kappa \ge 2\delta, $$ one may choose: $$ R_0>0 $$ so small that uniformly: $$ \boxed{ R_0^{-\kappa} \mathcal E(R_0) \le \frac12 R_1^{-\kappa} e_0. } \tag{15.3} $$ --- # 16. NEW THEOREM — Finite-Annulus PFET Compiler Gap ## Theorem 16.1 Under Section 15 there is: $$ c_{\rm PFET}>0 $$ such that every: $$ V\in \mathscr C_{\rm DSS} $$ satisfies: $$ \boxed{ \int_{R_0}^{R_1} \left( -\mathcal F(R) \right)_+ R^{-\kappa-1} dR \ge c_{\rm PFET}. } \tag{16.1} $$ A possible uniform value is: $$ \boxed{ c_{\rm PFET} = \frac{ e_0 }{ 2 (\alpha_{\max}+1) R_1^{\kappa_{\max}} }, } \tag{16.2} $$ after replacing the exponent factors by their compact-interval worst cases. ### Proof Integrate Theorem 9.1 from: $$ R_0 $$ to: $$ R_1. $$ Then: $$ (\alpha+1) \int_{R_0}^{R_1} (-\mathcal F) R^{-\kappa-1}dR = R_1^{-\kappa} \mathcal E(R_1) - R_0^{-\kappa} \mathcal E(R_0). $$ Use: $$ \mathcal E(R_1)\ge e_0 $$ and (15.3). The positive part dominates the signed integral. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED under the compact-class hypotheses}. } $$ --- # 17. Smooth-cutoff realization The radius-integrated flux functional is native. By Fubini/coarea, an integral of: $$ \mathcal F(R) $$ against a smooth compact radial weight can be rewritten as a spacetime integral of the Euler energy current against the gradient of one radial test function. Thus the annular aggregate may be compiled as a standard pressure--flux finite-window observable: $$ \boxed{ \iint \left( e+P \right) V\cdot\nabla\Phi dyds. } \tag{17.1} $$ The singular model weight: $$ R^{-\kappa-1} $$ is used only to derive the coercive lower bound. Once: $$ R_0>0 $$ is fixed, it can be replaced by a smooth equivalent weight on: $$ [R_0,R_1]. $$ Therefore no continuum of independent detectors is required. --- # 18. Translation and pressure gauges Pressure constants do not affect: $$ \mathcal F(R) $$ because: $$ \int_{\partial B_R} V\cdot n = 0. $$ Rigid spatial translations/center drift are handled before the DSS profile is placed in the fixed recurrent chart. If a moving center does not converge, it belongs to the transition/spatial-escape branch of DCRP-30. Thus the radial PFET gap is evaluated only after the declared same-parent center gauge has been fixed. --- # 19. Strict compact zero-PFET DSS branch excluded Suppose the DCRP strict same-parent DSS profile is: - compact/strong; - nonzero; - smooth on the core; -: $$ 1<\alpha<3/2; $$ - transition-parameter tight; - in the finite PFET compiler class. If the native pressure--flux observation vanishes on the entire declared annular aggregate: $$ \boxed{ \mathsf O_{\rm PFET}^{rad}=0, } \tag{19.1} $$ Theorem 16.1 gives a contradiction. Therefore: $$ \boxed{ \textbf{ strict compact DSS strong profile} \cap \ker \mathsf O_{\rm PFET}^{rad} = \varnothing. } \tag{19.2} $$ Status: $$ \boxed{ \textbf{PROVED after the radial PFET coordinate is declared in the finite compiler}. } $$ This is a genuine M-RIG exclusion of the strict compact DSS zero-observation kernel. --- # 20. Relation to the existing DSS literature The local-energy arguments of Chae--Tsai and Xue already show that pressure--velocity flux terms control the radial energy behavior of DSS profiles. DCRP-31 does not claim priority for the underlying local-energy mechanism. The project-specific contribution is to package the exact period-averaged radial identity into the MORP/DCRP obstruction compiler and to interpret it as the mandatory core-to-tail matching PFET of the same-parent Type-II branch. --- # 21. Far-tail pressure NO-GO versus matching-layer PFET The two new theorems are complementary: $$ \boxed{ \text{remote tail pressure} \to0 } \tag{21.1} $$ on every fixed core as the tail cutoff tends to infinity, but: $$ \boxed{ \text{finite matching-layer inward PFET} > 0. } \tag{21.2} $$ Thus the final coupling is not: > the infinite tail acts like a large distant pressure source. It is: > the DSS energy geometry forces a finite-radius physical transfer layer connecting the smooth core to the critical tail. This is a sharper mechanism statement. --- # 22. Critical raw scaling of the matching payment In the strict geometric Type-II recurrence: $$ \boxed{ \beta_{n+1} = q_\ast \beta_n + o(\beta_n), \qquad 0 Let: > > $$ > D_n > $$ > > be the same-parent strict DSS return chain with mandatory normalized inward PFET: > > $$ > \mathsf{PFET}_n > \ge > c_\ast>0. > $$ > > Then either: > > 1. the inward PFET produces a strict nonrecoverable return tax after one full DSS period; > 2. its replenishment requires a nonzero tail/scale transition carrier already counted in: > > $$ > \mathsf R_{\rm nat}; > $$ > > 3. the recurrence is exactly conservative in a critical tail channel, in which case classify the resulting equality solution and test it against known DSS/pressure/outgoing rigidity. The third branch is the equality-manifold route. This is the correct next attack. --- # 28. A candidate equality quantity Define: $$ \boxed{ G(R) = R^{-\kappa} \mathcal E(R). } \tag{28.1} $$ Then: $$ \boxed{ G'(R) = -(\alpha+1) \mathcal F(R) R^{-\kappa-1}. } \tag{28.2} $$ Thus: - inward PFET: $$ \mathcal F<0 $$ makes: $$ G'>0; $$ - outward PFET: $$ \mathcal F>0 $$ makes: $$ G'<0. $$ The critical tail corresponds to: $$ G(R) $$ approaching or oscillating around a positive scale-recurrent level. Therefore a full DSS return with zero net depletion requires an exact balance of the signed radial PFET in logarithmic radius. This suggests a scale-logarithmic transport ledger: $$ \boxed{ d\log R } $$ rather than a raw physical-energy sum. Whether this can generate a non-summable normalized return tax is the next unresolved calculation. --- # 29. Source-status audit ## Chae--Tsai The paper gives the Euler DSS scaling law, the periodic similarity equation, and local-energy identities involving: $$ |V|^3 + |P||V|. $$ It proves several DSS nonexistence criteria under velocity/vorticity integrability and decay assumptions. ## Xue The paper extends the DSS local-energy analysis to non-decaying profiles and gives pressure representation formulae appropriate to nonstandard spatial asymptotics. It proves the critical energy behavior: $$ R^{3-2\alpha} $$ in its admissible nontrivial profile classes. DCRP-31 uses this as a tail calibration and derives a conditional remote-pressure estimate from the Calderon--Zygmund kernel. ## Constantin--Ignatova--Vicol The outgoing self-similar guardrail: $$ \gamma\ge1/2 $$ continues to force the strict DCRP profile: $$ \gamma<1/2 $$ into non-outgoing/trapped Lagrangian behavior. This is compatible with the new matching-layer flux theorem and does not eliminate it. --- # 30. End state The tail-pressure proposal has been corrected: $$ \boxed{ \textbf{ infinite DSS tail} \not\Rightarrow \textbf{ large direct far-tail pressure on the core}. } $$ Under the critical energy envelope: $$ \boxed{ P_{>L} = O(L^{-2\alpha}), \qquad \nabla P_{>L} = O(L^{-1-2\alpha}). } $$ The unavoidable coupling is instead the exact radial PFET identity: $$ \boxed{ \mathcal F(R) = - \frac1{\alpha+1} R^{\kappa+1} \frac d{dR} \left[ R^{-\kappa} \mathcal E(R) \right]. } $$ A smooth core has: $$ R^{-\kappa}\mathcal E(R)\to0, $$ while a nonzero DSS profile has positive local energy at finite radius. Hence: $$ \boxed{ \int (-\mathcal F)_+ R^{-\kappa-1}dR > 0. } $$ Thus every nonzero strict compact DSS profile has a finite-radius inward physical pressure--kinetic matching flux. On a compact normalized class this becomes a uniform finite-annulus PFET gap. Therefore: $$ \boxed{ \textbf{ strict compact DSS zero-PFET branch is excluded}. } $$ But the raw physical payments may be geometrically summable: $$ \sum_n\beta_n<\infty. $$ The remaining problem is therefore not tail visibility. It is: $$ \boxed{ \textbf{ Same-Parent DSS PFET Return-Depletion / Critical Flux Summability. } } $$