# DCRP102 / X72-R85 — Backward Adjoint X-Test Dynamics, Oriented Pair-Copula Cone, and the No-Invariant-Sign-Cone Audit **Series:** Independent Navier–Stokes Research Series / X72 Bridge **Date:** 2026-08-20 **Status:** proof-development checkpoint / backward-adjoint copula-rigidity round **Immediate predecessor:** `NS_DCRP101_X72R84_JointPathYoung_SecondThirdMomentLock_2026-08-20.md` ## Primary internal dependencies - X72 Round37 — exact affine-response defect equation. - X72 Round38 — exact transport–Riesz pair/triple-increment identity. - DCRP65–68 — factorwise null closure, correlation frontier, two-stress geometry, self-lock integrability collapse. - DCRP95–96 — sign-coherent Kelvin slip / nematic second-moment lock. - DCRP99–101 — bounded-lag X–Kelvin word / finite Duhamel forcing / second–third moment path-copula lock. ## Fresh primary-source calibration - E. Hess-Childs, M. Rosenzweig, S. Serfaty, *Another look at regularity in transport-commutator estimates*, arXiv:2601.02326 (2026). Generic Riesz-type transport commutator estimates retain a sharp regularity burden; BMO does not generally replace the Lipschitz-gradient hypothesis, although defective estimates are available in almost-Lipschitz regimes. - E. Hess-Childs, M. Rosenzweig, S. Serfaty, *A sharp commutator estimate for all Riesz modulated energies*, arXiv:2511.13461 (2025). - R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier–Stokes Equations*, arXiv:2606.27560 (2026). At the critical exponent, bounded derivative-compatible velocity-increment defects yield cylindrical generalized Young profiles; this does not by itself furnish tightness for D101's enlarged finite-lag adjoint tuple. No full Navier–Stokes regularity theorem is claimed. --- # 0. Executive result D101 reduced the selected compact transport–Riesz/Kelvin branch to one finite-lag path profile carrying \[ \boxed{ \sigma_\Gamma\mathcal L_\Gamma(M^{(2)})\ge c_\Gamma>0 } \] and \[ \boxed{ \sigma_{\rm TR}\mathcal L_{\rm TR}(M^{(3)})\ge c_{\rm TR}>0. } \] The next question is whether the pulled-back X72 adjoint test has a sign/phase rigidity strong enough to collapse this mixed third-order copula. D102 proves that there is **no generic invariant sign cone** for the adjoint tensor. The X72 defect equation is \[ D_tE-\nu\Delta E=-\mathscr L_S[E]+\mathcal F_E, \] with \[ \mathscr L_S[E] = L_S(E)+2\mathcal T_0(S:E), \] \[ L_S(E) = ES+SE-\frac23(S:E)I. \] Writing \[ \partial_tE = \mathcal A_E(t)E+\mathcal F_E, \qquad \mathcal A_E = \nu\Delta-u\cdot\nabla-\mathscr L_S, \] the backward adjoint satisfies \[ \boxed{ -\partial_s\Phi = \nu\Delta\Phi + u\cdot\nabla\Phi - \mathscr L_S^*[\Phi]. } \tag{0.1} \] On trace-free symmetric tensors, \[ \boxed{ \mathscr L_S^*[\Phi] = L_S(\Phi) + 2S\,\mathcal T_0^*\Phi. } \tag{0.2} \] Along a forward material trajectory, \[ \boxed{ D_s\Phi = -\nu\Delta\Phi + \mathscr L_S^*[\Phi]. } \tag{0.3} \] Thus the pulled-back X test is dynamically rotated/stretched by local strain and a nonlocal Riesz scalar. Even the **local** part already prevents a generic detector half-space maximum principle. Take \[ G=\operatorname{diag}(1,-1,0), \] \[ \Phi_0=\operatorname{diag}(1,1,-2). \] Then \[ G:\Phi_0=0. \] For \[ S=\pm G, \] one has \[ L_S(\Phi_0)=\pm2G, \] so \[ \boxed{ G:L_S(\Phi_0)=\pm4. } \tag{0.4} \] The same boundary point of the detector half-space can be pushed to either side by admissible trace-free strain. Hence \[ \boxed{ \text{fixed-sign TR recurrence} \not\Rightarrow \text{generic pointwise sign preservation of }\delta\Phi_X. } \tag{0.5} \] What the fixed-sign TR branch **does** force is a recurrent **oriented pair-copula cone** on a regular pair-scale cylinder. After removing principal-value / scale concentration and path-profile escape, there is a fixed normalized pair cylinder on which \[ \boxed{ [\delta u\cdot\nabla K_0]:\delta\Phi_X\,\delta q } \] has fixed sign and uniformly positive pair-space occupancy. Equivalently, after fixing the sign of \(\delta q\), the tensor angle between \[ G_{xy}:=[\delta u\cdot\nabla K_0] \] and \[ \delta\Phi_X \] lies in one strict angular cone on a positive-measure pair set. The remaining compact survivor is therefore a **dynamically propagated oriented adjoint-copula state**. --- # 1. Exact adjoint of the local strain operator Work in \[ \mathrm{Sym}_0(3) \] with Frobenius pairing. For trace-free symmetric \(E,\Phi,S\), \[ L_S(E) = ES+SE-\frac23(S:E)I. \] Because \[ \Phi:I=0, \] \[ \langle\Phi,L_S(E)\rangle = \langle\Phi,ES+SE\rangle. \] By cyclicity of trace, \[ \langle\Phi,ES+SE\rangle = \langle E,S\Phi+\Phi S\rangle. \] Projecting back to trace-free tensors gives \[ \boxed{ L_S^*=L_S. } \tag{1.1} \] So the local strain operator is self-adjoint on \(\mathrm{Sym}_0(3)\). --- # 2. Adjoint of the Riesz scalar coupling Recall \[ \mathcal T_0 = \nabla^2(-\Delta)^{-1} + \frac13I, \] viewed as \[ \text{scalar}\to\mathrm{Sym}_0(3). \] Its Hilbert adjoint is \[ \boxed{ \mathcal T_0^*\Phi = (-\Delta)^{-1}\partial_i\partial_j\Phi_{ij} + \frac13\operatorname{tr}\Phi. } \tag{2.1} \] On trace-free tensors, \[ \boxed{ \mathcal T_0^*\Phi = (-\Delta)^{-1}\partial_i\partial_j\Phi_{ij}. } \tag{2.2} \] For \[ 2\mathcal T_0(S:E), \] \[ \begin{aligned} \langle\Phi,2\mathcal T_0(S:E)\rangle &= 2\langle\mathcal T_0^*\Phi,S:E\rangle \\ &= \langle E,2S\,\mathcal T_0^*\Phi\rangle. \end{aligned} \] Therefore ## Theorem D102.1 — Exact defect-adjoint strain operator \[ \boxed{ \mathscr L_S^*[\Phi] = L_S(\Phi) + 2S\,\mathcal T_0^*\Phi. } \tag{2.3} \] The adjoint contains a nonlocal scalar projection even when the terminal X detector is tensorial. --- # 3. Exact backward adjoint equation The forward equation is \[ \partial_tE = \nu\Delta E -u\cdot\nabla E -\mathscr L_S[E] +\mathcal F_E. \] Since \[ \nabla\cdot u=0, \] \[ (-u\cdot\nabla)^* = u\cdot\nabla. \] Hence ## Theorem D102.2 — Backward X72 adjoint \[ \boxed{ -\partial_s\Phi = \nu\Delta\Phi + u\cdot\nabla\Phi - L_S(\Phi) - 2S\,\mathcal T_0^*\Phi. } \tag{3.1} \] Along a forward material path, \[ D_s=\partial_s+u\cdot\nabla, \] so \[ \boxed{ D_s\Phi = -\nu\Delta\Phi + L_S(\Phi) + 2S\,\mathcal T_0^*\Phi. } \tag{3.2} \] In the inviscid/profile limit, \[ \boxed{ D_s\Phi = L_S(\Phi) + 2S\,\mathcal T_0^*\Phi. } \tag{3.3} \] --- # 4. Exact pair-increment adjoint equation Let \[ X(s),Y(s) \] be two forward material trajectories. Write \[ \delta\Phi = \Phi(X,s)-\Phi(Y,s), \] \[ \delta S = S(X,s)-S(Y,s), \] and \[ r=\mathcal T_0^*\Phi, \qquad \delta r=r(X)-r(Y). \] Define \[ B(S,\Phi) = S\Phi+\Phi S-\frac23(S:\Phi)I. \] Then \[ B(S_X,\Phi_X)-B(S_Y,\Phi_Y) = B(S_X,\delta\Phi) + B(\delta S,\Phi_Y), \] and \[ S_Xr_X-S_Yr_Y = S_X\delta r+\delta S\,r_Y. \] Therefore ## Theorem D102.3 — Material pair-adjoint increment equation \[ \boxed{ \begin{aligned} \frac d{ds}\delta\Phi ={}& -\nu\,\delta(\Delta\Phi) +B(S_X,\delta\Phi) +B(\delta S,\Phi_Y) \\ &+ 2S_X\delta r + 2\delta S\,r_Y. \end{aligned} } \tag{4.1} \] The pair increment is generated/rotated by: 1. backward diffusion / second-gradient mismatch; 2. action on the existing pair increment; 3. strain increment \(\delta S\); 4. nonlocal adjoint-Riesz increment \(\delta r\). Thus \(\delta\Phi_X\) is not a passive fixed detector tensor. --- # 5. Zero pair increment is not invariant If \[ \delta\Phi=0 \] at one time, (4.1) gives \[ \boxed{ \frac d{ds}\delta\Phi = -\nu\delta(\Delta\Phi) + B(\delta S,\Phi_Y) + 2S_X\delta r + 2\delta S\,r_Y. } \tag{5.1} \] Thus exact pairwise adjoint equality is immediately destroyed unless a nontrivial compatibility relation holds among: - \(\delta S\); - \(\delta r\); - viscous second gradients. --- # 6. No generic invariant detector half-space Fix a trace-free tensor detector \(G\). One might hope the adjoint evolution preserves \[ G:\Phi\ge0. \] This is false already for the local operator. Take \[ G=\operatorname{diag}(1,-1,0), \] \[ \Phi_0=\operatorname{diag}(1,1,-2). \] Then \[ G:\Phi_0=0. \] For \[ S=G, \] \[ L_S(\Phi_0)=2G, \] and \[ G:L_S(\Phi_0)=4>0. \] For \[ S=-G, \] \[ G:L_S(\Phi_0)=-4<0. \] Therefore ## Theorem D102.4 — No local invariant half-space There is no universal detector half-space \[ \{\Phi:G:\Phi\ge0\} \] preserved by the admissible local trace-free strain action. The full adjoint Riesz term adds further nonlocal mixing. --- # 7. Adjoint tensor-direction equation Whenever \[ \Phi\neq0, \] write \[ \widehat\Phi=\frac{\Phi}{|\Phi|}. \] Let \[ P_{\widehat\Phi^\perp} \] be orthogonal projection in the five-dimensional tensor space \(\mathrm{Sym}_0(3)\). From (3.2), ## Theorem D102.5 — Adjoint angular velocity \[ \boxed{ D_s\widehat\Phi = \frac1{|\Phi|} P_{\widehat\Phi^\perp} \left[ -\nu\Delta\Phi + L_S(\Phi) + 2S\mathcal T_0^*\Phi \right]. } \tag{7.1} \] Define \[ \boxed{ \Omega_\Phi = \frac1{|\Phi|} \left| P_{\widehat\Phi^\perp} \left[ -\nu\Delta\Phi + L_S(\Phi) + 2S\mathcal T_0^*\Phi \right] \right|. } \tag{7.2} \] If \[ \Omega_\Phi=0, \] the adjoint tensor direction is an instantaneous eigen-direction of the full defect-adjoint operator. --- # 8. Zero angular action normal form On an interval \(I\), if \[ \Omega_\Phi\equiv0, \] then \[ \boxed{ -\nu\Delta\Phi + L_S(\Phi) + 2S\mathcal T_0^*\Phi = \beta(s,x)\Phi } \tag{8.1} \] for some scalar \(\beta\). Thus zero adjoint angular action is the nonlocal tensor-ray condition \[ \boxed{ \Phi \text{ remains on one ray under the adjoint defect operator.} } \] Call this \[ \boxed{ \mathsf K_{\rm adj\mbox{-}eig}. } \] --- # 9. Transport–Riesz integrand as an angular observable At a pair \((x,y)\), define \[ \boxed{ G_{xy} = [\delta_{xy}u\cdot\nabla K_0(x-y)] \in\mathrm{Sym}_0(3). } \tag{9.1} \] Then the generic D101 finite-lag TR integrand is \[ \boxed{ h_{xy} = (G_{xy}:\delta\Phi_X)\,\delta q. } \tag{9.2} \] If all three factors are nonzero, write \[ \cos\theta_{xy} = \frac{G_{xy}:\delta\Phi_X} {|G_{xy}||\delta\Phi_X|}. \] Then \[ \boxed{ h_{xy} = |G_{xy}| |\delta\Phi_X| |\delta q| \operatorname{sgn}(\delta q) \cos\theta_{xy}. } \tag{9.3} \] The TR sign is therefore a joint amplitude–angle–scalar-sign condition. --- # 10. Regular pair-cylinder extraction The principal-value kernel is singular. D102 therefore does **not** infer pointwise sign directly from the total PV integral. Split the pair domain into normalized pair annuli \[ \sigma_j\le|x-y|\le\Sigma_j. \] If no fixed regular pair cylinder carries a definite fraction of the recurrent signed TR source, then the source is escaping through: - shrinking pair scale; - principal-value concentration; - tail pair scale; - path-profile lack of tightness. Record this as \[ \boxed{ R_{\rm PV/scale} \vee R_{\rm path/fib} \vee R_{\rm state} \vee R_{\rm crit}. } \tag{10.1} \] D102 studies the complementary regular pair-cylinder branch. --- # 11. Positive TR integral forces positive pair occupancy Normalize one selected regular pair cylinder to total pair measure \(1\). Assume compact bounds \[ |G_{xy}|\le M_G, \] \[ |\delta\Phi_X|\le M_\Phi, \] \[ |\delta q|\le M_q. \] Then \[ |h_{xy}| \le M:=M_GM_\Phi M_q. \] Suppose \[ \int h_{xy}\,d\mu_{\rm pair} \ge c_{\rm TR}>0. \] Set \[ A_+ = \{h_{xy}\ge c_{\rm TR}/2\}. \] Then \[ c_{\rm TR} \le M\mu(A_+) + \frac{c_{\rm TR}}2[1-\mu(A_+)]. \] Hence ## Theorem D102.6 — Positive pair occupancy \[ \boxed{ \mu(A_+) \ge \frac{c_{\rm TR}}{2M-c_{\rm TR}} >0. } \tag{11.1} \] Thus the TR source cannot be supported only by arbitrarily sparse pair states on a compact bounded-amplitude cylinder. --- # 12. Oriented angular cone On \(A_+\), \[ h_{xy}\ge c_{\rm TR}/2. \] Split \(A_+\) by the sign of \(\delta q\), and keep one sign cell with at least half the occupancy. Fix \[ \sigma_q=\operatorname{sgn}(\delta q). \] Then \[ \boxed{ \sigma_q\cos\theta_{xy} \ge \frac{c_{\rm TR}}{2M_GM_\Phi M_q} =: \eta_\theta>0 } \tag{12.1} \] on a positive-measure pair subset. Thus the fixed-sign TR branch forces the strict tensor-angle cone \[ \boxed{ \mathcal C_{\rm ang} = \left\{ (G,\Delta\Phi,c): \operatorname{sgn}(c) \frac{G:\Delta\Phi}{|G||\Delta\Phi|} \ge\eta_\theta \right\}. } \tag{12.2} \] This is the local geometric meaning of the third-order copula lock. --- # 13. Finite angular-cell recurrence Cover the compact normalized pair-state space by finitely many cells resolving: - pair direction \(z/|z|\); - normalized transport tensor \(G/|G|\); - normalized adjoint increment direction \(\delta\Phi/|\delta\Phi|\); - sign of \(\delta q\); - amplitude bands. Because \(\mathcal C_{\rm ang}\) has positive pair occupancy on every selected recurrent TR event, one cell carries a fixed fraction of that occupancy. Across the positive-density generation set, finite pigeonholing gives: ## Theorem D102.7 — Recurrent oriented adjoint pair cell There exists one fixed pair cell \(\mathfrak c_*\) such that: 1. \(\mathfrak c_*\subset\mathcal C_{\rm ang}\); 2. \(\mathfrak c_*\) recurs on a positive-density generation subsequence; 3. its pair-space occupancy is uniformly positive. Thus the D101 abstract third-order copula becomes an explicit recurrent oriented angular pair state. --- # 14. Why no static symmetry argument closes it A fixed-sign angular pair state is not ruled out by: - zero barycenter; - sign-symmetric velocity increments; - Kelvin second-moment lock; - pairwise marginal information. D101 already gave a parity-copula counterexample. D102 adds the dynamical fact that even at \(G:\Phi=0\), local strain may push the adjoint test to either side. So there is no generic fixed tensor reflection or half-space maximum principle forcing angular cancellation. The remaining lock is genuinely dynamical/path-dependent. --- # 15. Adjoint phase-lock versus angular action For a recurrent source cell, follow the pulled-back test over its fixed lag interval. Define the angular action \[ \boxed{ \mathcal A_{\rm adj} = \int_I\Omega_\Phi\,ds. } \tag{15.1} \] There are two exact possibilities: ### A. positive angular action \[ \boxed{ \mathcal A_{\rm adj}>0. } \tag{15.2} \] ### B. adjoint eigen-lock \[ \boxed{ \mathcal A_{\rm adj}=0 } \] and therefore \[ \boxed{ -\nu\Delta\Phi + L_S(\Phi) + 2S\mathcal T_0^*\Phi = \beta\Phi. } \tag{15.3} \] If a compact recurrent subfamily stays a positive distance from the eigen-lock set, continuity/compactness upgrades (15.2) to a uniform gap \[ \boxed{ \mathcal A_{\rm adj}\ge c_{\rm adj}>0. } \tag{15.4} \] Without that separation one must retain the eigen-lock limit as a genuine alternative. Therefore ## Theorem D102.8 — Adjoint phase dichotomy \[ \boxed{ \mathsf C_{2+3}^{\ell_*} \Longrightarrow \mathsf A_{\rm adj}>0 \vee \mathsf K_{\rm adj\mbox{-}eig} \vee R_{\rm PV/scale} \vee R_{\rm path/fib} \vee R_{\rm state} \vee R_{\rm crit}. } \tag{15.5} \] --- # 16. Zero-lag relation to D65–68 When \[ \ell_*=0 \] and \[ \Phi_X=E_p, \] Round38 pressure self-commutator cancellation reduces the test increment to the strain-cofactor sector. D65 closes every factorwise zero mechanism. D66 reduces exact silence to the two-stress \(4:1\) correlation manifold. D67–68 show the strongest aligned self-lock geometries are highly rigid: only two axisymmetric spectral modes avoid intrinsic cofactor self-rotation, and full gradient integrability removes those as nontrivial isotropic-covariance recurrent equality states. Thus the zero-lag branch already strongly favors real cofactor-shape action over a static tensor axis. D102 does **not** transplant that conclusion automatically to generic \(\ell_*\neq0\), because the pulled-back adjoint test is not the instantaneous cofactor. --- # 17. External commutator calibration Modern Riesz-commutator results reinforce the methodological choice here. Generic Riesz-type transport commutator control retains a sharp velocity-regularity burden; in general a BMO gradient bound cannot replace the standard Lipschitz-type control, though defective estimates exist near the almost-Lipschitz threshold. So D102 does not try to close the recurrent TR branch by a generic soft commutator estimate. Instead it exploits the special recurrent adjoint pair state. Likewise Yu's critical filtered-vorticity work supplies cylindrical Young profiles for velocity increments, but does not automatically furnish the enlarged adjoint/path tuple required here. Thus path-profile/fiber escape remains a real interface condition. --- # 18. Updated late compact normal form D101 gave \[ \boxed{ \mathsf C_{2+3}^{\ell_*}. } \] D102 refines it to the dynamically propagated oriented copula \[ \boxed{ \mathsf C_{\rm adj\mbox{-}copula}^{\ell_*} } \] with: 1. fixed Kelvin second-moment sign; 2. fixed lag; 3. fixed TR source sign; 4. fixed regular pair-scale cell; 5. positive occupancy of one oriented tensor-angle cone; 6. backward adjoint evolution governed by (3.1); 7. either positive adjoint angular action or an adjoint eigen-lock kernel. --- # 19. Status ledger ## PROVED this round ### D102-P1 — exact adjoint of the X72 strain/Riesz linear operator. ### D102-P2 — exact backward X72 adjoint equation. ### D102-P3 — exact material pair-increment adjoint equation. ### D102-P4 — zero pair increment is not generically invariant. ### D102-P5 — no universal local detector half-space is invariant under trace-free strain action. ### D102-P6 — exact adjoint tensor-direction/angular-velocity equation. ### D102-P7 — zero angular action is an explicit nonlocal adjoint eigen-lock condition. ### D102-P8 — on a regular compact pair cylinder, fixed positive TR source forces uniformly positive pair occupancy. ### D102-P9 — positive TR source forces a strict oriented tensor-angle cone after fixing \(\delta q\) sign. ### D102-P10 — finite pair-state pigeonholing yields one recurrent oriented angular cell. ### D102-P11 — generic finite-lag adjoint lock cannot be reduced to D65–68 instantaneous cofactor geometry without an additional adjoint-lock bridge. ### D102-P12 — the late compact path-copula survivor splits into positive adjoint angular action or one explicit adjoint eigen-lock normal form. --- # 20. What is NOT proved D102 does not prove: - the adjoint eigen-lock kernel is empty; - positive adjoint angular action has a finite global capacity; - a regular pair-scale cylinder exists if PV mass concentrates at vanishing pair scale; - Yu's velocity Young profile automatically lifts to the full adjoint path tuple; - generic finite-lag \(\Phi_X\) equals the instantaneous X72 cofactor/defect state; - the recurrent oriented pair cell is incompatible with Navier–Stokes dynamics; - global Navier–Stokes regularity. The remaining problem is now an **adjoint eigen-lock / angular-action rigidity problem**. --- # 21. STOP-D102 \[ \boxed{ \begin{minipage}{0.94\linewidth} The D101 Kelvin/TR second–third moment lock does not imply a static sign constraint on the pulled-back X72 test. The exact backward adjoint equation is \[ -\partial_s\Phi = \nu\Delta\Phi + u\cdot\nabla\Phi - L_S(\Phi) - 2S\mathcal T_0^*\Phi, \] and along forward material paths \[ D_s\Phi = -\nu\Delta\Phi + L_S(\Phi) + 2S\mathcal T_0^*\Phi. \] Its pair increment is driven by the existing adjoint increment, strain increment, nonlocal adjoint-Riesz increment, and viscous second gradients. There is no generic detector half-space maximum principle: at \(G:\Phi=0\), admissible local strains \(S=\pm G\) push the detector projection to opposite signs. Therefore fixed-sign transport–Riesz recurrence cannot be interpreted as pointwise adjoint sign preservation. On the complementary regular pair-scale branch, however, a fixed positive TR source and compact amplitude bounds force a positive-measure pair set on which the transport tensor \(G_{xy}=[\delta u\cdot\nabla K_0]\), the adjoint increment \(\delta\Phi_X\), and the sign of \(\delta q\) lie in one strict oriented angular cone. Finite pair-state pigeonholing yields one recurrent angular cell with uniformly positive occupancy. Following the pulled-back test across the fixed lag gives a new exact dichotomy: either the adjoint direction pays positive angular action, or it lies in the explicit nonlocal tensor-ray eigen-lock kernel \[ -\nu\Delta\Phi+L_S(\Phi)+2S\mathcal T_0^*\Phi=\beta\Phi. \] Thus the final compact survivor is no longer an abstract third-order copula; it is a dynamically propagated oriented adjoint-copula state. \end{minipage} } \] --- # 22. Next autonomous step ## DCRP103 / X72-R86 — Adjoint Eigen-Lock Compatibility / Tensor-Ray Rigidity **Working title** > **Can a Nonzero Trace-Free Backward X72 Test Remain on One Tensor Ray under \(L_S+2S\mathcal T_0^*\) while its pair increments stay inside the recurrent TR angular cone?** Primary tasks: 1. start from \[ -\nu\Delta\Phi+L_S(\Phi)+2S\mathcal T_0^*\Phi=\beta\Phi; \] 2. in the inviscid compact profile branch, diagonalize \(S\); 3. solve the five-dimensional tensor-ray algebra; 4. separate: - diagonal/coaxial adjoint rays; - off-diagonal shear rays; - nonlocal \(r=\mathcal T_0^*\Phi\) locking; 5. impose incompressibility and X72 carrier geometry; 6. test whether the recurrent TR angular cone can persist on each ray family; 7. if eigen-lock is excluded away from finitely many rays, obtain a compact-class angular-action gap; 8. otherwise isolate the finite tensor-ray normal forms. Desired endpoint: \[ \boxed{ \mathsf K_{\rm adj\mbox{-}eig} \Longrightarrow \text{finite tensor-ray normal forms} \vee R_{\rm state} \vee R_{\rm crit}. } \] **End checkpoint:** DCRP102 / X72-R85.