40 / C5-D: Spatial–Matrix Motif Compatibility, Strong-Middle Cones, and Quadratic/Pressure Convex-Hull Obstructions

C5-C proved that the scalar $E_0/E_1$ identities alone are not enough to exclude temporal compensation cycles. C5-D formally leaves the purely temporal, purely scalar route, and places the positive middle-strain direction, the local quadratic tensor $Q=S^2+\tfrac14\omega\otimes\omega-\tfrac14|\omega|^2I$, C4-J's seven-point cancellation witnesses, the local adjoint-core pressure mean, the shared far-field harmonic pressure matrix, and C3-S's convex-hull pressure geometry, all into the same finite-dimensional spatial-matrix problem. For a normalized (trace-free, Frobenius norm 1) strain direction $K$ with middle eigenvalue $k_2>0$, it defines the strong-middle shape parameter $\theta_K=k_2k_3=k_1^2-\tfrac12>0$, and the compressive-axis test tensor $H_K=e_1\otimes e_1-\tfrac{1+\theta_K}2I$ ($e_1$ being the eigenvector of the most negative eigenvalue $k_1$). The core result is C5-D.1, the Positive-Middle Cone to Quadratic Half-Space Theorem (Theorem 10.1): if the pointwise normalized strain direction lies within a strong-middle pointwise strain cone $\mathcal C_K$ of radius set by $\theta_K$ around $K$, then for any vorticity $\omega$ whatsoever (no vorticity-alignment assumption, no vorticity upper bound, no helicity-sign assumption needed), $H_K:Q\ge\tfrac{\theta_K}4(|S|^2+|\omega|^2)$ holds — so all normalized quadratic directions $Q/|Q|$ fall entirely into the same strict matrix half-space, with margin $\gamma_K=\theta_K/(4|H_K|)>0$. The round's most important result is C5-D.3, Seven-Point Zero-Barycenter Incompatibility: if the seven witness directions $U_i^\ast$ all lie simultaneously within this half-space, their convex combination cannot possibly equal zero — a direct contradiction. So a strong-middle pointwise cone and seven-point zero-barycenter cancellation cannot coexist within the same recurrent limit; the document explicitly stresses that this is the C5 series' first genuine finite-dimensional algebraic incompatibility: not a divergent norm, not an integral budget, not a temporal arrangement, but a purely finite-dimensional convex-geometric obstruction. A quantitative version (C5-D.4) further proves that even allowing some strain directions to leak outside the cone, as long as the leaking mass is not too large, the coefficient $\kappa_\chi^Q$ still has a positive lower bound; conversely, if the cancellation coefficient stays small, the leaking mass must have a positive lower bound — so for quadratic cancellation to survive, either the middle gap $\theta_K\to0$ must degenerate, or the strain direction must keep leaking out of the strong-middle cone. The document carefully distinguishes C3-S's average strain cone from the pointwise normalized strain cone used here — they are not the same object — and a bridge between them requires the relative perturbation $\eta_R^S$ to stay below a critical threshold (C5-D.5: average coherence plus small perturbation implies that seven-point cancellation is impossible on that core); if the perturbation is too large, combined with a Morrey-type estimate, cancellation instead forces a genuine higher-derivative strain-perturbation debt, connecting directly back to the derivative geometry of C3-V/W/X/Y. On pressure: once a strong-middle core holds, it excludes the cancellation branch, and C4-I's original trichotomy of cancellation, mean rotation, or pressure collapses here to just mean rotation or pressure concentration (C5-D.6/7, Oriented Pressure Re-entry). If multiple cores share the same harmonic far-field pressure matrix $F_\ast$, the test direction simplifies to the compressive-axis projection $G(e)=e\otimes e-\tfrac13I$, and C5-D.8, the Compressive-Axis Convex-Hull Pressure Obstruction (Theorem 40.1, an exact Carathéodory argument), proves that if zero lies within the convex hull of $\{G(e_i)\}$, then no single common $F_\ast$ can compensate all the cores simultaneously — because $\dim\operatorname{Sym}_0(3)=5$, at most six cores are needed to witness this; more simply, three mutually orthogonal compressive axes already directly give a zero barycenter (the Orthogonal-Triplet Pressure Obstruction, C5-D.9). The round concludes with a bidirectional cycle: a strong-middle cone implies a quadratic half-space, which excludes seven-point cancellation, which forces mean rotation or pressure re-entry, which, if a far-field pressure is shared, brings in compressive-axis convex-hull constraints; conversely, quadratic cancellation implies either middle-gap degeneration or directional dispersion. It formally hands off to C5-E, Strain-Direction Defect Measures, Middle-Gap Degeneration, and Derivative-Intermittency Closure, with 8 proof targets focused on the exact compactification of the strain-direction measure, middle-gap mass, directional-dispersion defect, and the interface with the derivative gate.