53 / C6-D: Geometry–Pressure Cycle Composition, Provenance Compatibility, and Signature-Return Tests

C6-A's second-ranked candidate $G\leftrightarrow P$ now undergoes the same cycle-composition audit, and it immediately turns out to need even more semantic correction than $H/F$: many arrows written in C5-D/F as $G\to P$ or $P\to G$ in fact all occur at the same moment, the same spatial core, the same compression axis, and the same local/far pressure decomposition — they are not sequential edges $G_n\to P_n\to G_{n+1}$, but a same-event compatibility relation $(G,P)_n\in\mathcal C_{GP}$. A genuine recurrent cycle would still need a separate temporal return map $\Phi_{GP}:\mathcal C_{GP,n}\dashrightarrow\mathcal C_{GP,n+1}$, which C5-D/F never proved. To this end, beyond C6-A's proof-status tags (I/C/N/E), a further edge dimension $\tau_e\in\{S,D,E\}$ is added (Static same-event relation, Dynamic cross-generation transition, External closure), and C6-D.1 Static-Edge Collapse Principle proves that same-event compatibility loops should be quotiented out before SCC extraction, and cannot be treated directly as cycle edges. Tracing what $G\to P$ actually produces: strong-middle geometry (via mean-strain evolution and the mean-stability gate) gives only the total localized mean pressure Hessian $P_\chi=\int\chi\nabla^2p$, not directly a far-field pressure — a much weaker premise than the old coarse $P\to G$ axis-locking route assumed. C6-D.2 Pressure-Provenance Split Lemma uses the Bradshaw–Tsai local pressure expansion to split it into $P_\chi^{loc}+P_\chi^{far}$; an oriented response guarantees only the disjunction $P_{\rm local}^{+}\vee P_{\rm far}^{+}$ — a cycle must first choose a provenance branch, and only the far branch is entitled to use the harmonic STF-matrix signature/axis-locking mechanism; the local branch cannot borrow the far-pressure obstruction mechanism. If the far branch genuinely dominates, C6-D.3 Far-Pressure Axis-Margin Theorem gives an exact negative-quadratic-form margin on the same core's compression axis; the signature trichotomy: one-negative $(-,+,+)$ produces narrow projective-cone axis-locking (C6-D.4 proves this is incompatible at the same event with Q-zero-barycenter/seven-point cancellation — but this is only same-event incompatibility, not a "future-event" theorem), two-negative $(-,-,+)$ gives only broader negative-band/planar geometry, not locking a single axis, with compatibility not equal to causality (C5-F never proved $(-,-,+)\Rightarrow$ Q cancellation or geometric-defect generation), and $\det F=0$ is the signature boundary. The round's most critical step (C6-D.5 Pressure-Alone Dynamic Return No-Go): the strain evolution equation simultaneously contains advection, viscosity, $S^2$, the vorticity quadratic term, and pressure — five terms — so the pressure Hessian alone cannot determine the future sign of $\lambda_2$, the middle gap, the compression axis, or Q-cancellation geometry — hence same-event P/G compatibility cannot be upgraded into a future-geometry edge without an explicit heredity/persistence theorem. C6-D.6 Joint-State Recurrence Requirement therefore proves that the old coarse $G\leftrightarrow P$ counts as a genuine dynamical cycle only if there exists a nonempty recurrent invariant set $\mathcal R_{GP}\subset\mathcal K_{GP}^{comp}$ that the Navier–Stokes evolution genuinely maps back into itself — C5-D/F has no such theorem at all, and the coarse $G/P$ dynamical cycle is formally ruled dead. Compressing the eight-dimensional reserve needed for recurrence (geometry, mean-rotation depletion, far capture, provenance coherence, signature distance, axis margin, far-matrix heredity, geometry heredity) into a vector $\mathbf R^{GP}$, C6-D.7 Finite GP Recurrence Bottleneck Theorem proves a dichotomy of the same type as C6-C's: any infinite candidate joint-recurrence sequence, after passing to a subsequence, is either uniformly hereditary/coherent or approaches one of nine named boundaries (middle-gap collapse, mean-rotation takeover, local-pressure takeover, provenance-compatibility cancellation, signature boundary, axis-margin collapse, far-field heredity collapse, geometry heredity collapse, pressure-regularity exit). The round's most important overall conclusion (Section 50): after the semantic refinement of C6-B/C/D, both of the two main candidates on C5-M's coarse graph — the generic $H/F$ cycle and the dynamical $G/P$ cycle — have now been formally removed, and the genuinely remaining physical candidates number only three: possibly isolated $T$, hereditary $GP_{\rm hereditary}$, and nonlinearly coherent $HF_{\rm nonlinear\ coherent}$ — a much smaller cycle frontier than C5-M's coarse graph suggested. Since $T$ is the only one of C6-A's three original candidates not yet given a composition audit, and since if $T$ cannot stay isolated all remaining physical survivor candidates may converge into the two joint coherent branches already typed, this formally hands off to C6-E — Temporal-to-Spatial Shared-Source Coupling, Isolation No-Go Tests, and the Fate of the $T$ Trap, with eight composition obligations (E1 defining a genuine temporal source carrier through E8 recomputing the cycle graph).