54 / C6-E: Temporal-to-Spatial Shared-Source Coupling, Isolation No-Go Tests, and the Fate of the T Trap

Of C6-A's three candidates, $H\leftrightarrow F$ has already been cut down by C6-B/C to $H_{\rm force}\to F_{\rm NL}^{+}\dashrightarrow H_{\rm force}$, and $G\leftrightarrow P$ has already been rewritten by C6-D into the hereditary joint state $(G,P)_{\rm joint}\dashrightarrow(G,P)_{\rm joint}$; C6-E formally audits the last one: $T$. C5-B/C already proved that a purely scalar temporal argument cannot by itself force same-time overlap; C6-E's question is no longer this old one, but rather: is the middle/operator temporal debt itself already the marginal of some spatial source measure? C6-E.1 Canonical Middle Spacetime Lift gives a precise, affirmative answer: the middle debt $m(t)=\int\lambda_2^+|S|^2dx$ itself naturally carries a spacetime probability measure $\Pi^M$ whose time marginal is exactly the middle-debt probability C5 already uses — the middle temporal state is itself already the marginal of a spatial source measure. C6-E.2 repeats the same construction for the operator's forward $H^1$-growth debt, using the local positive-growth capacity as the spatial density. Defining the temporal overlap $\Omega_T$ (between two time marginals) and the spacetime shared overlap $\Omega_{ST}$ (between two full spacetime measures), C6-E.4 Temporal Projection Contraction Theorem (total variation contracts under pushforward) proves $\Omega_{ST}\le\Omega_T$ — spacetime shared source is a strictly stronger condition than simultaneous occurrence in time. C6-E.5 uses an explicit abstract construction (identical time densities but spatial conditional distributions with disjoint supports) to prove that $\Omega_T=1$ can hold simultaneously with $\Omega_{ST}=0$ — a purely temporal marginal cannot logically prove a genuinely shared spatial source. If $\Omega_{ST}>0$ genuinely holds, one can build a shared middle–operator spacetime source probability $\Pi^\cap$, whose support has, at almost every point, both positive middle strain activity and positive local $H^1$ growth capacity simultaneously — strictly stronger than mere temporal overlap. Tracking the middle-gap variable $\vartheta(S)$ on the shared source, if non-degenerate mass survives, C6-E.6 Shared Directional-Cone Extraction Lemma (using compactness of the normalized strain-direction sphere together with a finite cover) forces some fixed-width directional cone to genuinely carry nonzero spacetime mass — a genuine shared strong-middle directional source. But the paper explicitly draws a boundary: this still does not give a pointwise core, mean-rotation depletion, pressure provenance, or Grujić–Xu sign geometry — it still needs additional core-scale localization (meaningful only relative to a legitimate reference scale, otherwise it routes into the legality class $\mathsf A$) and cross-generation heredity, neither of which yet exists. C6-E.7 Pure-Temporal State Completeness No-Go therefore proves: two entirely different spacetime source pairs can have identical time marginals, yet differ completely in spatial overlap, middle-gap geometry, directional concentration, core localization, and heredity — $T$ by itself does not contain enough state information to determine the physical source-coupling state; it is only a marginal label, not a complete physical state. Hence $T\overset{N}{\looparrowright}T$ as a complete physical self-cycle is formally ruled dead — the same shape of verdict as H/F and G/P — but it is not a global exclusion: the correct object becomes $TS_n\overset{\text{spacetime source heredity}}{\dashrightarrow}TS_{n+1}$. C6-E.8 Finite Temporal–Spatial Coupling Bottleneck Theorem gives a dichotomy of exactly the same type as C6-C/D: any infinite candidate must be either uniformly shared-source coherent, or approaching one of seven named boundaries (temporal-phase separation, spatial-source separation, operator-capacity cancellation/inflation, shared-middle-gap collapse — routed directly into the existing $G$ class — core-scale diffusion/multiplicity, source-heredity collapse, and scale/setup exit — routed into class $\mathsf A$). The round's most important overall conclusion: at this point all three of C6-A's original coarse candidates — $H/F$, $G/P$, $T$ — have been refined to the same form: $HF_{\rm coherent}$, $GP_{\rm hereditary}$, $TS_{\rm hereditary}$; after five rounds, no nontrivial recurrent PDE defect cycle has yet been genuinely certified, but the entire problem has been rewritten precisely into a finite number of well-defined combinatorial problems. Since all three are now individually typed into the same shape, the natural next step is no longer to keep subdividing each separately, but to try to build genuine cross-domain edges: can uniform TS coherence genuinely route into $GP$ or $HF$? Formally hands off to C6-F — Shared-Source Core Extraction, Spatiotemporal Heredity, and Cross-Domain Routing to GP/HF, with eight composition obligations (F1 reference-scale shared concentration through F8 recomputing the candidate graph).