55 / C6-F: Shared-Source Core Extraction, Spatiotemporal Heredity, and Cross-Domain Routing to GP/HF

C6-B–E have refined C6-A's three candidates into $HF_{\rm coherent}$, $GP_{\rm hereditary}$, $TS_{\rm hereditary}$; the key new object left by C6-E is the shared spacetime source probability measure $\Pi^\cap$, but temporal overlap $\Omega_T$ alone does not guarantee a genuinely shared spatial source — that needs $\Omega_{ST}>0$. C6-F's central question: if this TS shared source genuinely is uniformly non-degenerate, can it forever remain pure temporal bookkeeping, or must it genuinely enter $GP$, $HF$, or high-order forcing? C6-F.1 Shared Density Physical Domination Theorem gives a precise pointwise bridge: the same normalized shared-source density is simultaneously pointwise dominated by both the middle physical strain density and the positive local operator-growth capacity — no longer merely abstract probabilistic bookkeeping. But probability normalization by itself forgets absolute load $M_J,P_J$, so explicit absolute-load reserves $\rho_M,\rho_P$ are introduced. Via Fubini/an averaging principle, C6-F.2/3 genuinely extract the same instant $t_\ast$ and the same spatial region $E_\ast$ from the shared-core cylinder, at which both the middle strain density and the operator's positive growth capacity have simultaneous lower bounds — "the temporal-synchronization problem is genuinely solved at this extraction level." Since the middle state satisfies $\lambda_2^+/|S|\le1/\sqrt6$, C6-F.4 converts the middle lower bound directly into a genuine cubic strain-activity lower bound; C6-F.5/6 use Cauchy–Schwarz to convert the operator lower bound into a same-instant product cost $\|\mathcal Q_{SV}\|_{L^2(E_\ast)}\|\Delta S\|_{L^2(E_\ast)}$, and the Operator/Derivative Junction Dichotomy proves that the shared core must enter either nonlinear operator forcing (F-OP) or cubic strain activity (F-DER) — unconditionally: "uniform TS shared-source recurrence cannot remain purely spacetime bookkeeping forever." But genuinely entering $GP$ still lacks one key bridge: the extracted directional cone is source-weighted, not the Q/field-weighted cone that C5-D's strong-middle theorem genuinely needs — this is "the main gap in TS→GP composition"; a Source-to-Field Capture Gate (Q-weighted leakage $\epsilon_Q$) is defined, and if the leakage is below C5-D's quantitative threshold, combined with a mean-rotation-depletion reserve, C6-F.7 Conditional TS-Core → GP Theorem proves that the same event genuinely enters $GP$ (at least at the level of the joint total-pressure state; entering the finer signature/axis subtype further requires far-field dominance and legitimate provenance) — explicitly marked as conditional. The other branch: F-DER gives genuine third-order derivative activity, but this does not automatically equal $H$ — it still needs a derivative-realization theorem flag, component/sign geometry, and theorem-setup legality; if all hold, the dichotomy is: passing the spatial gate goes to external $\mathrm{REG}$, while failing throughout the whole admissible window gives genuine $H$ — "F-DER + theorem realization $\Rightarrow H\vee\mathrm{REG}$," not an unconditional $TS\to H$ edge. The paper also explicitly distinguishes same-instant extraction (existence of $t_\ast$) from genuine spacetime persistence (persisting over a positive-measure time set) — these differ, and recurrence still needs an independent heredity/concentration analysis. Compressing all the reserves needed for bridging (absolute load, shared overlap, core localization, middle gap/direction, source thickness, Q-field capture, mean rotation, pressure provenance, derivative realization, heredity, legitimate scale — 13 dimensions total) into a vector $\mathbf R^X$, C6-F.9 Finite Cross-Domain Bottleneck Theorem gives a dichotomy of exactly the same type as C6-C/D/E: uniform cross-domain coherence, or approaching one of eleven named boundaries. C6-F.10 Conditional Cross-Domain Routing Theorem is the round's climax — on the uniform branch: on the geometry–pressure side, $TS_{\rm core}\to GP$ (conditional); on the high-order side, $TS_{\rm core}\to F_{\rm OP}\vee F_{\rm DER}$, the former potentially entering $HF_{\rm coherent}$'s Duhamel-coherent re-entry, the latter potentially entering $H\vee\mathrm{REG}$ — this is the first typed bridge in the entire series that genuinely connects all three refined candidate families. It explicitly draws boundaries: C6-F does not prove "every TS recurrence enters GP or H" (the bridging reserves may degenerate), nor does it prove "GP/HF recurrence is impossible" — but it does exclude a stronger form of isolation: a shared, localized, non-degenerate, source-field-coherent, hereditary TS core that never enters any geometric/high-order interface — such a state already carries cubic strain, nonlinear operator/$D^3u$ toll, Q-field coherence, pressure response, or mean-rotation debt, so "pure TS" on the uniform branch is merely a bookkeeping projection, not an independent physical mechanism. The round's net effect: the three refined candidate families $TS$, $GP$, $HF$ are for the first time genuinely connected into a single graph via typed cross-domain edges, but the whole graph still has no closed recurrent SCC certified. Formally hands off to C6-G — Typed Cross-Domain Graph Rebuild, Joint-Node SCC Audit, and Minimal Boundary-Saturated Survivor Cycles, with eight composition obligations (G1 rebuilding the graph with new joint nodes through G8 the minimal survivor theorem).