Paper 56 carried thermodynamic standing per patch — a declared region with its own rest-frame field, the temperature tier stated as detailed balance/KMS with respect to the patch's own flow, no external frame consumed — but only on laboratory kinematics (flat benchmark; v/c ≲ 10^-8; a ≤ 10 m/s²), typing everything beyond out of its domain with the reason displayed. On curved patches the literature's standing license for temperature is a global timelike Killing structure: Tolman–Ehrenfest's lawful variation presupposes the global static flow, and the Kay–Wald uniqueness theorems live on bifurcate Killing horizons. This paper — the closing panel of the Shared-Frontier Program, consuming the charter Paper 57 defined — audits whether the patch gate itself extends, in two carefully distinguished moves, and closes the grouping at the panel level. Move one: extension and classification. The gate G-LEᶜov, c re-poses Paper 56's six components on declared weakly-curved static patch classes (the benchmark corner: Earth-surface static tube segments at Paper 52's frozen operating points), with the temperature tier carried by the patch pair (rhoₚ, alphaₜau) and the no-Killing criterion made executable: a two-part deformation test — ambient-metric families agreeing on the patch's causal neighborhood but not globally static, and Hadamard-state families agreeing on the patch's local data but differing in global boundary conditions and state selection — with the certified conclusion narrowed in print to what the test proves: no component consumes an explicit global metric/staticity input, and none consumes global state selection beyond the declared local form of the one at-risk input. That input, (hc3), is split and graded honestly: the response functional is defined (state class; the tube-pullback two-point function; a windowed stationary transform with the window-broadening row displayed), while the response control is a declared hypothesis — a declared bounded constant (Cᵣc ≤ 10³) bounding the curved correction at the declared orders, motivated by the locally covariant structure of Hadamard states and delivered by no consumed theorem, with a pre-committed withdrawal branch. The class predicate itself consumes local staticity on the patch's causal neighborhood only — the global qualifier is the thesis, not an input. The twelve-cell table becomes sixteen: the curved column is computed at the frozen corners under the displayed hypothesis list H-c, every cell claimed at the conditional strength "admitted under H-c" (Paper 56's own pattern), and the no-flip theorem — a conditional model-class result whose force is exactly the H-c list, said plainly — bounds every component's shift at the declared orders (epsilonN ≈ 1. 1×10^-18; tidal epsilonR ≈ 1. 7×10^-27) and closes the sixteen frozen cells against their displayed distances to the decision thresholds (≥ 10^-2 — thirteen orders above the corrections at the declared Cᵣc bound), with the general in-class case stated under an explicit margin hypothesis rather than silently assumed. Tolman–Ehrenfest enters as a two-patch comparison row — a 1. 1×10^-16 differential per metre of height, fourteen orders below the probe tolerance: on the declared class the patch reading and the Killing-licensed reading agree within every consumed tolerance — a check, never a derivation, never consumed. Move two: a boundary display, not an extension. Strong acceleration receives no verdict anywhere. What is published is the typed exclusion frontier: the class corner is declaration-limited, not tolerance-limited — the tolerance-parametric crossing scale a_* (epsilon) = 2 pi c omegac / ln (1 + 1/epsilon) (equivalently TU (a_*) = hbar omegac / kB ln (1 + 1/epsilon) ) sits seventeen-to-eighteen orders above the corner, log-robustly (a_* in 5×10¹8, 3×10¹9 m/s² across five orders of tolerance; TU (a_*) in 21, 120 mK straddling the licensed 73 mK, itself a member of the same formula at the resident-occupancy tolerance) — so the gap is the finding, and the two reasons the declarations stop at the corner are typed with their seats (the environment stipulation's validity domain; description-relative particle content), with no independent conceptual threshold computed or claimed. The verdict is scoped: (y1) – (y4) close on the declared extended classes — the extended gate with the two-part test; the boundary display; the curved column at classification grade; an exact flat-limit recovery of Paper 56's gate, all twelve cells and corner numbers unchanged — while (y5) (general curvature and strong fields; horizon-crossing patches; nonequilibrium terms as physics; the modular provenance of the patch flow; device grade; the background-independent frontier) is scored open and built elsewhere. FrontierStatus (patch) = scoped at model grade. No branch licenses a universal local temperature in curved spacetime. The closure ledger reports the sibling panels' verdicts as statuses, consumes neither, and names the program's residue once, each row owned by a named register conjunct. Nothing is adopted, calibrated, or scored; no band moves; the substantive Killing-licensed face is declined, not refuted.
Tomoyuki Uchida (Sun,) studied this question.