Theorem: VII-J-ST local Grade S release

For the declared conical geometry and finite permissions:

1. complete finite dependency graphs bound joint-output Heisenberg support and deterministic reference-assisted influence;

2. certified causal routes yield ${\mathrm{Inf}}^{-}\subseteq {\mathrm{Sch}}^{-}\subseteq {\mathrm{Phys}}^{-}$ for the full encoded–computed–decoded channel;

3. directional cycle polytopes have the finite-time remainder (eq:offset), and the finite potential certificate (eq:potential) records directional transients;

4. signalling stability has the constant $2d_{\diamond }$, and the finite Hamiltonian comparison has the reference-stable tail (eq:tail);

5. destructive clocks realize the complete instrument (eq:clock), with event-time and programme/history calibrations distinguished;

6. the benchmark (eq:benchmark-channel) is an exact, nontrivial, reference-preserving Target B realization entirely in the regular part, buffered through the prescribed incidence parameter and supported by the displayed resource ledger.

These statements constitute a nonempty local Grade S package.

Proof. Theorems thm:nosignal and thm:schedule prove (i). Theorem thm:physical proves (ii). Theorem thm:polytope and Proposition prop:potential prove (iii). Theorems thm:signal-stability and thm:tail, with Proposition prop:hamiltonian-errors, prove (iv). Theorem thm:clock and Proposition prop:calibration prove (v). Lemma lem:benchmark-causal, Theorem thm:benchmark, and Section sec:ledger prove (vi). These provide all clauses of Definition def:grade.