Theorem: Combined mathematical release

For the declared complete channel model and increasing-stencil family:

1. snapshot factorization, adjoint support, order-theoretic dependence, and the metric cone are proved;

2. legal ordering invariance holds for one finite network;

3. fixed word stencils and fixed finite periodic velocity models have their stated polyhedral obstructions;

4. changing stencils satisfy the uniform shape theorem, Euclidean cone convergence, zero-slack microscopic compatibility, and directional relay attainment;

5. the defined full-tail fixed-wire coarse theory has continuum domain of dependence, operational no-signalling, and sharp speed $c$;

6. under T it has a Poincare action and frame-invariant coarse speed, in every $D\ge 1$;

7. under C6${}^{\mathrm{obs}}$, algebraic microcausality passes to the matching observable limit;

8. unrestricted device resynchronization closure is incompatible with ordinary bidirectional relays.

The instrument limit, general transport construction, matching dynamical observable net, field content, and gravitational identification are not supplied.

Proof. Items (i)–(ii) are Lemma lem:comparison and Theorems thm:order–thm:foliation. Item (iii) is Theorem thm:fixedstencil and Proposition prop:polytope. Item (iv) is Theorems thm:shape, thm:coneconv, thm:relays, and Proposition prop:C2b. Item (v) is Theorem thm:coarseDOD, Proposition prop:coarseNS, and Corollary cor:coarsespeed. Items (vi)–(viii) are Theorem thm:action, Corollary cor:frame, Theorem thm:algmicro, and Theorem thm:withdrawal.