Theorem: Spectator-blind certified executions

Consider a finite execution of the rule types in A:contract:rules. Extend its initial state to an admissible joint state $\tilde{\rho }$ on $\Gamma N$ such that

Assume the complete blindness and extension clauses of A:ass:control, identical original locations, and legal separate spectator transport/storage.

Then an aligned extended execution is legal. At every step, the original reduced matter states, geometric descriptions, persistent signatures, certification decisions, original transports, and declared controller outputs agree.

Proof. Induct on steps. For each accessible observable,

Equal complete inputs therefore select the same rule, guard, support, and certificate.

A matter channel preserves the reduction identity by A:lem:restriction. A $G$-rule receives the same geometric/controller inputs and leaves internal matter unchanged. A coupling rule receives the same accessible functional, so A:thm:A2 gives the same geometric and certification output. Any common included matter channel again preserves the original reduction. Original transport and successor control agree. The separate-reservation clauses establish spectator legality without changing these decisions.