Theorem: Matched realization of interventions

A $K$-local replacement pair admits realizing families with:

1. the same substrate and partition schedule;

2. the same reserved interfaces, resource preparation, firing times, transport routes, and unchanged block channels;

3. different block channels only at the corresponding local vertex firings, whose complete footprints approach $K$;

4. the same discrete implementation of every common readout;

5. equal initial snapshots outside the declared discrete initial modifications, also jointly with a reference when that is required.

These conclusions persist under admitted finite compositions and recorded continuations.

Proof. The realization choices in Lemma lem:timing and Theorem thm:realization depend only on the common skeleton and its finite interface types. Choose them once for both circuits.

At an unchanged vertex, use the same complete local channel. At a replaced vertex, replace only its assigned complete instrument inside the same junction firing. Its input and output types are unchanged, so all transport permutations and storage rules remain the same. A different classical outcome changes the state of a represented controller; it does not change an undeclared scheduler.

Resource registers and route colours are initialized identically. A changed external input is placed only in its assigned local slots. The comparison encoding is tensorial on those slots with a common background; exterior reductions therefore agree whenever the logical exterior reductions agree.

A finite composite is realized by the same procedure on its common composite skeleton. Lemma lem:adaptive provides the fixed complete interface for recorded continuations. References are not read or modified.