Theorem: Matched locality survives completion

Suppose compatible pairs $(E_k,F_k)$ on a common skeleton differ only in fixed local replacement slots, with those slots' supports converging into a compact set $K$. Their completion limits form a $K$-local replacement pair and admit the matched realizing families of Theorem thm:matched.

Equality of exterior initial reductions, including a reference, also survives the limit.

Proof. Every unchanged local channel and coordinate is equal in the two members at each $k$. Their limits are equal by continuity of the coordinate and channel projections. The changed supports lie in $K$ in the limit. The resulting pair therefore meets Definition def:replacement; apply Theorem thm:matched.

Partial trace and classical marginalization are continuous. Equality of exterior initial reductions is a closed condition, also after tensoring the fixed comparison reference.