Theorem: Matched continuum domain of dependence

Let $E,F$ be a locally matched pair in the finite-circuit class or its local completion, with a common readout supported in a compact set $Y$. Suppose every changed operation and initial-input modification lies in compact $K$, and

Then the complete readout laws on $Y$ agree. The conclusion is reference preserving in the joint-reduction sense of Lemma lem:comparison.

Proof. Use Theorem thm:matched, or its completed version, to choose common discrete schedules, backgrounds, routes, and readouts. The terminal readout events and changed firing footprints converge to their declared supports.

The positive margin and shrinking footprints imply that, for all sufficiently large $n$, no changed firing can be an ancestor of any common readout. Indeed every such dependency path would obey the exact physical cone of Lemma lem:physicalcone, contradicting the positive limiting separation. Initial changed inputs are excluded by the same argument.

The exterior initial snapshots and the required ancestor channels therefore agree. Lemma lem:locality gives equal discrete joint readout laws. Their logical complete channels converge, in fact are exact in the ideal construction, so their limits agree. The same proof uses joint cone–reference reductions.