Theorem: Gauge-algebra continuum domain-of-dependence transfer

Under Assumption ass:continuum, suppose a compact modification support is at positive distance from the limiting past of the common finite report tuple. Then the two limiting joint report laws agree. The conclusion does not require spatial tensor factorization.

Proof. The positive margin, shrinking footprints, and uniform path bound place every changed device outside the report ancestors for all sufficiently fine scales. The matched background and included complete device actions agree. Theorem thm:matched, applied to the full record-retaining implementation, gives equality of the discrete joint report laws. Their limits are therefore equal.