Proposition: Off-locus agreement does not determine extension

Suppose two distribution-valued candidate maps agree after restriction to an open set $U$. They need not agree globally, or satisfy the same global reconstruction-null identities. A correction supported on a closed set $C$ cannot cancel a defect detected by a test supported away from $C$.

Proof. A nonzero distribution supported outside $U$ has zero restriction to $U$ and can be added without changing that restriction. Locally, a smooth omitted submanifold admits nonzero compactly supported delta-type distributions.

For the second statement, every distribution supported on $C$ annihilates a test supported away from $C$. Thus it cannot change a nonzero pairing of the defect with that test.