Proposition: Support of differences detected by agreement

If two distributions agree on an open domain $V\subset U$, their difference is supported in $U\setminus V$.

For the supported-sector reference of Theorem thm:reference, two extensions agreeing on $U\setminus N_U$ differ by a distribution supported in $N_U$.

For the full target, agreement on $U_{\mathrm{can}}$ alone gives support in $W\setminus U_{\mathrm{can}}$, not automatically in $N$.

Proof. The difference vanishes on every test compactly supported in $V$, which is exactly the asserted support statement. The two applications use their respective agreement domains.