Theorem: Conditional symmetry-preserving matching

Under ass:matching,

For every continuous linear test $\ell$ whose constructed low-energy comparison gives

the continuum observable recovers that prepared EFT prediction to the stated error.

Exact equality with a finite perturbative truncation does not follow unless its remainder vanishes. Nor does this observable comparison establish equality of complete instruments or quantum channels.

Proof. Continuity gives $Q{F}_{\Lambda }\to Q{F}_{\infty }$. The two limits in eq:wardmatching give $Q{F}_{\Lambda }\to 0$. Hausdorffness yields $Q{F}_{\infty }=0$.

Continuity of $\ell$ permits evaluation of the limit. The displayed remainder is precisely the controlled comparison supplied by U5; it is not identified with zero. No complete-instrument hypothesis is implied by a linear-observable estimate alone.