Theorem: Propagation of comparison allowances

For a fixed finite package, suppose implemented or reconstructed values obey

Set

Then

and

For any $\ell \in Y^{\ast }$, a strict inequality

certifies failure of exact selected matching.

If $\ell \circ L=0$, a strict inequality

instead certifies infeasibility for every $a\in X$.

Proof. Subtract the two residuals:

The triangle and operator-norm inequalities give (eq:residual-error). The triangle and reverse triangle inequalities give (eq:residual-interval). Duality transfers the error bound to $\ell$. For the final assertion, the source pairing remains nonzero and $\ell$ annihilates the entire comparator image; apply Theorem thm:feasibility.