Theorem: Exact finite local-Ward transfer certificate

Under Assumption ass:ward,

Consequently:

1. vanishing of the right side certifies exactly the named finite Ward residual;

2. if the individual tested defects vanish, every named residual vanishes;

3. if their operator norms are bounded respectively by $a_{raj,\nu }$, $b_{raj}$, and $c_{raj}$, then

   $$\Vert W_{raj}{\Vert }_{\mathrm{op}}\le \sum _{\nu }{a}_{raj,\nu }+b_{raj}+c_{raj};$$

4. the result extends to linear combinations of the declared tests and conformal Killing fields, with the corresponding linearly specified contacts.

The theorem does not assert untested Ward identities, a global net action, Einstein dynamics, or recovery of a code.

Proof. The distributional improvement identities give

Apply Proposition prop:current, substitute (eq:divdefect)–(eq:contactdefect), and evaluate on $f_{raj}$. This proves (eq:wardtransfer). The remaining claims follow by the triangle inequality and linearity.