Optional finite Ward residual: a separate interface

The finite local-Ward decomposition discussed in Oai282-A is a comparison, not a premise for Model 3 CheungOai282. Its algebraic content can be stated independently. On a flat patch, let finite-matrix-valued distributions $T_{\mu \nu }=T_{\nu \mu }$, $L$, $P_{\nu }$, $S$, and $C_X$ be defined on a common joint-test domain. Put

and

For a conformal Killing field, ${\partial }_{(\mu }{X}_{\nu )}={\sigma }_X{\eta }_{\mu \nu }$. Direct differentiation gives, for every admitted compact joint test $f$,

Indeed the improvement has zero divergence and trace $-\square L$; the conformal Killing identity evaluates the differentiated $X$ term. The triangle inequality gives the corresponding finite norm certificate.

All insertion products and contacts in this statement are inputs. Equation (eq:ward) supplies no missing product prescription, does not identify these matrices with $B_{\Lambda ,M}(J)$, and does not establish an untested Ward identity or Einstein dynamics. Number compression alone would not even provide the finite matrix space required by this optional certificate.