A fixed insertion and contact prescription

The certificate must keep operator identities and insertion identities distinct.

For each member $r$ of a finite list, let

be the current coordinate and the declared spectator coordinates. Supply matrix-valued distributions

on an open joint-test domain $U_r$. Their values lie in $M_q(\mathbb{C})$, for a specified finite probe space of dimension $q$.

They may be obtained from declared matrix elements or correlation insertions, but their product prescription is part of the input. The contact distributions $\mathcal{P},\mathcal{S},{\mathcal{C}}_X$ include all contact terms that the claimed identity requires.

If time-ordered products are used, differentiating a field inside time ordering is not automatically the same operation as differentiating the time-ordered distribution. The chosen derivative and contact prescription must justify the improved insertion below. Any discrepancy belongs in the contact ledger HollandsWald.