Proposition: Same-adjoint audit

There are two distinct legitimate conclusions.

1. From the weak inherited inputs Assumptions ass:carrier–ass:parent, the present manuscript has not verified (TC) for the inherited operator. Its domain-wide bridge remains conditional.

2. If the inherited response and actual transpose on all required compact covector sources are identified with (eq:Eflat) and its transpose, with the specified normalization and retained surface term, Proposition prop:flatTC verifies (TC), and Theorem thm:bridge clears the commutator gate for that identified realization.

In neither case is a time-ordered product constructed.

Proof. In case (i), the displayed trace identity for the same actual adjoint has not been supplied as an input or proved for that inherited realization. In case (ii), the identity and regularity have been proved by transposition, and the bridge theorem applies. The trace and support statements contain no binary time-ordered replacement kernels or old time-ordered map.