Abstract

We return to the joint-extension problem left open in XIII-C. The entries are the same switched metric–scalar vertex and the same admitted second-order composite. The carrier, action, off-shell lifts, and partial-diagonal ambiguity class are unchanged.

Exactly one route is attempted: a common-frequency regularization of the complete time-ordered graph factors, with their inherited tensor coefficients retained. We prove that the finite-regulator graph functionals are compact microcausal polynomials, that the regularization converges to the prescribed products on the microlocally admissible open sets, and that auxiliary compact localization changes have a vanishing remainder. We also prove that a partial-diagonal contact cannot change any prescription on the collision complement.

These results do not establish the remaining limit through the noncoincident characteristic configurations. The inherited wavefront cancellation prevents an application of the sufficient product theorem there, while neither divergence of the complete pair nor an unavoidable normalization incompatibility is proved. The earliest unfinished gate is therefore E1.

We formulate the subsequent transverse, overlap, and pushforward obligations without treating them as completed. A relative-coordinate lemma identifies a sufficient balanced endpoint once a joint extension exists; it does not provide existence. The coefficient linear in $\hslash$ retains the ending Unresolved. No Ward defect, interacting differential, or restriction quotient is constructed.