Abstract

We organize the causal-interface programme following the constructed sharp free partial product as four ordered, supported stages on the regular part of the prescribed I-B conical spacetime. The adapted IX-A-ST carrier and the IX-Free-2 coefficient assignment are explicit inherited inputs; neither is reinterpreted as time ordering.

The principal analytical result is a complete common-regulator evaluation of the imaginary part of the locked mixed kernel. A direct null-cone distribution is accompanied by a nonzero observation-spacelike contribution between the direct and reflected cones; the two cosine tails cancel beyond the reflected cone. Consequently the commutator vanishes on the entire strongly separated two-way class, while a nonzero observation-spacelike control is not, by itself, a causal obstruction.

We prove exact functional support for the quadratic slot and show that a boundary-trace certificate implies a null-sphere lower-support theorem for the sharp insertion. Those spheres suffice for the full retained two-way commutator bridge; a solid-ball lower-support assertion is unnecessary and is false in the specified flat realization. We also prove the scalar non-lift, an actual-tensor algebraic consistency criterion, conditional causal subtraction, and the one-order loss under normalization.

The potential usefulness is methodological: the results separate a genuine fixed-domain obstruction from an uncertified support inference and identify a precise operator certificate for the next analytical step. The underlying algebraic and kernel findings are attributed to the cited Cheung manuscripts; the present contribution is their supported conical organization, explicit hypothesis audit, and finite certificate interface. Verification of the trace certificate for an otherwise unspecified inherited adjoint remains open. No sharp time-ordered product, normalized retarded product, interacting quantum coefficient, or spacetime-emergence theorem is constructed.