E4: contact equations and old overlap

If a finite-scaling-degree point-extension theorem applies in dimension $n$, uniqueness holds below degree $n$; at higher degree the degree-preserving freedom is a finite sum of derivatives of delta of order at most $\lfloor \mathrm{sd}(u)-n\rfloor$ BF. Application to a submanifold requires the corresponding transverse and microlocal hypotheses BF. No such hypotheses or degrees have been evaluated here for the sharp Cauchy or reflected loci.

The parent identity

classifies its triangle finite part. It does not classify binary time-ordered ambiguities.

On every supplied old overlap one must compute

and solve for permitted contacts so that this difference is exactly zero. A smooth nonzero spacelike discrepancy away from the named contact set is not a contact.

All coefficients of ${\star }_c$, all supplied old free and time-ordered coefficients, and the parent parameter $c$ remain fixed. No subtraction inside ${\star }_H$ is authorized.

If source slots are included, differentiating

shows why the simultaneous prescription

cannot retain that fundamental-solution identity when $c_F\ne 0$. This excludes that simultaneous prescription; it does not exclude every possible sharp causal extension.