Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

Audit of the finite tests

Outcome and the remaining interacting problem › Audit of the finite tests · lines 1562–1597

Test	Object	Result in this manuscript
A	Smoothly smeared Cauchy and bulk linear fields	Cauchy covariance construction exists.
B	$B_{f}$ against bulk linear fields	Compact polynomial representatives exist.
C	$B_{f} ⋆_{H} B_{f^{'}}$	The kernel (eq:self) exists; the $w w$ terms vanish.
D	$B_{f} ⋆_{H} Q_{q}$ and reverse order	The finite contraction kernels exist.
E	$B_{f} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}}$	No extension of the fully contracted coefficient with the stated normal-topology continuity and old-product agreement.

Neither the vanishing of $H_{w w}$ nor success of Tests A–D is a closure theorem. Test E supplies the named obstruction.

The characteristic-cone alternative also fails for the actual insertion kernel by Proposition prop:char. Merely allowing timelike linear boundary conormals misses the same-orientation null pairs in the bilinear kernel.