Proposition: Why the witness result alone does not supply the certificate

A response identity obtained by a fixed partial witness contraction does not, as a matter of mathematical implication, supply a graph bridge certificate.

Proof. Proposition prop:partial-contraction gives distinct smooth compact kernels with identical partial witness data. Hence that identity alone does not determine the required full distribution.

Even canonical-domain agreement does not remove the descent issue, by Proposition prop:restriction-descent. An extension still requires existence and normalization; Proposition prop:affine-extension only classifies its differences if it exists, and Theorem thm:jetextension requires actual coefficient extensions. Finally, Proposition prop:jet-wavefront makes the output bound depend on those coefficients, not on the witness identity.

Thus none of the missing distributional identifications, descent conditions, or output obligations is a formal consequence of a partial witness equality.