Theorem: Sufficient full-map certificate

Suppose regularized complete grouped maps $F_{\delta }$ converge to a continuous $F:{\mathcal{P}}_{\mathrm{int}}\to {{\mathcal{D}}^{\prime }}_{\Gamma }(W)$, uniformly on bounded presentation sets. Assume:

1. $F_{\delta }(A)\to 0$ for every $A\in \ker \mathsf{Rec}$;

2. the limit agrees with canonical contraction on its proved domain;

3. the support and cone are those required for the complete graph;

4. all tensor coefficients, lower terms, and both $K_5$ placements are retained.

Then $F$ descends to a continuous full reconstructed-kernel map, with the stated reference, support, and wavefront.

Proof. The topology gives a distributional limit and Fourier decay outside $\Gamma$, hence its wavefront bound. The null-presentation condition gives descent. Support and reference agreement pass to the distributional limit.