Definition: Source-specific support certificates

For the retained chart $\mathcal{L}$, distinguish:

1. Witness certificate:

   $$P_{\epsilon }:=K_f+B_{\epsilon }⋐\mathcal{L}.$$

   Because $K_f\subset \{r\le 1/8\}$,

   $$P_{\epsilon }\subset \{r\le 1/4\}⋐U_C.$$

   This suffices to transport the numerical witness pairing, its compact gauge parameter, and its convolution dependence.

2. Full analytical certificate:

   $${\mathcal{R}}_{\epsilon }:=D_f\cup Q_{\epsilon }\cup ({\overline{U}}_C+B_{\epsilon })⋐\mathcal{L}.$$

   The required neighborhoods and the original time reserve are also retained. This sufficient certificate covers the causal computation, the convolution inputs, and the whole frozen supported gauge algebra used in its chart representation.

A certificate is a geometric inclusion for the actual retained chart. Compactness of the displayed sets alone does not prove their inclusion in that chart.