Remark: Supported detection needs a support certificate

Theorem thm:matter is a completion-level theorem for every such switch. To call its detecting test a retained laboratory test, one must also verify

and choose $\beta$ there. This follows, for example, when the vertical segment from $\Sigma$ to a plateau point lies in the retained reserve.

The host assumption alone does not place every transition of an arbitrary compact $g$ inside $\mathcal{L}$. No such support claim is silently added. The completion-level breaking and the supported version are thereby kept distinct.