cert_inhabited
plain-language theorem explainer
The milestone certificate type for domain coverage is inhabited: there exists a pair of facts that the domain cost vanishes on the diagonal and that the canonical threshold is strictly positive. Anyone citing the FinalModule_1399 structural certificate uses this existence result. The proof is a one-line term that packages the prebuilt certificate witness.
Claim. There exists a milestone certificate: a record asserting that the domain cost satisfies $\mathrm{cost}(r,r)=0$ for every nonzero real $r$, and that the canonical threshold $T$ obeys $T>0$.
background
FinalModule_1399 is a structural milestone module in the Recognition Science physics layer (Plan v7, 109th pass). It packages a domain-coverage certificate with zero sorry and zero axioms. The ambient imports pull in the RS cost calculus and the constants layer, so the cost object is the standard Recognition cost (the unique $J$ fixed by the Recognition Composition Law and T5 uniqueness), specialized here to a two-argument domain cost.
A milestone certificate is a structure with two fields: diagonal vanishing of the domain cost for every nonzero real argument, and strict positivity of a canonical threshold. Those two propositions are the entire content of the certificate; the present theorem only asserts that the type is nonempty.
proof idea
Term-mode existence proof. The certificate type is a structure, so Nonempty is witnessed by a single constructor application: the prebuilt term cert (assembled elsewhere in the module from the diagonal-cost identity and the positivity lemma for the canonical threshold) is inserted as the inhabitant. No tactics, no further lemmas.
why it matters
This is the inhabitance seal on the FinalModule_1399 structural certificate: once the type is known to be nonempty, downstream milestone and coverage arguments can assume a domain-cost certificate without rebuilding the diagonal and threshold facts. In the RS forcing chain the cost side sits on T5 ($J$-uniqueness) and the Recognition Composition Law; the threshold side is the local positivity gate for the domain-coverage milestone. The module itself reports status STRUCTURAL THEOREM with empty axiom and sorry counts, so this declaration closes the inhabitance obligation for that milestone rather than advancing a new physical prediction.
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