cert_inhabited
plain-language theorem explainer
The milestone-certificate type is inhabited: there exists a structural certificate packing diagonal vanishing of domain cost and positivity of the canonical threshold. Anyone exporting the FinalModule_1398 domain-coverage milestone cites this. The proof is a one-line term that hands over the already-built certificate witness.
Claim. The type of milestone certificates is nonempty: there exists a certificate consisting of the identity that domain cost vanishes on the diagonal ($\mathrm{cost}(r,r)=0$ for all $r\neq 0$) together with strict positivity of the canonical threshold.
background
This file is a Recognition Science structural certificate module for a domain-coverage milestone (Plan v7, 109th pass). The module status is structural theorem: zero sorry, zero axioms.
A milestone certificate is a structure with two fields. The first asserts that the domain cost of any pair $(r,r)$ is zero whenever $r\neq 0$. The second asserts that the canonical threshold is strictly positive. Domain cost is the local cost functional measuring recognition defect on the domain; the canonical threshold is the positive cutoff against which that cost is compared.
The concrete certificate object is assembled earlier in the same module from the sibling facts that establish diagonal vanishing and threshold positivity. This declaration only records that the resulting structure type is inhabited.
proof idea
Term-mode one-liner. The proof supplies the already-constructed certificate as an explicit inhabitant of the milestone-certificate structure, which discharges nonemptiness. No rewriting, case splits, or external lemmas are invoked.
why it matters
FinalModule_1398 sits in the physics layer as a structural (hygiene) milestone rather than a forcing-chain step. Inhabiting the certificate type packages the domain-cost diagonal identity and threshold positivity into a single exportable witness. Downstream code that needs a nonempty domain-coverage certificate can depend on this fact without reopening the underlying cost and threshold lemmas. It does not advance T5–T8 (J-uniqueness, phi, eight-tick, D=3) or the Recognition Composition Law; it locks local module completeness for the domain-cost side of the physics stack.
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