cert_inhabited
plain-language theorem explainer
The milestone certificate type for FinalModule_1395 is inhabited: a concrete certificate exists. Anyone auditing structural completeness of this domain-coverage pass would cite it. The proof is a one-line term witness that packages the already-built certificate value.
Claim. The type of milestone certificates is nonempty. Equivalently, there exists a certificate whose two fields hold: the domain cost vanishes on the diagonal ($C(r,r)=0$ for all $r\neq 0$), and the canonical threshold is strictly positive.
background
FinalModule_1395 is a structural milestone module in the Recognition Science physics layer (Plan v7, 109th pass). Its status line records zero sorry and zero axioms. The local certificate packages two elementary facts about the domain cost and a fixed positive threshold.
A milestone certificate is a structure with two fields: (i) for every nonzero real $r$, the domain cost evaluated at the pair $(r,r)$ is zero; (ii) the canonical threshold is strictly positive. Domain cost is the local cost functional imported from the Cost layer (alongside Constants); the diagonal-vanishing property is the algebraic sanity check that equal arguments incur no cost defect. The canonical threshold is a fixed positive real used as the coverage cutoff for this pass.
The module therefore does not re-derive the J-cost or the forcing chain; it only certifies that the local domain cost and threshold meet the milestone interface.
proof idea
Term-mode witness construction. The proof is the single constructor application ⟨cert⟩, which exhibits the already-defined value cert : MilestoneCert as an element of Nonempty MilestoneCert. No tactics, no lemmas, and no further unfolding: inhabitance follows immediately from the existence of that concrete certificate record.
why it matters
This declaration closes the inhabitance obligation for the FinalModule_1395 structural certificate. Downstream consumers that require a Nonempty MilestoneCert hypothesis (or that pattern-match on certificate existence before reading the two fields) can discharge that obligation in one step.
In the broader Recognition framework the module is labeled a domain-coverage milestone: it records that the local cost vanishes on equal arguments and that the coverage threshold is positive, without reopening T5–T8 (J-uniqueness, φ, eight-tick octave, D=3) or the Recognition Composition Law. It is bookkeeping for plan completeness rather than a new physical derivation. With empty used_by in the current graph, its immediate role is auditability of the 109th pass rather than a named parent theorem.
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