cert_inhabited
plain-language theorem explainer
The completeness certificate for the third forcing-chain package is inhabited: there exists a record packing diagonal vanishing of the domain cost, its nonnegativity on positive arguments, and positivity of the canonical threshold. Anyone assembling the T0–T8 structural theorem cites this existence fact. The proof is a one-line term that supplies the already-built witness `cert`.
Claim. There exists a certificate packing three facts: the domain cost vanishes on the diagonal ($\mathrm{domainCost}(r,r)=0$ for all $r\neq 0$), the domain cost is nonnegative for positive mass and energy arguments, and the canonical threshold is strictly positive.
background
This module closes a structural slice of the T0–T8 forcing chain: the complete derivation of physical constants from the unique J-cost. The local certificate type packages three elementary analytic properties of the domain cost and a fixed positive threshold used downstream in the chain.
The domain cost is the cost functional restricted to the mass–energy plane relevant to this completeness fragment. Its diagonal vanishing says matched arguments incur zero cost; nonnegativity is the usual positivity axiom for a cost. The canonical threshold is a positive real cutoff appearing in the completeness bookkeeping.
The structure ForcingChainComp3Cert is exactly the triple of those three propositions. Upstream, the module already constructs a concrete inhabitant of that structure; the present theorem only records that the type is nonempty.
proof idea
Term-mode one-liner. The proof is the anonymous constructor ⟨cert⟩ for Nonempty, where cert is the already-defined concrete value of type ForcingChainComp3Cert assembled earlier in the same module from the three component lemmas (diagonal vanishing, nonnegativity, threshold positivity). No tactics and no further lemmas are invoked.
why it matters
In the Recognition Science forcing chain, structural completeness is often stated as inhabitation of a certificate type rather than as a bare conjunction of lemmas. This theorem is the inhabitation half for the third completeness package: it lets later modules treat the three cost/threshold facts as a single inhabited certificate.
The module header frames the whole file as a structural theorem (zero sorry, zero axiom) for T0–T8 completeness from J. The three fields of the certificate sit under that umbrella: cost axioms feed the J-uniqueness and RCL side of the chain, while the positive threshold supports the discrete cutoffs that appear once phi and the eight-tick octave are forced.
No downstream consumers are recorded yet in the graph, so the immediate role is local packaging inside ForcingChainCompleteness3. The open question it does not touch is the full identification of every T0–T8 landmark with these three fields alone; that identification lives in sibling completeness modules.
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