RSSTDStructural009Cert
plain-language theorem explainer
A certificate type packaging three structural facts for Standard Model module 9: the domain cost vanishes on the diagonal, stays non-negative for positive mass and energy, and the canonical threshold is strictly positive. Downstream code cites it to obtain a single inhabited witness. The declaration is a pure structure definition; the three fields are discharged by sibling lemmas when the concrete certificate is built.
Claim. A certificate is a triple of properties: (i) for every nonzero real $r$, the domain cost of $(r,r)$ is zero; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
Module 9 sits in the Standard Model structural layer of Recognition Science. The local forcing narrative is the chain T5 (J-uniqueness) through T6 (phi forced), T7 (eight-tick octave), and T8 ($D=3$), recorded as a structural theorem with no sorry and no axioms.
Domain cost is the module-local cost functional on pairs of reals (mass and energy style arguments). It is expected to inherit non-negativity from the global recognition cost: upstream, any recognition event has nonnegative cost because $J$-cost is nonnegative on positive states. The diagonal vanishing condition encodes that equal arguments carry zero structural defect. The canonical threshold is the positive cutoff used to separate trivial from nontrivial structural configurations in this module.
proof idea
No proof body: the declaration is a structure (record type) whose three fields are propositions. Inhabitation is deferred to the sibling definition cert, which fills the fields by domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos. The nonempty witness theorem is then a one-constructor application of that concrete certificate.
why it matters
The structure is the typed interface for Structural Module 9. Downstream, cert assembles an explicit inhabitant and cert_inhabited records Nonempty of the certificate, closing the module's structural claim under the T5–T8 forcing narrative. Packaging diagonal vanishing, non-negativity, and a positive threshold keeps the Standard Model layer aligned with the global cost calculus (nonnegative $J$-cost on recognition events) without reopening the uniqueness or dimension steps. It is bookkeeping that makes the zero-sorry structural status checkable as a single inhabited certificate rather than three scattered lemmas.
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