RS2026State3Cert
plain-language theorem explainer
A three-field certificate packaging the 2026 Recognition Science state: diagonal domain cost vanishes, domain cost is nonnegative for positive measure and energy, and the canonical threshold is strictly positive. Foundation and audit code cite it as the inhabited witness that the cost/threshold layer is in place. It is a pure structure (no proof body); inhabitants fill the fields from sibling lemmas.
Claim. A certificate consists of three properties: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ 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
The module records a structural 2026 state-of-the-art certificate for Recognition Science: forcing chain T0–T8 complete, constants derived, no code sorries. This declaration is the Prop-bundle that packages three elementary cost/threshold facts used as the local audit surface.
Domain cost is the real-valued cost assigned to a measure–energy pair $(m,e)$ in the recognition calculus (imported from Cost and related foundation modules). The diagonal identity $C(r,r)=0$ for $r\neq 0$ is the fixed-point minimum of that cost. Nonnegativity for positive arguments is the continuous analogue of the event-level fact that every recognition event has nonnegative cost (upstream cost_nonneg: "The cost of any recognition event is non-negative," via $J$-cost nonnegativity).
The canonical threshold is the positive real cutoff used downstream as the acceptance floor for certified recognition; positivity is required so the certificate is not vacuous.
proof idea
No proof body: the declaration is a structure with three fields. An inhabitant is assembled by assigning the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos to those fields (see the downstream cert definition). Nonemptiness then follows by packaging that inhabitant.
why it matters
This structure is the typed witness that the 2026 foundation layer exposes a coherent cost-and-threshold interface. Downstream, cert builds a concrete value and cert_inhabited proves Nonempty RS2026State3Cert, closing the module’s audit claim that the state certificate is inhabited with zero sorry and zero axiom.
In the broader framework it sits under the completed forcing chain (T0–T8) and the $J$-cost calculus: diagonal vanishing and nonnegativity are the minimal sanity conditions on recognition cost before mass ladders, eight-tick structure, or constant derivations are invoked. It does not itself derive $\phi$, $D=3$, or $\alpha$; it only certifies that the cost/threshold primitives used by those layers are well-posed.
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