RSForcingChain012Cert
plain-language theorem explainer
Certificate structure bundling three structural facts for RS forcing-chain module 12: diagonal domain cost vanishes, domain cost is nonnegative on positive arguments, and the canonical threshold is positive. Downstream code cites the inhabited instance when it needs the module-12 package as a single object. Pure structure definition with no proof body.
Claim. A module-12 forcing certificate is a triple of properties: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ equals zero; (ii) for all positive reals $m$ and $e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
Module 12 sits in the Foundation forcing chain and records the dimensionless identity $J(\varphi)=\varphi-3/2\approx 0.11803$ as a structural theorem (zero sorry, zero axiom). The local cost object is a real-valued domain cost on pairs of positive scale parameters; on the diagonal it is required to vanish, matching the J-cost minimum at identity.
Upstream, recognition-event cost is already known to be nonnegative: any recognition event $e$ satisfies $0\le e.\mathrm{cost}$ by nonnegativity of the J-cost on positive states. The certificate lifts that nonnegativity pattern to the module's domain-cost function and pairs it with a strictly positive canonical threshold used as a comparison scale in the chain.
Sibling lemmas in the same module discharge the three fields: diagonal vanishing, domain-cost nonnegativity for positive arguments, and positivity of the canonical threshold.
proof idea
No proof body: this is a structure declaration. The three fields are Prop-valued requirements (diagonal vanishing of domain cost, nonnegativity of domain cost on positive pairs, positivity of the canonical threshold). Inhabitation is supplied separately by packing the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos into a single value of this type.
why it matters
Gives a single named package for the three structural obligations of RS forcing-chain module 12, so later modules can depend on one certificate rather than three free-floating lemmas. The immediate consumers are the concrete inhabitant cert and the theorem cert_inhabited proving the type is nonempty.
In the broader Recognition Science chain this sits with the early forcing steps that fix the J-cost geometry and the golden-ratio self-similarity scale: the diagonal-zero and nonnegativity fields echo T5 J-uniqueness and the cost minimum at identity, while the positive threshold anchors dimensionless comparisons such as $J(\varphi)=\varphi-3/2$. It does not itself prove the $J(\varphi)$ identity; it only certifies the cost/threshold scaffolding that module uses.
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