RSForcingChain009Cert
plain-language theorem explainer
Certificate structure for RS forcing-chain module 9, bundling three structural facts: diagonal domain cost vanishes, domain cost is nonnegative on positive reals, and the canonical threshold is positive. Downstream code cites it via the module's concrete inhabitant and the Nonempty witness. Pure structure definition with no proof body; the field proofs live in sibling lemmas.
Claim. A module-9 forcing certificate is a triple of 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
Module 9 of the RS forcing chain treats J-cost symmetry: recognition cost is ratio-symmetric, i.e. $J(x)=J(1/x)$. The module is marked structural (0 sorry, 0 axiom). In RS units the cost is the unique J-functional $J(x)=(x+x^{-1})/2-1$ forced at T5.
Domain cost is the local cost on pairs of positive reals used in this module; the diagonal condition encodes that equal arguments (unit ratio) sit at the cost minimum. The canonical threshold is a strictly positive scale cut packaged with the cost facts.
Upstream, ObserverForcing records that "the cost of any recognition event is non-negative," proved via Cost.Jcost_nonneg on the event state. That nonnegativity is the conceptual parent of the second certificate field.
proof idea
No proof: this is a structure declaration. It only names the three Prop fields a certificate must carry. Inhabitation is separate: the def cert fills the fields by domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos, and cert_inhabited wraps that as Nonempty.
why it matters
Gives a single typed bundle for the structural cost facts of forcing-chain module 9 (J-symmetry / ratio-symmetric recognition cost). Downstream, cert is the concrete inhabitant and cert_inhabited proves the type is nonempty, so later chain steps can assume the package rather than re-prove diagonal vanishing, nonnegativity, and threshold positivity.
In the broader RS forcing chain this sits under T5 J-uniqueness and the Recognition Composition Law: the diagonal zero and nonnegativity are the minimum and positivity half of the J-cost story, while the positive threshold is the local scale cut the module needs. It does not itself force $\varphi$, the eight-tick octave, or $D=3$.
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