IndisputableMonolith.Foundation.RS_Forcing_Chain_Module_009
Certificate pack for forcing-chain step 009: a domain cost with nonnegativity, a positive canonical threshold, and an inhabited RSForcingChain009Cert bundling them. Foundation auditors tracing T0–T8 uniqueness cite the certificate rather than reopening Cost. Content is definitional plus short positivity and equality lemmas over the Cost and Constants imports.
claimThe module defines a domain cost $C$ with $C\ge 0$, records its value at a distinguished point, introduces a canonical threshold $\tau>0$, and packages these facts into an inhabited certificate for Recognition Science forcing-chain step 009.
background
Recognition Science derives physics from one functional equation along the T0–T8 forcing chain. Cost is carried by the J-functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), developed in the Cost library. Constants supplies RS-native units, including the fundamental tick $\tau_0=1$.
This Foundation module is a thin certificate layer on those imports. It introduces a domain-level cost, an equality lemma fixing the cost at a named point, nonnegativity of that cost, and a canonical threshold proved positive. The threshold is a comparison scale for later chain steps, not a new physical constant.
Sibling names (domainCost, canonicalThreshold, RSForcingChain009Cert, cert_inhabited) mark a self-contained pack: definitions plus the lemmas needed to inhabit the certificate type.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost and the canonical threshold are introduced by definition. Nonnegativity and threshold positivity are short lemmas discharged from Cost/Constants facts (nonnegative J-type structure and positive RS scales). An equality lemma pins the cost at a distinguished argument. The certificate record is inhabited by assembling those lemmas; no multi-step tactic proof is required beyond that packaging.
why it matters in Recognition Science
Earns its place as step-009 scaffolding in the Foundation forcing-chain series: downstream chain or audit code can depend on RSForcingChain009Cert for domain-cost nonnegativity and a positive threshold without re-proving Cost lemmas. No used_by edges are recorded yet, so the module is presently a leaf certificate. It does not itself force $\phi$, the eight-tick octave, or $D=3$ (T6–T8); it only supplies local cost/threshold facts those later steps may assume when the full UnifiedForcingChain is wired.
scope and limits
- Does not prove J-uniqueness (T5) or force $\phi$ as self-similar fixed point (T6).
- Does not establish the eight-tick octave (T7) or $D=3$ spatial dimensions (T8).
- Does not derive mass-ladder rungs, $\alpha$ bounds, or Berry-threshold numerics.
- Does not claim more than nonnegativity of domain cost and positivity of the threshold.
- Does not record downstream consumers; used_by is empty in the graph.