IndisputableMonolith.Foundation.ForcingChainCompleteness3
Third completeness package for the Recognition Science forcing chain: it packages a domain cost, a positive canonical threshold, and an inhabited certificate that the local forcing obligations hold. Foundation authors cite it when assembling the T0–T8 chain into a single certified block. The module is mostly definitions plus short positivity and inhabitation proofs over Cost and Constants.
claimOn the RS cost side, a domain cost $C$ is fixed together with a canonical threshold $\theta>0$. The module supplies a completeness certificate asserting that the third block of forcing-chain obligations (cost nonnegativity, threshold positivity, and the associated equalities at evaluation points) holds, and that this certificate type is inhabited.
background
Recognition Science forces its discrete ontology from a single cost functional and a short chain of uniqueness theorems (T0–T8). The cost layer lives in Cost: the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) is the unique symmetric generator of the Recognition Composition Law. Constants supplies the RS-native tick $\tau_0=1$ and the golden ratio $\varphi$ fixed at T6.
This module sits in Foundation as the third completeness slice of that chain. It introduces a domain-level cost (evaluation and nonnegativity), a canonical numerical threshold with a positivity lemma, and a certificate record ForcingChainComp3Cert whose inhabitation witnesses that the local obligations are discharged. Upstream imports are only Mathlib, Constants, and Cost; no geometry or particle ladder is required here.
proof idea
Definition-heavy module with short supporting lemmas. Domain cost is declared and tied to pointwise evaluation; nonnegativity is inherited from the Cost layer. The canonical threshold is a concrete positive real; positivity is a one-line arithmetic fact. The certificate is a structure bundling those facts; inhabitation is by constructing a canonical instance from the lemmas already proved in-module. No deep tactic scripts: wrappers and structure packing over Cost/Constants.
why it matters in Recognition Science
Completeness modules turn the narrative forcing chain (J-uniqueness at T5, $\varphi$ at T6, eight-tick octave at T7, $D=3$ at T8) into machine-checkable certificates that downstream Foundation assemblies can require as hypotheses. Part 3 specifically locks the cost/threshold fragment so later chain glue does not re-prove nonnegativity or threshold positivity. No downstream edges are recorded yet on this page; the natural consumers are unified forcing-chain aggregators and any top-level RS consistency certificate that demands an inhabited ForcingChainComp3Cert.
scope and limits
- Does not prove full T0–T8 uniqueness; only the third completeness certificate slice.
- Does not derive J-uniqueness, $\varphi$, eight-tick structure, or $D=3$.
- Does not fix physical constants beyond what Constants and Cost already export.
- Does not supply particle masses, $\alpha$, or ladder rung formulas.
- Does not claim downstream consumers; used_by is empty on this page.