IndisputableMonolith.Chemistry.PolymerChainStatisticsFromJCost
Chemistry module that packages polymer-chain statistics as a nonnegativity-and-threshold certificate built from the RS J-cost. It defines a domain cost, a positive canonical threshold, and an inhabited PolyChainStatCert record. Polymer and soft-matter theorists in the RS stack would cite it when lifting J-cost bounds into chain-statistics claims. The file is mostly definitions plus elementary positivity lemmas over Cost and Constants.
claimThe module introduces a domain cost $C$ built from the RS $J$-cost, proves $C \ge 0$ and evaluates it at equality cases, fixes a canonical positive threshold $\theta > 0$, and packages these into an inhabited certificate $\mathrm{PolyChainStatCert}$ asserting the polymer-chain statistics bounds implied by $J$.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0 = 1$ used as the time unit for chain dynamics.
In the chemistry layer one lifts $J$ from dimensionless ratios to a domain cost on polymer configurations (end-to-end ratios, segment stretch factors, or similar positive reals). Nonnegativity of that cost and a fixed positive threshold then become the quantitative content of a "statistics certificate": chains whose cost stays below threshold obey the RS-predicted ensemble bounds.
The module sits downstream of the pure cost theory and upstream of any concrete polymer or soft-matter corollaries that need a named, inhabitable certificate rather than raw $J$ inequalities.
proof idea
Definition-heavy module, not a single deep theorem. It introduces domainCost from the imported $J$-cost, records evaluation-at-equality and nonnegativity as short lemmas, defines a canonicalThreshold with a positivity proof, then bundles the pieces into the structure PolyChainStatCert together with a canonical inhabitant cert (and cert_inhabited). Argument structure is: reuse Cost nonnegativity, fix a positive threshold constant, package.
why it matters in Recognition Science
Gives the chemistry domain a first-class, Lean-level certificate that polymer-chain statistics descend from $J$-cost rather than from an ad-hoc free-energy ansatz. That matches the RS program of deriving material and chemical regularities from the same T5 $J$-uniqueness and RCL that force $\phi$, the eight-tick octave, and $D=3$. No downstream used_by edges are recorded yet; the module is a leaf certificate factory intended for later polymer, rubber-elasticity, or coil-globule results that need an inhabitable PolyChainStatCert. It closes the scaffolding gap between pure Cost and named chemistry claims.
scope and limits
- Does not derive a microscopic Hamiltonian or Langevin equation for the chain.
- Does not compute numerical persistence lengths, radii of gyration, or experimental fits.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not treat branched, cross-linked, or active polymers.
- Does not connect yet to mass-ladder or $\alpha$ band results.