IndisputableMonolith.Physics.RS_PHY_Structural_006
Structural physics module that packages a domain-level cost functional and a positive canonical threshold into a named certificate (RS-PHY-Structural-006). Physicists cite it when they need a nonnegativity-checked cost on a domain together with a fixed positive cutoff. The module is mostly definitional: it introduces the cost and threshold, proves elementary positivity and evaluation lemmas, and inhabits a certificate record.
claimDefine a domain cost $C$ (built from the RS $J$-cost), prove $C \ge 0$ and the evaluation identity at equality cases, fix a canonical threshold $\theta > 0$, and package these facts as the inhabited certificate RS-PHY-Structural-006.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that $J$-cost; Constants supplies the RS-native tick $\tau_0 = 1$ and the golden-ratio ladder constants used elsewhere in the physics layer.
This module lifts the pointwise cost to a domain-level functional (domain cost) and introduces a single positive canonical threshold against which domain costs can be compared. The local setting is structural rather than dynamical: no evolution equation is solved here; the objects are static cost and cutoff data meant to be reused by later physics certificates.
Sibling lemmas record nonnegativity of the domain cost, its value under equality of arguments, and positivity of the threshold. Those facts are then bundled into the certificate record so downstream modules can depend on one inhabited Prop rather than on a scatter of lemmas.
proof idea
Definition-first module. Domain cost is introduced as a def in terms of the imported $J$-cost; nonnegativity and the on-equality evaluation are short algebraic or rewriting proofs from Cost. The canonical threshold is a positive real constant (positivity is a one-line inequality). The certificate type collects these statements; inhabitation is by assembling the already-proved lemmas into the record constructor. No deep tactic search or external analytic machinery.
why it matters in Recognition Science
Gives the physics layer a reusable, certificate-shaped handle on domain cost plus a fixed positive cutoff, aligned with the RS cost infrastructure (T5 $J$-uniqueness and the Cost module). Downstream used-by edges are empty in the current graph, so the module presently stands as a leaf structural package rather than a direct lemma of a named parent theorem. It is the natural place to hang later comparisons of domain costs against the Berry-scale or rung thresholds once those links are wired. Closes a small structural obligation (nonnegative domain cost, positive cutoff) that any continuum or lattice RS physics argument will need before quoting mass-ladder or eight-tick timing results.
scope and limits
- Does not derive the form of $J$ or the Recognition Composition Law; those are imported.
- Does not prove dynamical evolution, mass spectra, or coupling constants.
- Does not fix numerical units beyond RS-native conventions from Constants.
- Does not assert uniqueness of the canonical threshold among all positive cutoffs.
- Does not currently feed a named parent theorem in the dependency graph.