IndisputableMonolith.Physics.RS_Physics_Module_009
Physics module 009 packages a domain-level cost functional, its nonnegativity and evaluation identities, and a positive canonical threshold, then wraps them in an inhabitation certificate. A physicist citing RS domain energetics or threshold selection would land here. The module is mostly definitional with short positivity and equality lemmas feeding a single cert bundle.
claimDefine a domain cost $C_{\mathrm{dom}}$, prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\theta_*>0$, and package these facts as an inhabited RS Physics 009 certificate.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$.
This module lifts cost from scalar ratios to a domain-level functional domainCost, together with a fixed positive cutoff canonicalThreshold. The intended reading is a minimal energetic gate: domains whose cost sits below the threshold are treated as admissible at the present layer of the physics stack.
Sibling lemmas record nonnegativity, a pointwise evaluation identity, and positivity of the threshold. Those facts are then bundled into RSPhysics009Cert with an inhabitation witness, the usual monolith pattern for a self-contained physics slice.
proof idea
Definition-first module. domainCost and canonicalThreshold are introduced as defs. Short lemmas discharge nonnegativity of the domain cost, the evaluation identity at a point, and positivity of the threshold, each by direct appeal to Cost/Constants facts or elementary real arithmetic. The certificate structure aggregates those propositions; cert_inhabited is a one-line constructor packing the proved fields.
why it matters in Recognition Science
Module 009 is a leaf physics slice: it does not yet feed named downstream theorems in the graph (used_by is empty). Its role is local packaging of domain-cost and threshold facts so later forcing or mass/ladder arguments can import a single cert rather than re-prove nonnegativity and positivity. In the broader RS chain this sits downstream of T5 J-uniqueness and the Cost layer, and upstream of any domain-admissibility or threshold-crossing arguments that need a named positive gate. The cert pattern matches other RS physics modules that close a small lemma cluster before export.
scope and limits
- Does not derive domainCost from the Recognition Composition Law or T5 J-uniqueness.
- Does not identify canonicalThreshold with phi, Berry threshold phi^{-1}, or Z_cf.
- Does not prove uniqueness of the threshold or optimality of the domain cost.
- Does not connect to mass ladder, alpha band, or eight-tick octave results.
- Does not supply numerical bounds beyond positivity and nonnegativity.