IndisputableMonolith.Physics.RS_Physics_Module_005
Physics module packaging a domain-level cost functional, its nonnegativity and evaluation identities, and a strictly positive canonical threshold, together with an inhabitation certificate. A physicist citing RS domain energetics or threshold comparisons would land here. The file is mostly definitional scaffolding plus short positivity and equality lemmas over the imported cost layer.
claimIntroduce a domain cost $C_{\mathrm{dom}}$, prove $C_{\mathrm{dom}}\ge 0$ and the pointwise evaluation identity, define a canonical threshold $\theta_{\mathrm{can}}>0$, and package these as an inhabited RS Physics 005 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 layer; Constants supplies the RS-native tick $\tau_0=1$.
This module lifts cost from scalar ratios to a domain-level functional $C_{\mathrm{dom}}$, then isolates a single positive scale $\theta_{\mathrm{can}}$ used as a comparison threshold in later physics statements. Sibling lemmas record nonnegativity, a pointwise evaluation identity, and positivity of the threshold.
The certificate bundle (RSPhysics005Cert / cert / cert_inhabited) is the module's export surface: a single inhabited record that downstream physics files can assume rather than re-proving the elementary cost facts.
proof idea
Definition module with thin lemma wrappers. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos discharge the sign claims from the imported Cost nonnegativity and constant positivity; domainCost_at_eq is an evaluation identity. The certificate record is assembled and shown inhabited by packing those fields. No deep tactic proof; structure is def-plus-short-lemma.
why it matters in Recognition Science
Supplies the domain-cost and canonical-threshold primitives used when RS physics statements compare energies or gate processes against a fixed positive scale. Sits on Constants ($\tau_0$) and Cost (J-layer), and exposes an inhabited certificate so later modules need not reopen elementary sign and evaluation facts. No downstream edges are recorded in the graph yet; the natural consumers are higher physics certificates that quote domain energetics or threshold comparisons along the forcing chain (J-uniqueness, phi ladder, eight-tick structure).
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law.
- Does not fix numerical values of masses, alpha, or G.
- Does not prove dynamical evolution or field equations.
- Does not connect domainCost to the eight-tick octave or D=3.
- Does not assert experimental bounds; only formal nonnegativity and positivity.