IndisputableMonolith.Physics.RS_Physics_Module_011
Physics module 011 packages a domain cost functional, its evaluation identity and nonnegativity, and a strictly positive canonical threshold. It also exposes a small certificate bundle (RSPhysics011Cert) inhabiting those facts. Cite it when a later physics argument needs a nonnegative cost on a domain together with a fixed positive cutoff. The content is definitional plus short positivity/nonnegativity lemmas over the Cost and Constants imports.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, and fixes a canonical threshold $\theta_*>0$. It bundles these into an inhabited certificate record for downstream physics use.
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 T5 step of the unified forcing chain. The Cost import supplies that cost infrastructure; Constants supplies the RS-native tick $\tau_0=1$.
Module 011 specializes cost to a domain-level functional: a nonnegative real assigned to a domain configuration, together with a single positive canonical threshold used as a cutoff or acceptance level. Sibling names indicate an evaluation identity (when the domain configuration sits at the equality case of the cost) and elementary sign lemmas.
The certificate record is the usual RS pattern: a Prop-carrying structure that packages the named facts so later modules can assume one inhabited cert rather than re-proving local inequalities.
proof idea
Definition module with short supporting lemmas, not a long derivation. domainCost is introduced as a def; domainCost_at_eq and domainCost_nonneg discharge the evaluation identity and nonnegativity from the Cost layer. canonicalThreshold is a positive constant (canonicalThreshold_pos). RSPhysics011Cert / cert / cert_inhabited wrap those facts into an inhabited certificate. No deep tactic scripts are implied by the sibling list; expect direct unfolding and Cost/Constants lemmas.
why it matters in Recognition Science
Gives physics code a reusable nonnegative domain cost and a fixed positive threshold under one certificate, so later RS physics developments can cite a single inhabited pack rather than ad-hoc inequalities. It sits on Cost (J-cost) and Constants ($\tau_0$), aligning with the RS cost calculus that underpins mass ladders, the Berry threshold $\phi^{-1}$, and related cutoffs. No downstream used_by edges are recorded yet; the module is a local physics utility rather than a forcing-chain step (T0–T8).
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law.
- Does not fix numerical values of c, hbar, G, or alpha.
- Does not prove mass-ladder or eight-tick octave statements.
- Does not claim a unique physical interpretation of the threshold beyond positivity.
- Does not discharge global physics certificates outside this module's cert record.