IndisputableMonolith.Physics.RS_Physics_Module_010
Physics module 010 packages a domain-level cost functional built from the RS J-cost, together with a canonical positive threshold and a small certificate bundle. It records nonnegativity of the domain cost and positivity of the threshold in RS-native units. Anyone citing domain-cost comparisons or threshold gates in later physics layers would import this file. The content is mostly definitions plus short positivity/nonnegativity lemmas over the Cost and Constants imports.
claimOn the RS cost side, define a domain cost $C_{\mathrm{dom}}$ from the standard $J$-cost, prove $C_{\mathrm{dom}}\ge 0$ and $C_{\mathrm{dom}}(x)=C_{\mathrm{dom}}$ at the evaluation point used by the module, and fix a canonical threshold $\theta>0$. Bundle these facts into an inhabited certificate object for Physics module 010.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced at T5 of the unified chain. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$ and the golden-ratio ladder constants used elsewhere in the physics layer.
This module sits in the Physics domain and introduces a domain-scoped cost (a lift or specialization of $J$ to the module's domain) together with a single positive numerical gate called the canonical threshold. Sibling names indicate equality-at-evaluation, nonnegativity of the domain cost, and positivity of the threshold, then wrap them in a certificate type RSPhysics010Cert with an inhabited instance.
No external physics law is proved here; the file is a local definitions-and-bounds unit that later modules can import when they need a named nonnegative domain cost or a fixed positive cutoff.
proof idea
Definition-heavy module, not a single deep theorem. Domain cost is introduced as a def tied to the Cost layer; domainCost_at_eq is an evaluation identity; domainCost_nonneg is a short nonnegativity argument inherited from $J\ge 0$. Canonical threshold is a positive constant def; positivity is a one-line numerical or algebraic check. The certificate record packages these facts, and cert_inhabited supplies a canonical inhabitant so downstream code can take a cert as a hypothesis without reconstructing the bundle.
why it matters in Recognition Science
Module 010 is a small physics-side scaffolding unit: it names the domain cost and the canonical threshold that later RS physics developments can cite without re-deriving $J\ge 0$ or re-picking a cutoff. The inhabited certificate pattern matches other RS cert bundles, giving a single object that asserts the local cost and threshold facts together.
Used-by edges are empty in the current graph, so this file is a leaf provider rather than a step inside a named forcing theorem (T0–T8 live upstream in Foundation). Its role is infrastructural: keep domain-cost bookkeeping and the positive threshold in one importable place beside Constants and Cost, ready for mass-ladder, gating, or comparison lemmas that need a nonnegative domain cost and a fixed $\theta>0$.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those stay in Foundation/Cost.
- Does not fix numerical particle masses, alpha, or rung placements on the phi-ladder.
- Does not claim the canonical threshold equals any named physical constant (Berry gate, Z_cf, etc.).
- Does not discharge downstream physics theorems; used_by is currently empty.
- Does not treat multi-domain coupling or spacetime dimension (T8).