IndisputableMonolith.Physics.RS_Physics_Module_004
Physics module defining a domain cost functional and a positive canonical threshold used in RS recognition bounds. It packages nonnegativity of the cost, evaluation identities, and an inhabited certificate type for downstream physics checks. The content is mostly definitional with short positivity and equality lemmas over the Cost and Constants imports.
claimIntroduce a domain cost $C_{\mathrm{dom}}$ with $C_{\mathrm{dom}}\ge 0$, an evaluation identity at equality cases, and a canonical threshold $\theta>0$. Bundle these into an inhabited RS physics certificate for module 004.
background
Recognition Science measures mismatch with a nonnegative cost built from the unique $J$-functional forced by the Recognition Composition Law. The Cost import supplies that infrastructure; Constants supplies the RS-native tick $\tau_0=1$ and related units.
This module sits in the physics layer and specializes cost language to a domain cost: a scalar that scores how far a configuration sits from a recognition-compatible domain. A canonical threshold is fixed as a positive cutoff against which that cost is compared in later bounds or selection rules.
Sibling declarations name the cost, its nonnegativity, an on-equality evaluation lemma, positivity of the threshold, and a certificate record RSPhysics004Cert with an inhabited instance, so callers can treat the package as a single witness.
proof idea
Definition-first module. Core objects are defs (domain cost, canonical threshold, certificate structure). Supporting facts are short: nonnegativity of the cost, positivity of the threshold, and an evaluation identity at equality. The certificate is assembled as a structure with an inhabited instance so downstream code can obtain a default witness without extra construction.
why it matters in Recognition Science
Gives the physics layer a reusable domain-cost and threshold package tied to the RS cost calculus, rather than ad hoc cutoffs. No downstream edges are recorded in the graph yet, so it currently acts as a leaf certificate module for physics audits and for any later bound that needs $C_{\mathrm{dom}}\ge 0$ and $\theta>0$ in one place. It sits downstream of Constants and Cost only, keeping the forcing-chain landmarks (unique $J$, $\phi$, eight-tick structure) implicit via those imports rather than re-proved here.
scope and limits
- Does not derive the unique $J$-cost or the Recognition Composition Law.
- Does not force $\phi$, eight-tick period, or $D=3$.
- Does not compute particle masses, $\alpha$, or dimensional constants.
- Does not state a comparison theorem linking domain cost to the canonical threshold.
- Does not record downstream consumers in the current dependency graph.