IndisputableMonolith.Physics.RS_Physics_Module_002
Physics module defining a domain-level cost functional and a positive canonical threshold used in RS recognition bounds. It packages nonnegativity of the domain cost, equality at the identity, and a certificate type witnessing the module's core inequalities. Cite when assembling threshold comparisons or cost-based selection rules. Structure is definitional with short positivity and equality lemmas plus an inhabited certificate.
claimIntroduce a domain cost $C_{\mathrm{dom}}$ built from the RS cost $J$, prove $C_{\mathrm{dom}}\ge 0$ and $C_{\mathrm{dom}}(e)=0$ at the identity, and fix a canonical threshold $\theta_*>0$. Package these facts in a certificate record $\mathrm{RSPhysics002Cert}$.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law. The Cost import supplies that $J$-calculus; Constants supplies the RS-native tick $\tau_0=1$.
This module lifts pointwise $J$-cost to a domain-level cost $C_{\mathrm{dom}}$ suitable for physics selection and threshold arguments. The canonical threshold $\theta_*$ is a fixed positive scale against which domain costs are compared when deciding recognition events or admissible configurations.
The local setting is pure RS units ($c=1$, ladder in powers of $\varphi$), with no continuum spacetime structure assumed beyond the discrete recognition ledger.
proof idea
Definition module with thin lemmas. domainCost is introduced from the imported $J$-cost; domainCost_nonneg and domainCost_at_eq discharge nonnegativity and the identity zero by reduction to known $J$ properties. canonicalThreshold is a positive constant; canonicalThreshold_pos is the positivity witness. RSPhysics002Cert bundles the above into a Prop/record, and cert_inhabited exhibits a concrete inhabitant.
why it matters in Recognition Science
Supplies the domain-cost and threshold primitives that later physics certificates compare against when enforcing recognition bounds (Berry-scale thresholds, rung selection, admissible configurations). No downstream edges are recorded in the graph yet; the natural consumers are higher RS physics modules that need a nonnegative domain cost vanishing at equilibrium and a fixed positive cutoff. Anchors the cost side of the forcing chain (T5 $J$-uniqueness) inside concrete physics packaging rather than pure foundation.
scope and limits
- Does not derive $J$ uniqueness or the RCL; those live upstream in Cost/Foundation.
- Does not fix numerical values of masses, $\alpha$, or $G$; only domain cost and a threshold scale.
- Does not prove continuum field equations or spacetime dynamics.
- Does not claim the threshold equals $\varphi^{-1}$ or $Z_{\mathrm{cf}}$ without further identification.
- Does not list downstream consumers; used_by is currently empty.