IndisputableMonolith.Physics.RS_Physics_Module_012
Physics module 012 packages a domain cost functional built from the RS J-cost, proves it is nonnegative, and introduces a positive canonical threshold. It closes with an inhabited certificate record that bundles those facts. Anyone wiring cost bounds or threshold gates into later RS physics lemmas would cite this module. The content is mostly definitional with short algebraic nonnegativity and positivity arguments.
claimDefine a domain cost $C$ from the RS cost $J$, prove $C \ge 0$ and $C$ agrees with its pointwise evaluation, fix a canonical threshold $\theta > 0$, and package these into an inhabited certificate for RS Physics Module 012.
background
Recognition Science measures mismatch with the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that $J$; Constants supplies the RS-native tick $\tau_0 = 1$.
This module sits in the Physics domain and treats a domain-level cost assembled from $J$, together with a fixed positive threshold used as a gate. Sibling names indicate pointwise agreement of the domain cost with its evaluation map, nonnegativity of that cost, and positivity of the canonical threshold.
The certificate record is the module's public face: a single inhabited Prop/structure that downstream physics developments can require instead of re-proving the elementary cost facts.
proof idea
Definition-heavy module. Domain cost is introduced as a construction from the imported $J$-cost; equality with pointwise evaluation is a definitional or short rewriting fact. Nonnegativity follows from the known nonnegativity of $J$ (Cost). The canonical threshold is a positive constant (likely tied to RS scales such as $\varphi$-powers); positivity is a numeric or algebraic check. The certificate is a structure packing these lemmas, shown inhabited by assembling the proofs.
why it matters in Recognition Science
Module 012 is a local physics certificate layer: it freezes a domain cost and a positive threshold so later RS physics arguments can assume a uniform cost gate without re-deriving $J \ge 0$. No downstream edges are recorded yet, so it currently stands as a leaf certificate rather than a forced step in T0-T8. It still aligns with the broader program in which $J$-cost controls recognition mismatch, thresholds (e.g. Berry-scale $\varphi^{-1}$ elsewhere) gate physical events, and certificates make those facts reusable. Parent consumers would be any physics lemma that needs a nonnegative domain cost or a strict positive cutoff.
scope and limits
- Does not derive $J$ uniqueness or the Recognition Composition Law; those stay upstream in Cost/Foundation.
- Does not fix numerical ties of the threshold to $\varphi$, $\alpha$, or mass rungs.
- Does not prove dynamical evolution, spectra, or eight-tick timing.
- Does not claim downstream use; used_by is empty in the graph.
- Does not replace full physics modules beyond this cost/threshold certificate.