IndisputableMonolith.Physics.RenormalizationGroup2FromJCost
Packages a domain cost built from the RS J-functional and a two-scale renormalization-group fixed-point certificate. Anyone deriving continuum thresholds or scaling fixed points from cost geometry would cite it. The file is largely definitional: nonnegativity and positivity facts plus an inhabited certificate record over Constants and Cost.
claimFrom the RS cost $J(x)=(x+x^{-1})/2-1$, define a domain cost $C$ on positive scale ratios, prove $C\ge 0$, fix a canonical positive threshold $t_*>0$, and package an inhabited two-scale RG fixed-point certificate witnessing the fixed structure induced by $J$.
background
Recognition Science forces a unique symmetric cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) via the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$ and related units.
Renormalization here means coarse-graining scale ratios under that cost, not a QFT counterterm scheme. A domain cost is the cost evaluated on a chosen positive domain (scale window); the canonical threshold is the positive cutoff at which the cost geometry marks a fixed or matching scale.
The module sits in the Physics layer: it turns pure $J$-geometry into named objects a continuum or RG argument can quote without re-deriving nonnegativity each time.
proof idea
Definition-first module, not a deep derivation. It introduces the domain cost as a thin wrapper around $J$ on the working domain, records equality-at-evaluation and nonnegativity (inherited from $J\ge 0$ with equality at $1$), defines a canonical positive threshold, and bundles these into an RG fixed-point certificate structure with an inhabited instance. Supporting lemmas are short algebraic or positivity facts; there is no multi-step forcing argument inside the file.
why it matters in Recognition Science
Gives the Physics layer a named bridge from T5 $J$-uniqueness to two-scale RG language: domain cost, threshold, and a certificate record. Downstream graph edges are empty here, so the module is a leaf packaging step rather than a proved parent theorem. It supports later continuum, matching, or coupling-flow claims that need a cost-derived fixed-point witness without reopening the Cost API. Framework landmarks in play are T5 ($J$ unique) and the RCL that forces $J$; $\phi$-ladder mass formulas and the alpha band are not settled in this file.
scope and limits
- Does not derive the Callan-Symanzik or Wilsonian beta function from QFT Lagrangians.
- Does not prove uniqueness of the continuum limit or existence of interacting fixed points beyond the certificate packaging.
- Does not compute numerical running couplings or match to SM beta coefficients.
- Does not discharge T6–T8 (phi, eight-tick, D=3); those remain upstream forcing facts.
- Does not claim experimental RG observables; only cost-side certificate structure.