IndisputableMonolith.Physics.RenormGroupFixedFromPhi
Module packaging the claim that the renormalization-group fixed point of the Recognition cost is forced by the golden ratio φ. It defines a domain cost, its equilibrium value and nonnegativity, a positive canonical threshold, and an inhabited RG-fixed-point certificate. Physicists tracing mass ladders or coupling running to T6 cite this package. The argument is definitional plus short positivity and equilibrium lemmas over the Cost and Constants imports.
claimOn the Recognition cost $J$, a domain cost $C$ is defined with $C$ nonnegative, $C=0$ at equilibrium, and a positive canonical threshold $\theta_*>0$. An inhabited certificate asserts that the renormalization-group fixed point of this cost is the self-similar scale $\varphi$ (the unique positive solution of $\varphi=1+1/\varphi$).
background
Recognition Science forces the cost functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) by the Recognition Composition Law and the T5 uniqueness step. The golden ratio $\varphi$ then appears as the unique self-similar fixed point (T6). Constants supplies the RS-native tick $\tau_0=1$; Cost supplies $J$ and its elementary identities.
This physics module lifts those primitives to a renormalization-group reading: a domain cost measuring departure from equilibrium under scale change, together with a canonical positive threshold that marks when the fixed-point condition is met. The certificate type packages the fixed-point claim so downstream mass or coupling arguments can assume a single inhabited witness rather than re-deriving $\varphi$-fixation.
proof idea
Definition module with short supporting lemmas, not a deep tactic development. domainCost is introduced from the Cost import; domainCost_at_equilibrium and domainCost_nonneg record the zero-at-equilibrium and nonnegativity facts. canonicalThreshold and canonicalThreshold_pos fix a strictly positive cutoff. RGFixedPointCert is a structure bundling these; cert and cert_inhabited supply a concrete inhabited instance tying the fixed point to $\varphi$. No multi-step rewriting beyond the Cost/Constants identities is required.
why it matters in Recognition Science
Closes the physics-side reading of T6: the same $\varphi$ that is the self-similar fixed point of $J$ is the RG fixed point of the domain cost. Downstream mass-ladder and coupling-running arguments can cite the inhabited certificate instead of reopening uniqueness. The module sits in the Physics domain and currently has no recorded used_by edges, so it functions as a leaf package ready for import by rung formulas, $\alpha$-band bounds, or eight-tick octave running. It does not itself re-prove T5/T6; it consumes them via Cost and Constants.
scope and limits
- Does not re-prove J-uniqueness (T5) or φ-forcing (T6); assumes Cost/Constants.
- Does not derive running couplings or beta functions beyond the fixed-point certificate.
- Does not fix numerical values of masses, α, or G; only the φ fixed-point claim.
- Does not address spatial dimension D=3 or the eight-tick octave (T7/T8).
- Does not supply a used_by consumer; the certificate is currently a leaf export.