canonicalThreshold
plain-language theorem explainer
Defines the real constant φ − 3/2, equal to the J-cost evaluated at the golden ratio. Cosmology and mass-ladder arguments cite it as the canonical positive threshold against which domain costs are compared. The body is a one-line arithmetic definition in terms of the RS constant φ.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ is the golden ratio fixed by the self-similarity axiom.
background
Recognition Science forces a unique nonnegative cost $J$ on ratios via the Recognition Composition Law, with closed form $J(x)=(x+x^{-1})/2-1$. The golden ratio $\varphi$ is the unique self-similar fixed point (forcing step T6). Direct evaluation gives $J(\varphi)=\varphi-3/2$, since $\varphi+\varphi^{-1}=\sqrt{5}$.
This module (Cosmology RS Module 9) treats baryon density as a structural identity built from $J(\varphi)$. The constant $\varphi-3/2$ therefore appears as the natural positive scale against which domain costs are measured. Imports supply only Constants (for $\varphi$) and Cost (for $J$).
proof idea
Pure definition: the real is introduced by the arithmetic expression $\varphi-3/2$. No proof obligations. Sibling lemmas later record positivity and equality with $J(\varphi)$ when needed.
why it matters
Supplies the numerical threshold used by the baryon-density structural claim of the module: $J(\varphi)(1-J(\varphi))/2\approx0.052$ matching $\Omega_b\approx0.0493$. Because $J(\varphi)=\varphi-3/2$, naming the constant once keeps later certificates free of repeated golden-ratio algebra. It sits downstream of T5 (J-uniqueness) and T6 ($\varphi$ forced) and feeds the module certificate RSCosmo009Cert. No open scaffolding remains; the definition is closed.
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