canonicalThreshold
plain-language theorem explainer
The canonical threshold is the real constant φ − 3/2 used as an RS-native cutoff in Cosmology Module 5. Anyone checking the module’s r-tensor consistency certificate against the Planck bound cites it when domain costs are compared to a fixed positive level. It is a bare definitional constant: no proof, only the arithmetic expression in φ.
Claim. Define the canonical threshold by $\mathrm{canonicalThreshold} := \varphi - \tfrac{3}{2}$, where $\varphi$ is the golden ratio (the unique self-similar fixed point of Recognition Science).
background
Recognition Science forces the golden ratio φ as the unique self-similar fixed point (forcing step T6). Cosmology Module 5 is a structural package whose headline claim is that the predicted tensor-to-scalar ratio $r = 2/(44\varphi^2) \approx 0.0174$ lies strictly below the Planck ceiling $0.036$, with status STRUCTURAL THEOREM (zero sorry, zero axioms).
The module imports global Constants (φ) and Cost (the J-cost layer). Beside this threshold sit a domain-cost functional, its nonnegativity and pointwise evaluation lemmas, a positivity lemma for the threshold itself, and the certificate bundle that packages the consistency check. The threshold is the fixed real level those cost comparisons use.
proof idea
Definitional, not a proved statement. The body is the single arithmetic term φ − 3/2 in the reals; there are no tactics, no lemmas applied, and no proof obligations.
why it matters
Module 5’s job is to certify that the RS r-prediction is Planck-consistent. The canonical threshold names the φ-native cutoff against which domain costs are measured inside that certificate (siblings include positivity of the threshold and the inhabited cert). φ-scaled cutoffs recur across RS (Berry creation at φ⁻¹, dream fraction φ⁻³, mass-ladder rungs); this one is the cosmology-side companion for the r-tensor bound $2/(44\varphi^2)$. It does not itself close an open forcing step; it supplies the numeric anchor the structural cert needs.
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