canonicalThreshold
plain-language theorem explainer
The canonical threshold is the real constant φ − 3/2, with φ the golden-ratio fixed point of Recognition self-similarity. Authors of RS domain-cost or Z-ladder comparisons cite it as the structural cutoff. The declaration is a one-line definitional abbreviation, not a proved claim.
Claim. Define the canonical threshold by $\varphi - 3/2$, where $\varphi$ is the unique positive self-similar fixed point forced by the Recognition cost.
background
Recognition Science forces the golden ratio $\varphi$ as the unique positive solution of the self-similarity fixed-point equation (forcing chain T6). The native cost is $J(x)=(x+x^{-1})/2-1$. Constants live in RS-native units with $c=1$ and $\hbar=\varphi^{-5}$.
This module records the structural Z-boson mass match $M_Z\approx 91.2,\mathrm{GeV}\sim\varphi^{13}\cdot 0.175$. Sibling definitions introduce a nonnegative domain cost and evaluate it at equality points; a separate lemma records positivity of the present threshold.
proof idea
Definitional abbreviation only: the real is written as $\varphi$ minus three-halves. No tactics, no lemmas, no proof obligations.
why it matters
Supplies the elementary real cutoff used by the Module 11 certificate that the Z mass sits on the $\varphi$-ladder. The module is marked a structural theorem (zero sorry, zero axiom). The threshold itself is pure arithmetic in $\varphi$; positivity and domain-cost comparison lemmas feed certificate inhabitation. Framework landmarks: T6 ($\varphi$ forced) and the mass formula yardstick $\cdot\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$.
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