canonicalThreshold
plain-language theorem explainer
Defines the canonical threshold as the real constant φ − 3/2, with φ the golden ratio. Standard-Model structural arguments in RS cite it as a fixed numerical cutoff built only from the forced self-similar scale. Sibling positivity and domain-cost lemmas sit beside it in the same certificate package. The body is a one-line arithmetic definition, not a derived inequality.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ denotes the golden ratio.
background
Recognition Science forces the cost $J(x)=\frac12(x+x^{-1})-1$ (T5) and the golden ratio $\varphi$ as the unique self-similar fixed point (T6); spatial dimension $D=3$ is forced at T8. This module packages those structural claims for Standard Model use and advertises status STRUCTURAL THEOREM (0 sorry, 0 axiom).
The constant $\varphi-3/2$ is the natural positive gap between $\varphi\approx1.618$ and $3/2$. Sibling declarations establish a domain cost, its evaluation identity, nonnegativity, and positivity of this threshold, then assemble them into an inhabited structural certificate. Imports pull $\varphi$ from Constants and the cost layer from Cost; no further upstream lemmas are required for the definition itself.
proof idea
Pure definition: the real is introduced by the arithmetic expression $\varphi-3/2$. There is no proof body and no tactic script. Positivity is handled by the separate sibling lemma that shows the constant is strictly positive.
why it matters
Supplies a concrete, φ-only numerical cutoff inside the RS Standard Model structural layer. It coexists with domain-cost nonnegativity and the inhabited certificate that the module presents as a zero-sorry, zero-axiom structural package for $J$, $\varphi$, and $D=3$. Unlike dynamical landmarks (Berry threshold $\varphi^{-1}$, $Z_{\mathrm{cf}}=\varphi^5$, dream fraction $\varphi^{-3}$), this constant is definitional scaffolding for structural comparisons rather than a mass-ladder or coupling prediction. No downstream consumers are recorded in the graph; its role is local to the certificate assembly in this module.
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