canonicalThreshold
plain-language theorem explainer
The canonical threshold is the real constant φ − 3/2, with φ the golden-ratio fixed point of the Recognition Composition Law. Foundation and forcing-chain work cites it as the named bar for domain-cost comparisons in the first RS lattice module. The declaration is a one-line definition from the imported φ, not a proved inequality.
Claim. Let $\varphi$ be the golden ratio. The canonical threshold is the real number $\varphi - 3/2$.
background
Recognition Science takes the cost $J(x)=\frac12(x+x^{-1})-1$ (equivalently $\cosh(\log x)-1$) as the unique generator of physical predictions. The module is a structural fragment of that lattice: zero sorry, zero axioms, built on the imported Constants and Cost layers.
The scalar $\varphi$ is the self-similar fixed point forced by the Recognition Composition Law (forcing-chain step T6). It is the base of the mass ladder and the source of dimensionless RS ratios. Domain cost in this module is the restriction of $J$ used to compare lattice values against fixed numerical bars.
The threshold $\varphi-3/2$ is one such bar: a single named real against which nonnegativity and certificate lemmas in the same file can test.
proof idea
Pure definition: the symbol is bound to the arithmetic expression $\varphi-3/2$. No tactics, no lemmas, no proof term beyond the imported constant $\varphi$.
why it matters
Gives a stable name to the cut used by sibling facts in the same forcing-chain module (positivity of the threshold, domain-cost nonnegativity, and the module certificate). In the broader RS chain it sits after T5–T6 ($J$-uniqueness and $\varphi$) and before the eight-tick and $D=3$ steps. Downstream certificates can compare against one symbol instead of an inline expression, keeping the structural theorem surface small and axiom-free.
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