canonicalThreshold

Recognition Science forces a unique nonnegative cost $J$ on the positive reals. By the T5 uniqueness step, $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$). The golden ratio $\varphi$ is the self-similar fixed point of the forcing chain (T6), satisfying $\varphi=1+1/\varphi$ and hence $\varphi^{-1}=\varphi-1$.

Substituting into the closed form gives $J(\varphi)=\varphi-3/2\approx 0.11803$. This module (Mathematics RS Structural Module 2) records that number as a named constant: the golden-ratio recognition cost. Constants and Cost supply $\varphi$ and $J$; the present file only freezes the numerical threshold used by later positivity and certificate lemmas.

One-line definition: the constant is bound to the arithmetic expression $\varphi-3/2$. There is no tactic proof and no lemma application. The supporting identity $J(\varphi)=\varphi-3/2$ is not re-proved here; it follows upstream from the closed form of $J$ together with the golden-ratio relation $\varphi^{-1}=\varphi-1$ in the Cost and Constants modules.

Freezes the golden-ratio value of the unique J-cost as a structural constant of the mathematics layer. Sibling lemmas in the same module (positivity of the threshold, domain-cost comparisons, and the structural certificate) consume this name rather than re-expanding $\varphi-3/2$. In the forcing chain it sits at the T5–T6 junction: once $J$ is unique and $\varphi$ is forced, this is the irreducible recognition-cost scale. The module advertises zero sorry and zero axioms, so the definition is part of the closed structural core rather than scaffolding.