canonicalThreshold
plain-language theorem explainer
Defines the canonical materials threshold as φ − 3/2, which equals the J-cost of the golden ratio. Materials and structural RS work cite this as the recognition-cost floor at the self-similar fixed point. The body is a one-line real constant from the imported φ.
Claim. The canonical threshold is the real number $\varphi - 3/2$, equal to the J-cost $J(\varphi)$ at the golden ratio (approximately $0.11803$).
background
In Recognition Science the unique cost functional forced by the Recognition Composition Law is $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$). The golden ratio $\varphi$ is the self-similar fixed point of the forcing chain (T6). Substituting the identity $\varphi^{-1} = \varphi - 1$ yields $J(\varphi) = \varphi - 3/2$.
This module packages that value as the structural materials threshold: the minimal recognition cost on the $\varphi$-ladder at the fixed point. The surrounding file is marked STRUCTURAL THEOREM (zero sorry, zero axiom) and imports the global Constants and Cost layers for $\varphi$ and $J$.
proof idea
Definitional abbreviation: the real constant is written directly as $\varphi - 3/2$ using the imported golden-ratio symbol. No tactic proof; equality with $J(\varphi)$ is the algebraic identity recorded in the module documentation and used by sibling lemmas such as positivity of the threshold.
why it matters
Pins the numerical floor of the RS J-cost at the golden-ratio fixed point for the materials domain. Downstream structural certificates in the same module (threshold positivity, domain-cost nonnegativity, the RS_MAT_Structural_002 certificate) treat this constant as the canonical comparison value. It sits at the T5–T6 junction: J-uniqueness plus $\varphi$ as the forced self-similar point, giving the concrete materials scale $\approx 0.11803$ quoted in the module header.
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