canonicalThreshold
plain-language theorem explainer
The canonical materials threshold is the real constant φ − 3/2, with φ the golden-ratio fixed point of Recognition cost. Anyone certifying Mohs-scale or domain-cost bounds in the diamond module cites it as the structural cutoff. It is a bare definitional abbreviation, not a derived identity.
Claim. The canonical threshold is the real number $T := \varphi - 3/2$, where $\varphi$ is the unique positive self-similar fixed point forced by Recognition cost.
background
Recognition Science forces a unique positive self-similar scale $\varphi$ (T6) from the J-cost $J(x)=(x+x^{-1})/2-1$. The same $\varphi$ generates the mass ladder and the dimensionless constants in RS-native units.
This module is the structural materials layer for diamond on the Mohs scale: the claim is that $\varphi^5\approx 11.09$ sits next to the conventional Mohs value 10. The file imports Constants (for $\varphi$) and Cost (for J-cost infrastructure) and carries status STRUCTURAL THEOREM (0 sorry, 0 axiom).
The threshold $\varphi-3/2\approx 0.118$ is a small positive real used as a comparison gate for domain costs inside the same module.
proof idea
Definitional one-liner. The real constant is written as the difference of the imported golden-ratio constant and the rational $3/2$. No lemmas are applied and no proof obligations are generated.
why it matters
Gives the numerical gate that the materials certification stack compares against (sibling positivity and certificate declarations in the same module). The parent setting is the structural Mohs-diamond correspondence $\varphi^5\sim 10$, not a step of the T0–T8 forcing chain. The value itself is a materials-side convention that keeps domain-cost inequalities on the same scale as the phi-ladder; it is not forced by RCL or by J-uniqueness.
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