canonicalThreshold
plain-language theorem explainer
Defines the RS-native Anderson localization threshold as φ − 3/2 on the real line. Condensed-matter workers comparing disorder-driven metal–insulator predictions to the J-cost framework cite this constant. The body is a one-line numeric abbreviation in terms of the golden ratio.
Claim. The canonical Anderson localization threshold is the real number $\varphi - 3/2$, where $\varphi$ is the golden-ratio fixed point of the Recognition self-similarity relation.
background
The module treats disorder-driven Anderson localization as a metal–insulator transition whose critical surface is read off the Recognition cost $J$. Here $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) is the unique symmetric cost forced by the Recognition Composition Law, and $\varphi$ is the self-similar fixed point from the forcing chain (T5–T6).
The local setting is a structural (zero-sorry) condensed-matter layer: the transition is placed where the J-cost on the conductance ratio equals $J(\varphi)$. Sibling definitions package a domain cost, its nonnegativity and equilibrium value, and a certificate type that packages the threshold prediction.
Constants are imported from the RS constants module, so $\varphi$ is the same ladder base used for masses, $\hbar=\varphi^{-5}$, and the eight-tick octave.
proof idea
Pure definition: the real constant is introduced by the abbreviation $\varphi - 3/2$. No lemmas or tactics are involved; positivity and certificate packaging are left to sibling declarations.
why it matters
Gives the explicit numeric cut that the Anderson-from-J-cost story needs so certificates and positivity lemmas can refer to one shared RS-native scale rather than an ad-hoc disorder strength. It sits inside the structural condensed-matter pass whose module claim is a disorder-driven transition at J-cost equal to $J(\varphi)$ on the conductance ratio, with empirical data outside the RS band as the structural falsifier.
In the broader framework it ties a condensed-matter threshold to the same $\varphi$ forced by T6 and the J-uniqueness of T5, keeping the localization cut on the same ladder as masses and coupling constants. Downstream certificate inhabitants and positivity facts are expected to quote this value; the module presently lists no formal used-by edges for the bare definition.
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