spinGlassFreezingCert
plain-language theorem explainer
The spinGlassFreezingCert definition packages the positivity, numerical bands, and exact phi-multiplication relation for 3D Heisenberg and 2D Ising spin-glass freezing ratios into one certificate structure. Condensed-matter researchers testing frustration models would cite it when comparing measured Tg/Tc values against the Recognition Science phi-ladder predictions. The construction is a direct structure literal that invokes the four component theorems and the dimensional crossover lemma.
Claim. Let $r_{3D}$ be the 3D Heisenberg spin-glass freezing ratio and $r_{2D}$ the 2D Ising ratio. The certificate asserts $0 < r_{3D}$, $0.617 < r_{3D} < 0.622$, $0 < r_{2D}$, $0.37 < r_{2D} < 0.40$, and $r_{3D} = r_{2D} · ϕ$, where ϕ denotes the golden ratio.
background
In the Recognition Science treatment of condensed matter the freezing temperature ratio Tg/Tc for spin glasses is obtained from the phi-ladder of recognition costs. The 3D Heisenberg case realizes the gap-45 frustration sector with ratio 1/ϕ while the 2D Ising case deepens the frustration to 1/ϕ². The module imports the cellular automata step definition to ground local interaction rules, yet the ratio theorems rest on the arithmetic properties of phi established in Constants.
proof idea
The definition is a direct structure constructor. It supplies the fields ratio_3D_pos, ratio_3D_band, ratio_2D_pos, ratio_2D_band, and dimensional_crossover by referencing the corresponding theorems freezingRatio3D_pos, freezingRatio3D_band, freezingRatio2D_pos, freezingRatio2D_band, and dimensional_crossover. No additional reasoning steps are performed.
why it matters
This definition supplies the single canonical statement of the spin-glass freezing prediction. It directly supports the empirical comparison in the module documentation where CuMn data lies inside the predicted (0.61, 0.62) band. The construction closes the track-E3 derivation by packaging the component results into a form that can be cited as a unit when discussing phi-rational ratios across dimensions. It touches the open question of extending the same lattice analysis to other frustrated systems.
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