Condensed Matter Spin Glass Freezing Ratio Freezing Ratio2 D Pos

The 2D freezing ratio

A spin glass is a magnetic alloy, such as copper with a little manganese, whose atomic magnetic moments point in random directions and freeze into a disordered pattern at a temperature called T_g. A ferromagnet, by contrast, is a material like pure iron whose moments align, and it loses that alignment above its Curie temperature T_c. The ratio T_g/T_c measures how much more easily a spin glass freezes than a ferromagnet orders.

The Recognition Science framework models this ratio for a canonical two-dimensional Ising spin glass, a standard theoretical model where magnetic moments point only up or down on a square lattice. Its definition is T_g/T_c = 1/φ², where φ is the golden ratio, approximately 1.618. The declaration freezingRatio2D_pos establishes the basic fact that this ratio is a positive number: 0 < 1/φ². A companion theorem, freezingRatio2D_band, pins the value between 0.37 and 0.40, so the ratio sits near 0.38.

This 2D value is deeper frustration than the 3D case. In three dimensions, the framework predicts T_g/T_c = 1/φ, near 0.62. The dimensional_crossover theorem states that the 3D ratio equals the 2D ratio multiplied by φ, which in plain terms means moving from 2D to 3D adds exactly one step of the golden ratio to the freezing temperature. The framework's library of machine-checked formal theorems proves these identities as pure statements about the number 1/φ², independent of any particular material.

The empirical check is not part of the theorem. The docstring cites CuMn and AuFe data with ratios in a broad 0.60 to 0.65 range for 3D, and predicts a 2D band of 0.38 to 0.39, but the declaration itself only establishes the positivity of the defined number. The prediction that real 2D spin glasses will be found in that band is a hypothesis with a named falsifier: a survey of ten or more systems whose median ratio falls outside the 3D band at the 2σ level would refute it. The theorem guarantees the arithmetic, not the experiment.

theorem freezingRatio2D_pos : 0 < freezingRatio2D :=
  div_pos (by norm_num) (pow_pos phi_pos _)

/-- Numerical band: `T_g / T_c ∈ (0.37, 0.40)` for 2D Ising. -/
theorem freezingRatio2D_band :
    0.37 < freezingRatio2D ∧ freezingRatio2D < 0.40 := by
  unfold freezingRatio2D
  obtain ⟨h_phi2_lo, h_phi2_hi⟩ := phi_squared_bounds
  have hpos : (0 : ℝ) < phi^2 := by linarith
  have h_lo : (0.37 : ℝ) < 1 / phi^2 := by
    rw [lt_div_iff₀ hpos]
    nlinarith
  have h_hi : (1 / phi^2 : ℝ) < 0.40 := by
    rw [div_lt_iff₀ hpos]
    nlinarith
  exact ⟨h_lo, h_hi⟩

/-- The 3D-to-2D ratio of freezing ratios is exactly φ. This is the
    structural content of "going from 3D to 2D adds one φ-step of
    frustration." -/
theorem dimensional_crossover :
    freezingRatio3D = freezingRatio2D * phi := by
  unfold freezingRatio3D freezingRatio2D
  have hp : phi ≠ 0 := ne_of_gt phi_pos
  field_simp

What this page does not claim

The declaration does not assert that any real material has a measured T_g/T_c ratio of 1/φ². The theorem does not prove that the 2D Ising spin glass model is an accurate description of any physical alloy. The framework does not derive the value of the golden ratio from spin glass physics; it uses φ as an input constant.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/CondensedMatter/SpinGlassFreezingRatio.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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