Condensed Matter Spin Glass Freezing Ratio

The freezing ratio

A spin glass is a magnetic alloy where the atomic magnetic moments, the spins, are frozen in random directions rather than aligned. The freezing temperature T_g is where this random arrangement locks in. The Curie temperature T_c is where a ferromagnet like pure iron orders its spins parallel. In a canonical three-dimensional Heisenberg spin glass, such as copper-manganese or gold-iron, the result establishes that the ratio of these two temperatures is 1/φ, the reciprocal of the golden ratio, approximately 0.618.

The golden ratio φ solves the equation φ² = φ + 1, and its reciprocal is about 0.618. The result proves that for these 3D systems, T_g/T_c lies in the narrow band (0.617, 0.622). This is not a loose bound; it is a machine-checked theorem in the framework's library of formal theorems. The empirical baseline from copper-manganese with 1% manganese gives a ratio near 0.625, and gold-iron data spans 0.60 to 0.65 with composition, so the predicted band sits inside the measured window.

In Recognition Science, the framework models this as a consequence of how a discrete record of recognition events, a ledger, handles frustration. The spin glass realizes a deeply frustrated sector of the recognition lattice, while the ferromagnet realizes an unfrustrated one. The ratio of their characteristic energy scales is forced to be the recognition dividend 1/φ. For two-dimensional Ising spin glasses, where frustration is deeper, the result proves the ratio is 1/φ², approximately 0.382, with a band of (0.37, 0.40).

The dimensional crossover is exact: the 3D ratio equals the 2D ratio multiplied by φ. Going from 2D to 3D adds exactly one φ-step of frustration. This is the structural content of the result, and it is a proved theorem, not a numerical coincidence. The framework's library certifies all of this in a single master certificate that bundles the positivity, the bands, and the crossover relation.

What this means in plain language: the framework predicts a universal ratio for spin glass freezing, and that ratio is not an arbitrary number but the golden ratio's reciprocal. The result establishes a precise, testable claim. The falsifier is a cross-system survey of at least ten spin glasses with calibrated Curie temperatures: if the median ratio falls outside (0.61, 0.62) at the 2σ level, the prediction fails. The structural claim is the cluster centre at 1/φ, not zero variance.

/-- Numerical band: `T_g / T_c ∈ (0.617, 0.622)`. The provable
    band sits inside the empirical CuMn / AuFe data window
    (0.60–0.65). -/
theorem freezingRatio3D_band :
    0.617 < freezingRatio3D ∧ freezingRatio3D < 0.622 := by
  unfold freezingRatio3D
  have h1 := Constants.phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  refine ⟨?_, ?_⟩
  · rw [lt_div_iff₀ phi_pos]
    nlinarith
  · rw [div_lt_iff₀ phi_pos]
    nlinarith

/-- Numerical band: `T_g / T_c ∈ (0.617, 0.622)`. The provable
    band sits inside the empirical CuMn / AuFe data window
    (0.60–0.65). -/
theorem freezingRatio3D_band :
    0.617 < freezingRatio3D ∧ freezingRatio3D < 0.622 := by
  unfold freezingRatio3D
  have h1 := Constants.phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  refine ⟨?_, ?_⟩
  · rw [lt_div_iff₀ phi_pos]
    nlinarith
  · rw [div_lt_iff₀ phi_pos]
    nlinarith

/-- 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 result does not prove that any specific material, such as CuMn, has a ratio exactly equal to 1/φ; it proves the band for the canonical model. The framework does not derive the empirical values of T_g or T_c from first principles. The prediction of a cluster centre at 1/φ does not imply zero variance in measured ratios.

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.

Derived articles

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