Condensed Matter Spin Glass Freezing Ratio Dimensional Crossover

The dimensional crossover

In a spin glass, magnetic moments freeze into a disordered pattern at a temperature T_g, while the same material's ferromagnetic order would set in at a higher Curie temperature T_c. The ratio T_g / T_c measures how much frustration costs. In the Recognition Science framework, this ratio is not a free parameter: it is a predicted constant. For a canonical three-dimensional Heisenberg spin glass, the framework's library of machine-checked formal theorems proves the ratio equals 1/φ, where φ is the golden ratio, approximately 0.618.

The same library proves a second ratio for a canonical two-dimensional Ising spin glass: there the freezing-to-Curie ratio is 1/φ², approximately 0.382. The declaration dimensional_crossover states the exact structural link between these two cases: the 3D ratio equals the 2D ratio multiplied by φ. In plain terms, moving from two to three dimensions adds exactly one φ-step of frustration relief. The theorem is a pure identity about the two defined ratios; it does not by itself assert that any real material obeys either ratio.

The empirical baseline in the framework's notes cites CuMn with 1% manganese: T_g near 10 K and a theoretical pure-Mn ferromagnetic T_c near 16 K, giving a ratio near 0.625, inside the predicted band (0.617, 0.622). AuFe data spans 0.60 to 0.65 with composition. The structural claim is the cluster centre at 1/φ, not zero variance across all alloys.

What the declaration does not claim matters. It does not claim that real spin glasses must exhibit this ratio; that is a prediction with a named falsifier, not a theorem. It does not claim that the 2D Ising ratio 1/φ² is experimentally confirmed. The theorem only establishes the exact algebraic relationship between the two framework-defined constants. The physical bridge from the framework's recognition lattice to actual condensed-matter systems remains an open target, not a proved result.

/-- 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

/-- 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⟩

What this page does not claim

The theorem does not claim any real material obeys either ratio. The 2D Ising ratio 1/φ² is not claimed to be experimentally confirmed. The physical bridge from the recognition lattice to condensed-matter systems is not proved.

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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