Condensed Matter Spin Glass Freezing Ratio Freezing Ratio2 D Band

The 2D freezing ratio

A spin glass is a magnetic alloy, such as copper with a few percent of manganese, whose magnetic moments freeze into random directions rather than lining up. Its freezing temperature T_g is the point where those moments stop reorienting. A ferromagnet, by contrast, orders its moments into alignment at the Curie temperature T_c. The ratio T_g/T_c measures how far below the ordering temperature freezing happens. For a two-dimensional Ising spin glass, the Recognition Science framework derives that this ratio equals 1/φ², where φ is the golden ratio, approximately 1.618. That places the ratio near 0.382.

The declaration freezingRatio2D_band is a machine-checked theorem in the framework's library of formal theorems. It proves that 0.37 < 1/φ² < 0.40. The proof unfolds the definition of the ratio and uses known bounds on φ². This is a purely mathematical statement about the real number 1/φ². The framework also proves a companion theorem, dimensional_crossover, that the three-dimensional ratio 1/φ is exactly φ times the two-dimensional ratio 1/φ². The two ratios are not independent; they are linked by one factor of the golden ratio.

In Recognition Science, the framework models the freezing ratio as a consequence of its recognition lattice. The 2D Ising case realizes a deeper frustration than the 3D Heisenberg case, and that deeper frustration is expressed as the squared reciprocal of the golden ratio. The framework predicts that canonical 2D Ising spin glasses will have T_g/T_c in the band (0.37, 0.40). This is a prediction with a named falsifier: a survey of at least ten spin glasses with calibrated T_c values whose median falls outside (0.61, 0.62) for the 3D case would falsify the structural claim.

What the theorem does not claim is important. It does not prove that any real material has this ratio; that is an empirical question. The 3D band (0.617, 0.622) sits inside the measured CuMn and AuFe window of 0.60 to 0.65, but the 2D band has no measured counterpart in the pack. The theorem also does not derive the golden ratio itself; it assumes φ as a constant. The physical bridge, that a spin glass realizes the recognition lattice, is itself an open question in the framework.

/-- 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

/-- **SPIN-GLASS FREEZING ONE-STATEMENT.** Canonical 3D Heisenberg
spin glasses have `T_g / T_c = 1/φ ∈ (0.617, 0.622)`; canonical 2D
Ising spin glasses have `T_g / T_c = 1/φ² ∈ (0.37, 0.40)`; the
dimensional crossover from 2D to 3D adds exactly one φ-step. -/
theorem spin_glass_one_statement :
    (0.617 < freezingRatio3D ∧ freezingRatio3D < 0.622) ∧
    (0.37 < freezingRatio2D ∧ freezingRatio2D < 0.40) ∧
    freezingRatio3D = freezingRatio2D * phi :=
  ⟨freezingRatio3D_band, freezingRatio2D_band, dimensional_crossover⟩

What this page does not claim

No measurement of a real 2D spin glass is cited in the pack. The theorem does not derive the value of the golden ratio itself. The physical identification of a spin glass with the recognition lattice 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)

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