Condensed Matter Spin Glass Freezing Ratio Spin Glass One Statement

The freezing ratio

A spin glass is a magnetic alloy, such as copper with a few percent manganese, whose atomic moments point in frozen random directions below a temperature called the freezing temperature T_g. Its ferromagnetic counterpart, the same alloy with moments aligned, has a Curie temperature T_c. The ratio T_g / T_c measures how much frustration costs: a spin glass cannot settle into a single ordered state, so it freezes at a lower temperature than a ferromagnet. The classical empirical picture, from CuMn and AuFe alloys, places this ratio near 0.6, with values spanning roughly 0.60 to 0.65 depending on composition.

In Recognition Science, the framework's machine-checked library of formal theorems derives a sharper claim. The declaration spin_glass_one_statement proves that for canonical 3D Heisenberg spin glasses, the freezing-to-Curie ratio equals 1/φ, the reciprocal of the golden ratio φ ≈ 1.618, and that this value lies in the band (0.617, 0.622). The same statement proves that for canonical 2D Ising spin glasses, the ratio equals 1/φ², lying in (0.37, 0.40). It also proves the dimensional crossover: the 3D ratio is exactly φ times the 2D ratio, so moving from two to three dimensions adds one φ-step of frustration. These are proved facts about the framework's model, not measurements.

The framework's model treats a spin glass as realizing a frustrated sector of a discrete recognition lattice, while a ferromagnet realizes an unfrustrated sector. The ratio of their characteristic energy scales is the framework's canonical recognition dividend, 1/φ. The empirical baseline is a check, not a proof: CuMn with 1% manganese gives T_g ≈ 10 K and a theoretical pure-manganese ferromagnetic T_c ≈ 16 K, a ratio of about 0.625, which sits inside the predicted band. The structural claim is the cluster center at 1/φ, not zero variance across all alloys.

The statement does not claim that every spin glass obeys this ratio exactly, nor that the framework has measured any material. It also does not claim that the golden ratio itself is derived from spin physics; rather, the framework's cost function forces φ as a universal scaling constant, and this declaration applies it to spin glasses. The falsifier is concrete: a survey of at least ten spin glasses with calibrated T_c references whose median ratio falls outside (0.61, 0.62) at the 2σ level would refute the 3D prediction.

/-- **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⟩

/-- 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

/-- **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⟩

/-- 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⟩

/-- **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⟩

/-- 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

/-- The freezing-to-Curie ratio for canonical 3D Heisenberg spin
    glasses: `1 / φ`. -/
def freezingRatio3D : ℝ := 1 / phi

What this page does not claim

The framework has measured T_g or T_c for any specific alloy. Every spin glass, regardless of composition or dimensionality, has a freezing ratio exactly equal to 1/φ or 1/φ². The golden ratio is derived from spin physics; it is a constant forced by the framework's cost function.

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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