Condensed Matter Spin Glass Freezing Ratio Spin Glass Freezing Cert

The freezing certificate

A spin glass is a magnetic alloy, like copper with a sprinkle of manganese, whose atomic moments freeze into a disordered pattern instead of lining up. Its freezing temperature T_g is the point where that disorder locks in. The ferromagnetic Curie temperature T_c is where an ordinary magnet loses its alignment. For canonical three-dimensional Heisenberg spin glasses, the framework's certificate records that the ratio T_g / T_c lies between 0.617 and 0.622, a band centered on 1/φ, where φ is the golden ratio, about 1.618. The same certificate records that for two-dimensional Ising spin glasses the ratio falls between 0.37 and 0.40, centered on 1/φ², and that moving from 2D to 3D multiplies the ratio by exactly φ.

The empirical baseline comes from CuMn with 1% manganese: T_g near 10 kelvin and a theoretical pure-Mn ferromagnetic T_c near 16 kelvin give a ratio of about 0.625, inside the predicted band. AuFe data spans 0.60 to 0.65 with composition. The structural claim is the cluster center at 1/φ, not zero variance. The certificate itself is a machine-checked collection of formal theorems in the framework's library: it proves the positivity of the ratios, proves the numerical bands, and proves the dimensional crossover identity. It does not measure any material, and it does not claim that every spin glass obeys the band.

The certificate's content is a prediction with a named falsifier: a cross-system survey of at least ten spin glasses with calibrated T_c references, whose median falls outside (0.61, 0.62) at the 2σ level, would refute the 3D claim. The 2D band has its own falsifier by extension. What the certificate does not claim is that the golden-ratio band is a law of nature. It claims that the framework's model of recognition cost forces these ratios, and that available data sits inside the band. The distinction matters: a proved theorem about a defined ratio is not the same as a measured property of every alloy.

/-- 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

def spinGlassFreezingCert : SpinGlassFreezingCert where
  ratio_3D_pos := freezingRatio3D_pos
  ratio_3D_band := freezingRatio3D_band
  ratio_2D_pos := freezingRatio2D_pos
  ratio_2D_band := freezingRatio2D_band
  dimensional_crossover := dimensional_crossover

What this page does not claim

The certificate does not measure any real spin glass; it proves statements about a defined ratio. The certificate does not claim that all spin glasses obey the band, only that canonical 3D Heisenberg and 2D Ising classes do. The certificate does not derive the golden ratio from first principles in this file; it uses φ as a defined 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.

Derived articles

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