Condensed Matter Spin Glass Freezing Ratio Freezing Ratio3 D Band

The 3D freezing band

A spin glass is a magnetic alloy, such as copper with a sprinkle of manganese atoms, where the magnetic moments of the impurity atoms point in random directions and become locked into a disordered pattern when cooled. The freezing temperature T_g is where that locking happens. In a ferromagnet, the Curie temperature T_c marks where moments align. The Recognition Science declaration freezingRatio3D_band proves a narrow numerical statement: for a canonical three-dimensional Heisenberg spin glass, the ratio T_g/T_c lies strictly between 0.617 and 0.622.

That band is not pulled from a fit. It is a theorem in the framework's machine-checked library of formal theorems, derived from the definition that sets the ratio to 1/φ, where φ is the golden ratio, approximately 1.618. The proof uses only the known bounds on φ to show 1/φ falls inside the stated interval. The framework's account treats the spin glass as realizing a frustrated sector of its recognition lattice, while a ferromagnet realizes an unfrustrated sector; the ratio of their characteristic energy scales is the golden ratio's reciprocal.

The empirical baseline comes from CuMn with 1% manganese: T_g ≈ 10 K and a theoretical pure-manganese ferromagnetic T_c ≈ 16 K, giving a ratio near 0.625. Gold-iron data spans 0.60 to 0.65 depending on composition. The framework's provable band of (0.617, 0.622) sits inside this observed window, but the structural claim is that the cluster center is 1/φ, not that variance is zero.

In Recognition Science, this is a recognition event, a discrete record of a system settling into a state, and the ratio is forced by the framework's cost structure rather than chosen. The declaration does not claim that every spin glass obeys this ratio, nor that the empirical data prove the framework. It establishes a precise mathematical bound on a defined quantity, and offers a falsifiable prediction: a survey of at least ten spin glasses with calibrated T_c references should find the median ratio inside (0.61, 0.62) at the 2σ level.

/-- 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

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

/-- 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

What this page does not claim

This does not claim that all spin glasses obey the golden ratio, only a canonical 3D Heisenberg class. This does not claim the empirical data prove the framework; the data are a consistency check, not a theorem. This does not claim the framework derives the freezing temperature itself, only the ratio to the Curie temperature.

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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