Condensed Matter Spin Glass Freezing Ratio Freezing Ratio3 D Pos

The positivity theorem

A spin glass is a magnetic alloy, like copper with a little manganese dissolved in it, whose atomic magnets freeze into a disordered, frustrated pattern when cooled, rather than lining up as in an ordinary ferromagnet. The temperature at which this freezing happens is called T_g, and the Curie temperature T_c is the temperature at which the same material would order ferromagnetically if frustration were absent. The ratio T_g/T_c compares these two energy scales. In the Recognition Science framework, the declaration freezingRatio3D_pos is a theorem in the framework's machine-checked library of formal theorems establishing that a specific proposed value for this ratio is positive.

The proposed value is 1/φ, where φ is the golden ratio, approximately 1.618, so 1/φ is about 0.618. The declaration proves 0 < 1/φ, a fact that is true because both 1 and φ are positive. A companion theorem in the same library, freezingRatio3D_band, proves the stronger statement that this value lies in the interval (0.617, 0.622). The positivity theorem is the weaker of the two: it rules out the ratio being zero or negative, which is a necessary condition for the value to be a physically meaningful temperature ratio, since temperatures are positive. The band theorem is what connects the formal value to the empirical data window for real materials like CuMn and AuFe, which spans roughly 0.60 to 0.65.

The framework models the spin glass as realizing a particular frustrated sector of its recognition lattice, and the ferromagnet as realizing an unfrustrated sector, with the ratio of their characteristic energy scales being the canonical recognition dividend 1/φ. The structural claim is that the cluster center of measured ratios sits at 1/φ, not that the variance is zero. The positivity theorem itself carries no empirical content: it is a purely formal statement about the real number 1/φ. The empirical prediction, that canonical 3D Heisenberg spin glasses have T_g/T_c in (0.61, 0.62), is a separate hypothesis with a named falsifier: a cross-system survey of at least 10 spin glasses with calibrated T_c references whose median falls outside that band at the 2σ level.

What the theorem does not claim is just as important as what it proves. It does not prove that any real spin glass has this ratio; that is an empirical matter. It does not establish the band (0.617, 0.622), which is a separate theorem. And it does not assert that the framework's model of spin glasses is correct; the model is a definitional choice, and the positivity of 1/φ is a fact about arithmetic, not about magnetism.

theorem freezingRatio3D_pos : 0 < freezingRatio3D :=
  div_pos (by norm_num) phi_pos

/-- 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

/-- **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

The theorem does not prove that any real spin glass has this ratio. The theorem does not establish the numerical band (0.617, 0.622), which is a separate theorem. The theorem does not assert that the framework's model of spin glasses is correct.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: