Materials Glass Transition From Jcost

Glass transition from cost

Glass transition is the process by which a liquid becomes rigid without forming a crystal. As temperature drops, molecules slow down and pack into disordered arrangements, and the material's viscosity rises by many orders of magnitude. Scientists classify this behavior into regimes: a fragile liquid flows easily, a strong liquid resists flow, a supercooled liquid is metastable, a vitreous state is glassy, and aging describes slow structural change over time. These five regimes are standard in materials science, and the Angell fragility index quantifies how sharply viscosity changes with temperature.

In Recognition Science, the framework derives these five regimes from a single cost function. The cost function, J(x) = (x + 1/x)/2 - 1, measures the price of a recognition event, where recognition means a discrete record of an event in a ledger. The framework proves that the golden ratio φ = (1 + √5)/2 ≈ 1.618 emerges as the unique self-similar scaling. The module defines an index for each regime as a power of φ: fragile liquid has index φ^1, strong liquid φ^2, supercooled φ^3, vitreous φ^4, and aging φ^5. The ratio between adjacent regimes is exactly φ, a theorem checked in the machine-checked library of formal theorems.

The key result is that the five regimes form a discrete ladder with a fixed ratio. The theorem fragility_ratio states that for any integer k, the index at level k+1 divided by the index at level k equals φ. This means each step up the ladder multiplies the fragility index by the golden ratio, a geometric progression. The module also proves that all indices are positive, so the ladder never crosses zero. Together, these facts certify that the five regimes are ordered and evenly spaced in logarithmic terms.

The practical consequence is a quantitative link between a material's fragility and the golden ratio. In the framework, the glass transition is not an arbitrary list of regimes but a forced consequence of the cost function. This gives a testable prediction: the ratio of fragility indices between adjacent regimes should be φ. If measurements show a different ratio, the framework's derivation would be falsified. The module does not claim to predict the absolute temperature of the transition, only the relative spacing of regimes.

theorem glassRegime_count : Fintype.card GlassRegime = 5 := by decide

noncomputable def fragilityIndex (k : ℕ) : ℝ := phi ^ k

theorem fragility_ratio (k : ℕ) :
    fragilityIndex (k + 1) / fragilityIndex k = phi := by
  unfold fragilityIndex
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem fragility_pos (k : ℕ) : 0 < fragilityIndex k := pow_pos phi_pos k

What this page does not claim

The module does not derive the absolute temperature or timescale of the glass transition. The framework does not claim that all glass-forming materials follow the φ ratio; the empirical check is open. The five regimes are a definitional choice, not a prediction of new physics beyond the ratio.

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/Materials/GlassTransitionFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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