Chemistry Glass Transition Is Strong Glass

The strong glass definition

In materials science, a supercooled liquid that does not crystallize as it cools eventually becomes a glass, an amorphous solid. Glasses differ in how abruptly their viscosity rises as they approach the glass transition temperature Tg. A strong glass, such as silica (SiO₂) or germania (GeO₂), shows a gradual, nearly Arrhenius increase in viscosity. A fragile glass, such as o-terphenyl or many polymers, shows a sharp, non-Arrhenius jump. The standard quantitative measure is the fragility index m, a dimensionless number that tracks how far the viscosity deviates from the simple Arrhenius law. Strong glasses typically have m between 16 and 30, fragile glasses between 100 and 200, and intermediate materials fall in between.

The Recognition Science framework, a machine-checked library of formal theorems, models this classification with a single definition. Its declaration isStrongGlass takes a number m and returns the proposition that 16 ≤ m and m ≤ 30. That is the entire content of the declaration: it is a formal label, not a physical law. The framework also defines the complementary predicate isFragileGlass for the range 100 ≤ m ≤ 200. These two definitions mirror the conventional empirical brackets used in glass science. The framework's contribution is to make the classification precise inside its own formal system, so that later theorems can refer to it unambiguously.

What the declaration does not do is predict, measure, or derive the fragility index of any specific material. It does not say that silica has m = 20 or that a given polymer has m = 150. Those values come from experiment, not from the definition. The declaration also does not explain why a material is strong or fragile; it only labels a number once that number is known. The framework does offer a separate, unformalized hypothesis that fragility relates to an eight-beat relaxation period and to the golden ratio φ, with a proposed scaling law for relaxation time, but that hypothesis is not part of isStrongGlass itself.

In plain terms, isStrongGlass is a box with a label. The box is the interval from 16 to 30. The label says "strong." The framework's library proves only that the fragility proxy it defines is positive and decreasing, and that the Kauzmann ratio 2/3 is positive and less than one. Those are mathematical facts about the framework's own definitions, not measurements of any real glass. A reader who wants to know whether a particular liquid is strong must still consult the laboratory, not the declaration.

/-- Strong glass fragility range. -/
def isStrongGlass (m : ℝ) : Prop := fragilityMin ≤ m ∧ m ≤ 30

/-- Fragile glass fragility range. -/
def isFragileGlass (m : ℝ) : Prop := 100 ≤ m ∧ m ≤ fragilityMax

/-- Universality: fragility is strictly positive for all k. -/
theorem glass_univ (k : Nat) : fragility k > 0 := by
  dsimp [fragility, eight_beat_period]
  have hφpos : 0 < Constants.phi := Constants.phi_pos
  have ha_pos : 0 < (1 / Constants.phi) := div_pos one_pos hφpos
  exact pow_pos ha_pos _

/-- Fragility at k=1 is less than at k=0 (fragility decays).
    This follows because 0 < 1/φ < 1 and 16 > 8 implies (1/φ)^16 < (1/φ)^8. -/
theorem fragility_one_lt_zero : fragility 1 < fragility 0 := by
  dsimp [fragility, eight_beat_period]
  -- Use numerical verification
  have h1 : (1 / Constants.phi) ^ 16 < (1 / Constants.phi) ^ 8 := by
    have h_phi_pos := Constants.phi_pos
    have h_phi_gt_1 : Constants.phi > 1 := by
      have := Constants.phi_gt_onePointFive
      linarith
    -- 1/φ < 1 since φ > 1
    have h_base_lt_1 : 1 / Constants.phi < 1 := by
      rw [div_lt_one h_phi_pos]
      exact h_phi_gt_1
    have h_base_pos : 0 < 1 / Constants.phi := by positivity
    -- For 0 < x < 1, x^16 < x^8 (since 16 > 8)
    have : 16 > 8 := by norm_num
    exact pow_lt_pow_right_of_lt_one₀ h_base_pos h_base_lt_1 this
  exact h1

/-- Kauzmann ratio is positive. -/
theorem kauzmann_pos : kauzmannRatio > 0 := by
  simp only [kauzmannRatio]
  norm_num

/-- Kauzmann ratio is less than 1. -/
theorem kauzmann_lt_one : kauzmannRatio < 1 := by
  simp only [kauzmannRatio]
  norm_num

What this page does not claim

The declaration does not predict the fragility index of any real material. The declaration does not explain the physical origin of strong or fragile behavior. The framework's theorems about positivity and the Kauzmann ratio do not constitute measurements of real glasses.

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/Chemistry/GlassTransition.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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