Chemistry Glass Transition Kauzmann Lt One

The two-thirds rule

The glass transition is the temperature range where a supercooled liquid, cooled fast enough to avoid crystallizing, becomes an amorphous solid. A common observation is that this transition temperature, written Tg, sits at roughly two-thirds of the material's melting temperature, written Tm. This 'two-thirds rule' is a useful thumb rule for many glass-forming liquids, though it is not a law of physics.

In Recognition Science, the framework models this ratio as a fixed constant. Its machine-checked library of formal theorems, a collection of proofs verified by a computer, defines the Kauzmann ratio, the ratio Tg/Tm, as the number 2/3. The library then proves two basic facts about this definition: the ratio is positive, and it is less than one. The declaration kauzmann_lt_one is the second of these proofs, a formal statement that the number 2/3 is indeed smaller than 1.

The proof itself is a simple calculation. Since 2 is less than 3, the fraction 2/3 is less than 1. The machine-checked proof verifies this arithmetic step by step. This is a theorem about a definition, not a measurement. The library chooses the value 2/3 and then proves its basic properties. It does not derive the value from deeper principles, and it does not claim that every real glass obeys this ratio exactly.

The declaration establishes a formal, verified fact about a defined constant. Its value is that it anchors the two-thirds rule inside a larger, machine-checked framework. The framework's own falsification criteria state that the glass transition derivation would fail if the Kauzmann ratio deviated significantly from 2/3 in real materials. So kauzmann_lt_one is a small but necessary piece: it confirms the chosen ratio is a proper fraction, thus the definition stays consistent with the physical expectation that Tg is below Tm.

/-- Kauzmann ratio Tg/Tm ≈ 2/3. -/
def kauzmannRatio : ℝ := 2 / 3

/-- Kauzmann ratio is less than 1. -/
theorem kauzmann_lt_one : kauzmannRatio < 1 := 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

This does not claim that the two-thirds rule is derived from first principles within the framework; it is a definition. This does not claim that every real glass-forming liquid has a Kauzmann ratio exactly equal to 2/3. This does not claim the declaration proves anything about the physical mechanism of the glass transition.

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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