Chemistry Glass Transition Relaxation Pos

The relaxation time theorem

In the physics of glass, relaxation time is the time a material takes to respond to a change in temperature or stress. Near the glass transition, this time can grow by many orders of magnitude, turning a flowing liquid into a rigid solid. The standard scaling law writes this as τ = τ₀ × φⁿ, where τ₀ is a base time, φ is the golden ratio (about 1.618), and n is an exponent that depends on temperature.

Within the Recognition Science framework, the declaration relaxation_pos proves a simple property of this law: if the base time τ₀ is positive, then the relaxation time τ is positive for every integer n. The proof is a short chain of arithmetic. It uses the fact that the golden ratio is positive, so any power of it is positive, and the product of two positive numbers is positive.

The theorem is tagged as a machine-checked result in the framework's library of formal theorems. It does not prove that the scaling law is correct, that the golden ratio is the right base, or that any particular glass follows this pattern. It only establishes a consistency property: a positive starting time never produces a negative or zero relaxation time.

This matters because the scaling law is used to predict how viscosity rises as a liquid cools toward the glass transition. If the formula could produce a negative time, the model would be internally broken. The theorem rules out that failure mode, leaving the physical question of whether the law matches experiment to separate empirical checks.

/-- Relaxation time is positive. -/
theorem relaxation_pos (τ₀ : ℝ) (hτ : 0 < τ₀) (n : ℕ) :
    0 < relaxationTime τ₀ n := by
  dsimp [relaxationTime]
  apply mul_pos hτ
  exact pow_pos Constants.phi_pos n

What this page does not claim

The relaxation_pos theorem does not prove the φ-scaling law matches any measured glass. The theorem does not establish the value of the exponent n for any material. The theorem does not claim that positive relaxation times are physically sufficient for a valid glass model.

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

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: