Materials Glass Transition Temp From Jcost

The glass transition temperature

The glass transition temperature T_g is the temperature at which a liquid becomes a glass: a solid that is rigid but has no crystalline order. Cool a melt fast enough and the molecules freeze into a disordered arrangement instead of forming a crystal lattice. The transition is not a sharp phase change like boiling. It is a kinetic crossover, and its value depends on cooling rate. A common empirical rule ties T_g to the Kauzmann temperature T_K, the temperature at which the entropy of the supercooled liquid would equal that of the crystal, through the ratio T_K ≈ T_g × φ^(-1), where φ is the golden ratio.

In Recognition Science, the framework models this crossover with a cost function J(x) = (x + 1/x)/2 - 1, which measures the price of a mismatch between two quantities. The module defines a domain cost as J(m/e), where m and e are two real numbers. The library establishes three general facts about this cost: it is zero when m equals e, it is never negative when m and e are positive, and the canonical threshold phi - 3/2 is positive. These are the only results the Lean code establishes. The module defines a certificate structure that packages these three facts, and establishes that this certificate exists.

What the module does not do is connect these facts to glass. The definition of domainCost uses J(m/e) without specifying what m and e mean for a glass-forming material. The docstring records the intended research direction: at T_g, the viscosity is one golden-ratio step above the liquid minimum, so J(η/η_0) = J(φ). Fragile glasses diverge at T_K below T_g on an Angell plot. But none of that is established in the Lean code. The module is a template, shared verbatim with 2383 sibling modules, that establishes only the universal properties of the cost function.

In plain language: the machine-checked library establishes that the cost function has the three basic properties it must have for any sensible measure of mismatch, and it defines a threshold constant. The physical claim about glass transition temperatures is a research note, not a theorem. What is established is the skeleton: a cost function that vanishes at equality, is nonnegative for positive inputs, and has a positive canonical threshold. The flesh, the specific definitions of m and e in terms of viscosity or temperature, remains open.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

structure GlassTgCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The module does not establish that the glass transition temperature equals any specific value. The module does not establish that the cost function applies to viscosity or any other glass property. The physical relationship T_K ≈ T_g × φ^(-1) is a research note, not a proved theorem.

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

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