Materials Glass Transition From Jcost Fragility Pos

The positivity guarantee

The fragility index is a number that orders how a liquid's viscosity responds to cooling near its glass transition. In the Recognition Science framework, that index takes values on a ladder of powers of the golden ratio, φ ≈ 1.618. The declaration fragility_pos proves a simple but load-bearing fact: for every integer step on that ladder, the index is strictly greater than zero. It is a theorem, checked by the framework's machine-checked library of formal theorems, with no unproved axioms in its derivation.

Why does positivity matter? The index sits in ratios that compare adjacent regimes, and a ratio is only meaningful when the denominator is not zero and the sign is stable. The theorem guarantees that the ladder never dips to zero or below, so the ratio of consecutive indices is always well defined. That ratio, in turn, is exactly the golden ratio again, a separate theorem in the same file. Positivity is the floor that keeps the whole ladder usable.

The framework models five canonical glass regimes, from fragile liquid through aging, and defines the index as φ^k for integer k. Since φ is positive and a positive number raised to any integer power is positive, the proof is short. But the declaration is not a claim about real glass. It says nothing about whether actual liquids follow this ladder, how well the model matches measured viscosities, or whether the five regimes are physically exhaustive. Those are empirical or modeling questions, not settled by this theorem.

What the reader can take away: within the framework, the fragility ladder is internally consistent and never breaks at zero. That is a small but necessary piece of a larger structure, and knowing exactly what it does and does not say is what makes the framework's claims auditable.

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

inductive GlassRegime where
  | fragileLiquid
  | strongLiquid
  | supercooled
  | vitreous
  | aging
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

No claim that real glass transitions follow this ladder; that would require empirical comparison. No claim that the five regimes are physically exhaustive; they are a modeling choice. No claim that the fragility index is measurable directly; it is a defined quantity.

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)

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: