Materials Grain Growth3 From Jcost

Grain growth and the cost function

In materials science, grain growth describes how the average size of crystals in a polycrystalline material increases during annealing. The classical empirical relation is D^n ~ D_0^n + K*t, where D is the average grain size at time t, D_0 is the initial size, K is a temperature-dependent rate constant, and n is the growth exponent. For normal grain growth, n is typically measured between 2 and 3; for abnormal growth, n can be as high as 4. The exponent n encodes the underlying mechanism: diffusion through the lattice, through grain boundaries, or through the vapor phase each give a different n.

In Recognition Science, the framework derives a unique cost function J(x) = (x + 1/x)/2 - 1 from five plain conditions, and this function appears throughout the framework's models. The module grain growth3 from jcost defines a domain cost as J(m/e), where m and e are real numbers. The machine-checked library of formal theorems proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and the constant phi - 3/2 (where phi is the golden ratio) is positive. These are theorems about the cost function itself, not about grain growth.

In Recognition Science, the framework's research note suggests that the growth exponent n equals the configuration dimension D, and for normal grain growth that dimension is 3. The idea is that the same cost function that forces three spatial dimensions might also force the growth exponent. But the module defines m and e without reference to any physical quantity, so it proves nothing specific to grain growth. The note records where the idea was meant to go, not a result.

What the module does establish, in plain language, is that the cost function has the three properties needed for a sensible energy-like quantity: zero at equilibrium, nonnegative away from it, and a positive threshold constant. These are general facts that would hold for any subject if m and e were defined in that subject's terms. The step from these general facts to a statement about grain growth requires a definition of m and e in terms of grain size and time, which the module does not provide.

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)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove that the grain growth exponent is 3. The module does not define m and e in terms of grain size or time. The framework's research note is not a 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/Grain_Growth3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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