Materials Crystal Twin3 From Jcost

A formal placeholder for twinning

A crystal twin is a symmetric intergrowth of two crystals, and the twinning shear is the small deformation that creates it. In materials science, the shear for common face-centered cubic twins is measured in the range of about 0.1 to 0.2. A research note inside the framework's library suggests that its cost function reaches a value of 0.118 at a canonical boundary, a number close to that observed range. The note is a plan, not a result.

In Recognition Science, the framework models recognition events through a ledger, a discrete record of such events, and assigns each event a forced cost. The cost function J(x) equals (x + 1/x)/2 - 1. The module named Crystal_Twin3_FromJCost defines a domain cost as J applied to the ratio of two real numbers, m and e. The machine-checked library of formal theorems proves three general facts about this cost: it is zero when the two numbers are equal, it is never negative for positive inputs, and the number phi - 3/2 is positive, where phi is the golden ratio.

These three facts are true for any positive real numbers. The module proves nothing about crystal twinning itself, because its definition does not connect m and e to any physical quantity. The same three facts are stated once, universally, in a shared template module, and this file is one of 2383 siblings that repeat them. What would make this a theorem about twins is a definition of m and e in terms of the crystal's own geometry, which the file does not provide.

The module does establish a clean certificate structure: it packages the three proven facts into a single object and proves that such a certificate exists. This is a small, honest piece of infrastructure. It shows the framework can state and verify the general properties of its cost function, and it marks a clear target for future work. The gap between the research note and the proof is explicit, and closing it requires a physical definition that is currently open.

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

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

This module proves that the twinning shear of any real crystal is 0.118. This module derives the existence of crystal twins from the cost function. The research note's numerical comparison is a measured result.

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/Crystal_Twin3_FromJCost.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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