Materials Radiation Effect From Jcost

Radiation damage thresholds

Radiation damage in materials begins when an incoming particle knocks an atom off its lattice site, creating a vacancy and an interstitial. The minimum energy needed for this displacement is the threshold displacement energy, typically tens of electron-volts. In silicon it is about 25 eV. A material with a higher threshold is more radiation-hard, because it takes a more energetic strike to create the defect.

The Recognition Science framework proposes a specific way to compute this threshold from its universal cost function. The cost function J(x) = (x + 1/x)/2 - 1 measures the recognition cost of a ratio x. The framework's proposal is that the threshold equals J(phi)^(-1) times the displacement energy, where phi is the golden ratio. Numerically, J(phi)^(-1) is about 8.47, so for a 25 eV displacement energy the predicted threshold is about 212 eV per displaced atom. This number is a research note in the source file, not a proved theorem.

What the machine-checked library of formal theorems does prove is more modest. The module defines a cost function domainCost (a per-domain recognition cost) as J(m/e) for two positive real numbers m and e. It proves three general facts: the cost is zero when m equals e, it is nonnegative for all positive inputs, and the constant phi - 3/2 is positive. These are true for any positive m and e, because the definition does not tie m and e to any specific physical quantities.

In Recognition Science, the framework models the radiation-hardness threshold as J(phi)^(-1) times the displacement energy, choosing the golden ratio as the self-similar scaling constant. The framework's library shows that the cost function has the three basic properties above, but it proves nothing specific about radiation damage. The step from the general cost function to a physical threshold would require defining m and e in terms of atomic properties, and that definition is not present in the module.

The practical consequence is clear: the 212 eV figure is a hypothesis, not a derivation. The formal results guarantee only that the cost function is well-behaved, not that it describes crystals. A reader who wants a radiation-hardness number should treat 212 eV as a prediction to test, not a proved value.

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)

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

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

What this page does not claim

The 212 eV figure is not proved by the module. The module does not define m and e in physical terms. No claim is made that the golden ratio threshold matches any experiment.

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/Radiation_Effect_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.

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