Materials Radiation Hardening From Jcost

Radiation hardening as a cost

Radiation hardening is the increase in a material's yield stress after it is bombarded by neutrons or ions. The classic empirical picture is that this increase scales with the square root of the defect density: more displaced atoms mean a stiffer, more brittle material. The Recognition Science framework's ledger, a discrete record of recognition events, enters this picture through a proposed formula: the hardening increment Δσ equals the yield stress σ_yield times the square root of J(φ), where J is the framework's cost function and φ is the golden ratio. Numerically that factor is about 0.344, which would put the predicted hardening at roughly 10 to 50 percent of yield stress, matching the observed range.

The machine-checked library of formal theorems, however, proves something much narrower. Its module defines a domain cost as J(m/e), the cost of recognizing a ratio of two real numbers. Three facts are proved: this cost is zero when the two numbers are equal, it is never negative for positive inputs, and the constant φ − 3/2 is positive. These are general properties of the cost function, not results about materials. The library's own docstring says so explicitly: it proves nothing specific to radiation hardening, because the definition of the domain cost does not reference defect density, lattice sites, or yield stress.

In Recognition Science, the framework models the hardening claim as a hypothesis. The formula J(φ) × lattice_sites is a proposed identification, not a derived theorem. What would turn the module into a theorem about its subject is a definition of m and e in the material's own terms, for instance m as a measure of displaced atoms and e as a measure of original lattice sites. Without that definition, the formal content is a template shared verbatim with 2383 sibling modules, each one a placeholder for a subject-specific argument.

The practical consequence for a reader is a clean separation: the formal library certifies the cost function's basic shape, and the radiation-hardening formula is a research note recording where the idea was meant to go. The square-root scaling and the 0.344 factor are a prediction with a named falsifier: a measurement of hardening that does not track the square root of defect density would falsify it. The formal theorems stand independently of that prediction, and the prediction stands independently of the formal theorems.

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 radiation-hardening formula Δσ = σ_yield × √J(φ) is a proved theorem. The module establishes any connection between the cost function and actual defect densities. The 10 to 50 percent hardening range is a formal 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/RadiationHardeningFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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