Materials Magnetostriction3 From Jcost

Magnetostrictive strain

Magnetostriction is the tendency of a ferromagnetic material to change its dimensions when it is magnetized. A nickel rod, for instance, shrinks by a few dozen parts per million when placed in a strong field. The effect is small, usually 10 to 100 parts per million, but it is the basis for sonar transducers, precision actuators, and vibration sensors. Engineers quote it as a strain λ_s, the fractional change in length.

In the Recognition Science framework, the module materials-magnetostriction3-from-jcost applies a universal cost function to the ratio of two quantities, m and e. The cost function, written J(x) = (x + 1/x)/2 − 1, is the unique function that satisfies five plain conditions such as symmetry and a forced composition law. It measures how far a ratio is from unity: it vanishes when the two quantities are equal, and it is always nonnegative for positive inputs.

What the module proves is deliberately modest. It establishes three general facts about J(m/e): the cost is zero when m equals e, the cost is never negative when both inputs are positive, and the golden-ratio threshold φ − 3/2 is positive. These are universal properties of the cost function itself, not claims about any specific material. The module defines a certificate structure that packages these three facts together, and it proves that such a certificate exists.

The research note attached to the module records where the idea was meant to go: a magnetostrictive strain of about 118 parts per million, which sits at the upper end for nickel, or about 11.8 parts per million for typical structural metals. Those numbers come from evaluating J at the golden ratio and scaling by 1000 or 100 ppm. But the Lean code proves nothing specific to magnetostriction, because the module never defines what m and e mean in a material's own terms. The physical identification remains a research note, not a theorem.

What a reader can take away is the boundary between a proved result and a hoped-for application. The cost function's behavior at equality and its nonnegativity are machine-checked facts. Whether those facts describe nickel's strain is a separate question, one that requires a definition of the two quantities in the language of magnetism and elasticity. That definition is the missing piece.

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]

theorem cert_inhabited : Nonempty Magnetostriction3Cert := ⟨cert⟩

What this page does not claim

The module does not prove that any real material has magnetostrictive strain of 118 ppm or any other value. The module does not define m and e in terms of magnetic or elastic properties. The module does not derive magnetostriction from the cost function; it only proves general properties of J.

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/Magnetostriction3_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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