Materials Magnetoelectric3 From Jcost

Magnetoelectric coupling

A magnetoelectric material is one in which an applied magnetic field produces an electric polarization, and an applied electric field produces a magnetization. The strength of this conversion is the magnetoelectric coupling coefficient, often written α_ME. In a composite material, the effect is a product of two separate properties: the piezoelectric coefficient d_33, which converts mechanical strain into electric charge, and the piezomagnetic coefficient e_33, which converts magnetic field into strain. The coefficient α_ME roughly equals the product of these two, divided by the permittivity of free space ε_0. For a typical ceramic with e_33 around 10 C/m² and d_33 around 400 pm/V, this estimate gives α_ME near 0.05 V/(m·Oe), a structural estimate rather than a measured value.

In the Recognition Science framework, the cost of recognition is a forced function J(x) = (x + 1/x)/2 − 1. The framework's machine-checked library of formal theorems proves that any cost function meeting five plain conditions must equal this J. The library then applies J to ratios: a module called magnetoelectric3 defines a domain cost as J(m/e), where m and e are two positive real numbers. The module proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the golden-ratio threshold φ − 3/2 is positive. These are properties of the ratio m/e, not of any specific material.

What the module does not do is define what m and e mean for a magnetoelectric material. The docstring records a research note: α_ME ~ J(φ) · e_33 · d_33 / ε_0, which with the numbers above gives about 0.05 V/(m·Oe). But the Lean code itself never introduces e_33, d_33, or ε_0. The module is one of 2383 siblings that share the same universally quantified content, all citing a single template. To turn this into a theorem about magnetoelectric coupling, someone would need to define m and e in the material's own terms, for example m as a magnetic field strength and e as an electric field strength.

The distinction matters for what the framework can honestly claim. The three proved facts are general: any ratio m/e has zero cost at equality, nonnegative cost for positive inputs, and a positive threshold at φ − 3/2. The magnetoelectric estimate is a structural guess, not a theorem. The library proves the cost function's properties; the application to a specific material remains a research note.

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 module proves any fact specific to magnetoelectric materials. The estimate α_ME ~ 0.05 V/(m·Oe) is a measured or derived value. The golden-ratio threshold φ − 3/2 has any demonstrated physical meaning for magnetoelectric coupling.

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/Magnetoelectric3_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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