Materials Magnetic Anisotropy3 From Jcost

Magnetic anisotropy from J-cost

Magnetocrystalline anisotropy energy is the energy cost of magnetizing a crystal along a hard direction instead of an easy one. In iron, the leading constant K1 is measured at about 48 kJ/m³. The framework Recognition Science models this energy as a cost function J applied to a ratio of two quantities, m and e, which stand for the magnetization and the energy scale of the crystal. The model's formula is K1 = J(phi) * (mu_0 * M_s^2 / 2), where phi is the golden ratio, mu_0 is the vacuum permeability, and M_s is the saturation magnetization.

Plugging in iron's numbers gives about 2.36 × 10⁵ J/m³, which is roughly five times the measured value. The match is not precise, but the order of magnitude is consistent. The framework's library, a machine-checked collection of formal theorems, proves three general facts about the cost function: it vanishes when m equals e, it is never negative for positive inputs, and the golden ratio minus 1.5 is positive. These facts are universal properties of J, not specific to magnetism.

In Recognition Science, the cost function, a forced measure of recognition effort, is defined as J(m/e). The certificate structure MagAnis3Cert bundles the three proved facts into a single object, showing they hold together. What the framework does not do is define m and e in terms of the crystal's atomic structure. Without that definition, the framework proves nothing specific to magnetic anisotropy; it is a template awaiting a physical interpretation.

The framework's contribution here is a scaling law: the cost function forces the golden ratio as a natural threshold, and the anisotropy energy inherits that scale. The numerical agreement with iron is a research note, not a theorem. The framework's real content is the universal properties of J, which hold for any positive inputs and any ratio, not just for magnets.

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 framework does not prove any fact specific to magnetic anisotropy, because m and e are not defined in the crystal's own terms. The numerical agreement with iron is a research note, not a theorem, and the model overestimates K1 by a factor of about five. The framework does not derive the fine-structure constant or any other coupling constant.

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/Magnetic_Anisotropy3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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