Materials Demagnetization Factor3 From Jcost

Demagnetization factor and the cost ledger

The demagnetization factor N describes how a material's shape affects its internal magnetic field. A sphere has N = 1/3, meaning its internal field opposes the applied field by one third of the magnetization. A long needle has N near zero, because its elongated geometry lets the field align easily along its length. These values are classical results from magnetostatics, derived by solving boundary conditions for each shape.

The standard formula for a general ellipsoid along a principal axis is N = (a b c / 2) ∫₀^∞ ds / ((s + a²) √((s + a²)(s + b²)(s + c²))), where a, b, c are the semiaxes. For a sphere, a = b = c, and the integral evaluates to 1/3. For a prolate spheroid with a ≫ b = c, the integral approaches 0 as the aspect ratio grows. This integral form dates to the 19th century work of Poisson and Maxwell on magnetic induction in ellipsoids.

In Recognition Science, the framework models the demagnetization factor through a cost function, a measure of how far a ratio of two quantities is from unity. Its library defines domainCost as J(m/e), where J is the forced cost function J(x) = (x + 1/x)/2 - 1, and m and e are shape-related parameters. The module proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive, where phi is the golden ratio. These are properties of the cost function itself, not of any specific material.

The framework's research note connects this to demagnetization: for a sphere, N = 1/configDim = 1/3, matching the classical value. For a prolate spheroid, the note suggests N ≈ J(phi)/D = 0.118/3 ≈ 0.039 for a needle shape, consistent with elongated geometry. This is a research direction, not a proved theorem. The formal module proves only the three general cost properties, because it defines domainCost without specifying what m and e mean in magnetic terms.

The practical upshot: the machine-checked library guarantees that any cost function of this form behaves sensibly, vanishing at equality and staying nonnegative. What it does not yet do is prove that the specific ratio m/e for a needle equals the classical demagnetization factor. That step requires defining m and e from the material's geometry, which remains open.

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]

What this page does not claim

The module does not prove that any specific material's demagnetization factor equals J(m/e). The needle value 0.039 is a research note, not a formal theorem. The classical sphere value 1/3 is not derived from the cost function in this module.

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/Demagnetization_Factor3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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