Materials Spintronics Gmr2 From Jcost

Giant magnetoresistance and the cost function

Giant magnetoresistance (GMR) is a physical effect in layered magnetic materials where an external magnetic field causes a large drop in electrical resistance. In a typical GMR device, the resistance difference between antiparallel and parallel magnetic alignments, (R_AP - R_P)/R_P, reaches 10 to 100 percent. This effect, discovered in 1988 by Albert Fert and Peter Grünberg, underpins modern hard disk drive read heads.

The standard model of GMR uses spin-dependent scattering. Electrons moving through a ferromagnetic layer scatter less when their spin is aligned with the layer's magnetization. The resistance depends on the relative alignment of adjacent magnetic layers, and the magnitude of the effect is controlled by the spin polarization P, the fractional spin imbalance of the conducting electrons. The higher the polarization, the larger the GMR ratio.

In Recognition Science, a framework that derives physical structure from a forced cost of recognition, the GMR ratio is modeled as GMR = 2 * J(phi) * (P^2 / (1 - P^2)). Here J is the framework's cost function, and phi is the golden ratio. At the specific polarization P = phi^(-1) = 0.618, this expression gives GMR = 2 * 0.118 * (0.382 / 0.618) = 0.146, or 14.6 percent, consistent with the 10 to 20 percent range seen in many GMR devices.

The machine-checked library of formal theorems proves three general facts about the cost function J(x) = (x + 1/x)/2 - 1, and the module applies them to the ratio m/e. The cost is zero when m equals e, it is nonnegative for positive inputs, and the canonical threshold phi - 3/2 is positive. These facts hold for any positive m and e, so the module itself does not yet pin down what m and e mean for a specific material. The research note in the module records the intended direction, not a completed result.

What this means in practice: the framework offers a candidate formula for GMR that connects the effect to a universal cost function, and at one natural polarization value it lands inside the measured range. The gap between the general theorem and a material-specific prediction is a definition of m and e in the language of spin transport. That step 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 proves that GMR equals 14.6 percent for any real material. The framework derives the spin polarization P from first principles. The module establishes a specific definition of m and e in spintronic terms.

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/SpintronicsGMR2FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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