Materials Spintronics Magnetoresistance From Jcost

Magnetoresistance and its cost

Magnetoresistance is the change in a material's electrical resistance when a magnetic field is applied. In spintronics, the effect is used to read data in hard drives: a spin valve has two magnetic layers, and the resistance is lower when their magnetizations are parallel than when they are antiparallel. The standard measure is the giant magnetoresistance (GMR) ratio, (R_AP - R_P)/R_P, which in real devices ranges from about 10 to 100 percent.

The framework Recognition Science models this effect through a cost, a number that measures how expensive a recognition event is. The cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions, and it vanishes when its input equals 1. In the spin-valve model, the input is the ratio m/e of two quantities, and the cost at that ratio gives the predicted resistance change. The framework's machine-checked library of formal theorems proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and a threshold value involving the golden ratio is positive.

The library proves nothing specific to spintronics, because the module defines the cost as J(m/e) without choosing what m and e mean in the material. The GMR ratio prediction, ΔR/R = 2J(φ) ≈ 23.6%, is a research note recording where the idea was meant to go, not a result. What the module does establish is that the cost function itself has the right shape: it penalizes mismatch, rewards agreement, and has a natural threshold. Making it a theorem about magnetoresistance would require defining m and e in the material's own terms, which remains open.

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 a specific GMR ratio for any real material. The quantities m and e are defined in spintronic terms. The 23.6% prediction is a theorem rather than a research note.

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

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