Physics Spin Torque From Jcost

Spin torque and the cost threshold

Spin transfer torque is a physical effect in which an electric current, passed through a thin magnetic layer, transfers angular momentum to a neighboring magnet and can flip its magnetization. This is how modern magnetic memory writes bits: a current pulse, not a magnetic field, switches the storage element. The critical current density needed to switch is the key engineering quantity, and it depends on material parameters such as the magnetization saturation M_s, the layer thickness t, and the spin polarization P of the current.

The standard expression for this critical current density is J_c ~ J(phi) * e * M_s * t / (hbar * P), where e is the electron charge and hbar is the reduced Planck constant. The factor J(phi) is a dimensionless threshold that depends on the angle phi between the magnetizations. In Recognition Science, this factor is not a free parameter. It is identified with the framework's universal cost function, evaluated at the ratio of two quantities m and e, so that J(phi) = Jcost(m/e).

The framework's cost function is forced by five plain conditions: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The proved result is that any cost function satisfying these must equal J(x) = (x + 1/x)/2 - 1. In the spin torque setting, the cost is defined as Jcost(m/e), and the machine-checked library of formal theorems proves three general facts about it: it vanishes when m equals e, it is nonnegative for positive inputs, and the golden-ratio-based threshold phi - 3/2 is positive.

What this means in plain language is that the framework's cost function provides a candidate for the recognition threshold in spin torque switching. The library proves that this cost is zero at the balanced point m = e, never negative for positive arguments, and that the canonical threshold is a positive number. It does not, however, prove anything specific to spin torque itself, because the cost is defined as Jcost(m/e) without connecting m and e to physical quantities like magnetization or current. The physical bridge remains a research note, not a theorem.

The practical consequence is a prediction: if the recognition threshold J(phi) in the spin current exceeds the magnetization barrier, switching occurs. The framework's cost function supplies a concrete, derived form for that threshold, which could be tested against measured critical currents. The three proved facts are necessary conditions for any such threshold, but they do not by themselves establish the spin torque connection.

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 library does not prove any physical fact specific to spin torque, only general properties of the cost function. The identification of J(phi) with Jcost(m/e) is a research note, not a derived result. No prediction of a numerical critical current density is made here.

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

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