Physics Spin Hall Effect From Jcost

Spin Hall conductivity and the cost function

The spin Hall effect is a transport phenomenon in solid-state physics where an applied electric current generates a transverse flow of electron spin, even in the absence of a magnetic field. First predicted by M. I. Dyakonov and V. I. Perel in 1971 and later understood through spin-orbit coupling, the effect produces a spin accumulation at the sample edges. The intrinsic spin Hall conductivity, σ_SH, measures the efficiency of this spin-current generation and is typically expressed in units of (ℏ/e) S/cm. For canonical metals, measured values fall in the range of 100 to 1000 (ℏ/e) S/cm, a scale set by the unit conductance.

In Recognition Science, the framework models the conductivity using a single universal cost function. The cost is a measure of mismatch between a measured value and an expected value, defined as J(x) = (x + 1/x)/2 - 1. The framework defines a domain cost for the spin Hall effect as J(measured / expected), where measured is the observed conductivity and expected is the theoretical prediction. The framework proves three general facts about this cost: it equals zero when the measured value matches the expected value, it is never negative for positive inputs, and the threshold φ - 3/2 is positive. These are properties of the cost function itself, not specific to the spin Hall effect.

The framework's research note proposes that the intrinsic spin Hall conductivity scales as σ_SH ≈ e/ℏ × J(φ)/a², where a is the lattice constant and φ is the golden ratio. This gives a predicted range of 100-1000 (ℏ/e) S/cm for canonical metals, consistent with the measured scale. However, this scaling is a research note, not a proved result. The formal library contains no definition of m and e in terms of spin Hall physics; it proves only the general cost properties. What would make the library a theorem about the spin Hall effect is a definition of measured and expected in the subject's own terms.

The library establishes a general template: any physical quantity can be assigned a cost by comparing it to an expected value. The cost vanishes at equilibrium, is nonnegative, and has a positive threshold related to the golden ratio. These are the only claims the machine-checked library proves. The spin Hall effect provides a concrete domain where this template could apply, but the connection remains a research direction, not a formal result.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (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 spin Hall conductivity formula is a research note, not a proved result. The library proves nothing specific to the spin Hall effect beyond the general cost properties. No measured spin Hall conductivity value is compared against the framework's prediction in the pack.

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

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