Physics Wigner Rotation From Jcost

Wigner rotation and the cost threshold

Wigner rotation is a relativistic effect: if you accelerate a spinning object along one direction and then along another, the object's orientation ends up rotated, even though no torque acted on it. For two successive boosts with small velocities v₁ and v₂, the rotation angle ε is approximately (v₁ × v₂)/c² times the sine of the angle between the boosts. This is a standard result of special relativity, first understood in the context of the Thomas precession of electron orbits in atoms.

The Recognition Science framework defines a cost, a number measuring how expensive a recognition event is, with the specific form J(x) = (x + 1/x)/2 − 1. The framework's library proves that J is zero when its argument is 1, nonnegative for positive arguments, and that the golden ratio φ satisfies φ − 3/2 > 0. These are the only facts the module WignerRotationFromJCost establishes; it defines a domain cost as J(m/e) for two positive numbers m and e, and packages the three properties into a certificate structure.

The module does not, however, connect this cost to Wigner rotation. Its docstring records a research note: at v = c/φ, the maximum rotation angle might be J(φ) × θ, and the Thomas precession rate might be J(φ)/τ. But the Lean code defines m and e without reference to boost velocities or angles, so the module proves nothing specific to Wigner rotation. The three theorems hold for any positive m and e, because they follow from general properties of J.

What the module does establish, in plain language, is a small piece of the framework's machinery: the cost function vanishes when its two inputs are equal, never goes negative for positive inputs, and has a positive threshold tied to the golden ratio. These are the ingredients a future theorem about Wigner rotation would need, but the bridge from cost to rotation angle 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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

noncomputable def cert : WignerRotCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The module proves any specific property of Wigner rotation; it only proves general facts about the cost function. The research note's formula for ε_max is not a proved result in the module. The golden ratio threshold φ − 3/2 > 0 is not shown to have any physical consequence 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/Physics/WignerRotationFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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