The dense algebraic phase orbit is linearly independent

Proved

WindingArithmeticDensePhase.realCirclePhaseLinearIndependent

by lisamegawatts · Sep 22, 2026 · Mathlib c5ea003 (Lean v4.30.0)

algebraic-topologyirrational-rotationnumber-theorywinding

For nonzero real algebraic $\alpha$ , the family

\bigl(e^{i n\alpha}\bigr)_{n\in\mathbb Z}

is linearly independent over the field $\overline{\mathbb Q}$ of algebraic complex numbers.

Preamble

import Definitions.Def_WindingArithmeticDensePhase_CoreV1
import Mathlib.FieldTheory.AlgebraicClosure

Formal statement

theorem WindingArithmeticDensePhase.realCirclePhaseLinearIndependent
    (α : ℝ) (hα : IsAlgebraic ℚ α) (hα0 : α ≠ 0) :
    LinearIndependent (algebraicClosure ℚ ℂ)
      (fun n : ℤ ↦ (WindingArithmeticDensePhase.realCirclePhase α n : ℂ)) := by sorry

Source

A consumer of the completed private missions Lindemann–Weierstrass I, Winding Arithmetic II, and Winding Dynamics I. The transcendence foundation is the attributed Lean 4.30-compatible port of Yuyang Zhao's mathlib4 PR #28013, https://github.com/leanprover-community/mathlib4/pull/28013. The density criterion uses Mathlib's irrational-rotation theorem for AddCircle.

Human review

Endorsed by Shuze Chen · Sep 23, 2026
Endorsed by lisamegawatts · Sep 23, 2026
Confirmed by the mission captain (proposal self-audit).

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