Nonzero algebraic winding phases form a dense Circle orbit

Proved

WindingArithmeticDensePhase.algebraicRealCirclePhaseDense

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

algebraic-topologyirrational-rotationnumber-theorywinding

If $\alpha\in\mathbb R$ is nonzero and algebraic over $\mathbb Q$ , then $n\mapsto e^{i n\alpha}$ has dense range in $S^1$ .

Preamble

import Definitions.Def_WindingArithmeticDensePhase_CoreV1

open Function

Formal statement

theorem WindingArithmeticDensePhase.algebraicRealCirclePhaseDense
    (α : ℝ) (hα : IsAlgebraic ℚ α) (hα0 : α ≠ 0) :
    DenseRange (WindingArithmeticDensePhase.realCirclePhase α) := 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).

View graph