A nonzero algebraic angle has irrational rotation ratio

Proved

WindingArithmeticDensePhase.algebraicAngleDivTwoPiIrrational

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

\frac{\alpha}{2\pi}\notin\mathbb Q.

This is the arithmetic nonresonance input for the dense integer rotation orbit.

Preamble

import Definitions.Def_WindingArithmeticDensePhase_CoreV1
import Theorems.Thm_transcendental_pi
import Mathlib.RingTheory.Algebraic.Integral
import Mathlib.RingTheory.Localization.Integral
import Mathlib.NumberTheory.Real.Irrational

Formal statement

theorem WindingArithmeticDensePhase.algebraicAngleDivTwoPiIrrational
    (α : ℝ) (hα : IsAlgebraic ℚ α) (hα0 : α ≠ 0) :
    Irrational (α / (2 * Real.pi)) := 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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