Circle-loop phases agree exactly when winding agrees

Proved

WindingArithmeticDensePhase.circleLoopRealPhaseEqIffWinding

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

algebraic-topologyirrational-rotationnumber-theorywinding

For nonzero real algebraic $\alpha$ and based Circle loops $\gamma,\delta$ ,

e^{i\alpha\operatorname{wind}(\gamma)}=e^{i\alpha\operatorname{wind}(\delta)}\quad\Longleftrightarrow\quad\operatorname{wind}(\gamma)=\operatorname{wind}(\delta).

Winding is the canonical zero-based lift-endpoint integer.

Preamble

import Definitions.Def_WindingArithmeticDensePhase_CoreV1
import Definitions.Def_WindingDynamics_CoreV1

Formal statement

theorem WindingArithmeticDensePhase.circleLoopRealPhaseEqIffWinding
    (α : ℝ) (hα : IsAlgebraic ℚ α) (hα0 : α ≠ 0)
    (γ δ : WindingDynamics.CircleLoop) :
    WindingArithmeticDensePhase.realCirclePhase α
        (WindingDynamics.circleWinding γ) =
      WindingArithmeticDensePhase.realCirclePhase α
        (WindingDynamics.circleWinding δ) ↔
      WindingDynamics.circleWinding γ = WindingDynamics.circleWinding δ := 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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