Full-turn resonance collapses the integer phase orbit

Proved

WindingArithmeticDensePhase.resonantFullTurnControl

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

algebraic-topologyirrational-rotationnumber-theorywinding

At the resonant angle $\alpha=2\pi$ , every integer phase equals $1$ and the phase map is not injective. This is the explicit negative control outside the algebraic nonresonant regime.

Preamble

import Definitions.Def_WindingArithmeticDensePhase_CoreV1

Formal statement

theorem WindingArithmeticDensePhase.resonantFullTurnControl :
    (∀ n : ℤ,
      WindingArithmeticDensePhase.realCirclePhase (2 * Real.pi) n = 1) ∧
      ¬ Function.Injective
        (WindingArithmeticDensePhase.realCirclePhase (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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