weinstein_conjecture_3sphere

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Weinstein conjecture: Every Reeb vector field on a compact contact manifold has at least one closed orbit. Proved for the 3-sphere (Viterbo 1987), all compact contact 3-manifolds (Taubes 2007 via Seiberg-Witten theory). Open in higher dimensions.

Why this node was retired

The posted statement is

import Mathlib

theorem weinstein_conjecture_3sphere
    (alpha : EuclideanSpace ℝ (Fin 4) → EuclideanSpace ℝ (Fin 4))
    (hcontact : ∀ x : EuclideanSpace ℝ (Fin 4), ‖x‖ = 1 →
      ∑ i : Fin 4, alpha x i ≠ 0)
    (hreeb : ∀ x : EuclideanSpace ℝ (Fin 4), ‖x‖ = 1 →
      ∑ i : Fin 4, (alpha x i) ^ 2 = 1) :
    ∃ (gamma : ℝ → EuclideanSpace ℝ (Fin 4)) (T : ℝ),
      0 < T ∧ (∀ t, ‖gamma t‖ = 1) ∧
      gamma T = gamma 0 := by
  sorry

The curve is not required to be an integral curve of a Reeb field; the contact and Reeb hypotheses are unrelated scalar conditions.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Take a smooth contact 1-form α on S³ with α∧dα nowhere zero. Define its Reeb vector field R by α(R)=1 and dα(R,·)=0. Require a nonconstant smooth γ:ℝ→S³ with γ′(t)=R(γ(t)) and γ(t+T)=γ(t) for all t, for some T>0.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.