andrews_curtis_conjecture

⚠️ 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.

Andrews–Curtis conjecture (1965): Any balanced presentation of the trivial group can be simplified to empty using AC-moves. No counterexample known.

Why this node was retired

The posted statement is

import Mathlib

theorem andrews_curtis_conjecture (n : ℕ) (hn : 1 ≤ n)
    (rels : Fin n → FreeGroup (Fin n))
    (htrivial : ∀ (G : Type*) [Group G] (phi : FreeGroup (Fin n) →* G),
      (∀ i, phi (rels i) = 1) → Function.Surjective phi → Subsingleton G) :
    ∃ (m : ℕ) (seq : Fin (m + 1) → Fin n → FreeGroup (Fin n)),
      seq 0 = rels ∧ seq ⟨m, Nat.lt_succ_self m⟩ = fun _ => 1 := by
  sorry

The sequence is constrained only at its two endpoints; allowed Andrews–Curtis moves are absent.

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

For a balanced presentation ⟨x₁,…,xₙ | r₁,…,rₙ⟩ of the trivial group, conjecture a finite sequence of balanced relator tuples from (r₁,…,rₙ) to the standard generator tuple (x₁,…,xₙ), with every step inversion of a relator, multiplication by another relator, or conjugation of one relator (and optional permutations as part of the chosen equivalent convention). Do not use the all-identity tuple as the terminal presentation.

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.