beraha_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.

Beraha's conjecture: The Beraha numbers Bₙ = 2 + 2cos(2π/n) are limits of zeros of chromatic polynomials of planar graphs. Connected to the 4-color theorem and quantum groups. B₅ = golden ratio + 1; B₇ ≈ 3.25. Open in full generality.

Why this node was retired

The posted statement is

import Mathlib

theorem beraha_conjecture (k : ℕ) (hk : 2 ≤ k) (n : ℕ) (hn : 3 ≤ n) :
    ∀ (G : SimpleGraph (Fin n)) (_ : DecidableRel G.Adj),
      G.Connected →
      ¬(∀ (col : Fin n → Fin 4), ∀ v w : Fin n, G.Adj v w → col v ≠ col w) →
      ∃ (p : Polynomial ℝ),
        p.eval (2 + 2 * Real.cos (2 * Real.pi / k)) = 0 := by
  sorry

The polynomial witness is arbitrary, need not be nonzero or chromatic, and no limiting sequence is required. The hypothesis also negates a universal coloring claim rather than existence of a coloring.

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

Provisional statement matching the prose: for each intended Beraha number B_k, there is a sequence of finite planar graphs G_j and zeros z_j of their chromatic polynomials with z_j→B_k. Specify the intended k range and graph restrictions from an authoritative version of Beraha’s conjecture; those cannot be recovered reliably from this target. Merely requiring p≠0 is insufficient because p=X−B_k also works.

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.