unique_games_conjecture

⚠️ Retired — specification defect

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

The unique games conjecture (Khot 2002): For every ε, δ > 0 there exists k such that it is NP-hard to distinguish (1−ε)-satisfiable from (1−δ)-satisfiable unique 2-CSP instances with k labels. Neither proved nor disproved.

Why this node was retired

The posted statement is

import Mathlib

theorem unique_games_conjecture :
    ∀ eps delta : ℝ, 0 < eps → 0 < delta →
    ∃ (k : ℕ), 1 ≤ k ∧
    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj]
      (constraints : Fin n → Fin n → ℕ → ℕ)
      (assignment : Fin n → ℕ),
      (1 - eps) * G.edgeFinset.card ≤
        {p : Fin n × Fin n | G.Adj p.1 p.2 ∧
          constraints p.1 p.2 (assignment p.1) = assignment p.2 ∧
          assignment p.1 < k ∧ assignment p.2 < k}.ncard →
      ∃ (assign' : Fin n → ℕ) (_ : ∀ v, assign' v < k),
        (1 - delta) * G.edgeFinset.card ≤
          {p : Fin n × Fin n | G.Adj p.1 p.2 ∧
            constraints p.1 p.2 (assign' p.1) = assign' p.2}.ncard := by
  sorry

Constraints are arbitrary natural-valued functions, ε can exceed1, and an existence claim replaces a complexity-hardness gap.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For every ε,δ∈(0,1), seek a finite alphabet size k for which distinguishing optimum≥1−ε from optimum≤δ is NP-hard for edge constraints that are permutations of Fin k.

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.