The Unique Games Conjecture

Theorem 1.1: for every fixed $\varepsilon,\delta\in(0,1/2)$ there are $s\ge1$ and a deterministic polynomial-time reduction from 3SAT to explicit unweighted, simple bipartite Unique Games over $K=\mathbb F_2^s$ whose constraints are translations $a(v)=a(u)+c_e$ , with value at least $1-\varepsilon$ on satisfiable formulas and at most $\delta$ on unsatisfiable ones. This is the conjecture in its usual edge-value form. The route keeps the Barak-Kothari-Steurer degree-two matrix shortcode and its Khot-Minzer-Safra inverse theorem and changes the noise through a nonlinear map to push completeness to $1-\varepsilon$ . Section 7 transfers known UG-hardness reductions (cut, covering, CSP, ordering, deletion, clustering) to NP-hardness; those reductions are credited to their authors. Companion manuscripts give direct NP-hardness proofs for Max-Cut beyond $\alpha_{GW}$ , Vertex Cover below factor $2$ , and every constant factor for Min-UnCut and directed feedback vertex set.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Unique Games Conjecture · Unique Games Conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
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Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The Unique Games Conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript, read against Khot's conjecture as cited (Khot 2010 survey, Conjecture 2.5; translation form via KKMO Corollary 13). The proof was not refereed. Lean: formalization.yaml lists OAI.UniqueGamesTheorem.theorem11 (lean/OAI/Computability/UniqueGames/Theorem.lean, comparator ComparatorChallenges/UniqueGamesTheorem.lean). Its statement was read: for all real 0 < eps, delta < 1/2 there is a reduction from binary-encoded 3SAT to nonempty simple bipartite translation Unique Games over a fixed alphabet identified with F_2^s, computable in polynomial time by a Mathlib TM2 machine, with completeness at least 1-eps and soundness at most delta. That is the headline claim, with no extra hypotheses; permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The Max-Cut and Vertex Cover companions also have Lean main results (OAI.OptimalMaxCut.main, OAI.VertexCover.every_fixed_factor_below_two) whose statements match their headlines; also not rebuilt. The paper names Hastad's parity hardness and the Khot-Minzer-Safra Grassmann expansion theorem as external inputs; since theorem11 carries no hypotheses, a passing comparator check would mean those inputs are formalized too, which was not audited here.

Source ↗

The Unique Games Conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

A Direct Proof of Optimal Max-Cut Hardness

paper · pending