Result 105, Theoretical computer science

Perfect completeness for 2-to-1 games

Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational δ∈(0,1)\delta\in(0,1), it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map.

Lean formalization Proof

The bigger picture

Why it matters

A labeling puzzle can be perfectly solvable yet computationally hard to distinguish from one where almost every constraint must fail. The manuscript reports this sharp gap for 2-to-1 games, a problem about satisfying tightly structured constraints.

What changes?

A 2-to-1 game assigns labels to variables on two sides; each constraint maps left labels to right labels, with exactly two preimages per right label. Its value is the largest fraction of constraints satisfied. The manuscript claims NP-hardness of distinguishing value one from value at most delta, for every fixed rational delta strictly between zero and one. The finite label alphabets depend only on delta. Instances are explicit, unweighted multisets of constraints, allowing repetitions.

What does that help mathematicians do?

Perfect completeness means the hard distinction starts with instances where every constraint can be satisfied, not merely almost every constraint. Unless P equals NP, the claimed result rules out a polynomial-time test that always separates these fully satisfiable instances from those with value at most the chosen delta. Researchers therefore cannot rely on exact feasibility or the two-preimage structure to make this distinction efficiently.

Are there practical applications?

For optimization, the result identifies a limit on worst-case guarantees for maximizing satisfied constraints. It does not supply a faster solver or establish difficulty on typical real-world inputs. Its immediate value is foundational: showing that even fully feasible instances with this restricted constraint structure can be hard to distinguish from severely unsatisfiable ones.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Perfect completeness for 2-to-1 games

September 23, 2026 53 pages Main result formalized in Lean

We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational δ∈(0,1)\delta\in(0,1), it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Perfect-completeness-for-2-to-1-games-September-23-2026,
  author = {{OpenAI}},
  title = {{Perfect completeness for 2-to-1 games}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf}{OAI:Perfect-completeness-for-2-to-1-games-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/105.md.

Perfect completeness for 2-to-1 games

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result establishes perfect-completeness hardness for 2-to-1 games. For every fixed rational 0<δ<10<\delta<1, a deterministic polynomial-time reduction from binary 3SAT produces a nonempty unweighted game of value exactly 11 on satisfiable inputs and at most δ\delta on unsatisfiable inputs. Each constraint projection has exactly two preimages for each output label. The alphabet depends only on δ\delta, and the runtime is measured in the original input bit length.

Comparator links

Result Comparator statement
Perfect-completeness reduction for 2-to-1 games PerfectCompleteness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.