Result 136, Theoretical computer science

A quasilinear PCP theorem for PPAD

Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length N(log⁡N)O(1)N(\log N)^{O(1)} such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.

Proof

The bigger picture

Why it matters

The manuscript reports a way to encode endpoint-search problems so that even an approximately correct assignment with some failed checks still reveals an endpoint. The encoding increases size only by a fixed power of the logarithm of the input size.

What changes?

The claimed reduction runs deterministically in polynomial time. It transforms End-of-Line, an endpoint search in compactly described directed paths, into a generalized circuit: a collection of numerical gate constraints. For input binary length N, the circuit's total binary length is at most N times a fixed power of log N. Fixed positive rational constants set the allowed numerical error at each satisfied gate and the fraction of gates permitted to fail. Neither constant depends on N.

What does that help mathematicians do?

Any rational assignment with polynomial encoding length meeting these tolerances yields an endpoint in polynomial time, regardless of which gates fail. Such assignments are guaranteed to exist with one fixed polynomial bound on encoding length, so the claim does not rely on impossible numerical witnesses. This gives researchers a robust representation of endpoint search: violations scattered through a fixed fraction of local constraints do not prevent recovery of a solution.

Are there practical applications?

The immediate value is foundational, in understanding PPAD search problems through robust local verification. The reported quasilinear size bound keeps this representation close to the original input size while tolerating numerical error and failed constraints. It does not by itself provide a fast algorithm for finding the required assignment or solving the original search problem.

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

The PCP-for-PPAD conjecture: a quasilinear reduction

September 25, 2026 147 pages

We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko, Papadimitriou, and Rubinstein. There are fixed positive rational constants ε and δ and a deterministic polynomial-time reduction that transforms an End-of-Line instance of binary length N into a generalized circuit of total binary length N(log⁡N)O(1)N(\log N)^{O(1)}. From any rational assignment of polynomial encoding length that ε-satisfies all but a δ fraction of the gates, a solution to the original End-of-Line instance can be recovered in polynomial time, regardless of which gates fail. Such assignments always exist, with one fixed polynomial bound on their encoding length.

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Cite (BibTeX) 
@misc{OAI:The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026,
  author = {{OpenAI}},
  title = {{The PCP-for-PPAD conjecture: a quasilinear reduction}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf}{OAI:The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026}},
  year = {2026}
}

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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.