A quasilinear PCP theorem for PPAD

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Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length \(N(\log N)^{O(1)}\) 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.

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The paper states

Theorem 1.3 (Quasilinear PCP for PPAD)

In The PCP-for-PPAD conjecture: a quasilinear reduction, 25 September 2026

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There are fixed rational constants

0<\eps<\tfrac1{10},\qquad 0<\delta<1,

a fixed positive integer A, a fixed nonnegative integer b, a fixed polynomial p with nonnegative integer coefficients, and deterministic algorithms R,D with the following properties.

(i)
Every generalized circuit in Definition 1.2 has an acceptable rational assignment of encoding length at most p(M).
(ii)
For every End-of-Line instance I of length N, the output R(I) is a generalized circuit whose complete encoding length satisfies
M\leq A N\bigl\lceil\log_2(N+2)\bigr\rceil^b.
(iii)
The construction R runs in time polynomial in N, with one fixed polynomial bound.
(iv)
For every acceptable assignment x of R(I), the algorithm D(I,x) returns a solution of I in time polynomial in N plus the encoding length of x, with one fixed polynomial bound.

All constants, polynomials, and algorithms are chosen once. They use no advice depending on the input length and no randomized construction.

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The PCP-for-PPAD conjecture: a quasilinear reduction

25 September 2026, current edition.

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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 \(N(\log N)^{O(1)}\). 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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34 references in The PCP-for-PPAD conjecture: a quasilinear reduction

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