A Simple Algebraic Proof of the PCP Theorem

Prashanth Amireddy; Amik Raj Behera; Srikanth Srinivasan; Madhu Sudan; Sophus Valentin Willumsgaard

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arXiv · 2608.17429

A Simple Algebraic Proof of the PCP Theorem

Prashanth Amireddy·Amik Raj Behera·Srikanth Srinivasan·Madhu Sudan·Sophus Valentin Willumsgaard

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Abstract

We give the simplest known algebraic proof of the PCP theorem, involving only ingredients like code concatenation, polynomial interpolation, and polynomial multiplication. Specifically, we prove that graph 3-coloring has a polynomial-sized proof that can be verified by a verifier tossing logarithmically many coins and querying a constant number of bits in the proof. In particular, our proof does not involve any PCP compositions; notably, it does not invoke the NP-completeness of any fixed problem, such as SAT or 3-coloring, in the construction of the verifier. The main innovation in our work is a clean, coding theoretic, way to encode univariate polynomials that allows us to implement ``low-degree testing'' using just a constant number of bits of queries. Insights from recent attempts to simplify the PCP proof by the authors (STOC 2026) and Goldreich (ECCC 2025) allow us to observe that low-degree was the key bottleneck in converting previous algebraic constructions of the PCP verifier into a constant query PCP. Thus, by overcoming this bottleneck, we get the full PCP verifier using elementary and self-contained steps. As concrete support for the claimed simplicity, we include the full pseudocode of the PCP verifier, assuming finite field arithmetic, and a full description of the completeness (aka ``honest'') prover, assuming multivariate polynomial arithmetic including interpolation and evaluation, that fit in about a page each.

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Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, Madhu Sudan, Sophus Valentin Willumsgaard. 2026-08-18. A Simple Algebraic Proof of the PCP Theorem. https://arxiv.org/abs/2608.17429

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