AI 中文总结
本研究提出PCP定理的最简代数证明，通过编码论方法实现低次测试以克服关键瓶颈，给出PCP验证者及诚实证明者的完整伪代码与描述。
AI 中文摘要
我们给出了目前已知的PCP定理的最简单代数证明，仅涉及码级联、多项式插值和多项式乘法等要素。具体而言，我们证明了图的3着色问题存在一个多项式规模的证明，该证明可由仅投掷对数个硬币、查询证明中常数个比特的验证者进行验证。值得注意的是，我们的证明不涉及任何PCP组合；尤其在验证者的构造中，我们没有调用SAT或3着色等任何固定问题的NP完全性。本研究的主要创新点在于一种简洁的、编码论的单变量多项式编码方式，该方式使我们能够仅使用常数个查询比特来实现“低次测试”。借助作者（STOC 2026）与Goldreich（ECCC 2025）近期简化PCP证明的尝试中的见解，我们发现低次测试是将PCP验证者的先前代数构造转化为常数查询PCP的关键瓶颈。因此，通过克服这一瓶颈，我们利用基础且自包含的步骤得到了完整的PCP验证者。作为对所声称简单性的具体支撑，我们包含了PCP验证者的完整伪代码（基于有限域算术），以及完整的完备性（即“诚实”）证明者的完整描述（基于包含插值和求值的多元多项式算术），两者各约占一页篇幅。
英文摘要
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.
