The P versus NP problem asks whether every problem in NP, whose membership can be verified in polynomial time given a suitable certificate, can be decided by a deterministic Turing machine in polynomial time. In this paper, we present a proof that P = NP by constructing a deterministic polynomial-time algorithm for NP problems based on a graph-based computation framework. We introduce a structured computation model in which the transitions of a deterministic Turing machine are incrementally realized in the corresponding computation graph via edge extensions. Each extension step enforces a local feasibility condition that preserves consistency with valid NP verification paths across all possible certificates, ensuring that the maintained computation graph remains feasible at every stage. The total number of extension steps is polynomially bounded in the input size, and each step can be verified in polynomial time. As a result, the entire graph construction process runs in deterministic polynomial time and decides NP problems without enumerating certificates. This provides a direct and constructive resolution of the P versus NP question. Our result has significant implications for cryptography, combinatorial optimization, and artificial intelligence, where NP-complete problems play a central role.

翻译：P与NP问题探讨的是：对于NP类中的每个问题，若存在合适的证明时其成员资格可在多项式时间内验证，那么是否也能在确定性图灵机上于多项式时间内判定。本文通过构建一种基于图计算框架的确定性多项式时间算法来处理NP问题，从而证明了P = NP。我们引入了一种结构化计算模型，其中确定性图灵机的状态转移通过边扩展的方式逐步实现在对应的计算图中。每个扩展步骤强制执行一个局部可行性条件，该条件确保在所有可能的证明下均与有效的NP验证路径保持一致，从而保证所维护的计算图在每一阶段均保持可行性。扩展步骤的总数在输入规模上是多项式有界的，且每一步均可在多项式时间内验证。因此，整个图构建过程以确定性多项式时间运行，并能在不枚举证明的情况下判定NP问题。这为P与NP问题提供了一个直接且构造性的解答。我们的结果对密码学、组合优化及人工智能等领域具有重要影响，其中NP完全问题扮演着核心角色。