The P versus NP problem asks whether every language verifiable in polynomial time can also be decided in deterministic polynomial time. In this paper, we present a constructive proof that P = NP by introducing a universal, graph-based deterministic framework applicable to all NP problems without requiring reduction to an NP-complete problem. We model computational transitions as edges within a unified graph structure, where edges correspond to the steps of a deterministic verifier Turing machine for all possible certificates. Due to the overlap of edges among computation paths, the total cardinality of the edge set remains polynomially bounded. A key feature of our approach is that each extension step enforces global consistency via a local infeasibility trimming tool. This mechanism systematically preserves valid NP paths that lead to the target edge under polynomial verification, ensuring the graph remains globally feasible at every stage without explicit enumeration. This represents a paradigm shift from searching over exponential certificates to the incremental extension of verified edges. Since our construction decides NP problems in deterministic polynomial time, it provides a direct resolution to the P versus NP question.

翻译：P与NP问题询问：是否每个能在多项式时间内验证的语言，也能在确定性多项式时间内判定。本文通过引入一种通用的、基于图的确定性框架，提出P=NP的构造性证明，该框架适用于所有NP问题，且无需规约至NP完全问题。我们将计算转移建模为统一图结构中的边，其中边对应于确定性验证图灵机针对所有可能证书所执行的步骤。由于计算路径间存在边重叠，边集的总基数保持多项式有界。本方法的一个关键特征在于：每个扩展步骤通过局部不可行性修剪工具强制全局一致性。该机制系统性地保留在多项式验证下通向目标边的有效NP路径，确保图在每一阶段无需显式枚举即可保持全局可行性。这标志着从搜索指数级证书到增量扩展已验证边的范式转变。由于我们的构造可在确定性多项式时间内判定NP问题，因此为P与NP问题提供了直接解答。