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 via a deterministic polynomial-time algorithm based on a graph-based computation framework. We introduce a model where edges correspond to deterministic Turing machine transitions. Because these execution steps form paths sharing the initial edge, the total number of edges is polynomially bounded due to overlapping edges across the entire certificate space. By shifting the verification focus from searching over exponential certificates to incremental edge extensions through verification of their validity,we reduce the complexity of the verification process to polynomial size. A key feature of our approach is that each extension step enforces global consistency via a local in-feasibility trimming tool. This mechanism systematically preserves consistent NP paths to the edge under verification, ensuring the graph remains globally feasible at every stage without explicit enumeration of certificates. Both extension and trimming steps are strictly bounded by a polynomial in the input size. Consequently, our construction decides NP problems in deterministic polynomial time, providing a direct resolution to the P versus NP question.

翻译：P与NP问题探讨的是：是否所有能在多项式时间内验证的语言，也都能在确定性多项式时间内判定？本文通过一种基于图计算框架的确定性多项式时间算法，给出了P = NP的构造性证明。我们引入了一个模型，其中边对应于确定性图灵机的转移。由于这些执行步骤形成了共享初始边的路径，且整个证书空间中的边存在重叠，因此边的总数受多项式界限约束。通过将验证焦点从搜索指数级证书转移到通过验证其有效性来逐步扩展边，我们将验证过程的复杂度降低至多项式规模。我们方法的一个关键特征是：每个扩展步骤都通过局部不可行性剪枝工具来强制执行全局一致性。该机制系统性地保留与待验证边一致的多项式NP路径，确保图在每一阶段都保持全局可行性，而无需显式枚举证书。扩展与剪枝步骤均严格受输入规模的多项式界限约束。因此，我们的构造可在确定性多项式时间内判定NP问题，为P与NP问题提供了直接解答。