A Formal and Spectral view of P ≠ NP via Decision Tree Lower Bounds and Coq Verification

This work provides a rigorous and formally verified proof that P ≠ NP by combining three independent methods: decision tree complexity, cyclic XOR-SAT hardness, and spectral analysis. We show that any algorithm solving SAT must explore an exponential number of paths, rendering polynomial-time solutions impossible. A new class of XOR-SAT problems is introduced that systematically resists heuristic and probabilistic solvers. Spectral graph theory is used to demonstrate that the Laplacian eigenvalues of NP problems diverge exponentially, unlike those of P problems. The proof is implemented and formally verified using the Coq proof assistant, ensuring mathematical rigor and logical completeness. Accompanying simulations confirm the expected gap in complexity between NP and P instances. This result resolves one of the most fundamental open problems in theoretical computer science. Included are two documents:– A human-readable exposition of the argument– A machine-verifiable Coq formalization of the complete proof本研究结合决策树复杂度（decision tree complexity）、循环异或可满足性难度（cyclic XOR-SAT hardness）与谱分析（spectral analysis）三种独立方法，给出了P≠NP的严格形式化验证后的证明。我们证明，任何求解布尔可满足性问题（SAT）的算法都必须探索指数级数量的路径，使得多项式时间求解方案无从实现。本文引入了一类全新的异或可满足性问题（XOR-SAT），这类问题能够系统性地抵御启发式与概率型求解器。研究借助谱图理论（spectral graph theory）证明，NP问题的拉普拉斯特征值（Laplacian eigenvalues）呈指数级发散，而P问题的拉普拉斯特征值则无此特性。 该证明通过Coq证明助手（Coq proof assistant）实现并完成形式化验证，保障了数学严谨性与逻辑完备性。配套的仿真实验验证了NP与P类问题之间预期的复杂度差距。本研究结果解决了理论计算机科学领域最核心的未解难题之一。 本次公开的数据集包含两份文档：一份为面向人类读者的论证阐述文档，另一份为可被机器验证的完整证明的Coq形式化文档。