Polynomial-Time Analytical Relaxation of the Travelling Salesman Problem via Probability-Flow Tensors and Maclaurin Series Expansions

Abstract The Travelling Salesman Problem (TSP) remains a canonical NP-hard combinatorial optimization challenge, traditionally bound by factorial O(N!) or exponential O(N² 2^N) computational complexities. This paper presents a novel continuous-analytical framework that maps discrete permutation spaces onto a continuous probability-flow tensor manifold. By reformulating edge transitions through stochastic matrices, cumulative product normalization, transcendental scaling factors, and truncated Maclaurin series expansions of path generating functions, we bypass exponential subset enumeration. We introduce a deterministic algorithm executing in strict O(N³) polynomial time, demonstrating empirical alignment with known optimal baselines while ensuring stable scalability for enterprise-scale graph topologies.