A Proof of Impossibility of a Polynomial-Time Solution of the Traveling Salesman Problem on the Self-Referral Structure of the Global Optimal Tour

This paper presents an information-theoretic proof demonstrating that the general Traveling Salesman Problem (TSP) cannot be solved in polynomial time by any Deterministic Turing Machine ($P \neq NP$). By formalizing an adversarial worst-case model where edge weights are mutually independent and identically distributed random variables, we establish that local evaluations yield zero mutual information regarding global optimality, precluding any macroscopic algebraic shortcuts. Furthermore, we prove that even optimal dynamic programming strategies like the Bellman-Held-Karp algorithm are structurally bounded by an exponential state space $\Omega(2^n)$ because independent sub-tours cannot be losslessly compressed or factored by any deterministic state transition function.