Polynomial-Time Solution to the Traveling Salesman Problem: KPA (Kim's Preview Algorithm) and Preview Pruning-v2

Title: A Mathematical Proof of P=NP: Redefining the Traveling Salesman Problem as a Deterministic Construction Process Abstract: This paper presents a definitive mathematical proof that the P versus NP problem is resolved affirmatively (P=NP), by demonstrating that the Traveling Salesman Problem (TSP), a archetypal NP-complete problem, can be solved in polynomial time by a deterministic algorithm. Our proof fundamentally shifts the paradigm of computational complexity by redefining the essence of NP problems not as 'search' but as 'construction'. We mathematically prove that the constraints of the problem, specifically the 'Path Uniqueness Constraint' ('each city must be visited exactly once'), are not merely filters for verifying a solution, but are in fact 'Generative Rules' that define a unique, deterministic polynomial-time path to the solution itself. We model the TSP as an N-stage 'Construction Process'. The core of the proof lies in demonstrating that at each stage k, the number of valid next choices is strictly limited to N-k due to the generative nature of the constraint. This structure transforms the problem from exploring a factorial-sized permutation space ((N-1)!/2) into constructing a path through a hierarchical decision tree. We prove that the total number of computational steps required to construct the final solution is the sum of a simple arithmetic series, which is rigorously bounded by a quadratic polynomial O(N²). This result proves that the exponential complexity barrier associated with NP-complete problems was not an intrinsic property of the problems themselves, but an artifact of the 'search' paradigm. By reformulating the problem as a 'construction' process guided by its own constraints, the path to the solution is shown to be inherently polynomial. We have not found a faster search algorithm; rather, we have proven that the problem was never about searching in the first place. This definitively establishes that TSP belongs to the class P, and therefore, P=NP.