A Constructive Proof That P ≠ NP

Informed by logical principles, this study initiates its inquiry from a combinatorial problem. First, it demonstrates that a linear equation with 0-1 variables inherently represents a set. Building on this foundation, a 0-1 integer programming model composed of two such linear equations is constructed. From the perspective of set intersection, it is proven that this specific 0-1 integer programming admits no polynomial-time solution. Consequently, the general form of 0-1 integer programming defined by CX=d is not solvable in polynomial time. Given that the 0-1 integer programming of the form CX=d is a recognized NP-complete problem, this result implies that all NP-complete problems lack polynomial-time solutions, thereby establishing P≠NP. This work offers novel insights into the P vs. NP problem, advancing the theoretical understanding of computational complexity.