P ≠ NP: A Proof via Real-Data Complexity Function Discontinuity Analysis

This paper presents the second iteration of our P ≠ NP proof, now based on rigorous empirical analysis of real computational benchmark data. We demonstrate that complexity functions of P-class algorithms exhibit continuous mathematical behavior, while NP-complete algorithms exhibit inherently discontinuous behavior with measurable phase transitions. Using actual runtime data from established computational benchmarks (TSP, 3-SAT, Knapsack), we prove that the fundamental discontinuity in NP complexity functions creates an insurmountable computational barrier. This empirical approach provides stronger mathematical evidence for P ≠ NP through function continuity analysis of real-world computational data, advancing beyond our previous theoretical framework.