Hybrid Complexity Growth in NP-Complete Problems: Analytical Evidence for P ≠ NP via the Begüm Function

The present study investigates the cumulative complexity growth in canonical NP-complete problems using the Begüm function, a hybrid infinite series recently introduced in the literature [4]. Analytical evaluation demonstrates that, for Boolean Satisfiability (SAT), Hamiltonian Path, and Travelling Salesman Problem (TSP), the value of the Begüm function increases super-polynomially as a function of problem size. Numerical simulations further confirm these theoretical findings, demonstrating that the Begüm function’s value rises exponentially or factorially with n, in contrast to polynomial-time behavior. The results provide supporting analytical and computational evidence for the hypothesis that P ≠ NP, without claiming a formal proof. The Begüm function is proposed as a novel tool for hybrid modeling of complexity in theoretical computer science.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl