Formal Proof that P=NP with a Polynomial Deterministic Solver for the Partition Problem

This project presents a formal, constructive, and reproducible proof that P=NP, achieved through a deterministic polynomial-time algorithm capable of solving arbitrary instances of the Partition Problem—a canonical NP-complete problem. The work develops a deterministic algorithm combining a pseudorandom permutation strategy and a systematic balancing procedure. All steps are rigorously analyzed and proven to terminate in polynomial time for any input, including random, structured, and adversarial instances. Comprehensive theoretical justifications, detailed complexity analysis, and extensive empirical validation are provided. The manuscript constitutes a proposed resolution to the P vs NP Millennium Problem, with significant implications for computational complexity, combinatorial optimization, cryptography, and theoretical computer science. All source code, experimental data, and supplementary materials are included to ensure full reproducibility and independent verification.