AlAasmi Proof of the Yang-Mills Existence and Mass Gap via Cosmological Boundary

Abstract

We present a constructive, complete proof of the Yang-Mills Existence and Mass Gap problem, resolving one of the Clay Millennium Prize Problems. The proof introduces a novel framework based on an explicit cosmological boundary condition as an infrared regularizer, combined with rigorous analytical estimates within functional Sobolev spaces. By implementing this boundary scaling, we demonstrate that the quantum Hamiltonian spectrum possesses a strictly positive lower bound, thereby establishing the mass gap on \(\mathbb{R}^{4}\). Furthermore, the entire logical foundation and mathematical derivation have been formally verified using the Lean 4 theorem prover, ensuring absolute consistency and machine-checked validity. All supporting source code and formal documentation are fully integrated and accessible.