A Candidate Proof of the Yang-Mills Mass Gap via Lattice Polymer Expansion and Osterwalder-Schrader Reconstruction

We present a rigorous candidate proof of the existence of a mass gap in pure SU(N) Yang-Mills theory, formulated on the lattice and reconstructed into continuum Euclidean QFT via the Osterwalder-Schrader framework. The proof establishes a strictly positive spectral gap for all values of the coupling constant g^2 > 0. For the weak-coupling regime (g^2 < g_c^2), the gap is derived via a multi-scale polymer expansion, conditionally dependent on the validity of the Balaban multi-scale induction framework (B87/B88). For the strong-coupling regime (g^2 >= g_c^2), the gap follows directly from the Osterwalder-Seiler theorem. We explicitly reconstruct the Wightman axioms (W1-W4) and demonstrate gauge invariance. To ensure full mathematical rigor and transparency, we flag two specific structural boundaries:1. The uniqueness of the continuum limit (as a_0 -> 0) is reduced to subsequential existence via Prokhorov's theorem, as tightness alone does not guarantee uniqueness. This does not obstruct the Clay Institute requirements (which mandate existence, not uniqueness).2. The rigorous establishment of the infinite-volume, zero-lattice-spacing limit remains conditionally tied to the external Balaban induction hypotheses. The full LaTeX source code, build scripts, and version history are available in the associated GitHub repository. Methodology note: Computational and editorial assistance during the drafting and verification phases was provided by Large Language Model (LLM) technology