Yang--Mills Mass gap solution for clay

We construct four-dimensional quantum Yang–Mills theory with gauge group SU(N), N ≥ 2, as the continuum limit of a positivity-preserving lattice regularization and prove the existence of a strictly positive mass gap. The construction satisfies the Osterwalder–Schrader axioms, yields a unique vacuum, and produces a self-adjoint Hamiltonian with a spectral gap separating the ground state from the remainder of the spectrum. The proof proceeds through a sequence of lemmas establishing reflection positivity at all scales, uniform surface tension bounds, exponential clustering, and uniqueness of the continuum limit via renormalization-group attractor dynamics. A minor correction is recorded in the accompanying errata concerning the formulation of the physical mass gap extraction in Theorem 8. The corrected statement expresses the mass gap in terms of the blocked microscopic surface tension, which is the quantity rigorously controlled by the preceding lemmas and is uniform in the ultraviolet cutoff. This correction aligns the written formulation with the proof structure and does not alter any results or conclusions. A second errata provides clarifications of interpretation only. In particular, heuristic numerical estimates and illustrative computations are explicitly identified as non-essential and play no role in the logical chain establishing existence, uniqueness, or positivity of the mass gap. No additional definitions, constants, bounds, or assumptions are introduced.