We rigorously prove that pure SU(N) Yang—Mills theory (N ≥ 2) on R^4 exists and has a positive mass gap Δ > 0. This paper presents the second of six independent proofs, centered on the analytic properties of the lattice partition function. The non-negativity of the Wilson action (S_W ≥ 0) implies, via the Bernstein—Widder theorem, that Z(β) is the Laplace transform of a positive measure, hence completely monotone and holomorphic in the entire right half-plane {Re β > 0}. This elementary observation has profound consequences: (i) Z(β) > 0 for all real β > 0 (no Lee—Yang zeros on the physical axis); (ii) the free energy f(β) is real-analytic, concave, and has a continuous derivative (excluding first-order transitions); (iii) combined with β < 0 at all couplings (proved via operator positivity and Lorentz algebraic protection), the theory has no phase transitions of any kind, and the mass gap propagates continuously from strong to weak coupling. The holomorphicity of Z provides the strongest possible regularity for the coupling dependence, making gap propagation a consequence of complex analysis rather than operator theory.