Spectral Gap and Thermodynamic Limit for SU(N) Lattice Yang-Mills Theory via Log-Sobolev Inequalities and Complete Analyticity

We present two parallel results for SU(N) pure gauge lattice Yang-Mills in four Euclidean dimensions, at fixed lattice spacing eta > 0 and weak coupling g0 <= g*, both conditional on a single shared input, the Dobrushin-Shlosman complete analyticity condition (H-CA): (A) a log-Sobolev inequality Ent(f^2) <= (2/rho) E(f,f) with rho > 0 independent of L, via Cesi's quasi-factorisation seeded by a Bakry-Emery/Holley-Stroock local LSI; and (B) a spectral gap m_gap >= m0 > 0 for the Osterwalder-Seiler Hamiltonian, via Dobrushin clustering and reflection positivity. The two outputs are logically parallel; neither implies the other here (the bridge lemma remains open, Remark 6.3). Version 2 corrects the status of the shared input: v1 declared (H-CA) verified from Balaban's infrastructure; per the audited chain (2602.0053/0054 v2) that route is conditional on (H-DOB-blk)+(H-P0) and valid only in the volume window L <= exp(C/g0^2). Accordingly all results are stated in two regimes: windowed (audit-backed conditional) and all-volume (under the strictly stronger bare hypothesis (H-CA)_infty, required for the thermodynamic limit, Theorem C). What is unconditional and machine-verified: the curvature computation Ric_SU(N) = N/4, the Holley-Stroock block-seed arithmetic, the variance/entropy decompositions and the failure of pointwise inheritance, the Dobrushin contraction machinery and its window arithmetic, the clustering => transfer-gap lemma on explicit operators, and the convergence of Cesi's geometric factor. All bounds remain explicit in N, g0, eta.