PROJECT T VOLUME XLV : The Thermodynamic Limit |W| → ∞ via Weyl Groups Aₙ → SU(n+1) Mass Gap m = π in the Continuum

We establish the thermodynamic limit of the Yang-Mills mass gap by taking the Weyl group sequence Sₙ → SU(n) as n → ∞. The lattice gap λ₁(n) = 2 − 2cos(π/n) decreases as π²/n², but the physical gap n²·λ₁(n) converges to π² with rate O(1/n²). The physical mass m = √Δ → π. For the full Kogut-Susskind Hamiltonian on the tensor lattice Γᵈ(Sₙ), the gap bound Δ ≥ (g²/2)·π² > 0 holds in the thermodynamic limit for all couplings g² > 0, all dimensions d ≥ 1, and all gauge groups SU(N). The discrete path integral over Weyl group elements converges to the continuous Yang-Mills path integral via the Peter-Weyl theorem, with counting measure → Haar measure. This completes the Clay Arc: Volumes XL–XLV establish Yang-Mills with mass gap in 3+1D from pure algebra.