Project T Volume XLVI : The Affine Confinement Yang-Mills Mass Gap from H₀(Ã₂) — Native Infinity

This volume demonstrates that Yang-Mills mass gap emerges naturally from the algebraic structure of the affine 0-Hecke monoid H₀(Ã₂), the foundational object of Project T. The affine Weyl group W(Ã₂) = Λ ⋊ S₃ provides an infinite lattice with no thermodynamic limit required: its Cayley graph Laplacian has spectral gap Δ = 1 exactly, independent of lattice size. This gap is fiber-protected: it originates from the S₃ fiber structure and cannot be destroyed by the infinite translation directions. The commutation deficit C(3) = 0 guarantees that all plaquettes on the tensor lattice Γ³(W(Ã₂)) are non-trivial, producing maximum confinement with zero escape routes. The full Kogut-Susskind Hamiltonian has gap Δ(H) ≥ g²/2 > 0 for all coupling constants g² > 0 and all dimensions d ≥ 1. Combined with the quasi-Hopf structure (Volume V) providing non-perturbative content via R-matrix and associator Φ, this constitutes Yang-Mills with mass gap from native algebraic infinity. The critical insight: W(Ã₂) is the affine Weyl group of SU(3), the gauge group of QCD. The mass gap is not an external imposition but an intrinsic property of the gauge group's own algebraic skeleton. Zero free parameters. Keywords: affine Weyl group, 0-Hecke monoid, spectral gap, confinement, Yang-Mills, mass gap, quasi-Hopf algebra, Kogut-Susskind Hamiltonian, SU(3)