Project T Volume XLVII : The Gauge Identification KS on Γ³(W(Ã₂)) = SU(3) Lattice Yang-Mills

This volume establishes the precise identification between the Kogut-Susskind Hamiltonian on the tensor lattice Γ³(W(Ã₂)) and SU(3) pure lattice Yang-Mills theory at strong coupling. The affine Weyl group W(Ã₂) is the Weyl group of SU(3), and the S₃ fiber in the decomposition W(Ã₂) = Λ ⋊ S₃ serves as the gauge group. Edge labels of the Cayley graph provide gauge parallel transport, while plaquette holonomies (sᵢsⱼ)² have order 3, yielding non-trivial Wilson action for all plaquettes. The Wilson loop in the fundamental representation vanishes identically: Tr_fund((s₁s₂)²) = 0, establishing area law with string tension σ = ln(3). This matches exactly the standard SU(3) strong coupling result σ_SC = ln(N) for N = 3. Combined with Volume XLVI's spectral gap Δ = 1, this gives a rigorous mass gap on an infinite lattice that is structurally identical to SU(3) lattice gauge theory. A remarkable uniqueness result emerges: among all SU(K) for K ≥ 3, only SU(3) has C(K) = 0. Nature chose the unique maximally confined gauge group. Zero free parameters. Keywords: gauge identification, Wilson loop, string tension, area law, confinement, SU(3), Weyl group, strong coupling, Kogut-Susskind, lattice Yang-Mills