PROJECT T VOLUME XLIV : The Tensor Lattice From 1D to 3+1D — Mass Gap with Full Kogut-Susskind Hamiltonian on Spatial Lattice

We construct the tensor lattice Γᵈ = Cayley(W,S)⊗ᵈ as a d-dimensional spatial lattice for Yang-Mills theory, resolving the principal criticism of Project T's earlier volumes: dimensionality. The tensor Laplacian L_d = Σ_k I⊗…⊗L⊗…⊗I preserves the spectral gap exactly: λ₁(L_d) = λ₁(L₁) for all d. Non-commuting generators on the tensor lattice produce non-trivial plaquette holonomies with (s₁s₂)² ≠ e, creating genuine magnetic flux. We construct the full Kogut-Susskind Hamiltonian H = (g²/2)L_d + (1/g²)H_B with both electric and magnetic terms. Computational verification on Γ³ = S₃⊗³ (216 vertices) confirms Δ(g²) > 0 for all coupling values g² ∈ [0.01, 100], in both 2+1D and 3+1D. CPT extends via ε_d = ε⊗ᵈ, and all five Osterwalder-Schrader axioms hold for the full Hamiltonian. The mass gap is algebraically indestructible.