ALGEBRAIC RESOLUTION Volume I : Quantum Lattice Confinement From the 0-Hecke Monoid to the Mass Gap

This volume establishes the foundational results connecting the 0-Hecke monoid H₀(Ã₂) to quantum confinement through a chain of purely algebraic theorems. Starting from the affine Coxeter group W(Ã₂) with three generators {a, b, c} arranged in a triangle, we derive the commutation deficit C(K) = K(K−3)/2 and prove that K = 3 is the unique value among K ≥ 3 for which C(K) = 0 — perfect confinement with zero charge leakage. We construct the Cayley graph of S₃ = W(A₂), compute its Laplacian L = 2I − A explicitly as a 6×6 integer matrix, and verify the complete spectrum {0, 1, 1, 3, 3, 4} through six eigenpairs with linear independence confirmed by nonzero determinant. The spectral gap Δ = 1 is topologically protected by C(3) = 0. The plaquette holonomy (s₁s₂)² has order 3 with zero fixed points, forcing the Wilson loop area law with string tension σ = ln(3). Every theorem is machine-verified in Lean 4 with zero sorry statements. Every numerical claim is cross-validated in Python. Zero free parameters. Keywords: 0-Hecke monoid, commutation deficit, spectral gap, confinement, Cayley graph, Laplacian, mass gap, Lean 4