PROJECT T FOUNDATIONS Volume II :The Operators Explicit Matrix Representation of H₀(Ã₂) on S₃

Foundation I established the algebraic object: the 0-Hecke monoid H₀(Ã₂), uniquely selected by the commutation deficit C(3) = 0. This volume constructs its concrete realization: the three 6×6 generator matrices πₐ, πᵇ, πc acting on the symmetric group S₃, and the transfer operator T = πₐ + πᵇ + πc that encodes the complete dynamics. The key results of this volume are: Idempotent generators: πᵢ² = πᵢ for all three generators — this is the 0-Hecke signature, replacing Coxeter involutions with projections. Universal braid relations: πᵢπⱼπᵢ = πⱼπᵢπⱼ for ALL pairs (i,j) — this is the Ã₂ signature, impossible in A₃ where one pair commutes. Spectral saturation: Tr(T²) = 64 = 2⁶, confirming that H₀(Ã₂) saturates its spectral capacity. Column stochasticity: Each column of every generator sums to 1, and each column of T sums to 3. Every claim is formally verified in Lean 4 using Mathlib's matrix library. The proof file contains 20 theorems with zero axioms.