PROJECT T Volume XXVII Quantization of the Vibrational Field: Emergence of Three Generations from the Triple Hopf Structure

We construct the quantum field theory of vibrational modes on the affine triangle Ã₂, starting from the three generators T₁, T₂, T₀ of the 0-Hecke monoid H₀(Ã₂) acting on ℂ[S₃]. All three generators are the true affine sorting operators (rank 3 each), preserving the exact ℤ₃ symmetry of the Dynkin diagram. The deformed coproduct Δᵢ(T) = Tᵢ⊗I + I⊗Tᵢ − Tᵢ⊗Tᵢ, with deformation parameter λ = −1 uniquely forced by idempotence, defines a quasi-Hopf structure on ℂ³⁶. The central result is the Coproduct Amplification Theorem: for all three generator pairs (i,j), the squared Frobenius norm of the coproduct commutator equals K² = 9 times the monoid commutator norm. The ℤ₃ symmetry is exact at every level: all three ratios are identical. The idempotent Fock space (dim = 2³ = 8, Pauli from T² = T) selects exactly 4 confined states via ℤ₃ charge conservation, yielding the energy spectrum Eₙ = 3n for n = 0,1,2,3. No fourth generation exists. Every constant is forced by the algebra. All 38 theorems are verified in Lean 4 via native_decide.