PROJECT T Volume XXV : Jordan Structure of H₀(A₃): Quadratic Growth Law, Poincaré Duality, and the Reinforced No-Go Theorem

We carry out a complete structural analysis of the transfer operator T = π₁ + π₂ + π₃ of the 0-Hecke algebra H₀(A₃) acting on the group algebra ℂ[S₄]. The Jordan normal form exhibits the block partition [3,3,3,1,1] at both interior eigenvalues λ = 1 and λ = 2. Extending to H₀(An) for n = 1,...,5, we discover the Quadratic Growth Law: the maximum Jordan block size at eigenvalue λ equals b(n,λ) = 1 + λ(n−λ), yielding a global nilpotency order of 1 + ⌊n²/4⌋ — quadratic in the Coxeter rank, not linear as previously conjectured. We establish Poincaré duality by the sign character: Vλ ⊗ ε ≅ Vn−λ as Sn+1-modules, verified through explicit irreducible decompositions for S₄ and S₅. We prove that T does not preserve Solomon’s descent algebra, and show that the quadratic growth of Jordan complexity strengthens the no-go theorem for α ≈ 1/137 from spectral data.