VOLUME XXI : Spectral Saturation A Classification Theorem for 0-Hecke Algebras with Proof that A₂ and A₃ are the Unique Saturated Root Systems

We prove that among all irreducible finite root systems, the spectral saturation condition Tr(T²) = 2|Φ⁺| holds if and only if the root system is A₂ or A₃. Here T = ∑ Ls is the transfer operator of the 0-Hecke monoid H₀(W) acting by left multiplication on the Weyl group W, and Φ⁺ denotes the set of positive roots. The proof rests on three pillars: (1) a theorem that the fixed-point count f(w) = |{(i,j) : sisjw = w}| equals des(w)² for all w in any symmetric group, proved by case analysis on generator adjacency; (2) an arithmetic argument showing that the resulting equation (n+1)!(3n²+n+2)/12 = 2n(n+1)/2 has exactly two integer solutions n = 2, 3, because the prime 5 enters (n+1)! at n = 4 and can never be cancelled; and (3) direct computation verifying that no other classical or exceptional type (B, D, G₂, H₂, I₂(m)) saturates. The systems A₂ and A₃ form a saturation window — the unique interval of ranks where the spectral second moment of the Eulerian distribution matches the binary capacity of the root system. A₃ is the maximal element of this window, providing a structural selection principle for the K = 3 framework of Project T.