On the Uniqueness of Rank K = 3 in Affine 0-Hecke Structures

We establish that K = 3 is the unique rank for which the affine 0-Hecke monoid H₀(Ã_{K-1}) admits a non-trivial topological phase transition to a finite boundary structure. The proof relies on three independent convergences: (1) the commutation count K(K−3)/2 vanishes only for K = 3; (2) the triple coincidence F₁(K) = F₂(K) = F₃(K) = 33 occurs uniquely at K = 3; (3) the quasi-Hopf structure remains irreducible only for K = 3. These results provide an algebraic foundation for the threefold replication of fermion generations and the confinement mechanism in gauge theories.