where γ γ is an arbitrary branching ratio (greater than or equal to 2), t ≡ ≡ tanh ⁡ ⁡ J , τ τ ≡ ≡ t 2 , J ≡ ≡ β β ϵ ϵ (with ϵ ϵ representing the nearest-neighbor interaction energy) and there are k (→ ∞ in the thermodynamic limit) generations in each of the tree branches (forming the closed tree architecture as shown in the given closed Cayley tree diagram.) The sum in the last term can be shown to converge uniformly and rapidly (i.e. for z → ∞, it remains finite) yielding a continuous and monotonous function, establishing that, for γ γ greater than or equal to 2, the free energy is a continuous function of temperature T. Further analysis of the free energy indicates that it exhibits an unusual discontinuous first derivative at the critical temperature (Krizan, Barth & Glasser (1983), Glasser & Goldberg (1983).)