(α,β)-Membrane Graphs: Cycle Balance and Coupling Decoupling as Independent Axes of Graph Organization

We introduce a two-parameter family of graphs dened by the simultaneous satisfaction of two orthogonal constraints: cycle balance (the ratio of observed to expected short-cycle quotient matches the conguration model prediction) and monotone decoupling (vertex degree and local clustering coecient are rank-uncorrelated). For parameters α,β ≥0, the class M(α,β) consists of all simple graphs whose cycle balance parameter deviates from unity by at most α and whose Spearman degree clustering correlation is bounded by β. We establish the analytic foundations using exact cycle-count expectations from the configuration model, eliminating the need for simulation-based null models. We prove that vertex-transitive graphs with triangles lie in M+(α,0) for computable α; that Erdos-Rényi random graphs converge to the center (1,0) in the regime np→∞, np2 →0; and that regular graphs of girth at least ve form the singular boundary of the extended class M+. Explicit witness pairs demonstrate that the two axes are functionally independent. We study the behavior of the membrane parameters under graph products, formalize the singular boundary through cage graphs and the Erdos-Gyárfás conjecture, and dene the suppression spectrum as a higher-order generalization connecting the framework to the Ihara zeta function.