Orthogonal Differentiation in Constrained Networks: Cycle Homogeneity and Coupling Homogeneity as Independent Axes of Network Organization

We introduce a two-dimensional phase space for network topology defined by two orthogonal invariants: Cycle Homogeneity (CH), measuring the uniformity of the cycle spectrum relative to a degree-preserving null model, and Coupling Homogeneity (KH), measuring the statistical independence between node degree and local clustering. Using the developmental connectome of Caenorhabditis elegans (Witvliet et al. 2021; 8 timepoints, n = 187–222 neurons) and the complete census of cubic vertex-transitive graphs with girth 6 (58,438 graphs from the Erdős–Gyárfás conjecture classification), we demonstrate the following five results.
First, CH and KH are empirically orthogonal (ρ = 0.31, p = 0.46). Second, CH is stationary across development (CV = 2.4%, 8 stages) at CH ≈ 0.78, representing active suppression of generic clustering (Z(p3) = +18 to +46 against the Maslov–Sneppen null model). Third, KH drifts monotonically from 0 to −0.52 (R2 = 0.872, p < 10−3). Fourth, degree-preserving rewiring destroys both invariants (Iφ(CH) = 0.176, Iφ(KH) = 2.24). Fifth, extremal graphs from the Potocnik–Vidali census occupy a singular point where both axes collapse to zero variance.
Ontogenesis is a vertical trajectory in (CH, KH) space: development differentiates coupling, not cycles. We define three membrane types (biological, algebraic, destructive), reformulate the Gradient-Balance Principle for two dimensions, and derive five testable predictions. The use of corr(k, σ) as a developmental order parameter and the CH/KH decomposition over ontogenesis are both novel.