A Diagnostic Framework for Structural Regimes in Majority-Minority Field Dynamics

Scalar hierarchy metrics conflate structurally distinct field configurations. Three configurations of a 100-node field — an intact minority cluster, a Lévy-driven transient spike, and a hub-forced bifurcation — each produce H≈0. 33, yet differ mechanistically and require different interventions. We introduce a hierarchy vector (C, A, D) — Centralization, Asymmetry, Dominance — as a diagnostic framework for structural regimes in RCR (Recursion-Collapse-Recombination) dynamical systems. The central formal result (Proposition 1) is that no weighted linear scalar of (C, A, D) can replace the vector: every such scalar produces at least 43 structurally conflated timesteps across a 500-step trajectory, while H (the natural scalar) conflates 110. Complementing this, six pre-registered robustness claims hold across seeds, partition methods, parameter perturbations, and adversarial conditions. The C–A anti-correlation (r=−0. 677±0. 032) and D near-independence (MI<0. 14) survive ±10% parameter perturbation and four partition methods (93–97% agreement). The analytical D ceiling (D=hub/ (N·|R|ₘaj+hub) ) matches observation within 7% — the result is predictability of the ceiling, not mere smallness of D. The framework is computable on any real-valued field R (t) without presupposing RCR architecture. Practitioners working with any continuous-valued ABM can apply (C, A, D) directly using the three formulas in Section 3, with no dependence on RCR architecture or parameter choices. Across a 500-step trajectory, three structural phases are identified algorithmically from (C, A, D) alone, with phase boundaries stable across 10 seeds. Core results are cross-validated under a second RCR architecture (Model 5) and, in the cross-model validation of Section 8, against Hegselmann–Krause, Deffuant, and noisy bounded confidence — confirming that (C, A, D) correctly identifies distinct structural regimes across architecturally unrelated continuous-opinion dynamics models, including noise-sustained minority persistence absent from scalar measures.