Many coordinated systems from coupled oscillators to institutions, ecosystems, neural populations, and multi-agent artificial intelligence all face a structural tension: strong global coordination tends to produce uniform collective behavior, but uniformity can be achieved either by genuine balanced participation or by suppressing the very heterogeneity that makes the system adaptive. This paper formalizes that tension as a single abstract dynamical condition: the equi-coherent state, in which population variance in local participation vanishes and asks when such a state is the unique stable outcome of a coupled system. We define a substrate-independent class of dynamical systems (participation systems) subject to three rate conditions, and we prove, in full and self-contained detail, that a variance-penalized Kuramoto model with Lorentzian heterogeneity exhibits: (i) a unique non-trivial fixed point r ∗ =p1 − 2∆/K for coupling above a critical value Kc = 2∆; (ii) local asymptotic stability of the zero-variance state for equalization pressure γ above a critical threshold γc = ∆r∗; and (iii) a continuous, second-order phase transition at γc with mean-field exponent β = 1. We then state, and clearly mark as conjectural, two further claims: that the threshold’s stability properties extend to a broad class of substrate-dependent penalty functionals (universality), and that effective equalization pressure can propagate across coupled systems that share no microscopic correspondence (cross-substrate propagation), via an epistemic-depth mechanism that we derive explicitly. Every claim in this document is labeled Proven, Partial, or Conjectural, and no claim is labeled more strongly than its actual proof status warrants. We close with a substrate-correspondence table illustrating —but not claiming to have empirically validated — how the formalism specializes across physics, neuroscience, governance, ecology, and artificial intelligence, and with a consolidated list of open problems whose resolution would convert this from a dynamical model into a general selection principle.
Christopher Chorkey (Fri,) studied this question.