Collective Phase Transitions beyond Individual Saturation

This paper develops a mathematically explicit route by which individually saturated intelligences can still open new performance frontiers by forming collectives, under strictly intrinsic (“no-meta”) evaluation constraints. We work in a layered observation–semantic framework in which each agent has an individual observation space, a set of stable semantic phases, and a scalar value functional grounded in persistence and resource flows. Operational saturation is formalised via an εsatεₛₐₜεsat-condition: within a fixed design and parameter set, no further local tuning can improve an agent’s value by more than εsatεₛₐₜεsat. On top of these saturated individuals we construct a group-intelligence layer modelled by Gibbs measures on interaction graphs subject to law-like locality and resource constraints. Collective semantic phases are identified with extremal Gibbs measures, and a scalar collective value is defined as a monotone transform of a bounded local order parameter. The Collective Phase Theorem shows that, whenever a classical phase transition exists and the order parameter is calibrated against the individual benchmark, there are collective phases whose per-agent value strictly exceeds any individually attainable value—even though all agents are already saturated in their local design space. The second part of the paper asks whether saturated agents can tell, from their own finite-radius and finite-time observation streams, whether they are currently in a “good” or “bad” collective phase. Under stationary exponential mixing and a non-zero margin in a bounded local statistic, a Local Evaluability Theorem guarantees simple tests with exponentially decaying misclassification probability. A complementary Local Impossibility Theorem shows that if good and bad phases share all finite local space–time marginals, then no local decision rule can beat random guessing. We discuss implications for multi-agent reinforcement learning, large language model collectives, human–AI group intelligence, and self-splitting strategies inside a single saturated system, and we outline limitations arising from graph structure, mixing times, and the deliberate restriction to a single scalar value axis.