A complete graph Kn, where every node connects to every other. Each node (purple) is an agent whose internal model traverses a continuous cycle of belief updating FEP — SO(2) dynamics CRR. Evidence accumulates along the agent's statistical manifold until the accumulated arc length spans the manifold's full geodesic extent: C = 2π. Then the current regime is exhausted and the agent must reorganise. This is a belief update.
Each edge (teal) is a Markov blanket FEP — a statistical boundary that alternates between two regimes of influence, Z₂ dynamics CRR. Evidence accumulates across a bistable boundary until C = π, then the boundary flips. In both cases, C · Ω = 1 at the update threshold — structurally analogous to the Cramér-Rao bound FEP.
The gold glow is the edge of criticality CRR B(C,Ω) = exp(C/Ω) · (1 − C·Ω), peaking at C* − Ω: one fluctuation-width before the belief update. The system is most sensitive to perturbation — most poised, most responsive — at the edge, not at the transition itself. This is peak information capacity.
Watch the coherence trace below the network. The Z₂ channel (teal) ruptures frequently at threshold π; the SO(2) channel (purple) ruptures rarely at threshold 2π. The gain-frequency trade-off is visible: many small sensory updates, few large prior updates. Both channels process equal total precision-gain per unit time. The fast clock ticks more often but learns less per tick, and the totals balance exactly.
With ε = 0, agents accumulate evidence independently and the frequency ratio hovers near 2 (the topological ratio). Raise the entrainment slider and each belief update delivers a precision-weighted prediction error FEP of ε · Ω to connected agents. Agents near their own threshold are more susceptible: the prediction error is amplified by (1 + 3p²), where p is the proportion of evidence accumulated.
At maximum entrainment, watch the frequency ratio collapse from ~2 to ~1. This is a phase transition: the independent accumulation that produces the 2:1 ratio is overwhelmed by coupling. Every rupture triggers its neighbours, the channels lock together, and the entire network oscillates as one. The two clocks synchronise.
Each blanket is a Z₂ process with variance ΩZ₂ = 1/π, exactly twice the agent's ΩSO(2) = 1/2π. The blanket's higher variance means it is more permeable — it must be, to mediate between agent and environment. The agent's lower variance means tighter precision at the expense of longer evidence accumulation.
The quadratic explosion: each new agent must form a blanket with every existing agent. Person 4 adds three blankets. Person 10 adds nine. Total blankets = n(n−1)/2. At K₃ blankets equal agents; after that, blankets dominate forever.
The Kuramoto order parameter Φ = |1/n · Σ exp(iθˇ)|, where θ = 2π · C/C* is each agent's evidence-accumulation phase. Φ = 1 when all agents are synchronised. Φ ≈ 1/√n when independent.
ACTIVE INFERENCE INSTITUTE · TEMPORAL GRAMMAR · 2026
Beyond ~20 agents, the network crosses a threshold. At K₅₀ there are 1,225 Markov blankets and no agent can attend to each one. The system's dynamics — its free energy landscape FEP — are dominated by the collective, not by any particular boundary.
Blankets scale as n(n−1)/2. At K₂₀ there are 190. At K₅₀ there are 1,225. At K₁₅₀ there are 11,175. The ratio of blankets to agents is (n−1)/2: K₁₅₀ has 74.5 blankets per agent. Each agent is embedded in 149 simultaneous statistical boundaries. No generative model can track this. The group manages itself.
Robin Dunbar proposed ~150 as the cognitive limit for stable social relationships. At K₁₅₀, individual autonomy is 1/150 ≈ 0.7%. The collective carries 99.3% of the system's variance. Beyond this, the blankets are so dominant that you need hierarchical structure — nested Markov blankets FEP — to maintain any meaningful individual contribution.
With ε > 0 at large n, each belief update delivers ε · Ω to connected agents, amplified by susceptibility. At K₁₅₀, each agent has 149 blankets. When several blankets flip in succession, the cumulative prediction error pushes the agent past its own threshold. Its belief update propagates outward through all 149 of its blankets. The cascade appears as a wave of red — and the frequency ratio collapses toward 1.
ACTIVE INFERENCE INSTITUTE · TEMPORAL GRAMMAR · 2026