Flag Game finds agent swarms flip from belief collapse to belief polarization as population grows

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arXiv 2609.19124 (2026-09-16) builds a deliberately tiny testbed for collective belief in agent swarms: a hidden country flag is ground truth, each bounded agent sees only a private crop, and agents exchange beliefs and weigh peer evidence. Even at that scale it reproduces non-monotonic scaling of performance with population size, gains from social-awareness prompting and team diversity, and strong organizational-structure effects. The mechanism they identify is a transition, with collective belief collapse at small populations turning into polarization as the population grows, and that polarization driving the large-population performance decline while creating belief diversity. They introduce social circuit attribution to predict which agent and which view drives the outcome, verify it by causal patching, and fall back to a statistical-mechanical theory once interventions lose power at scale.