A Unified Information-Geometric Framework for Collective Belief Dynamics in Multi-Agent LLMs

Communication among LLM agents sometimes adds information and sometimes collapses a population onto a wrong consensus. We isolate one step of this question: a population of beliefs is pooled and is also updated on common evidence, and we ask when the order matters. Logarithmic ($e$-) pooling commutes with a common Bayes update, whereas arithmetic ($m$-) pooling reallocates the social weights in proportion to the evidence normalizers; we characterize the exact zero set of this order gap, bound it, and show that it is zero or fourth order near consensus. Because the order gap is not a quality ordering, we score both orderings against an independently declared finite observation model: update-then-pool has higher expected log loss in all 18 cells (exact enumeration), and at matched observation budget no single rule is best across six information structures. Stated as operators on an information-geometric map of predictive states, the same language also closes the second moments of sampled communication, which includes the sampled convex-combination rule used to model memetic drift. A preregistered one-step transfer test on native Qwen2.5 receivers failed its gate, so we do not claim that native agents follow these operators.