Response-Aware Coarse-Graining in Summary-Mediated Opinion Dynamics

RESUMO

Interacting populations increasingly respond to compressed representations generated from their own collective state. We ask which information such an endogenous coarse-graining must preserve in order to reproduce the collective dynamics. The question has two complementary answers for continuous opinions with attraction, indifference, and repulsion. Inside any strict $K$-bloc response cell where every represented stance attracts every bloc, the dynamics reduce exactly to translation and disagreement. The represented distribution's shape cancels from all disagreement modes, which form a homogeneously anchored weighted signed-Laplacian system, while represented-mean error acts only on translation. This reduction yields a geometry-dependent stability boundary and shows that prevalence bias produces secular drift at marginal stability only through its projection onto the neutral eigenspace. Outside that common linear-response regime, threshold geometry makes distributional information observable. The induced force discrepancy is a behavior-defined integral probability metric; restricted force probes identify generic finite empirical discussions, and for finitely supported candidates uniform force error controls total-variation and Wasserstein error while support reduction incurs a nonzero distortion floor. A cumulative-discrepancy characterization further explains why transport-small summaries can remain behaviorally inaccurate at response thresholds. Mode-resolved integrations, projected particle simulations, finite-size ensembles, and threshold-crossing experiments test these two regimes. As a deployment consequence, broadcast summary error survives as collective noise whereas independently personalized error self-averages. The model is stylized and not empirically calibrated; generated discussion summaries motivate the general problem of endogenous, response-aware coarse-graining.