{"paper":{"title":"Critical numerosity in collective behavior","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["nlin.AO"],"primary_cat":"math.PR","authors_text":"Jacob Calvert","submitted_at":"2023-02-03T18:53:17Z","abstract_excerpt":"Natural collectives, despite comprising individuals who may not know their numerosity, can exhibit behaviors that depend sensitively on it. This paper proves that the collective behavior of number-oblivious individuals can even have a critical numerosity, above and below which it qualitatively differs. We formalize the concept of critical numerosity in terms of a family of zero--one laws and introduce a model of collective motion, called chain activation and transport (CAT), that has one.\n  CAT describes the collective motion of $n \\geq 2$ individuals as a Markov chain that rearranges $n$-elem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.01919","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2302.01919/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}