{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:ZMS6LACEC4ESREBAADPOLWKU4M","short_pith_number":"pith:ZMS6LACE","schema_version":"1.0","canonical_sha256":"cb25e58044170928902000dee5d954e3383047a7fce37f783c3b6ab90ebc47b0","source":{"kind":"arxiv","id":"2302.01919","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2302.01919","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-02-03T18:53:17Z","cross_cats_sorted":["nlin.AO"],"title_canon_sha256":"31eb6f1ecb2207e9f07dc03597e93c00ce641a66769f00207ed47b498295bde0","abstract_canon_sha256":"7c0e781c893646e482a5476d296f25d1e0c3e392f6ae6263ee30344fe3b77c17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:38:37.930252Z","signature_b64":"/LbjHhQExltYPSv0NwWs/l3m8WArhgvy7ujPQAf0EtQ+suQFtMsjP7zZW8OWIdhAI3k36XVykP7/3d1lGwHxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb25e58044170928902000dee5d954e3383047a7fce37f783c3b6ab90ebc47b0","last_reissued_at":"2026-07-05T05:38:37.929857Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:38:37.929857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2302.01919","created_at":"2026-07-05T05:38:37.929916+00:00"},{"alias_kind":"arxiv_version","alias_value":"2302.01919v1","created_at":"2026-07-05T05:38:37.929916+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.01919","created_at":"2026-07-05T05:38:37.929916+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZMS6LACEC4ES","created_at":"2026-07-05T05:38:37.929916+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZMS6LACEC4ESREBA","created_at":"2026-07-05T05:38:37.929916+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZMS6LACE","created_at":"2026-07-05T05:38:37.929916+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M","json":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M.json","graph_json":"https://pith.science/api/pith-number/ZMS6LACEC4ESREBAADPOLWKU4M/graph.json","events_json":"https://pith.science/api/pith-number/ZMS6LACEC4ESREBAADPOLWKU4M/events.json","paper":"https://pith.science/paper/ZMS6LACE"},"agent_actions":{"view_html":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M","download_json":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M.json","view_paper":"https://pith.science/paper/ZMS6LACE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2302.01919&json=true","fetch_graph":"https://pith.science/api/pith-number/ZMS6LACEC4ESREBAADPOLWKU4M/graph.json","fetch_events":"https://pith.science/api/pith-number/ZMS6LACEC4ESREBAADPOLWKU4M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M/action/storage_attestation","attest_author":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M/action/author_attestation","sign_citation":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M/action/citation_signature","submit_replication":"https://pith.science/pith/ZMS6LACEC4ESREBAADPOLWKU4M/action/replication_record"}},"created_at":"2026-07-05T05:38:37.929916+00:00","updated_at":"2026-07-05T05:38:37.929916+00:00"}