{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WAYBXNSJ6ABMXUQ2BWOLXEIINK","short_pith_number":"pith:WAYBXNSJ","schema_version":"1.0","canonical_sha256":"b0301bb649f002cbd21a0d9cbb91086abb86d1102a0bbdf6d422a67cff48554d","source":{"kind":"arxiv","id":"2606.31391","version":1},"attestation_state":"computed","paper":{"title":"Radial Transform Extremality for the Siblings of the Coupon Collector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christopher D. Long","submitted_at":"2026-06-30T09:19:30Z","abstract_excerpt":"In the siblings version of the coupon collector, a main collector stops when every coupon type has appeared once. Duplicates are passed successively to siblings, and $U_j^N$ denotes the number of empty spaces in the $j$th collector's album at the main completion time. We prove finite-$N$ radial transform strengthenings of the uniform-probability extremality principle. For every $N\\ge2$, every $j\\ge2$, every positive nonuniform probability vector $p$, and the ray $p(\\theta)=u+\\theta(p-u)$ from the uniform vector $u$, the full probability generating function $\\mathbb{E}_{p(\\theta)}z^{U_j^N}$ is "},"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":"2606.31391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-30T09:19:30Z","cross_cats_sorted":[],"title_canon_sha256":"801a5de2d58faf538eb335ce186b3c4cbeee45197fa55f47064b5de224fa7196","abstract_canon_sha256":"8fc2f1c3d32fbdc3c0fffa53ed537bc0ed3dff61d3425771db835a896e4c91d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T01:18:01.676053Z","signature_b64":"GraolsG24vcbGlKmF0W+nm7NmsL/NQRke5tOto1NBo1XIQ7F/HaekUn9Y7gZk263CmMllOALBDQhQJRiGPyBDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0301bb649f002cbd21a0d9cbb91086abb86d1102a0bbdf6d422a67cff48554d","last_reissued_at":"2026-07-01T01:18:01.675636Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T01:18:01.675636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Radial Transform Extremality for the Siblings of the Coupon Collector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christopher D. Long","submitted_at":"2026-06-30T09:19:30Z","abstract_excerpt":"In the siblings version of the coupon collector, a main collector stops when every coupon type has appeared once. Duplicates are passed successively to siblings, and $U_j^N$ denotes the number of empty spaces in the $j$th collector's album at the main completion time. We prove finite-$N$ radial transform strengthenings of the uniform-probability extremality principle. For every $N\\ge2$, every $j\\ge2$, every positive nonuniform probability vector $p$, and the ray $p(\\theta)=u+\\theta(p-u)$ from the uniform vector $u$, the full probability generating function $\\mathbb{E}_{p(\\theta)}z^{U_j^N}$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31391","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/2606.31391/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":"2606.31391","created_at":"2026-07-01T01:18:01.675693+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.31391v1","created_at":"2026-07-01T01:18:01.675693+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.31391","created_at":"2026-07-01T01:18:01.675693+00:00"},{"alias_kind":"pith_short_12","alias_value":"WAYBXNSJ6ABM","created_at":"2026-07-01T01:18:01.675693+00:00"},{"alias_kind":"pith_short_16","alias_value":"WAYBXNSJ6ABMXUQ2","created_at":"2026-07-01T01:18:01.675693+00:00"},{"alias_kind":"pith_short_8","alias_value":"WAYBXNSJ","created_at":"2026-07-01T01:18:01.675693+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.08551","citing_title":"Weighted-threshold Coupon Collection","ref_index":29,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK","json":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK.json","graph_json":"https://pith.science/api/pith-number/WAYBXNSJ6ABMXUQ2BWOLXEIINK/graph.json","events_json":"https://pith.science/api/pith-number/WAYBXNSJ6ABMXUQ2BWOLXEIINK/events.json","paper":"https://pith.science/paper/WAYBXNSJ"},"agent_actions":{"view_html":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK","download_json":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK.json","view_paper":"https://pith.science/paper/WAYBXNSJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.31391&json=true","fetch_graph":"https://pith.science/api/pith-number/WAYBXNSJ6ABMXUQ2BWOLXEIINK/graph.json","fetch_events":"https://pith.science/api/pith-number/WAYBXNSJ6ABMXUQ2BWOLXEIINK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK/action/storage_attestation","attest_author":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK/action/author_attestation","sign_citation":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK/action/citation_signature","submit_replication":"https://pith.science/pith/WAYBXNSJ6ABMXUQ2BWOLXEIINK/action/replication_record"}},"created_at":"2026-07-01T01:18:01.675693+00:00","updated_at":"2026-07-01T01:18:01.675693+00:00"}