{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:2F6WACSBKDGH4NQQDBIPXRJAYX","short_pith_number":"pith:2F6WACSB","schema_version":"1.0","canonical_sha256":"d17d600a4150cc7e36101850fbc520c5c96228de9f0705922993fcddbf30bd84","source":{"kind":"arxiv","id":"2507.08646","version":2},"attestation_state":"computed","paper":{"title":"Additive sumset sizes with tetrahedral differences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Melvyn B. Nathanson","submitted_at":"2025-07-11T14:50:34Z","abstract_excerpt":"Experimental calculations suggest that the $h$-fold sumset sizes of 4-element sets of integers are concentrated at $h$ numbers that are differences of tetrahedral numbers. In this paper it is proved that these \"popular\" sumset sizes always exist. Explicit $h$-adically defined sets are constructed for each of these numbers."},"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":"2507.08646","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-11T14:50:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4035c71e6b5ace7afd672aeddfdc53be079bcfaba124de048e7335d78a09237c","abstract_canon_sha256":"9d7a630cae44bbd446b20933cdec2b2ef7c128776cd97a6d2ff979e63bac0b1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:28.478913Z","signature_b64":"1bWT0SOyoCwmBLzuvVS3yMGxx7Nsp2pGJY+JVheKVYXz3+JNhO72P7quXZ/WCB/QR7gXsAg8cJOgy+lsdwt8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d17d600a4150cc7e36101850fbc520c5c96228de9f0705922993fcddbf30bd84","last_reissued_at":"2026-07-05T12:00:28.478427Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:28.478427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additive sumset sizes with tetrahedral differences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Melvyn B. Nathanson","submitted_at":"2025-07-11T14:50:34Z","abstract_excerpt":"Experimental calculations suggest that the $h$-fold sumset sizes of 4-element sets of integers are concentrated at $h$ numbers that are differences of tetrahedral numbers. In this paper it is proved that these \"popular\" sumset sizes always exist. Explicit $h$-adically defined sets are constructed for each of these numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.08646","kind":"arxiv","version":2},"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/2507.08646/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":"2507.08646","created_at":"2026-07-05T12:00:28.478485+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.08646v2","created_at":"2026-07-05T12:00:28.478485+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.08646","created_at":"2026-07-05T12:00:28.478485+00:00"},{"alias_kind":"pith_short_12","alias_value":"2F6WACSBKDGH","created_at":"2026-07-05T12:00:28.478485+00:00"},{"alias_kind":"pith_short_16","alias_value":"2F6WACSBKDGH4NQQ","created_at":"2026-07-05T12:00:28.478485+00:00"},{"alias_kind":"pith_short_8","alias_value":"2F6WACSB","created_at":"2026-07-05T12:00:28.478485+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/2F6WACSBKDGH4NQQDBIPXRJAYX","json":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX.json","graph_json":"https://pith.science/api/pith-number/2F6WACSBKDGH4NQQDBIPXRJAYX/graph.json","events_json":"https://pith.science/api/pith-number/2F6WACSBKDGH4NQQDBIPXRJAYX/events.json","paper":"https://pith.science/paper/2F6WACSB"},"agent_actions":{"view_html":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX","download_json":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX.json","view_paper":"https://pith.science/paper/2F6WACSB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.08646&json=true","fetch_graph":"https://pith.science/api/pith-number/2F6WACSBKDGH4NQQDBIPXRJAYX/graph.json","fetch_events":"https://pith.science/api/pith-number/2F6WACSBKDGH4NQQDBIPXRJAYX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX/action/storage_attestation","attest_author":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX/action/author_attestation","sign_citation":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX/action/citation_signature","submit_replication":"https://pith.science/pith/2F6WACSBKDGH4NQQDBIPXRJAYX/action/replication_record"}},"created_at":"2026-07-05T12:00:28.478485+00:00","updated_at":"2026-07-05T12:00:28.478485+00:00"}