{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KP7FFSU5OAT6W4UDH3LQPH4OMV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c3b9028542f44573426a6515ba026cf2e3dc24c4ef93938b072ec849070abd76","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-09-27T11:59:02Z","title_canon_sha256":"2632f8dd4aeb9d8be5adbbf41f143c2aafb3ead45616284077198e2bd6db26bc"},"schema_version":"1.0","source":{"id":"1109.5847","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5847","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5847v2","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5847","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"pith_short_12","alias_value":"KP7FFSU5OAT6","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"KP7FFSU5OAT6W4UD","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"KP7FFSU5","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:5cc8ba3dc19e4c231c5fb3f3dad24360bfaa09f33e93c6416944c68178e045a1","target":"graph","created_at":"2026-05-18T03:24:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We revisit the problem of computing the spreading and covering numbers. We show a connection between some of the spreading numbers and the number of non-negative integer 2x2 matrices whose entries sum to d, and we construct an algorithm to compute improved upper bounds for the covering numbers.","authors_text":"Adam Van Tuyl, Ben Babcock","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-09-27T11:59:02Z","title":"Revisiting the spreading and covering numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5847","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a94b37a62c75a9ecb8a386e65975e55808bdd5835f4b3c857d4d74123243a2ee","target":"record","created_at":"2026-05-18T03:24:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c3b9028542f44573426a6515ba026cf2e3dc24c4ef93938b072ec849070abd76","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-09-27T11:59:02Z","title_canon_sha256":"2632f8dd4aeb9d8be5adbbf41f143c2aafb3ead45616284077198e2bd6db26bc"},"schema_version":"1.0","source":{"id":"1109.5847","kind":"arxiv","version":2}},"canonical_sha256":"53fe52ca9d7027eb72833ed7079f8e6574d3817d55aab6e57dee0aa2f64d91d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53fe52ca9d7027eb72833ed7079f8e6574d3817d55aab6e57dee0aa2f64d91d8","first_computed_at":"2026-05-18T03:24:42.018204Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:24:42.018204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6S0Jdcd5hXw6ghpgl1seT3rzd4HvRN5A59RN5w3xH+QQxcuJl6/ljiamKchCQeX6/LZKgA9kFRSy4NvxKL8AAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:24:42.018803Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.5847","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a94b37a62c75a9ecb8a386e65975e55808bdd5835f4b3c857d4d74123243a2ee","sha256:5cc8ba3dc19e4c231c5fb3f3dad24360bfaa09f33e93c6416944c68178e045a1"],"state_sha256":"6779dd83080f737dc6cc19c506fafbe9e2ade80d4a8822bc2ec97cf3f60250b3"}