{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:W4JXED2QIKB6HI32FXBKWINHSB","short_pith_number":"pith:W4JXED2Q","canonical_record":{"source":{"id":"1206.1553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-06-07T16:54:19Z","cross_cats_sorted":[],"title_canon_sha256":"222e12454208bfd6573a7a685e7e52ed1cb359e2df029df852bdbf9236ddec15","abstract_canon_sha256":"2b7165cc2e1db34a5422a581cba8580fe859326e27ba8a04b5350599bcf8fa3d"},"schema_version":"1.0"},"canonical_sha256":"b713720f504283e3a37a2dc2ab21a7904231e5d9b340fb7c54b1329c2f9a3ae6","source":{"kind":"arxiv","id":"1206.1553","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1553","created_at":"2026-05-18T03:54:04Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1553v1","created_at":"2026-05-18T03:54:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1553","created_at":"2026-05-18T03:54:04Z"},{"alias_kind":"pith_short_12","alias_value":"W4JXED2QIKB6","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"W4JXED2QIKB6HI32","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"W4JXED2Q","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:W4JXED2QIKB6HI32FXBKWINHSB","target":"record","payload":{"canonical_record":{"source":{"id":"1206.1553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-06-07T16:54:19Z","cross_cats_sorted":[],"title_canon_sha256":"222e12454208bfd6573a7a685e7e52ed1cb359e2df029df852bdbf9236ddec15","abstract_canon_sha256":"2b7165cc2e1db34a5422a581cba8580fe859326e27ba8a04b5350599bcf8fa3d"},"schema_version":"1.0"},"canonical_sha256":"b713720f504283e3a37a2dc2ab21a7904231e5d9b340fb7c54b1329c2f9a3ae6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:04.835667Z","signature_b64":"05sFqKL9DUPWdvE0oUQ3FbxUUJCtSknqlGjHKO+jvl0GP7G+AnNCUZPRwN0sAiNsPBopAlwiOED63tPCQth4DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b713720f504283e3a37a2dc2ab21a7904231e5d9b340fb7c54b1329c2f9a3ae6","last_reissued_at":"2026-05-18T03:54:04.834730Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:04.834730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.1553","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:54:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sXrOqwvp45wWa5mcBiczUgvMxu4oyzIJX+potiV0KGqfZOwmicMWo9u7eVwNvu36StGaXd+ZJp0VZKsGKjhOBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T07:28:09.949757Z"},"content_sha256":"ecd0fcb80530ca8083170b306f45b47953583ba6a8b0bf7965716af83449ddb6","schema_version":"1.0","event_id":"sha256:ecd0fcb80530ca8083170b306f45b47953583ba6a8b0bf7965716af83449ddb6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:W4JXED2QIKB6HI32FXBKWINHSB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tight Lower Bounds for Unequal Division","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Andrew Lohr","submitted_at":"2012-06-07T16:54:19Z","abstract_excerpt":"Alice and Bob want to cut a cake; however, in contrast to the usual problems of fair division, they want to cut it unfairly. More precisely, they want to cut it in ratio $(a:b)$. (We can assume gcd(a,b)=1.) Let f(a,b) be the number of cuts will this take (assuming both act in their own self interest). It is known that f(a,b) \\le \\ceil{lg(a+b)}. We show that (1) for all a,b, f(a,b) \\ge lg(lg(a+b)) + (2) for an infinite number of (a,b), f(a,b) \\le 1+lg(lg(a+b)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1553","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:54:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q/ZTaywIioQKRiLN7G5GwUaN18RYRRbIoU2XNylbQwH0lcM7O6tEzYhCjk3iS/Gd9K9uH/j8iI4Pdnj/LMetCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T07:28:09.950104Z"},"content_sha256":"f51ef0047e8699fa0c4222e7f01e33adaa8f7b2d348f780de8b5bd0d3febed46","schema_version":"1.0","event_id":"sha256:f51ef0047e8699fa0c4222e7f01e33adaa8f7b2d348f780de8b5bd0d3febed46"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W4JXED2QIKB6HI32FXBKWINHSB/bundle.json","state_url":"https://pith.science/pith/W4JXED2QIKB6HI32FXBKWINHSB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W4JXED2QIKB6HI32FXBKWINHSB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T07:28:09Z","links":{"resolver":"https://pith.science/pith/W4JXED2QIKB6HI32FXBKWINHSB","bundle":"https://pith.science/pith/W4JXED2QIKB6HI32FXBKWINHSB/bundle.json","state":"https://pith.science/pith/W4JXED2QIKB6HI32FXBKWINHSB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W4JXED2QIKB6HI32FXBKWINHSB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:W4JXED2QIKB6HI32FXBKWINHSB","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":"2b7165cc2e1db34a5422a581cba8580fe859326e27ba8a04b5350599bcf8fa3d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-06-07T16:54:19Z","title_canon_sha256":"222e12454208bfd6573a7a685e7e52ed1cb359e2df029df852bdbf9236ddec15"},"schema_version":"1.0","source":{"id":"1206.1553","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1553","created_at":"2026-05-18T03:54:04Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1553v1","created_at":"2026-05-18T03:54:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1553","created_at":"2026-05-18T03:54:04Z"},{"alias_kind":"pith_short_12","alias_value":"W4JXED2QIKB6","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"W4JXED2QIKB6HI32","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"W4JXED2Q","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:f51ef0047e8699fa0c4222e7f01e33adaa8f7b2d348f780de8b5bd0d3febed46","target":"graph","created_at":"2026-05-18T03:54:04Z","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":"Alice and Bob want to cut a cake; however, in contrast to the usual problems of fair division, they want to cut it unfairly. More precisely, they want to cut it in ratio $(a:b)$. (We can assume gcd(a,b)=1.) Let f(a,b) be the number of cuts will this take (assuming both act in their own self interest). It is known that f(a,b) \\le \\ceil{lg(a+b)}. We show that (1) for all a,b, f(a,b) \\ge lg(lg(a+b)) + (2) for an infinite number of (a,b), f(a,b) \\le 1+lg(lg(a+b).","authors_text":"Andrew Lohr","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-06-07T16:54:19Z","title":"Tight Lower Bounds for Unequal Division"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1553","kind":"arxiv","version":1},"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:ecd0fcb80530ca8083170b306f45b47953583ba6a8b0bf7965716af83449ddb6","target":"record","created_at":"2026-05-18T03:54:04Z","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":"2b7165cc2e1db34a5422a581cba8580fe859326e27ba8a04b5350599bcf8fa3d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2012-06-07T16:54:19Z","title_canon_sha256":"222e12454208bfd6573a7a685e7e52ed1cb359e2df029df852bdbf9236ddec15"},"schema_version":"1.0","source":{"id":"1206.1553","kind":"arxiv","version":1}},"canonical_sha256":"b713720f504283e3a37a2dc2ab21a7904231e5d9b340fb7c54b1329c2f9a3ae6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b713720f504283e3a37a2dc2ab21a7904231e5d9b340fb7c54b1329c2f9a3ae6","first_computed_at":"2026-05-18T03:54:04.834730Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:04.834730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"05sFqKL9DUPWdvE0oUQ3FbxUUJCtSknqlGjHKO+jvl0GP7G+AnNCUZPRwN0sAiNsPBopAlwiOED63tPCQth4DA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:04.835667Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.1553","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ecd0fcb80530ca8083170b306f45b47953583ba6a8b0bf7965716af83449ddb6","sha256:f51ef0047e8699fa0c4222e7f01e33adaa8f7b2d348f780de8b5bd0d3febed46"],"state_sha256":"8fb20f82df1ebcef607eb3224e4d3800952142db47429f9160ec734f455a7fbd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tI1OUJr6zdacSW70r5+83QgvdHRU5AbTxMOkGSpncWjssyHbmpmjUHyNMwTQ6g4I4gA/MTjh/HoS5UhkVAowAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T07:28:09.952341Z","bundle_sha256":"d12a810e7c337d779f4b687a2a23751d662c068a50c860617d39650a687cb156"}}