{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:LSHJNYVKBD6C5QJZGQRQV6PQJI","short_pith_number":"pith:LSHJNYVK","canonical_record":{"source":{"id":"1212.0836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-12-04T19:40:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"25fb51dfae8c2e5f5f5133eb138c0aa76a6dad457667dea33c112cd9957c2b6b","abstract_canon_sha256":"bc405624cc2d5cafb139aa44d441f2a04fcf6aebd99bfc32c6cbb71248969ca0"},"schema_version":"1.0"},"canonical_sha256":"5c8e96e2aa08fc2ec13934230af9f04a146f31777be25c31930a3fb5603dfb81","source":{"kind":"arxiv","id":"1212.0836","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.0836","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"arxiv_version","alias_value":"1212.0836v1","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0836","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"pith_short_12","alias_value":"LSHJNYVKBD6C","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"LSHJNYVKBD6C5QJZ","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"LSHJNYVK","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:LSHJNYVKBD6C5QJZGQRQV6PQJI","target":"record","payload":{"canonical_record":{"source":{"id":"1212.0836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-12-04T19:40:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"25fb51dfae8c2e5f5f5133eb138c0aa76a6dad457667dea33c112cd9957c2b6b","abstract_canon_sha256":"bc405624cc2d5cafb139aa44d441f2a04fcf6aebd99bfc32c6cbb71248969ca0"},"schema_version":"1.0"},"canonical_sha256":"5c8e96e2aa08fc2ec13934230af9f04a146f31777be25c31930a3fb5603dfb81","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:15.100900Z","signature_b64":"gHNYuYP8v69qdp6SDuSF/IPN5tfee8ayFgCXZFD6Dbt/yJ82hK/rPR5VnFwYuFRDsVVmB+YGrGz46qR7PAmIBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c8e96e2aa08fc2ec13934230af9f04a146f31777be25c31930a3fb5603dfb81","last_reissued_at":"2026-05-18T03:39:15.100308Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:15.100308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.0836","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":"2026-05-18T03:39:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6m9eR5ofKTbPMp7UxLfzeUJLapu4ARWqM6UIVYz7c7slUanOIEJ03gc1U6PAK3VsB75rV+HPvcdbCFvR5mw4Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T21:49:37.988909Z"},"content_sha256":"5f1371de271f90c200d72550b3a15c233a20a1c351d30870842d9af50497e89e","schema_version":"1.0","event_id":"sha256:5f1371de271f90c200d72550b3a15c233a20a1c351d30870842d9af50497e89e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:LSHJNYVKBD6C5QJZGQRQV6PQJI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Improved Lower Bound for Stack Sorting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Luke Schaeffer","submitted_at":"2012-12-04T19:40:54Z","abstract_excerpt":"We consider the problem of sorting elements on a series of stacks, introduced by Tarjan and Knuth. We improve the asymptotic lower bound for the number of stacks necessary to sort $n$ elements to $0.561 \\log_2 n + O(1)$. This is the first significant improvement since the previous lower bound, $1/2 \\log_2 n + O(1)$, was established by Knuth in 1972."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0836","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":"2026-05-18T03:39:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Eh4owBc0wH/3wXjOOqhZwxNzc2vCZtonTjtswHjNGscDyvVqE/i/Htb22YJn5mrt3vpgwZydDjDZNDgCJNICA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T21:49:37.989261Z"},"content_sha256":"a5d09fcef9d0e13a60cd54041de3f2e7aa440391609f4b4229baaa982669e349","schema_version":"1.0","event_id":"sha256:a5d09fcef9d0e13a60cd54041de3f2e7aa440391609f4b4229baaa982669e349"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI/bundle.json","state_url":"https://pith.science/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI/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":"2026-06-04T21:49:37Z","links":{"resolver":"https://pith.science/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI","bundle":"https://pith.science/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI/bundle.json","state":"https://pith.science/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LSHJNYVKBD6C5QJZGQRQV6PQJI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:LSHJNYVKBD6C5QJZGQRQV6PQJI","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":"bc405624cc2d5cafb139aa44d441f2a04fcf6aebd99bfc32c6cbb71248969ca0","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-12-04T19:40:54Z","title_canon_sha256":"25fb51dfae8c2e5f5f5133eb138c0aa76a6dad457667dea33c112cd9957c2b6b"},"schema_version":"1.0","source":{"id":"1212.0836","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.0836","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"arxiv_version","alias_value":"1212.0836v1","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0836","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"pith_short_12","alias_value":"LSHJNYVKBD6C","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"LSHJNYVKBD6C5QJZ","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"LSHJNYVK","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:a5d09fcef9d0e13a60cd54041de3f2e7aa440391609f4b4229baaa982669e349","target":"graph","created_at":"2026-05-18T03:39:15Z","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 consider the problem of sorting elements on a series of stacks, introduced by Tarjan and Knuth. We improve the asymptotic lower bound for the number of stacks necessary to sort $n$ elements to $0.561 \\log_2 n + O(1)$. This is the first significant improvement since the previous lower bound, $1/2 \\log_2 n + O(1)$, was established by Knuth in 1972.","authors_text":"Luke Schaeffer","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-12-04T19:40:54Z","title":"An Improved Lower Bound for Stack Sorting"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0836","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":"sha256:5f1371de271f90c200d72550b3a15c233a20a1c351d30870842d9af50497e89e","target":"record","created_at":"2026-05-18T03:39:15Z","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":"bc405624cc2d5cafb139aa44d441f2a04fcf6aebd99bfc32c6cbb71248969ca0","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-12-04T19:40:54Z","title_canon_sha256":"25fb51dfae8c2e5f5f5133eb138c0aa76a6dad457667dea33c112cd9957c2b6b"},"schema_version":"1.0","source":{"id":"1212.0836","kind":"arxiv","version":1}},"canonical_sha256":"5c8e96e2aa08fc2ec13934230af9f04a146f31777be25c31930a3fb5603dfb81","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c8e96e2aa08fc2ec13934230af9f04a146f31777be25c31930a3fb5603dfb81","first_computed_at":"2026-05-18T03:39:15.100308Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:15.100308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gHNYuYP8v69qdp6SDuSF/IPN5tfee8ayFgCXZFD6Dbt/yJ82hK/rPR5VnFwYuFRDsVVmB+YGrGz46qR7PAmIBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:15.100900Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.0836","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5f1371de271f90c200d72550b3a15c233a20a1c351d30870842d9af50497e89e","sha256:a5d09fcef9d0e13a60cd54041de3f2e7aa440391609f4b4229baaa982669e349"],"state_sha256":"8d949e0ef3c0328e89b6e59fad0af296c2a6eb50e9168930cfede83879b8f08d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GyGcXLBkFjFgRfiug2rBBLelbcK+wCJZW1e/c9OlKXap7POlQ/hh37RZJ2iW9w5vEQ8qju+s+Qit9Dee7wXwDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T21:49:37.991301Z","bundle_sha256":"42a68be46f0503dcf3921c9698546a4c562828494057c7582b451b183d4b4533"}}