{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ALNEEQYMEDRRBXL647XANH4PNQ","short_pith_number":"pith:ALNEEQYM","canonical_record":{"source":{"id":"1911.02569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-06T19:00:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea76f09e4a51e9930521e013e687eb8378c4c51c7c05944355ff6ff9e0f481f3","abstract_canon_sha256":"2ab60098a94a31b98e73adfb2e9a9b90d5c8dcad14d7c32ed740af341aeb72bd"},"schema_version":"1.0"},"canonical_sha256":"02da42430c20e310dd7ee7ee069f8f6c2924d4b150544162bea78872db5918dd","source":{"kind":"arxiv","id":"1911.02569","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.02569","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"arxiv_version","alias_value":"1911.02569v1","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.02569","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"pith_short_12","alias_value":"ALNEEQYMEDRR","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"pith_short_16","alias_value":"ALNEEQYMEDRRBXL6","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"pith_short_8","alias_value":"ALNEEQYM","created_at":"2026-07-05T00:17:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ALNEEQYMEDRRBXL647XANH4PNQ","target":"record","payload":{"canonical_record":{"source":{"id":"1911.02569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-06T19:00:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea76f09e4a51e9930521e013e687eb8378c4c51c7c05944355ff6ff9e0f481f3","abstract_canon_sha256":"2ab60098a94a31b98e73adfb2e9a9b90d5c8dcad14d7c32ed740af341aeb72bd"},"schema_version":"1.0"},"canonical_sha256":"02da42430c20e310dd7ee7ee069f8f6c2924d4b150544162bea78872db5918dd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:17:39.424801Z","signature_b64":"QstAq5fIt9c13a8e0LFK/QZw5RL1YllWiosAhQXI1P6rnEYGYvcVbr6673Nn96Epm3tqmpS2U6gH//brPY6ZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02da42430c20e310dd7ee7ee069f8f6c2924d4b150544162bea78872db5918dd","last_reissued_at":"2026-07-05T00:17:39.424347Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:17:39.424347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1911.02569","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":"2026-07-05T00:17:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pwAkyTszf5/XU+eQN+SnjZL5E1Rh7ZUbiA5cUYG1vT5bqQAb7jyI/ubTsTqRp/Ii4ohd+F3x6IbtzDTA1pn9Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T03:21:25.400920Z"},"content_sha256":"dd7d724013e763483d40e2f001fce0cd10120909bee849ee12cee28643fe479b","schema_version":"1.0","event_id":"sha256:dd7d724013e763483d40e2f001fce0cd10120909bee849ee12cee28643fe479b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ALNEEQYMEDRRBXL647XANH4PNQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimum weight disk triangulations and fillings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Eyal Lubetzky, Itai Benjamini, Yuval Peled","submitted_at":"2019-11-06T19:00:00Z","abstract_excerpt":"We study the minimum total weight of a disk triangulation using vertices out of $\\{1,\\ldots,n\\}$, where the boundary is the triangle $(123)$ and the $\\binom{n}3$ triangles have independent weights, e.g. $\\mathrm{Exp}(1)$ or $\\mathrm{U}(0,1)$.\n  We show that for explicit constants $c_1,c_2>0$, this minimum is $c_1 \\frac{\\log n}{\\sqrt n} + c_2 \\frac{\\log\\log n}{\\sqrt n} + \\frac{Y_n}{\\sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $\\frac14\\log n + O_P(\\sqrt{\\log n}) $ vertices.\n  Moreover, for disk triangulations that are canonical, in th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.02569","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/1911.02569/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:17:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vd3OXlhpPCOb7ZNMcq8c20iOqwjeNy5ME/MwiEbIhOS7n8TfcKOhITWWow+8fpeOlUZeT2TnV3wxR231QPAOAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T03:21:25.401560Z"},"content_sha256":"1680de94a21f08439fa60cef6b48d1080b18442f4612b0581131127b1b533929","schema_version":"1.0","event_id":"sha256:1680de94a21f08439fa60cef6b48d1080b18442f4612b0581131127b1b533929"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ALNEEQYMEDRRBXL647XANH4PNQ/bundle.json","state_url":"https://pith.science/pith/ALNEEQYMEDRRBXL647XANH4PNQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ALNEEQYMEDRRBXL647XANH4PNQ/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":"2026-08-11T03:21:25Z","links":{"resolver":"https://pith.science/pith/ALNEEQYMEDRRBXL647XANH4PNQ","bundle":"https://pith.science/pith/ALNEEQYMEDRRBXL647XANH4PNQ/bundle.json","state":"https://pith.science/pith/ALNEEQYMEDRRBXL647XANH4PNQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ALNEEQYMEDRRBXL647XANH4PNQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ALNEEQYMEDRRBXL647XANH4PNQ","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":"2ab60098a94a31b98e73adfb2e9a9b90d5c8dcad14d7c32ed740af341aeb72bd","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-06T19:00:00Z","title_canon_sha256":"ea76f09e4a51e9930521e013e687eb8378c4c51c7c05944355ff6ff9e0f481f3"},"schema_version":"1.0","source":{"id":"1911.02569","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.02569","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"arxiv_version","alias_value":"1911.02569v1","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.02569","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"pith_short_12","alias_value":"ALNEEQYMEDRR","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"pith_short_16","alias_value":"ALNEEQYMEDRRBXL6","created_at":"2026-07-05T00:17:39Z"},{"alias_kind":"pith_short_8","alias_value":"ALNEEQYM","created_at":"2026-07-05T00:17:39Z"}],"graph_snapshots":[{"event_id":"sha256:1680de94a21f08439fa60cef6b48d1080b18442f4612b0581131127b1b533929","target":"graph","created_at":"2026-07-05T00:17:39Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1911.02569/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the minimum total weight of a disk triangulation using vertices out of $\\{1,\\ldots,n\\}$, where the boundary is the triangle $(123)$ and the $\\binom{n}3$ triangles have independent weights, e.g. $\\mathrm{Exp}(1)$ or $\\mathrm{U}(0,1)$.\n  We show that for explicit constants $c_1,c_2>0$, this minimum is $c_1 \\frac{\\log n}{\\sqrt n} + c_2 \\frac{\\log\\log n}{\\sqrt n} + \\frac{Y_n}{\\sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $\\frac14\\log n + O_P(\\sqrt{\\log n}) $ vertices.\n  Moreover, for disk triangulations that are canonical, in th","authors_text":"Eyal Lubetzky, Itai Benjamini, Yuval Peled","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-06T19:00:00Z","title":"Minimum weight disk triangulations and fillings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.02569","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:dd7d724013e763483d40e2f001fce0cd10120909bee849ee12cee28643fe479b","target":"record","created_at":"2026-07-05T00:17:39Z","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":"2ab60098a94a31b98e73adfb2e9a9b90d5c8dcad14d7c32ed740af341aeb72bd","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-06T19:00:00Z","title_canon_sha256":"ea76f09e4a51e9930521e013e687eb8378c4c51c7c05944355ff6ff9e0f481f3"},"schema_version":"1.0","source":{"id":"1911.02569","kind":"arxiv","version":1}},"canonical_sha256":"02da42430c20e310dd7ee7ee069f8f6c2924d4b150544162bea78872db5918dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"02da42430c20e310dd7ee7ee069f8f6c2924d4b150544162bea78872db5918dd","first_computed_at":"2026-07-05T00:17:39.424347Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:17:39.424347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QstAq5fIt9c13a8e0LFK/QZw5RL1YllWiosAhQXI1P6rnEYGYvcVbr6673Nn96Epm3tqmpS2U6gH//brPY6ZDw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:17:39.424801Z","signed_message":"canonical_sha256_bytes"},"source_id":"1911.02569","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd7d724013e763483d40e2f001fce0cd10120909bee849ee12cee28643fe479b","sha256:1680de94a21f08439fa60cef6b48d1080b18442f4612b0581131127b1b533929"],"state_sha256":"a04f4342b586410aab05d1e0797b27661e01a28a416b2a244216f2ce895d5697"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lsWs0VPAB2QZFWQoVJAhz9Njdgl09bcyaxeOv9PRVgD8awg4LijwqF3JKjCBlzAdx8zr4tecrAlgyg23PkV7Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T03:21:25.406438Z","bundle_sha256":"6da3c3dc0b8b8f3b6ea0259421f76bb2eab992f6691c2100c846b93df1096a07"}}