{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:VKE3JGQOFDWK7SGPO6HY3H347W","short_pith_number":"pith:VKE3JGQO","canonical_record":{"source":{"id":"1209.3541","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-17T02:20:15Z","cross_cats_sorted":[],"title_canon_sha256":"e420f7babdcacccf8215757441bfe0faa917527e94782bcb7d10d65285890469","abstract_canon_sha256":"cf867048cebbcdec8c2a180e9ef01bd1524250b6dd0ffb7aaa6ce13d68f1e53e"},"schema_version":"1.0"},"canonical_sha256":"aa89b49a0e28ecafc8cf778f8d9f7cfdb16849f9203a9cfea131cdb7ed0e1139","source":{"kind":"arxiv","id":"1209.3541","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3541","created_at":"2026-05-18T01:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3541v1","created_at":"2026-05-18T01:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3541","created_at":"2026-05-18T01:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"VKE3JGQOFDWK","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VKE3JGQOFDWK7SGP","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VKE3JGQO","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:VKE3JGQOFDWK7SGPO6HY3H347W","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3541","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-17T02:20:15Z","cross_cats_sorted":[],"title_canon_sha256":"e420f7babdcacccf8215757441bfe0faa917527e94782bcb7d10d65285890469","abstract_canon_sha256":"cf867048cebbcdec8c2a180e9ef01bd1524250b6dd0ffb7aaa6ce13d68f1e53e"},"schema_version":"1.0"},"canonical_sha256":"aa89b49a0e28ecafc8cf778f8d9f7cfdb16849f9203a9cfea131cdb7ed0e1139","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:54:27.521945Z","signature_b64":"IWp1rN8lgRLwD/lInaRzrj3Hcx1STMwOtpuhrzjwgzwD7mNuqYv2mU71Y59ryQwgceogpM9pya4Um8683fjHDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa89b49a0e28ecafc8cf778f8d9f7cfdb16849f9203a9cfea131cdb7ed0e1139","last_reissued_at":"2026-05-18T01:54:27.521401Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:54:27.521401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3541","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":"2026-05-18T01:54:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P+SExsRSKGYVvvL+t64Jx3wNaOiir5v8gyr1s3jzt3TBT1S45VgJV+yyxGLkdV4kaUkk8k+SkTjaWJ9cgQYBBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T05:13:33.443479Z"},"content_sha256":"0cf6d4ee5433cde666eec000f6ac7f21b7f9886c92263c70b2bc07a11da96271","schema_version":"1.0","event_id":"sha256:0cf6d4ee5433cde666eec000f6ac7f21b7f9886c92263c70b2bc07a11da96271"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:VKE3JGQOFDWK7SGPO6HY3H347W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Optimality of Functionals over Triangulations of Delaunay Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Herbert Edelsbrunner, Nikolay P. Dolbilin, Oleg R. Musin","submitted_at":"2012-09-17T02:20:15Z","abstract_excerpt":"In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3541","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":"2026-05-18T01:54:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jvyG9KJ3HRBRf/g560A+uXjvu48R7uQBcO43KVpGpsLPUEDRlQpMQN2zfnwoRS+hQUk5G2M7w2auviEeCC7JBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T05:13:33.444060Z"},"content_sha256":"069ab3c94fe955f4f5357064a1f94d99d84a36e72e467261e83b28187072a821","schema_version":"1.0","event_id":"sha256:069ab3c94fe955f4f5357064a1f94d99d84a36e72e467261e83b28187072a821"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VKE3JGQOFDWK7SGPO6HY3H347W/bundle.json","state_url":"https://pith.science/pith/VKE3JGQOFDWK7SGPO6HY3H347W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VKE3JGQOFDWK7SGPO6HY3H347W/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":"2026-06-23T05:13:33Z","links":{"resolver":"https://pith.science/pith/VKE3JGQOFDWK7SGPO6HY3H347W","bundle":"https://pith.science/pith/VKE3JGQOFDWK7SGPO6HY3H347W/bundle.json","state":"https://pith.science/pith/VKE3JGQOFDWK7SGPO6HY3H347W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VKE3JGQOFDWK7SGPO6HY3H347W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:VKE3JGQOFDWK7SGPO6HY3H347W","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":"cf867048cebbcdec8c2a180e9ef01bd1524250b6dd0ffb7aaa6ce13d68f1e53e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-17T02:20:15Z","title_canon_sha256":"e420f7babdcacccf8215757441bfe0faa917527e94782bcb7d10d65285890469"},"schema_version":"1.0","source":{"id":"1209.3541","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3541","created_at":"2026-05-18T01:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3541v1","created_at":"2026-05-18T01:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3541","created_at":"2026-05-18T01:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"VKE3JGQOFDWK","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VKE3JGQOFDWK7SGP","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VKE3JGQO","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:069ab3c94fe955f4f5357064a1f94d99d84a36e72e467261e83b28187072a821","target":"graph","created_at":"2026-05-18T01:54:27Z","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":"In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the plane, then for infinite sets the density of this functional attains its minimum also on the Delaunay triangulations.","authors_text":"Herbert Edelsbrunner, Nikolay P. Dolbilin, Oleg R. Musin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-17T02:20:15Z","title":"On the Optimality of Functionals over Triangulations of Delaunay Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3541","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":"sha256:0cf6d4ee5433cde666eec000f6ac7f21b7f9886c92263c70b2bc07a11da96271","target":"record","created_at":"2026-05-18T01:54:27Z","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":"cf867048cebbcdec8c2a180e9ef01bd1524250b6dd0ffb7aaa6ce13d68f1e53e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-17T02:20:15Z","title_canon_sha256":"e420f7babdcacccf8215757441bfe0faa917527e94782bcb7d10d65285890469"},"schema_version":"1.0","source":{"id":"1209.3541","kind":"arxiv","version":1}},"canonical_sha256":"aa89b49a0e28ecafc8cf778f8d9f7cfdb16849f9203a9cfea131cdb7ed0e1139","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa89b49a0e28ecafc8cf778f8d9f7cfdb16849f9203a9cfea131cdb7ed0e1139","first_computed_at":"2026-05-18T01:54:27.521401Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:54:27.521401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IWp1rN8lgRLwD/lInaRzrj3Hcx1STMwOtpuhrzjwgzwD7mNuqYv2mU71Y59ryQwgceogpM9pya4Um8683fjHDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:54:27.521945Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3541","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0cf6d4ee5433cde666eec000f6ac7f21b7f9886c92263c70b2bc07a11da96271","sha256:069ab3c94fe955f4f5357064a1f94d99d84a36e72e467261e83b28187072a821"],"state_sha256":"1945d9d6b2fd71e96925c0d56d652048d92c18d7e5e5d859e59f550aeeed15fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ny24pm5BN/5IOcSilaWCWJ+qy2GM9pEcxPMXMXnKeBSeZpqsbYay7hqB8OhzZSlC8FIQ5W5YbIHQE16gmg8lCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T05:13:33.446735Z","bundle_sha256":"adeb577d8cedd8b3b6a878ab31c80dc726b91334f4d0feae57406e41b16c15e5"}}