{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:6MKRQGE5N43AFPUKCMFLKA6PFT","short_pith_number":"pith:6MKRQGE5","canonical_record":{"source":{"id":"math/0603097","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-03-03T17:59:43Z","cross_cats_sorted":[],"title_canon_sha256":"42555ac2025fc30487100333ba1531ed3e349186b25bce08f92ee9c27247af21","abstract_canon_sha256":"b8e7533996790f983c11b5296da7bfeec9727d3cff90c5b62f82fd74ae891341"},"schema_version":"1.0"},"canonical_sha256":"f31518189d6f3602be8a130ab503cf2cdd0b11e225e067481187d8a4c938d28d","source":{"kind":"arxiv","id":"math/0603097","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0603097","created_at":"2026-05-18T03:13:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0603097v2","created_at":"2026-05-18T03:13:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603097","created_at":"2026-05-18T03:13:20Z"},{"alias_kind":"pith_short_12","alias_value":"6MKRQGE5N43A","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6MKRQGE5N43AFPUK","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6MKRQGE5","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:6MKRQGE5N43AFPUKCMFLKA6PFT","target":"record","payload":{"canonical_record":{"source":{"id":"math/0603097","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2006-03-03T17:59:43Z","cross_cats_sorted":[],"title_canon_sha256":"42555ac2025fc30487100333ba1531ed3e349186b25bce08f92ee9c27247af21","abstract_canon_sha256":"b8e7533996790f983c11b5296da7bfeec9727d3cff90c5b62f82fd74ae891341"},"schema_version":"1.0"},"canonical_sha256":"f31518189d6f3602be8a130ab503cf2cdd0b11e225e067481187d8a4c938d28d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:20.404525Z","signature_b64":"D9IfZDXUb71H9UUTCR31zAXv8bANsAu3SMc7aNNikChZjJCVLL8MezAGhWRJSMeFuLp2lH/Q21pUbNou+c6/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f31518189d6f3602be8a130ab503cf2cdd0b11e225e067481187d8a4c938d28d","last_reissued_at":"2026-05-18T03:13:20.403952Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:20.403952Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0603097","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w86Czqd5dlFr/BJhrmg37IIpjRF0WGV5lJ/3fxJfQSuepLQDZbO1gdoHtSQQXz/6mM6fD9gtVLc1bgp7sQFRCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T10:14:03.435311Z"},"content_sha256":"b18eac7c919bd0b82d2a0fca4bb65d8dbcb1170dfddee7511d5482bf1e290a9e","schema_version":"1.0","event_id":"sha256:b18eac7c919bd0b82d2a0fca4bb65d8dbcb1170dfddee7511d5482bf1e290a9e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:6MKRQGE5N43AFPUKCMFLKA6PFT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A variational principle for weighted Delaunay triangulations and hyperideal polyhedra","license":"","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Boris A. Springborn","submitted_at":"2006-03-03T17:59:43Z","abstract_excerpt":"We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603097","kind":"arxiv","version":2},"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":"2026-05-18T03:13:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pFsTno5aGxSt66ges3UjpQlGfXOwchf8OUkGyzlLiC3r7BFyMMN+5yloJIYj+u4jmaSKubQtS8VR41lugOghBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T10:14:03.436146Z"},"content_sha256":"96167918dd1364d87a4fde36441f85aadb6387272a3a85df672cd6cf00dd9065","schema_version":"1.0","event_id":"sha256:96167918dd1364d87a4fde36441f85aadb6387272a3a85df672cd6cf00dd9065"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6MKRQGE5N43AFPUKCMFLKA6PFT/bundle.json","state_url":"https://pith.science/pith/6MKRQGE5N43AFPUKCMFLKA6PFT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6MKRQGE5N43AFPUKCMFLKA6PFT/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":"2026-06-09T10:14:03Z","links":{"resolver":"https://pith.science/pith/6MKRQGE5N43AFPUKCMFLKA6PFT","bundle":"https://pith.science/pith/6MKRQGE5N43AFPUKCMFLKA6PFT/bundle.json","state":"https://pith.science/pith/6MKRQGE5N43AFPUKCMFLKA6PFT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6MKRQGE5N43AFPUKCMFLKA6PFT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:6MKRQGE5N43AFPUKCMFLKA6PFT","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":"b8e7533996790f983c11b5296da7bfeec9727d3cff90c5b62f82fd74ae891341","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2006-03-03T17:59:43Z","title_canon_sha256":"42555ac2025fc30487100333ba1531ed3e349186b25bce08f92ee9c27247af21"},"schema_version":"1.0","source":{"id":"math/0603097","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0603097","created_at":"2026-05-18T03:13:20Z"},{"alias_kind":"arxiv_version","alias_value":"math/0603097v2","created_at":"2026-05-18T03:13:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603097","created_at":"2026-05-18T03:13:20Z"},{"alias_kind":"pith_short_12","alias_value":"6MKRQGE5N43A","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6MKRQGE5N43AFPUK","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6MKRQGE5","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:96167918dd1364d87a4fde36441f85aadb6387272a3a85df672cd6cf00dd9065","target":"graph","created_at":"2026-05-18T03:13:20Z","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 use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, w","authors_text":"Boris A. Springborn","cross_cats":[],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2006-03-03T17:59:43Z","title":"A variational principle for weighted Delaunay triangulations and hyperideal polyhedra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603097","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:b18eac7c919bd0b82d2a0fca4bb65d8dbcb1170dfddee7511d5482bf1e290a9e","target":"record","created_at":"2026-05-18T03:13:20Z","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":"b8e7533996790f983c11b5296da7bfeec9727d3cff90c5b62f82fd74ae891341","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2006-03-03T17:59:43Z","title_canon_sha256":"42555ac2025fc30487100333ba1531ed3e349186b25bce08f92ee9c27247af21"},"schema_version":"1.0","source":{"id":"math/0603097","kind":"arxiv","version":2}},"canonical_sha256":"f31518189d6f3602be8a130ab503cf2cdd0b11e225e067481187d8a4c938d28d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f31518189d6f3602be8a130ab503cf2cdd0b11e225e067481187d8a4c938d28d","first_computed_at":"2026-05-18T03:13:20.403952Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:20.403952Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D9IfZDXUb71H9UUTCR31zAXv8bANsAu3SMc7aNNikChZjJCVLL8MezAGhWRJSMeFuLp2lH/Q21pUbNou+c6/Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:20.404525Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0603097","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b18eac7c919bd0b82d2a0fca4bb65d8dbcb1170dfddee7511d5482bf1e290a9e","sha256:96167918dd1364d87a4fde36441f85aadb6387272a3a85df672cd6cf00dd9065"],"state_sha256":"a4bca483f9402dbbf69447ef6ce0047f29295b5116c89c10ccdecf11b05ea532"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oBelo4ScyNCGSm4G2nfBf0Wc941+IJu8rKNo3lJZ9qoW0es/FpuEMBdnaptervakbXfHClt0qs22OaJWpCfiAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T10:14:03.440222Z","bundle_sha256":"49f2bef12654878733ddb90527e375ae8d26bd4e82af0193b09af2de6b5ef857"}}