{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:PGU66PEXVZAVQVZ5OZ2MYX3FNO","short_pith_number":"pith:PGU66PEX","schema_version":"1.0","canonical_sha256":"79a9ef3c97ae4158573d7674cc5f656b9789398d6213247ca4f3ab75d4c8a828","source":{"kind":"arxiv","id":"2607.05383","version":1},"attestation_state":"computed","paper":{"title":"Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Evanthia Papadopoulou, Nicolau Oliver, Sang Won Bae","submitted_at":"2026-07-06T17:56:13Z","abstract_excerpt":"Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set $S$ of $n$ colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-$k$ abstract color Voronoi diagram is at most $4k(n-k)-2n$, and present an iterative construction algorithm"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.05383","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2026-07-06T17:56:13Z","cross_cats_sorted":[],"title_canon_sha256":"e3bf1d23ff8c830d76befe71016f41f3c708457c0908f9f33df93f93d0e8bcfc","abstract_canon_sha256":"2e9b8ed35e891ed439a0debd901ee9711d1012c3dbb3653a3b40d3a9e8d5ee8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T03:19:25.598110Z","signature_b64":"F7xzZA4nGIJAQSdvPnL2pRWLMn6Rvg7muplNS4Mla1vQlFuutk7X2hWxOQtgSC9+gXZZb7fgIayjSn2WobM0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"79a9ef3c97ae4158573d7674cc5f656b9789398d6213247ca4f3ab75d4c8a828","last_reissued_at":"2026-07-07T03:19:25.597539Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T03:19:25.597539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Abstract Color Voronoi Diagrams and Circular Sequences of Color Permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Evanthia Papadopoulou, Nicolau Oliver, Sang Won Bae","submitted_at":"2026-07-06T17:56:13Z","abstract_excerpt":"Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites and metrics. In this paper, we formulate higher-order abstract color Voronoi diagrams of a set $S$ of $n$ colored abstract sites, simultaneously considering all concrete instances under their umbrella. We prove that the number of vertices in the order-$k$ abstract color Voronoi diagram is at most $4k(n-k)-2n$, and present an iterative construction algorithm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05383","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/2607.05383/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.05383","created_at":"2026-07-07T03:19:25.597631+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.05383v1","created_at":"2026-07-07T03:19:25.597631+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05383","created_at":"2026-07-07T03:19:25.597631+00:00"},{"alias_kind":"pith_short_12","alias_value":"PGU66PEXVZAV","created_at":"2026-07-07T03:19:25.597631+00:00"},{"alias_kind":"pith_short_16","alias_value":"PGU66PEXVZAVQVZ5","created_at":"2026-07-07T03:19:25.597631+00:00"},{"alias_kind":"pith_short_8","alias_value":"PGU66PEX","created_at":"2026-07-07T03:19:25.597631+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO","json":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO.json","graph_json":"https://pith.science/api/pith-number/PGU66PEXVZAVQVZ5OZ2MYX3FNO/graph.json","events_json":"https://pith.science/api/pith-number/PGU66PEXVZAVQVZ5OZ2MYX3FNO/events.json","paper":"https://pith.science/paper/PGU66PEX"},"agent_actions":{"view_html":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO","download_json":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO.json","view_paper":"https://pith.science/paper/PGU66PEX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.05383&json=true","fetch_graph":"https://pith.science/api/pith-number/PGU66PEXVZAVQVZ5OZ2MYX3FNO/graph.json","fetch_events":"https://pith.science/api/pith-number/PGU66PEXVZAVQVZ5OZ2MYX3FNO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO/action/storage_attestation","attest_author":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO/action/author_attestation","sign_citation":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO/action/citation_signature","submit_replication":"https://pith.science/pith/PGU66PEXVZAVQVZ5OZ2MYX3FNO/action/replication_record"}},"created_at":"2026-07-07T03:19:25.597631+00:00","updated_at":"2026-07-07T03:19:25.597631+00:00"}