{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:PSBRWMZ5XGYSNPGSKJBJFQQYKR","short_pith_number":"pith:PSBRWMZ5","canonical_record":{"source":{"id":"1208.5045","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2012-08-17T19:28:36Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"347f47da1349e3c45c76410b1bca8118f02cef6b6289a2a68d9aed207b86e524","abstract_canon_sha256":"445da685e5bb20fe3dcbe1d39fed11936b2a29dcc47db6865232614cb8b5f67b"},"schema_version":"1.0"},"canonical_sha256":"7c831b333db9b126bcd2524292c2185471b3b25c02a117e8f7af41a7ec19b3c5","source":{"kind":"arxiv","id":"1208.5045","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5045","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5045v1","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5045","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"pith_short_12","alias_value":"PSBRWMZ5XGYS","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PSBRWMZ5XGYSNPGS","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PSBRWMZ5","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:PSBRWMZ5XGYSNPGSKJBJFQQYKR","target":"record","payload":{"canonical_record":{"source":{"id":"1208.5045","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2012-08-17T19:28:36Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"347f47da1349e3c45c76410b1bca8118f02cef6b6289a2a68d9aed207b86e524","abstract_canon_sha256":"445da685e5bb20fe3dcbe1d39fed11936b2a29dcc47db6865232614cb8b5f67b"},"schema_version":"1.0"},"canonical_sha256":"7c831b333db9b126bcd2524292c2185471b3b25c02a117e8f7af41a7ec19b3c5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:47:52.024041Z","signature_b64":"sEH24dagAXNHZUmx4WoaHBSgBxamV3qVrIxlSJoi2mhWkO1gX1pFgK1eGTmxuRlc2Ud6Z11BA5rEL3sLQxOXCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c831b333db9b126bcd2524292c2185471b3b25c02a117e8f7af41a7ec19b3c5","last_reissued_at":"2026-05-18T03:47:52.023605Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:47:52.023605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.5045","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":"2026-05-18T03:47:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3DHXsvoxv77EaNyaewqc2GWB/wa1uSC4E0M9CCH36Tyg5YYIOXx3up93n7XgOMzp9frNgnq8zhcKcqrn2h6ABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T05:48:31.214379Z"},"content_sha256":"d18a7e70bea5190b04808bbd1bcd7126ee644c67cf96aaddef8347adf4b54e9c","schema_version":"1.0","event_id":"sha256:d18a7e70bea5190b04808bbd1bcd7126ee644c67cf96aaddef8347adf4b54e9c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:PSBRWMZ5XGYSNPGSKJBJFQQYKR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the existence of a neutral region","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"cs.CG","authors_text":"Daniel Reem","submitted_at":"2012-08-17T19:28:36Z","abstract_excerpt":"Consider a given space, e.g., the Euclidean plane, and its decomposition into Voronoi regions induced by given sites. It seems intuitively clear that each point in the space belongs to at least one of the regions, i.e., no neutral region can exist. As simple counterexamples show this is not true in general, but we present a simple necessary and sufficient condition ensuring the non-existence of a neutral region. We discuss a similar phenomenon concerning recent variations of Voronoi diagrams called zone diagrams, double zone diagrams, and (double) territory diagrams. These objects are defined "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5045","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":"2026-05-18T03:47:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"as9scLu2LE7lEZCobQeUCnFQOJF7Q0AdtohipEwYwzI5RajRyGx1RmjT3g0sCtJjndSM0hgY9pzUUZyazSBOCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T05:48:31.214724Z"},"content_sha256":"49a5971ce7427d5fcc335920af58bb806dc345dca3e6843291784dc2e6eaa5eb","schema_version":"1.0","event_id":"sha256:49a5971ce7427d5fcc335920af58bb806dc345dca3e6843291784dc2e6eaa5eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR/bundle.json","state_url":"https://pith.science/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR/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":"2026-06-30T05:48:31Z","links":{"resolver":"https://pith.science/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR","bundle":"https://pith.science/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR/bundle.json","state":"https://pith.science/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PSBRWMZ5XGYSNPGSKJBJFQQYKR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:PSBRWMZ5XGYSNPGSKJBJFQQYKR","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":"445da685e5bb20fe3dcbe1d39fed11936b2a29dcc47db6865232614cb8b5f67b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2012-08-17T19:28:36Z","title_canon_sha256":"347f47da1349e3c45c76410b1bca8118f02cef6b6289a2a68d9aed207b86e524"},"schema_version":"1.0","source":{"id":"1208.5045","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5045","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5045v1","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5045","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"pith_short_12","alias_value":"PSBRWMZ5XGYS","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"PSBRWMZ5XGYSNPGS","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"PSBRWMZ5","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:49a5971ce7427d5fcc335920af58bb806dc345dca3e6843291784dc2e6eaa5eb","target":"graph","created_at":"2026-05-18T03:47:52Z","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":"Consider a given space, e.g., the Euclidean plane, and its decomposition into Voronoi regions induced by given sites. It seems intuitively clear that each point in the space belongs to at least one of the regions, i.e., no neutral region can exist. As simple counterexamples show this is not true in general, but we present a simple necessary and sufficient condition ensuring the non-existence of a neutral region. We discuss a similar phenomenon concerning recent variations of Voronoi diagrams called zone diagrams, double zone diagrams, and (double) territory diagrams. These objects are defined ","authors_text":"Daniel Reem","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2012-08-17T19:28:36Z","title":"On the existence of a neutral region"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5045","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:d18a7e70bea5190b04808bbd1bcd7126ee644c67cf96aaddef8347adf4b54e9c","target":"record","created_at":"2026-05-18T03:47:52Z","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":"445da685e5bb20fe3dcbe1d39fed11936b2a29dcc47db6865232614cb8b5f67b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2012-08-17T19:28:36Z","title_canon_sha256":"347f47da1349e3c45c76410b1bca8118f02cef6b6289a2a68d9aed207b86e524"},"schema_version":"1.0","source":{"id":"1208.5045","kind":"arxiv","version":1}},"canonical_sha256":"7c831b333db9b126bcd2524292c2185471b3b25c02a117e8f7af41a7ec19b3c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c831b333db9b126bcd2524292c2185471b3b25c02a117e8f7af41a7ec19b3c5","first_computed_at":"2026-05-18T03:47:52.023605Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:47:52.023605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sEH24dagAXNHZUmx4WoaHBSgBxamV3qVrIxlSJoi2mhWkO1gX1pFgK1eGTmxuRlc2Ud6Z11BA5rEL3sLQxOXCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:47:52.024041Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.5045","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d18a7e70bea5190b04808bbd1bcd7126ee644c67cf96aaddef8347adf4b54e9c","sha256:49a5971ce7427d5fcc335920af58bb806dc345dca3e6843291784dc2e6eaa5eb"],"state_sha256":"485bc0ec1744bdf73e762008ec203a6d20bca0da0242bcbf4b541c3771adb5fc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"71HRLf2MG1TEtKB7+kZEWITvgMB5Aumppl10MCoFOu40oSq1+IJn83vqgYcvkfXZwajDX4rmVb4UnIYSbJWoDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T05:48:31.216611Z","bundle_sha256":"953966ce45ddaa9bc6f2e6e7bffeb77453a7917c847de7d759270c844a983784"}}