{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:4V7XMK5CG4QCLQL3T4QQ6NWY7S","short_pith_number":"pith:4V7XMK5C","canonical_record":{"source":{"id":"1504.03921","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.DG","submitted_at":"2015-04-15T14:10:02Z","cross_cats_sorted":[],"title_canon_sha256":"fc67bff736fe0b5297c6060a571d08e6eed1cdcd8ffb4c0431d9b147cbea33b3","abstract_canon_sha256":"39d30a289e5d88610a95df0d3f3a43b44ccd58ced1b9544430d0579f404477de"},"schema_version":"1.0"},"canonical_sha256":"e57f762ba2372025c17b9f210f36d8fc8bf3a2e05cf8ee04ea4bd16189e8da02","source":{"kind":"arxiv","id":"1504.03921","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03921","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03921v3","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03921","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"pith_short_12","alias_value":"4V7XMK5CG4QC","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4V7XMK5CG4QCLQL3","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4V7XMK5C","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:4V7XMK5CG4QCLQL3T4QQ6NWY7S","target":"record","payload":{"canonical_record":{"source":{"id":"1504.03921","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.DG","submitted_at":"2015-04-15T14:10:02Z","cross_cats_sorted":[],"title_canon_sha256":"fc67bff736fe0b5297c6060a571d08e6eed1cdcd8ffb4c0431d9b147cbea33b3","abstract_canon_sha256":"39d30a289e5d88610a95df0d3f3a43b44ccd58ced1b9544430d0579f404477de"},"schema_version":"1.0"},"canonical_sha256":"e57f762ba2372025c17b9f210f36d8fc8bf3a2e05cf8ee04ea4bd16189e8da02","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:12.828721Z","signature_b64":"Ikhmr22Mfugx509rqxJUMNBNQMF1iWceFcWfR2Ml0Kz5inEfyE6NhbNtEy2l3esTm6xZ055C4yo/OBIozongBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e57f762ba2372025c17b9f210f36d8fc8bf3a2e05cf8ee04ea4bd16189e8da02","last_reissued_at":"2026-05-18T01:17:12.828064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:12.828064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.03921","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p5UwiNEmm6YbNoTr+q6F7fqNdFvLcJbGYwttZPxomkiLYw7FEA3Zv+ilx0Feyccz6u5NcYUauUba2shSpf13CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T00:47:36.611135Z"},"content_sha256":"9a872170e207c27bfd2f66bfc8cad0a971968fb0163a47dd6630efc08d82719b","schema_version":"1.0","event_id":"sha256:9a872170e207c27bfd2f66bfc8cad0a971968fb0163a47dd6630efc08d82719b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:4V7XMK5CG4QCLQL3T4QQ6NWY7S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Co-Points of Rays are Cut Points of Upper Level Sets for Busemann Functions","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Sorin V. Sabau","submitted_at":"2015-04-15T14:10:02Z","abstract_excerpt":"We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, and other properties follow immediately from the known results about the cut locus. We point out that some of our findings, in special the relation of co-point set to the upper lever sets, are new even for Riemannian manifolds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03921","kind":"arxiv","version":3},"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":"2026-05-18T01:17:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9ROTUZZahdRfp7a6CP/1BYtmYPk2V/ATEU3zrEw2OHcgAbD7gEeLGBuxedcPoaZGPzQhY5ooQwXzAGNx8A6CCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T00:47:36.611521Z"},"content_sha256":"f79c533cec34220fa2dd24c5936073a6216d33f9efa400088d492809986967d8","schema_version":"1.0","event_id":"sha256:f79c533cec34220fa2dd24c5936073a6216d33f9efa400088d492809986967d8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S/bundle.json","state_url":"https://pith.science/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S/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":"2026-07-04T00:47:36Z","links":{"resolver":"https://pith.science/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S","bundle":"https://pith.science/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S/bundle.json","state":"https://pith.science/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4V7XMK5CG4QCLQL3T4QQ6NWY7S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4V7XMK5CG4QCLQL3T4QQ6NWY7S","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":"39d30a289e5d88610a95df0d3f3a43b44ccd58ced1b9544430d0579f404477de","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.DG","submitted_at":"2015-04-15T14:10:02Z","title_canon_sha256":"fc67bff736fe0b5297c6060a571d08e6eed1cdcd8ffb4c0431d9b147cbea33b3"},"schema_version":"1.0","source":{"id":"1504.03921","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03921","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03921v3","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03921","created_at":"2026-05-18T01:17:12Z"},{"alias_kind":"pith_short_12","alias_value":"4V7XMK5CG4QC","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4V7XMK5CG4QCLQL3","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4V7XMK5C","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:f79c533cec34220fa2dd24c5936073a6216d33f9efa400088d492809986967d8","target":"graph","created_at":"2026-05-18T01:17:12Z","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 show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, and other properties follow immediately from the known results about the cut locus. We point out that some of our findings, in special the relation of co-point set to the upper lever sets, are new even for Riemannian manifolds.","authors_text":"Sorin V. Sabau","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.DG","submitted_at":"2015-04-15T14:10:02Z","title":"The Co-Points of Rays are Cut Points of Upper Level Sets for Busemann Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03921","kind":"arxiv","version":3},"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":"sha256:9a872170e207c27bfd2f66bfc8cad0a971968fb0163a47dd6630efc08d82719b","target":"record","created_at":"2026-05-18T01:17:12Z","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":"39d30a289e5d88610a95df0d3f3a43b44ccd58ced1b9544430d0579f404477de","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.DG","submitted_at":"2015-04-15T14:10:02Z","title_canon_sha256":"fc67bff736fe0b5297c6060a571d08e6eed1cdcd8ffb4c0431d9b147cbea33b3"},"schema_version":"1.0","source":{"id":"1504.03921","kind":"arxiv","version":3}},"canonical_sha256":"e57f762ba2372025c17b9f210f36d8fc8bf3a2e05cf8ee04ea4bd16189e8da02","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e57f762ba2372025c17b9f210f36d8fc8bf3a2e05cf8ee04ea4bd16189e8da02","first_computed_at":"2026-05-18T01:17:12.828064Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:12.828064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ikhmr22Mfugx509rqxJUMNBNQMF1iWceFcWfR2Ml0Kz5inEfyE6NhbNtEy2l3esTm6xZ055C4yo/OBIozongBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:12.828721Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.03921","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a872170e207c27bfd2f66bfc8cad0a971968fb0163a47dd6630efc08d82719b","sha256:f79c533cec34220fa2dd24c5936073a6216d33f9efa400088d492809986967d8"],"state_sha256":"e4915b063dc62dce95fc3d70330a3f5b3233ef6e1e7b33cec39232d4a2e79e2a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Duf410xJDUa7PN7gPkmFFT7+8eiMyWYGiMHThDC+eHtynv5iKiVHcrjr+MLtn/uOAqFrSJy0Lj291jVmG4h8Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T00:47:36.614739Z","bundle_sha256":"3d084fbfaafec8c6c2f90fefb5e4acd28c7e5aadc8fc926cf97a714c0b9c7b51"}}