{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:UEVPC5ZOLVABAGNH5KDRVABDHN","short_pith_number":"pith:UEVPC5ZO","canonical_record":{"source":{"id":"1207.3668","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-16T13:08:23Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"eae92ed33f65c400e92d6eaf12da09a9c8f3d0eb4cacc73988f2c58fa5dba48f","abstract_canon_sha256":"d622c912ccdc1feb1c10e6269c3b29289af876705142c78a43a86b864ba2efbd"},"schema_version":"1.0"},"canonical_sha256":"a12af1772e5d401019a7ea871a80233b60dfb926d2e8da8b4afe020961c50cc3","source":{"kind":"arxiv","id":"1207.3668","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3668","created_at":"2026-05-18T03:50:10Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3668v2","created_at":"2026-05-18T03:50:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3668","created_at":"2026-05-18T03:50:10Z"},{"alias_kind":"pith_short_12","alias_value":"UEVPC5ZOLVAB","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"UEVPC5ZOLVABAGNH","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"UEVPC5ZO","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:UEVPC5ZOLVABAGNH5KDRVABDHN","target":"record","payload":{"canonical_record":{"source":{"id":"1207.3668","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-16T13:08:23Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"eae92ed33f65c400e92d6eaf12da09a9c8f3d0eb4cacc73988f2c58fa5dba48f","abstract_canon_sha256":"d622c912ccdc1feb1c10e6269c3b29289af876705142c78a43a86b864ba2efbd"},"schema_version":"1.0"},"canonical_sha256":"a12af1772e5d401019a7ea871a80233b60dfb926d2e8da8b4afe020961c50cc3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:10.310047Z","signature_b64":"YeycNnhm0id1WL584d+XUd0m0rUXosNnxQJqxAjrB29FN5k4ctLxpGeBzmwiQ/rMEQ8K4CPuH4USo8HUKZ3xAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a12af1772e5d401019a7ea871a80233b60dfb926d2e8da8b4afe020961c50cc3","last_reissued_at":"2026-05-18T03:50:10.309349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:10.309349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.3668","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":"2026-05-18T03:50:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tf6/kvs6XpmAAnOZRxW1R85ERqfB1lsTvonq5P74gCTHXZPxQxYFRZF5ZzCxzYnwPc2vrmqDxsCsDIc7+qO/BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:40:43.535995Z"},"content_sha256":"f79993267fb33f6502777e6944692c1a00ecf09b289664bebd02851a19678103","schema_version":"1.0","event_id":"sha256:f79993267fb33f6502777e6944692c1a00ecf09b289664bebd02851a19678103"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:UEVPC5ZOLVABAGNH5KDRVABDHN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Toponogov's comparison theorem for Alexandrov spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.MG","authors_text":"Urs Lang, Viktor Schroeder","submitted_at":"2012-07-16T13:08:23Z","abstract_excerpt":"In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3668","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":"2026-05-18T03:50:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8ZdhP8opc8u9THxBp+H16WdUKrjpospZkVxhb2+vE65RODGEkj6lGTF0mrLplE3DLqIUM/Tdh6LxBX6PpCAwDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:40:43.536346Z"},"content_sha256":"efda82634976f8ecbb43383b0af0ab9f228423a4d9f27df8c83c75e2c1af7845","schema_version":"1.0","event_id":"sha256:efda82634976f8ecbb43383b0af0ab9f228423a4d9f27df8c83c75e2c1af7845"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UEVPC5ZOLVABAGNH5KDRVABDHN/bundle.json","state_url":"https://pith.science/pith/UEVPC5ZOLVABAGNH5KDRVABDHN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UEVPC5ZOLVABAGNH5KDRVABDHN/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":"2026-06-12T01:40:43Z","links":{"resolver":"https://pith.science/pith/UEVPC5ZOLVABAGNH5KDRVABDHN","bundle":"https://pith.science/pith/UEVPC5ZOLVABAGNH5KDRVABDHN/bundle.json","state":"https://pith.science/pith/UEVPC5ZOLVABAGNH5KDRVABDHN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UEVPC5ZOLVABAGNH5KDRVABDHN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:UEVPC5ZOLVABAGNH5KDRVABDHN","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":"d622c912ccdc1feb1c10e6269c3b29289af876705142c78a43a86b864ba2efbd","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-16T13:08:23Z","title_canon_sha256":"eae92ed33f65c400e92d6eaf12da09a9c8f3d0eb4cacc73988f2c58fa5dba48f"},"schema_version":"1.0","source":{"id":"1207.3668","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3668","created_at":"2026-05-18T03:50:10Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3668v2","created_at":"2026-05-18T03:50:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3668","created_at":"2026-05-18T03:50:10Z"},{"alias_kind":"pith_short_12","alias_value":"UEVPC5ZOLVAB","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"UEVPC5ZOLVABAGNH","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"UEVPC5ZO","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:efda82634976f8ecbb43383b0af0ab9f228423a4d9f27df8c83c75e2c1af7845","target":"graph","created_at":"2026-05-18T03:50:10Z","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 expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut.","authors_text":"Urs Lang, Viktor Schroeder","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-16T13:08:23Z","title":"On Toponogov's comparison theorem for Alexandrov spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3668","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:f79993267fb33f6502777e6944692c1a00ecf09b289664bebd02851a19678103","target":"record","created_at":"2026-05-18T03:50:10Z","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":"d622c912ccdc1feb1c10e6269c3b29289af876705142c78a43a86b864ba2efbd","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-16T13:08:23Z","title_canon_sha256":"eae92ed33f65c400e92d6eaf12da09a9c8f3d0eb4cacc73988f2c58fa5dba48f"},"schema_version":"1.0","source":{"id":"1207.3668","kind":"arxiv","version":2}},"canonical_sha256":"a12af1772e5d401019a7ea871a80233b60dfb926d2e8da8b4afe020961c50cc3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a12af1772e5d401019a7ea871a80233b60dfb926d2e8da8b4afe020961c50cc3","first_computed_at":"2026-05-18T03:50:10.309349Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:10.309349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YeycNnhm0id1WL584d+XUd0m0rUXosNnxQJqxAjrB29FN5k4ctLxpGeBzmwiQ/rMEQ8K4CPuH4USo8HUKZ3xAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:10.310047Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.3668","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f79993267fb33f6502777e6944692c1a00ecf09b289664bebd02851a19678103","sha256:efda82634976f8ecbb43383b0af0ab9f228423a4d9f27df8c83c75e2c1af7845"],"state_sha256":"2f9a10772cfe458699afb7168bdd3ac96bc5472aa201866eff3e532cbfaae119"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nb6JETgyrRmeiFr/g3cfcNuSMC9hFGjTPbJtsZM10emTJOUWoq83x6lUBDDwQ7TPgZkCu2r70MFqU3lEMNB6BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T01:40:43.538144Z","bundle_sha256":"a175592bf2c3bc5390e7a63852ad9dc3014a0d3138313a554e6716a6f1b25cb8"}}