{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YWR2UBD6FHGK7C26QJ5NSLQTA7","short_pith_number":"pith:YWR2UBD6","canonical_record":{"source":{"id":"1210.5603","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-10-20T11:10:12Z","cross_cats_sorted":[],"title_canon_sha256":"e5b2284bb31c90107f29c271ac7e428fb17bf52bc4b55f8f23b8272daf5627af","abstract_canon_sha256":"f4d25a87fc209930f5503ae44576ead73ef35090ead70568c24c228ab6b0a2f2"},"schema_version":"1.0"},"canonical_sha256":"c5a3aa047e29ccaf8b5e827ad92e1307e968cff303e2610df4e2bd888fa761ed","source":{"kind":"arxiv","id":"1210.5603","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5603","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5603v1","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5603","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"pith_short_12","alias_value":"YWR2UBD6FHGK","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"YWR2UBD6FHGK7C26","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"YWR2UBD6","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YWR2UBD6FHGK7C26QJ5NSLQTA7","target":"record","payload":{"canonical_record":{"source":{"id":"1210.5603","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-10-20T11:10:12Z","cross_cats_sorted":[],"title_canon_sha256":"e5b2284bb31c90107f29c271ac7e428fb17bf52bc4b55f8f23b8272daf5627af","abstract_canon_sha256":"f4d25a87fc209930f5503ae44576ead73ef35090ead70568c24c228ab6b0a2f2"},"schema_version":"1.0"},"canonical_sha256":"c5a3aa047e29ccaf8b5e827ad92e1307e968cff303e2610df4e2bd888fa761ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:43.561011Z","signature_b64":"eOOAqpYHajjg+0smNbQHCO6OOk7ZYr0JapJJxD3yrt5f/BVhAbYZfM8XQvOKJ0zcBa05P73s3Jmwua8AleX0DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5a3aa047e29ccaf8b5e827ad92e1307e968cff303e2610df4e2bd888fa761ed","last_reissued_at":"2026-05-18T03:42:43.560574Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:43.560574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.5603","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":"2026-05-18T03:42:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xxwsFWTpNcPF5tcqdRwYaT4pkFhciOpsD/dmaJ6FNyqbWoiOvtvSVcJeWEoip5izQnNeGQaPNaIUzg1ebKJHDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T14:09:22.894707Z"},"content_sha256":"a89160883e395c88a17b40d6b6da577953ba5712f908be0d6f6a9a7a18791fb4","schema_version":"1.0","event_id":"sha256:a89160883e395c88a17b40d6b6da577953ba5712f908be0d6f6a9a7a18791fb4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YWR2UBD6FHGK7C26QJ5NSLQTA7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"One Dimensional T.T.T Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Daniel Lowengrub","submitted_at":"2012-10-20T11:10:12Z","abstract_excerpt":"In this paper we analyze the relationship between o-minimal structures and the notion of \\omega -saturated one dimensional t.t.t structures. We prove that if removing any point from such a structure splits it into more than one definably connected component then it must be a one dimensional simplex of a finite number of o-minimal structures. In addition, we show that even if removing points doesn't split the structure, additional topological assumptions ensure that the structure is locally o-minimal. As a corollary we obtain the result that if an \\omega -saturated one dimensional t.t.t structu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5603","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":"2026-05-18T03:42:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GXEt4tEO7vJsl4szoMp8hR91AKtJVm8E/0AjOWkvfCdjaGTNJMhC0nAfmQ47f4jwSPZaTAtfYLXRdXujcAh/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T14:09:22.895049Z"},"content_sha256":"b2bc153f0f9c94ad97c19141b74c9f9a605199a547e4495722a754a7546dbc9f","schema_version":"1.0","event_id":"sha256:b2bc153f0f9c94ad97c19141b74c9f9a605199a547e4495722a754a7546dbc9f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7/bundle.json","state_url":"https://pith.science/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7/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":"2026-06-25T14:09:22Z","links":{"resolver":"https://pith.science/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7","bundle":"https://pith.science/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7/bundle.json","state":"https://pith.science/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YWR2UBD6FHGK7C26QJ5NSLQTA7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YWR2UBD6FHGK7C26QJ5NSLQTA7","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":"f4d25a87fc209930f5503ae44576ead73ef35090ead70568c24c228ab6b0a2f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-10-20T11:10:12Z","title_canon_sha256":"e5b2284bb31c90107f29c271ac7e428fb17bf52bc4b55f8f23b8272daf5627af"},"schema_version":"1.0","source":{"id":"1210.5603","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5603","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5603v1","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5603","created_at":"2026-05-18T03:42:43Z"},{"alias_kind":"pith_short_12","alias_value":"YWR2UBD6FHGK","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"YWR2UBD6FHGK7C26","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"YWR2UBD6","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:b2bc153f0f9c94ad97c19141b74c9f9a605199a547e4495722a754a7546dbc9f","target":"graph","created_at":"2026-05-18T03:42:43Z","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 paper we analyze the relationship between o-minimal structures and the notion of \\omega -saturated one dimensional t.t.t structures. We prove that if removing any point from such a structure splits it into more than one definably connected component then it must be a one dimensional simplex of a finite number of o-minimal structures. In addition, we show that even if removing points doesn't split the structure, additional topological assumptions ensure that the structure is locally o-minimal. As a corollary we obtain the result that if an \\omega -saturated one dimensional t.t.t structu","authors_text":"Daniel Lowengrub","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-10-20T11:10:12Z","title":"One Dimensional T.T.T Structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5603","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:a89160883e395c88a17b40d6b6da577953ba5712f908be0d6f6a9a7a18791fb4","target":"record","created_at":"2026-05-18T03:42:43Z","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":"f4d25a87fc209930f5503ae44576ead73ef35090ead70568c24c228ab6b0a2f2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-10-20T11:10:12Z","title_canon_sha256":"e5b2284bb31c90107f29c271ac7e428fb17bf52bc4b55f8f23b8272daf5627af"},"schema_version":"1.0","source":{"id":"1210.5603","kind":"arxiv","version":1}},"canonical_sha256":"c5a3aa047e29ccaf8b5e827ad92e1307e968cff303e2610df4e2bd888fa761ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c5a3aa047e29ccaf8b5e827ad92e1307e968cff303e2610df4e2bd888fa761ed","first_computed_at":"2026-05-18T03:42:43.560574Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:43.560574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eOOAqpYHajjg+0smNbQHCO6OOk7ZYr0JapJJxD3yrt5f/BVhAbYZfM8XQvOKJ0zcBa05P73s3Jmwua8AleX0DA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:43.561011Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.5603","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a89160883e395c88a17b40d6b6da577953ba5712f908be0d6f6a9a7a18791fb4","sha256:b2bc153f0f9c94ad97c19141b74c9f9a605199a547e4495722a754a7546dbc9f"],"state_sha256":"bb794b8ac333398d3014747d2a605bfdfe6e69210dbc630225bbaeaee229aab9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IONx4V0GJUESdn3aPJRLpHZOaHmXXOTq3lnbpJjoYQN7fsVtl462/AWG+GQNLSBPx7ajm2paH3q3uPUFUC6PDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T14:09:22.897117Z","bundle_sha256":"eab644f515d24aa43985bfafa351773b3ab3de3bf80469bb5841a894be733f02"}}