{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PSNB3XQFMCWZCBBY66L5PMR4WB","short_pith_number":"pith:PSNB3XQF","canonical_record":{"source":{"id":"1509.07101","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-23T19:23:45Z","cross_cats_sorted":[],"title_canon_sha256":"badf772fe1eae2f1ab7901d74e1c49c9c6ecde7a35ef02dbfc98b123315d8c27","abstract_canon_sha256":"b80a30ba99636ba61c60d7cb7ff97614dadb6987bd8dcdd1f035866a14b4d212"},"schema_version":"1.0"},"canonical_sha256":"7c9a1dde0560ad910438f797d7b23cb07d4610f8f2242e51291864ffae7f2d8b","source":{"kind":"arxiv","id":"1509.07101","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07101","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07101v4","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07101","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"PSNB3XQFMCWZ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PSNB3XQFMCWZCBBY","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PSNB3XQF","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PSNB3XQFMCWZCBBY66L5PMR4WB","target":"record","payload":{"canonical_record":{"source":{"id":"1509.07101","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-23T19:23:45Z","cross_cats_sorted":[],"title_canon_sha256":"badf772fe1eae2f1ab7901d74e1c49c9c6ecde7a35ef02dbfc98b123315d8c27","abstract_canon_sha256":"b80a30ba99636ba61c60d7cb7ff97614dadb6987bd8dcdd1f035866a14b4d212"},"schema_version":"1.0"},"canonical_sha256":"7c9a1dde0560ad910438f797d7b23cb07d4610f8f2242e51291864ffae7f2d8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:34.829707Z","signature_b64":"qvcd8k/3lul9ba3+IfkTKzlYvyYEmOIJCMl95Vf+Y8nY0+DrHEnnSNBVcAhn+C2JMTh9cztiq6NZ+Lous/5mCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c9a1dde0560ad910438f797d7b23cb07d4610f8f2242e51291864ffae7f2d8b","last_reissued_at":"2026-05-18T01:12:34.829186Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:34.829186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.07101","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P7+GYIfvbDlMSsywAcL9Do9FU/dOUDN/WjpINvBQ+aIYiTMPwW4anfFO4mEud3jKE9o0btWcAHHxN/Pm4SvBBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T04:41:56.247266Z"},"content_sha256":"9ddb34fe47fdc62d4069cc41229c6064fb868ce75d7fea0f6c204e87d28c32ca","schema_version":"1.0","event_id":"sha256:9ddb34fe47fdc62d4069cc41229c6064fb868ce75d7fea0f6c204e87d28c32ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PSNB3XQFMCWZCBBY66L5PMR4WB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generic finiteness of minimal surfaces with bounded Morse index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alessandro Carlotto","submitted_at":"2015-09-23T19:23:45Z","abstract_excerpt":"Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07101","kind":"arxiv","version":4},"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":"2026-05-18T01:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"llznNpd4lAOLUF2mdA0ZwfdKiA84MCKL+n7yOHsK2k9E1PqQqf9W20VIgzi2GhuKYiii5il6t1oMsyBna4nnBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T04:41:56.247636Z"},"content_sha256":"a261028781025c844717b0c175de59357d18fa4ab2ae08316685bb9147db3de8","schema_version":"1.0","event_id":"sha256:a261028781025c844717b0c175de59357d18fa4ab2ae08316685bb9147db3de8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PSNB3XQFMCWZCBBY66L5PMR4WB/bundle.json","state_url":"https://pith.science/pith/PSNB3XQFMCWZCBBY66L5PMR4WB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PSNB3XQFMCWZCBBY66L5PMR4WB/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":"2026-06-24T04:41:56Z","links":{"resolver":"https://pith.science/pith/PSNB3XQFMCWZCBBY66L5PMR4WB","bundle":"https://pith.science/pith/PSNB3XQFMCWZCBBY66L5PMR4WB/bundle.json","state":"https://pith.science/pith/PSNB3XQFMCWZCBBY66L5PMR4WB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PSNB3XQFMCWZCBBY66L5PMR4WB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PSNB3XQFMCWZCBBY66L5PMR4WB","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":"b80a30ba99636ba61c60d7cb7ff97614dadb6987bd8dcdd1f035866a14b4d212","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-23T19:23:45Z","title_canon_sha256":"badf772fe1eae2f1ab7901d74e1c49c9c6ecde7a35ef02dbfc98b123315d8c27"},"schema_version":"1.0","source":{"id":"1509.07101","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07101","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07101v4","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07101","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"PSNB3XQFMCWZ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PSNB3XQFMCWZCBBY","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PSNB3XQF","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:a261028781025c844717b0c175de59357d18fa4ab2ae08316685bb9147db3de8","target":"graph","created_at":"2026-05-18T01:12:34Z","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":"Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not ha","authors_text":"Alessandro Carlotto","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-23T19:23:45Z","title":"Generic finiteness of minimal surfaces with bounded Morse index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07101","kind":"arxiv","version":4},"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":"sha256:9ddb34fe47fdc62d4069cc41229c6064fb868ce75d7fea0f6c204e87d28c32ca","target":"record","created_at":"2026-05-18T01:12:34Z","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":"b80a30ba99636ba61c60d7cb7ff97614dadb6987bd8dcdd1f035866a14b4d212","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-09-23T19:23:45Z","title_canon_sha256":"badf772fe1eae2f1ab7901d74e1c49c9c6ecde7a35ef02dbfc98b123315d8c27"},"schema_version":"1.0","source":{"id":"1509.07101","kind":"arxiv","version":4}},"canonical_sha256":"7c9a1dde0560ad910438f797d7b23cb07d4610f8f2242e51291864ffae7f2d8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c9a1dde0560ad910438f797d7b23cb07d4610f8f2242e51291864ffae7f2d8b","first_computed_at":"2026-05-18T01:12:34.829186Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:34.829186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qvcd8k/3lul9ba3+IfkTKzlYvyYEmOIJCMl95Vf+Y8nY0+DrHEnnSNBVcAhn+C2JMTh9cztiq6NZ+Lous/5mCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:34.829707Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.07101","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ddb34fe47fdc62d4069cc41229c6064fb868ce75d7fea0f6c204e87d28c32ca","sha256:a261028781025c844717b0c175de59357d18fa4ab2ae08316685bb9147db3de8"],"state_sha256":"727f5df9b8256d7af6ddc49501b2cbb2bd2997e7159cb49347d464d13790334f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ymR4uAtTiZ8LvZNk9uNNOYXpRm7DOTHCMsFKK1kjznO3DOwLioaVbD7AXgVCoEokmJcl5FAZJDQh5vhrWFp0BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T04:41:56.249587Z","bundle_sha256":"73fd9915598de9281fc3ae1cf07f99a173d2b9f8c51f36b33bf4814dcb735f34"}}