{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:IU3X3G4LJ7AUBIP5KWFAKYCAYZ","short_pith_number":"pith:IU3X3G4L","canonical_record":{"source":{"id":"2101.08217","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-01-20T17:01:02Z","cross_cats_sorted":[],"title_canon_sha256":"c33d335e36ac48402e1dbc645c16b04f0ba1b5bea9afcb69d6cd0b5d847d675c","abstract_canon_sha256":"388f18f0507cc5dd117d397c87facc403f6e9b1e80ac5297675960eae1f62d20"},"schema_version":"1.0"},"canonical_sha256":"45377d9b8b4fc140a1fd558a056040c6455aac5f14901459ea395f66bc643fa4","source":{"kind":"arxiv","id":"2101.08217","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.08217","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"arxiv_version","alias_value":"2101.08217v4","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.08217","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"pith_short_12","alias_value":"IU3X3G4LJ7AU","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"pith_short_16","alias_value":"IU3X3G4LJ7AUBIP5","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"pith_short_8","alias_value":"IU3X3G4L","created_at":"2026-07-05T10:21:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:IU3X3G4LJ7AUBIP5KWFAKYCAYZ","target":"record","payload":{"canonical_record":{"source":{"id":"2101.08217","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-01-20T17:01:02Z","cross_cats_sorted":[],"title_canon_sha256":"c33d335e36ac48402e1dbc645c16b04f0ba1b5bea9afcb69d6cd0b5d847d675c","abstract_canon_sha256":"388f18f0507cc5dd117d397c87facc403f6e9b1e80ac5297675960eae1f62d20"},"schema_version":"1.0"},"canonical_sha256":"45377d9b8b4fc140a1fd558a056040c6455aac5f14901459ea395f66bc643fa4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:21:53.277343Z","signature_b64":"pnrDMAIq8yvS8vQHqftvrTDBT3w4GR5VxYb0Y/Ig9+wROlzIHzNv62tVtgfrfSMTderSyRyiQxUflcvRcIdaDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45377d9b8b4fc140a1fd558a056040c6455aac5f14901459ea395f66bc643fa4","last_reissued_at":"2026-07-05T10:21:53.276865Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:21:53.276865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2101.08217","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":"2026-07-05T10:21:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JxME/n9IolTeeWdWMquyfyzmLoRIyzTJJ5mzx2WffJTDr7z2fa5uhJ/8EgQ005zMq6jX+jlEcPQFzXYXabGjBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:07:11.641294Z"},"content_sha256":"c6cbce31c0429078263e1a29fd3d2b0f4454cb4241b1d968035e0f630b6f2121","schema_version":"1.0","event_id":"sha256:c6cbce31c0429078263e1a29fd3d2b0f4454cb4241b1d968035e0f630b6f2121"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:IU3X3G4LJ7AUBIP5KWFAKYCAYZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rational lines on smooth cubic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Stephen McKean","submitted_at":"2021-01-20T17:01:02Z","abstract_excerpt":"We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a sufficient criterion for each of these line counts in terms of the Galois theory of the base field."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.08217","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.08217/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:21:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NtVMKu/VcdFVyBv6L+/8uS0Ww6JGarL7rT4uK7f80egry1mFakDy/qeppyz/ePQlOOmsGRxdtXc6HK6kMMOGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:07:11.641680Z"},"content_sha256":"ab1d4b65d35503add60dc303da96626c3fd04199c9531844b7741934bc6cbc94","schema_version":"1.0","event_id":"sha256:ab1d4b65d35503add60dc303da96626c3fd04199c9531844b7741934bc6cbc94"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ/bundle.json","state_url":"https://pith.science/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ/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":"2026-07-07T08:07:11Z","links":{"resolver":"https://pith.science/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ","bundle":"https://pith.science/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ/bundle.json","state":"https://pith.science/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IU3X3G4LJ7AUBIP5KWFAKYCAYZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:IU3X3G4LJ7AUBIP5KWFAKYCAYZ","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":"388f18f0507cc5dd117d397c87facc403f6e9b1e80ac5297675960eae1f62d20","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-01-20T17:01:02Z","title_canon_sha256":"c33d335e36ac48402e1dbc645c16b04f0ba1b5bea9afcb69d6cd0b5d847d675c"},"schema_version":"1.0","source":{"id":"2101.08217","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2101.08217","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"arxiv_version","alias_value":"2101.08217v4","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.08217","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"pith_short_12","alias_value":"IU3X3G4LJ7AU","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"pith_short_16","alias_value":"IU3X3G4LJ7AUBIP5","created_at":"2026-07-05T10:21:53Z"},{"alias_kind":"pith_short_8","alias_value":"IU3X3G4L","created_at":"2026-07-05T10:21:53Z"}],"graph_snapshots":[{"event_id":"sha256:ab1d4b65d35503add60dc303da96626c3fd04199c9531844b7741934bc6cbc94","target":"graph","created_at":"2026-07-05T10:21:53Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2101.08217/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a sufficient criterion for each of these line counts in terms of the Galois theory of the base field.","authors_text":"Stephen McKean","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-01-20T17:01:02Z","title":"Rational lines on smooth cubic surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.08217","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:c6cbce31c0429078263e1a29fd3d2b0f4454cb4241b1d968035e0f630b6f2121","target":"record","created_at":"2026-07-05T10:21:53Z","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":"388f18f0507cc5dd117d397c87facc403f6e9b1e80ac5297675960eae1f62d20","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-01-20T17:01:02Z","title_canon_sha256":"c33d335e36ac48402e1dbc645c16b04f0ba1b5bea9afcb69d6cd0b5d847d675c"},"schema_version":"1.0","source":{"id":"2101.08217","kind":"arxiv","version":4}},"canonical_sha256":"45377d9b8b4fc140a1fd558a056040c6455aac5f14901459ea395f66bc643fa4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45377d9b8b4fc140a1fd558a056040c6455aac5f14901459ea395f66bc643fa4","first_computed_at":"2026-07-05T10:21:53.276865Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:21:53.276865Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pnrDMAIq8yvS8vQHqftvrTDBT3w4GR5VxYb0Y/Ig9+wROlzIHzNv62tVtgfrfSMTderSyRyiQxUflcvRcIdaDw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:21:53.277343Z","signed_message":"canonical_sha256_bytes"},"source_id":"2101.08217","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6cbce31c0429078263e1a29fd3d2b0f4454cb4241b1d968035e0f630b6f2121","sha256:ab1d4b65d35503add60dc303da96626c3fd04199c9531844b7741934bc6cbc94"],"state_sha256":"7d59b47395972b5abf0697db28441c4997f8b8bba7e7c8009d675dc3b4010f30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UyK4ZCr0Jo3G8OAHj2D1+LKJMx4Pk87/ywJBtVw7QhZbKSTUTNUgNG6LpxVaY5kNY6I+2/uqdassYPT4+SjcAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T08:07:11.643716Z","bundle_sha256":"99b9caecb5e0aa2e27cb6a92e7a276568fc2bf8a3ed136d2d0586b3e200a1067"}}