{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:Z45YE7OBP6G3X7XY64UXBIP4LB","short_pith_number":"pith:Z45YE7OB","canonical_record":{"source":{"id":"1511.00315","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-11-01T22:03:23Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e9472e1268551ca0a5d8322535f5fd0335b6add7d339ac0fa6c39865131158dd","abstract_canon_sha256":"ae0dc234b6ac0ed38405aa636be966add2d09210db8f41649181eac7da79e629"},"schema_version":"1.0"},"canonical_sha256":"cf3b827dc17f8dbbfef8f72970a1fc585c4ef5b252f68dbad1ea908e34e48735","source":{"kind":"arxiv","id":"1511.00315","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.00315","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"1511.00315v3","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00315","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"Z45YE7OBP6G3","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z45YE7OBP6G3X7XY","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z45YE7OB","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:Z45YE7OBP6G3X7XY64UXBIP4LB","target":"record","payload":{"canonical_record":{"source":{"id":"1511.00315","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-11-01T22:03:23Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e9472e1268551ca0a5d8322535f5fd0335b6add7d339ac0fa6c39865131158dd","abstract_canon_sha256":"ae0dc234b6ac0ed38405aa636be966add2d09210db8f41649181eac7da79e629"},"schema_version":"1.0"},"canonical_sha256":"cf3b827dc17f8dbbfef8f72970a1fc585c4ef5b252f68dbad1ea908e34e48735","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:40.122525Z","signature_b64":"4mpLw8sJtNnoZbY+l7ZbfZDce2IJfpo++J1hjxyj7fciFKQAeMgwfVHPzT5IK/NAst17Mq5taftekzQlmHv+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf3b827dc17f8dbbfef8f72970a1fc585c4ef5b252f68dbad1ea908e34e48735","last_reissued_at":"2026-05-18T00:38:40.121839Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:40.121839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.00315","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":"2026-05-18T00:38:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P2b7uMGDSrR854GjbFhCKy5V8f8ZTAKXM+Otbw0+UbmSdZzCqAzZLm7WA3Pj3H4ef3z8TxhSKveDhqlDbdHjDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T21:07:04.522422Z"},"content_sha256":"0b468e42199affa7350d772934d52a3b79c310d6cc868bdeac51eadedd6938f7","schema_version":"1.0","event_id":"sha256:0b468e42199affa7350d772934d52a3b79c310d6cc868bdeac51eadedd6938f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:Z45YE7OBP6G3X7XY64UXBIP4LB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Four-Dimensional Algebraic Tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Nicole Lemire","submitted_at":"2015-11-01T22:03:23Z","abstract_excerpt":"The study of the birational properties of algebraic $k$-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Th\\'el\\`ene and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors and later Saltman in the eighties determined necessary and sufficient conditions to determine when an algebraic torus is stably rational, re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00315","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":"2026-05-18T00:38:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LnSo513lsBjYZmwXypazVyyiZeSGeo45qUPQlXIq5yRKIyUzpQ8jqfKEbsneU1LbmXjJz5PtS32M9p5BdJVDCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T21:07:04.523079Z"},"content_sha256":"e6228f2c4b4bea08bde60b3f1cc6a420a357853e651de1a83ab2036a18889190","schema_version":"1.0","event_id":"sha256:e6228f2c4b4bea08bde60b3f1cc6a420a357853e651de1a83ab2036a18889190"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z45YE7OBP6G3X7XY64UXBIP4LB/bundle.json","state_url":"https://pith.science/pith/Z45YE7OBP6G3X7XY64UXBIP4LB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z45YE7OBP6G3X7XY64UXBIP4LB/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":"2026-05-28T21:07:04Z","links":{"resolver":"https://pith.science/pith/Z45YE7OBP6G3X7XY64UXBIP4LB","bundle":"https://pith.science/pith/Z45YE7OBP6G3X7XY64UXBIP4LB/bundle.json","state":"https://pith.science/pith/Z45YE7OBP6G3X7XY64UXBIP4LB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z45YE7OBP6G3X7XY64UXBIP4LB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Z45YE7OBP6G3X7XY64UXBIP4LB","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":"ae0dc234b6ac0ed38405aa636be966add2d09210db8f41649181eac7da79e629","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-11-01T22:03:23Z","title_canon_sha256":"e9472e1268551ca0a5d8322535f5fd0335b6add7d339ac0fa6c39865131158dd"},"schema_version":"1.0","source":{"id":"1511.00315","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.00315","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"arxiv_version","alias_value":"1511.00315v3","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00315","created_at":"2026-05-18T00:38:40Z"},{"alias_kind":"pith_short_12","alias_value":"Z45YE7OBP6G3","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z45YE7OBP6G3X7XY","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z45YE7OB","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:e6228f2c4b4bea08bde60b3f1cc6a420a357853e651de1a83ab2036a18889190","target":"graph","created_at":"2026-05-18T00:38:40Z","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":"The study of the birational properties of algebraic $k$-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Th\\'el\\`ene and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors and later Saltman in the eighties determined necessary and sufficient conditions to determine when an algebraic torus is stably rational, re","authors_text":"Nicole Lemire","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-11-01T22:03:23Z","title":"Four-Dimensional Algebraic Tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00315","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":"sha256:0b468e42199affa7350d772934d52a3b79c310d6cc868bdeac51eadedd6938f7","target":"record","created_at":"2026-05-18T00:38:40Z","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":"ae0dc234b6ac0ed38405aa636be966add2d09210db8f41649181eac7da79e629","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-11-01T22:03:23Z","title_canon_sha256":"e9472e1268551ca0a5d8322535f5fd0335b6add7d339ac0fa6c39865131158dd"},"schema_version":"1.0","source":{"id":"1511.00315","kind":"arxiv","version":3}},"canonical_sha256":"cf3b827dc17f8dbbfef8f72970a1fc585c4ef5b252f68dbad1ea908e34e48735","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf3b827dc17f8dbbfef8f72970a1fc585c4ef5b252f68dbad1ea908e34e48735","first_computed_at":"2026-05-18T00:38:40.121839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:40.121839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4mpLw8sJtNnoZbY+l7ZbfZDce2IJfpo++J1hjxyj7fciFKQAeMgwfVHPzT5IK/NAst17Mq5taftekzQlmHv+Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:40.122525Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.00315","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b468e42199affa7350d772934d52a3b79c310d6cc868bdeac51eadedd6938f7","sha256:e6228f2c4b4bea08bde60b3f1cc6a420a357853e651de1a83ab2036a18889190"],"state_sha256":"a154c9e87470528c9a63bafa2c2c55a1872ac586ffa41af159e0e4f1da4c634e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HcDDMUpwQ50d9PDpBrV5ThblsHjAowiKN/5Glli++yFfCdfh0MzSlGp5k6kDIdB+yHyWwRk3Awx/FHqD5K1gCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T21:07:04.526558Z","bundle_sha256":"276c209ae1e7c50ce8df7c7e8f721382b107323736f75dd490624cd0a8277b0e"}}