{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:MXMGPHXTGTDJ2MN5AVYTCEMWCT","short_pith_number":"pith:MXMGPHXT","canonical_record":{"source":{"id":"2412.18830","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-12-25T08:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"3d27ac958dd4ab775c57b7aab994d12067bfc37d4a79e6156969b1dc3072929e","abstract_canon_sha256":"efd385ae05b09982e4984dd427b602415377be886577beda76133e0564852867"},"schema_version":"1.0"},"canonical_sha256":"65d8679ef334c69d31bd057131119614d83aa36a32682cc2c4cb0b0012c6185c","source":{"kind":"arxiv","id":"2412.18830","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.18830","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"2412.18830v2","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.18830","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"MXMGPHXTGTDJ","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"pith_short_16","alias_value":"MXMGPHXTGTDJ2MN5","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"pith_short_8","alias_value":"MXMGPHXT","created_at":"2026-07-05T11:59:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:MXMGPHXTGTDJ2MN5AVYTCEMWCT","target":"record","payload":{"canonical_record":{"source":{"id":"2412.18830","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-12-25T08:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"3d27ac958dd4ab775c57b7aab994d12067bfc37d4a79e6156969b1dc3072929e","abstract_canon_sha256":"efd385ae05b09982e4984dd427b602415377be886577beda76133e0564852867"},"schema_version":"1.0"},"canonical_sha256":"65d8679ef334c69d31bd057131119614d83aa36a32682cc2c4cb0b0012c6185c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:59:40.795097Z","signature_b64":"KJFJvDMQwEYVmITgNw56sPX0ti8gyFcsAPx9S+H2DNSMwOowxuLPiwctM1odnn/QHRTU4U7xolZKfA7I1DgJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65d8679ef334c69d31bd057131119614d83aa36a32682cc2c4cb0b0012c6185c","last_reissued_at":"2026-07-05T11:59:40.794661Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:59:40.794661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.18830","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":"2026-07-05T11:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"slRrAYZz1rLSd2PD3blx4XenFf3Go+mfmNhQIlDg8UHqlzA1UFTCfvBYbg+tcu3KPRdTWOxop3pSqsCZgW2VCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T02:42:26.431511Z"},"content_sha256":"d2c6b590be91bac9cc79475b326b5a6c738509bbbe2e2c879c4159b266c96637","schema_version":"1.0","event_id":"sha256:d2c6b590be91bac9cc79475b326b5a6c738509bbbe2e2c879c4159b266c96637"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:MXMGPHXTGTDJ2MN5AVYTCEMWCT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Calabi-Yau pairs of complexity two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Joaqu\\'in Moraga, Jos\\'e Ignacio Y\\'a\\~nez","submitted_at":"2024-12-25T08:36:37Z","abstract_excerpt":"A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface $X$ is cluster type if and only if $X$ has only $A$-type singularities and either $\\mathrm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.18830","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.18830/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":"2026-07-05T11:59:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w4P6UYliHJa/Lrsbgpwtr9Z4OLCdw9Dq8YosGy64Y8aMXCa4k0QYWKIxEgZz4hN514S93Um1sibJyeiDo5+iAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T02:42:26.432052Z"},"content_sha256":"e1f3aacee513242f1da70ff2144813bce161dc64c12ddf70cb015235a6cbed7b","schema_version":"1.0","event_id":"sha256:e1f3aacee513242f1da70ff2144813bce161dc64c12ddf70cb015235a6cbed7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT/bundle.json","state_url":"https://pith.science/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT/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":"2026-08-21T02:42:26Z","links":{"resolver":"https://pith.science/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT","bundle":"https://pith.science/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT/bundle.json","state":"https://pith.science/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MXMGPHXTGTDJ2MN5AVYTCEMWCT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:MXMGPHXTGTDJ2MN5AVYTCEMWCT","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":"efd385ae05b09982e4984dd427b602415377be886577beda76133e0564852867","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-12-25T08:36:37Z","title_canon_sha256":"3d27ac958dd4ab775c57b7aab994d12067bfc37d4a79e6156969b1dc3072929e"},"schema_version":"1.0","source":{"id":"2412.18830","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.18830","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"arxiv_version","alias_value":"2412.18830v2","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.18830","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"pith_short_12","alias_value":"MXMGPHXTGTDJ","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"pith_short_16","alias_value":"MXMGPHXTGTDJ2MN5","created_at":"2026-07-05T11:59:40Z"},{"alias_kind":"pith_short_8","alias_value":"MXMGPHXT","created_at":"2026-07-05T11:59:40Z"}],"graph_snapshots":[{"event_id":"sha256:e1f3aacee513242f1da70ff2144813bce161dc64c12ddf70cb015235a6cbed7b","target":"graph","created_at":"2026-07-05T11:59:40Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2412.18830/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface $X$ is cluster type if and only if $X$ has only $A$-type singularities and either $\\mathrm","authors_text":"Joaqu\\'in Moraga, Jos\\'e Ignacio Y\\'a\\~nez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-12-25T08:36:37Z","title":"Calabi-Yau pairs of complexity two"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.18830","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:d2c6b590be91bac9cc79475b326b5a6c738509bbbe2e2c879c4159b266c96637","target":"record","created_at":"2026-07-05T11:59:40Z","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":"efd385ae05b09982e4984dd427b602415377be886577beda76133e0564852867","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-12-25T08:36:37Z","title_canon_sha256":"3d27ac958dd4ab775c57b7aab994d12067bfc37d4a79e6156969b1dc3072929e"},"schema_version":"1.0","source":{"id":"2412.18830","kind":"arxiv","version":2}},"canonical_sha256":"65d8679ef334c69d31bd057131119614d83aa36a32682cc2c4cb0b0012c6185c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"65d8679ef334c69d31bd057131119614d83aa36a32682cc2c4cb0b0012c6185c","first_computed_at":"2026-07-05T11:59:40.794661Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:59:40.794661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KJFJvDMQwEYVmITgNw56sPX0ti8gyFcsAPx9S+H2DNSMwOowxuLPiwctM1odnn/QHRTU4U7xolZKfA7I1DgJCg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:59:40.795097Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.18830","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2c6b590be91bac9cc79475b326b5a6c738509bbbe2e2c879c4159b266c96637","sha256:e1f3aacee513242f1da70ff2144813bce161dc64c12ddf70cb015235a6cbed7b"],"state_sha256":"458a6b5dac402792d2444631f86ee528f86e54e241a69fe73fd410a4ddc82c79"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UU8AC7svYeC+ndof9WQuBF3Ses5AoguGMUzVL8Mkig0CYhLNtSCpuZdJ74RdEUjuO3wX8Q/HBGYypYOhWgTNDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T02:42:26.437817Z","bundle_sha256":"2d0d1d216d126f8f124c34290bc007e1059ed533bded74c620428f06c29ad2d0"}}