{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:X6VJZBCHCEYGA4IBBKWNXHYTKW","short_pith_number":"pith:X6VJZBCH","canonical_record":{"source":{"id":"2602.03722","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-03T16:43:42Z","cross_cats_sorted":["math.AG","math.GT"],"title_canon_sha256":"db42e068cd006e29b4dfc5288d8c56db3330eec83bbdf8590c503c5ae3825d00","abstract_canon_sha256":"4e034636cb944c531fbbdceffdcb5f57cbf13219759617633a2027056620c769"},"schema_version":"1.0"},"canonical_sha256":"bfaa9c844711306071010aacdb9f1355bd05beea2e1cef99b9a2dbfad525fc1f","source":{"kind":"arxiv","id":"2602.03722","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.03722","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"arxiv_version","alias_value":"2602.03722v2","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.03722","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"pith_short_12","alias_value":"X6VJZBCHCEYG","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"pith_short_16","alias_value":"X6VJZBCHCEYGA4IB","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"pith_short_8","alias_value":"X6VJZBCH","created_at":"2026-07-07T02:17:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:X6VJZBCHCEYGA4IBBKWNXHYTKW","target":"record","payload":{"canonical_record":{"source":{"id":"2602.03722","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-03T16:43:42Z","cross_cats_sorted":["math.AG","math.GT"],"title_canon_sha256":"db42e068cd006e29b4dfc5288d8c56db3330eec83bbdf8590c503c5ae3825d00","abstract_canon_sha256":"4e034636cb944c531fbbdceffdcb5f57cbf13219759617633a2027056620c769"},"schema_version":"1.0"},"canonical_sha256":"bfaa9c844711306071010aacdb9f1355bd05beea2e1cef99b9a2dbfad525fc1f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:17:18.013223Z","signature_b64":"LUJeG4wIXgEHQgoRnRIF1wSpYkHiqfonkLu0Zj8T8wxGuTGDwe7sGOHgTIROSTAAajs5XWVT0FM8EJAEqw4SAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfaa9c844711306071010aacdb9f1355bd05beea2e1cef99b9a2dbfad525fc1f","last_reissued_at":"2026-07-07T02:17:18.012190Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:17:18.012190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.03722","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":"2026-07-07T02:17:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0SBLXRM66B81D9z7byJXD1Op1flmy8zuvMGRNlPPOL1ZU1n6wWTPb0jx7uqot4paWPufm+BHn3KVmYLWgapYAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:21:27.033542Z"},"content_sha256":"de47babb8b9596d6fa65bd6430cef6562e98d7a4ef10ab20aeb67c94ae8cfb46","schema_version":"1.0","event_id":"sha256:de47babb8b9596d6fa65bd6430cef6562e98d7a4ef10ab20aeb67c94ae8cfb46"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:X6VJZBCHCEYGA4IBBKWNXHYTKW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parity of $k$-differentials in genus zero and one","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.GT"],"primary_cat":"math.NT","authors_text":"Dawei Chen, Evan Chen, Jujian Zhang, Kenny Lau, Ken Ono","submitted_at":"2026-02-03T16:43:42Z","abstract_excerpt":"Here we completely determine the spin parity of $k$-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.03722","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/2602.03722/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":"2026-07-07T02:17:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GXXd6xMgnjc6VRLqgkIErZQd10ZcvCYV6XWMVCTXodSaXoQPIlYvhGHLlPPze8qII9VuJfgKq4iTNQmd+oDTCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:21:27.034050Z"},"content_sha256":"1562f11895f98cfdba09423b3a027cdb5efedd93213fc6d004700a43b2d5b150","schema_version":"1.0","event_id":"sha256:1562f11895f98cfdba09423b3a027cdb5efedd93213fc6d004700a43b2d5b150"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW/bundle.json","state_url":"https://pith.science/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW/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":"2026-08-03T16:21:27Z","links":{"resolver":"https://pith.science/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW","bundle":"https://pith.science/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW/bundle.json","state":"https://pith.science/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X6VJZBCHCEYGA4IBBKWNXHYTKW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:X6VJZBCHCEYGA4IBBKWNXHYTKW","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":"4e034636cb944c531fbbdceffdcb5f57cbf13219759617633a2027056620c769","cross_cats_sorted":["math.AG","math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-03T16:43:42Z","title_canon_sha256":"db42e068cd006e29b4dfc5288d8c56db3330eec83bbdf8590c503c5ae3825d00"},"schema_version":"1.0","source":{"id":"2602.03722","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.03722","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"arxiv_version","alias_value":"2602.03722v2","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.03722","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"pith_short_12","alias_value":"X6VJZBCHCEYG","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"pith_short_16","alias_value":"X6VJZBCHCEYGA4IB","created_at":"2026-07-07T02:17:18Z"},{"alias_kind":"pith_short_8","alias_value":"X6VJZBCH","created_at":"2026-07-07T02:17:18Z"}],"graph_snapshots":[{"event_id":"sha256:1562f11895f98cfdba09423b3a027cdb5efedd93213fc6d004700a43b2d5b150","target":"graph","created_at":"2026-07-07T02:17:18Z","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/2602.03722/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Here we completely determine the spin parity of $k$-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appen","authors_text":"Dawei Chen, Evan Chen, Jujian Zhang, Kenny Lau, Ken Ono","cross_cats":["math.AG","math.GT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-03T16:43:42Z","title":"Parity of $k$-differentials in genus zero and one"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.03722","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:de47babb8b9596d6fa65bd6430cef6562e98d7a4ef10ab20aeb67c94ae8cfb46","target":"record","created_at":"2026-07-07T02:17:18Z","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":"4e034636cb944c531fbbdceffdcb5f57cbf13219759617633a2027056620c769","cross_cats_sorted":["math.AG","math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-02-03T16:43:42Z","title_canon_sha256":"db42e068cd006e29b4dfc5288d8c56db3330eec83bbdf8590c503c5ae3825d00"},"schema_version":"1.0","source":{"id":"2602.03722","kind":"arxiv","version":2}},"canonical_sha256":"bfaa9c844711306071010aacdb9f1355bd05beea2e1cef99b9a2dbfad525fc1f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfaa9c844711306071010aacdb9f1355bd05beea2e1cef99b9a2dbfad525fc1f","first_computed_at":"2026-07-07T02:17:18.012190Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:17:18.012190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LUJeG4wIXgEHQgoRnRIF1wSpYkHiqfonkLu0Zj8T8wxGuTGDwe7sGOHgTIROSTAAajs5XWVT0FM8EJAEqw4SAg==","signature_status":"signed_v1","signed_at":"2026-07-07T02:17:18.013223Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.03722","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de47babb8b9596d6fa65bd6430cef6562e98d7a4ef10ab20aeb67c94ae8cfb46","sha256:1562f11895f98cfdba09423b3a027cdb5efedd93213fc6d004700a43b2d5b150"],"state_sha256":"403f7f108f79475924e057a5cfb307f7015a066e6c1eb455383e033a3021db07"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qHU4XKAUfIBz72NCkF4wEBJtJUm1V8FGa4Jc1MlfD+l4vUfdp0hQCxdB+wzq8mSr6Nasqvce0roFF/bltZ2rDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T16:21:27.039260Z","bundle_sha256":"73a96d5009296354fda7d1c93e737dddc2aba9a2d74d327f0120146755a85442"}}