{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VYKNFWPAJ32PJSPXGNPROQCLWQ","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":"157c3bad7cc33214e6c0387effa28e1839b1e2ef077d51444204bf226509eeca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-02T04:39:20Z","title_canon_sha256":"5fd6c7e216cc8c2397c0ef42b2dca2642b66afce6c59fa3811a8694fe0ced750"},"schema_version":"1.0","source":{"id":"2607.01695","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01695","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01695v1","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01695","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"pith_short_12","alias_value":"VYKNFWPAJ32P","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"pith_short_16","alias_value":"VYKNFWPAJ32PJSPX","created_at":"2026-07-03T01:17:26Z"},{"alias_kind":"pith_short_8","alias_value":"VYKNFWPA","created_at":"2026-07-03T01:17:26Z"}],"graph_snapshots":[{"event_id":"sha256:80ab6ba9be8415a50eadd43e053af21b9eb43cf73d19a589adda40b0752ac5eb","target":"graph","created_at":"2026-07-03T01:17:26Z","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/2607.01695/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\\chi=(\\frac{\\cdot}{p})$, with $\\chi(0)=0$. For $a\\in\\mathbb F_p^\\times$ define \\[\n  D_a(x)=\\det_{1\\le i,j\\le n}(x+\\chi(i^2-aj)),\n  \\qquad\n  D_a^{(0)}(x)=\\det_{0\\le i,j\\le n}(x+\\chi(i^2-aj)). \\] We prove \\[\n  D_a(0)=0\n  \\quad\\Longleftrightarrow\\quad\n  p\\equiv 3 \\pmod 4\n  \\quad\\text{and}\\quad\n  \\chi(a n!)=1. \\] For $p\\equiv3\\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\\lfloor(p+1)/8\\rfloor}$. If $\\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. If $\\chi","authors_text":"Fei Liu, Hong-Ge Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-02T04:39:20Z","title":"A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01695","kind":"arxiv","version":1},"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":"sha256:257ed5a058853414b9e587e21527c6311722891279d86cbb5b21f4181eabc28a","target":"record","created_at":"2026-07-03T01:17:26Z","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":"157c3bad7cc33214e6c0387effa28e1839b1e2ef077d51444204bf226509eeca","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-02T04:39:20Z","title_canon_sha256":"5fd6c7e216cc8c2397c0ef42b2dca2642b66afce6c59fa3811a8694fe0ced750"},"schema_version":"1.0","source":{"id":"2607.01695","kind":"arxiv","version":1}},"canonical_sha256":"ae14d2d9e04ef4f4c9f7335f17404bb4051fbf7212ae8412893ae68825076fdd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ae14d2d9e04ef4f4c9f7335f17404bb4051fbf7212ae8412893ae68825076fdd","first_computed_at":"2026-07-03T01:17:26.977598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:26.977598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0en7j19u5SQHkvJwoxRQ7iXQIT4Cz3gji1F/WJ/RSYyFJY7VE2XN55w6UiwKW/iHgKG37/xdxQo/EgSV6ygWCg==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:26.977992Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01695","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:257ed5a058853414b9e587e21527c6311722891279d86cbb5b21f4181eabc28a","sha256:80ab6ba9be8415a50eadd43e053af21b9eb43cf73d19a589adda40b0752ac5eb"],"state_sha256":"f156160f35318eb783f1a60d409bc02c935a1f7a623651b7093eff917077d64d"}