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The first one resolves the \\(p\\equiv1\\pmod4\\) part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor \\([\\chi(j-k+1)]_{0\\le j,k<(p-1)/2}\\). The core evaluation is \\[ \\det H=\\leg{2}{p}(b'_p-a'_p),\\qquad U^TH^{-1}U= \\begin{pmatrix} \\leg{2}{p}\\dfrac{pb'_p-a'_p}{b'_p-a'_p}&1\\\\[2mm] \\dfrac{b'_p-a'_p-1}{b'_p-a'_p}&1 \\end{pmatrix}, \\] where \\(U=(\\mathbf1,\\eta)\\) and \\(\\eta_j=\\chi(j)\\). The proof uses Vsemirnov's factorisation"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.19517","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-19T08:13:57Z","cross_cats_sorted":[],"title_canon_sha256":"a8d90db0073e58fc594b0e77c80564485fa3139d9c8ed9ad89d8d35aac15dec3","abstract_canon_sha256":"a96a2d38df9e532ee62afac0cd8b858e2fc2b909c333916ed6f76f6bf2fe4472"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:49.605890Z","signature_b64":"i6JG/dvFLAlZ2k41KMmP8XR1oHwDD7MFeljEpQmaeuBe8L4019v07BpRIafHKyFRsMcxX00zj/M6kXGmb59IDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d843979bd15591b15501d3f89ef9f5ccec5beabca49ed4dc13a392cfc957910","last_reissued_at":"2026-05-20T01:05:49.605097Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:49.605097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two determinant evaluations in Sun's conjectures involving Legendre symbols","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-19T08:13:57Z","abstract_excerpt":"We prove two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. The first one resolves the \\(p\\equiv1\\pmod4\\) part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor \\([\\chi(j-k+1)]_{0\\le j,k<(p-1)/2}\\). The core evaluation is \\[ \\det H=\\leg{2}{p}(b'_p-a'_p),\\qquad U^TH^{-1}U= \\begin{pmatrix} \\leg{2}{p}\\dfrac{pb'_p-a'_p}{b'_p-a'_p}&1\\\\[2mm] \\dfrac{b'_p-a'_p-1}{b'_p-a'_p}&1 \\end{pmatrix}, \\] where \\(U=(\\mathbf1,\\eta)\\) and \\(\\eta_j=\\chi(j)\\). The proof uses Vsemirnov's factorisation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19517","kind":"arxiv","version":1},"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/2605.19517/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.19517","created_at":"2026-05-20T01:05:49.605229+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.19517v1","created_at":"2026-05-20T01:05:49.605229+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19517","created_at":"2026-05-20T01:05:49.605229+00:00"},{"alias_kind":"pith_short_12","alias_value":"DWCDS6N5CVMR","created_at":"2026-05-20T01:05:49.605229+00:00"},{"alias_kind":"pith_short_16","alias_value":"DWCDS6N5CVMRWFKQ","created_at":"2026-05-20T01:05:49.605229+00:00"},{"alias_kind":"pith_short_8","alias_value":"DWCDS6N5","created_at":"2026-05-20T01:05:49.605229+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT","json":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT.json","graph_json":"https://pith.science/api/pith-number/DWCDS6N5CVMRWFKQDU7YT347LT/graph.json","events_json":"https://pith.science/api/pith-number/DWCDS6N5CVMRWFKQDU7YT347LT/events.json","paper":"https://pith.science/paper/DWCDS6N5"},"agent_actions":{"view_html":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT","download_json":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT.json","view_paper":"https://pith.science/paper/DWCDS6N5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.19517&json=true","fetch_graph":"https://pith.science/api/pith-number/DWCDS6N5CVMRWFKQDU7YT347LT/graph.json","fetch_events":"https://pith.science/api/pith-number/DWCDS6N5CVMRWFKQDU7YT347LT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT/action/storage_attestation","attest_author":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT/action/author_attestation","sign_citation":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT/action/citation_signature","submit_replication":"https://pith.science/pith/DWCDS6N5CVMRWFKQDU7YT347LT/action/replication_record"}},"created_at":"2026-05-20T01:05:49.605229+00:00","updated_at":"2026-05-20T01:05:49.605229+00:00"}