{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:GQHAHAZPLUYTKKQE2FTFD55YAN","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":"19223b8a4fa5fcebefaa278ab14ca2f17eed1ae97f4f2f1d26cb260638b1acb8","cross_cats_sorted":["math.QA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-08-25T11:57:49Z","title_canon_sha256":"9322322717cea13353758039c4e5a1b7c44affc4417d09e02d81ff2ab51199ee"},"schema_version":"1.0","source":{"id":"2508.18334","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.18334","created_at":"2026-05-18T03:09:34Z"},{"alias_kind":"arxiv_version","alias_value":"2508.18334v3","created_at":"2026-05-18T03:09:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.18334","created_at":"2026-05-18T03:09:34Z"},{"alias_kind":"pith_short_12","alias_value":"GQHAHAZPLUYT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"GQHAHAZPLUYTKKQE","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"GQHAHAZP","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:9e2e210b0f8de9fd4c5701163fb28e8ac20c04db0cd73e17e860dfe108fbfa9b","target":"graph","created_at":"2026-05-18T03:09:34Z","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":"We study $\\eta$-correction terms in the Kauffman bracket skein algebra of the once-punctured torus $K_t(\\Sigma_{1,1})$. While the Frohman--Gelca product-to-sum rule gives an explicit multiplication formula on the closed torus, the once-punctured torus introduces correction terms in the ideal $(\\eta)$. We give a closed formula for the Chebyshev-threaded family generated by the primitive determinant-two pair \\[ P_n=T_n((1,2))\\cdot(1,0). \\] The correction $\\epsilon_n$ has an explicit Chebyshev expansion whose coefficients factor as geometric sums in $t^{\\pm4}$ and whose terms are governed by a pa","authors_text":"Nelson A. Colon Vargas","cross_cats":["math.QA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-08-25T11:57:49Z","title":"Closed Formulas for $\\eta$-Corrections in the Once Punctured Torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.18334","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":"sha256:112906547eef8c5cd7cc68916ca193c7a84599558a01f69769c68ac5110a1eae","target":"record","created_at":"2026-05-18T03:09:34Z","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":"19223b8a4fa5fcebefaa278ab14ca2f17eed1ae97f4f2f1d26cb260638b1acb8","cross_cats_sorted":["math.QA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2025-08-25T11:57:49Z","title_canon_sha256":"9322322717cea13353758039c4e5a1b7c44affc4417d09e02d81ff2ab51199ee"},"schema_version":"1.0","source":{"id":"2508.18334","kind":"arxiv","version":3}},"canonical_sha256":"340e03832f5d31352a04d16651f7b8037971417ac1fd26975ed76ada5750841a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"340e03832f5d31352a04d16651f7b8037971417ac1fd26975ed76ada5750841a","first_computed_at":"2026-05-18T03:09:34.818278Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:34.818278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WzdbWAnFbWLYzAYMDFmo4PI2kFelgkrJQrp4bTqtUXwXdPZ87KUCNt3NPTt84+aocT7shLrRmEgITHWhQ9FmDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:34.818962Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.18334","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:112906547eef8c5cd7cc68916ca193c7a84599558a01f69769c68ac5110a1eae","sha256:9e2e210b0f8de9fd4c5701163fb28e8ac20c04db0cd73e17e860dfe108fbfa9b"],"state_sha256":"287d8d954022b523b00938ecaad708fd442cf5d2b329fae11f52a3221e1893da"}