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By analyzing the locus admitting quasi-periodic solutions, we construct two fundamental algebraic curves:\n  (i) The generalized Lam'e curve (GLC), $\\mathcal{Y}_{\\mathbf{n}, \\mathbf{p}}$, which lies in an affine bundle over $\\operatorname{Sym}^n E$ for total weight $n:=\\sum n_i \\in \\mathbb{Z}_{\\geq 0}$ and parametrizes generalized Hermite--Halphen ansatz solutions.\n  (ii) The log-free curve, $V_{\\mathbf{n}, \\mathbf{p"},"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":"2604.21880","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-04-23T17:23:20Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"9e86185d2a8b36ba59dd9729fa8d38ca7088cbf7494d755f408e05f9c85ce264","abstract_canon_sha256":"f57175116f7940738aa5d824329e7122cf475bea7a848245430ccd86c0290a2b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:50.589291Z","signature_b64":"2eBapMd8MJ2pFDPK7On9xXK7KAN5Bx6rMsAyJTVadOl3qqP/WvMWGBjXUTfsendmm/WxxPIFQEPBHIHY7Jz4Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da414bd8a61bc7ec5eb577a1800c3336efe828885356ed29ba19284f6ecdec70","last_reissued_at":"2026-06-03T01:05:50.588832Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:50.588832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A theory of generalized Lam\\'e curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Generalized Lamé curves parametrize quasi-periodic solutions to elliptic equations with multiple poles and prove the Treibich conjecture for up to four symmetric pairs.","cross_cats":["math.CA"],"primary_cat":"math.AG","authors_text":"Chin-Lung Wang, Po-Sheng Wu, You-Cheng Chou","submitted_at":"2026-04-23T17:23:20Z","abstract_excerpt":"We study the generalized Lam'e equation (GLE) on an elliptic curve $E$ with multiple regular singularities $\\mathbf{p} = (p_i)_{i = 1}^r$ of weights $\\mathbf{n} = (n_i)_{i = 1}^r$. By analyzing the locus admitting quasi-periodic solutions, we construct two fundamental algebraic curves:\n  (i) The generalized Lam'e curve (GLC), $\\mathcal{Y}_{\\mathbf{n}, \\mathbf{p}}$, which lies in an affine bundle over $\\operatorname{Sym}^n E$ for total weight $n:=\\sum n_i \\in \\mathbb{Z}_{\\geq 0}$ and parametrizes generalized Hermite--Halphen ansatz solutions.\n  (ii) The log-free curve, $V_{\\mathbf{n}, \\mathbf{p"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the Treibich conjecture stated for r=2 extra symmetric pairs, as well as its generalizations for r ≤ 4. We construct the generalized Lamé curve Y_n(p;τ) which lies in an affine bundle over Sym^n E and parametrizes generalized Hermite-Halphen ansatz solutions, and we prove that the log-free curve V_n(p;τ) is a reduced curve.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The restriction to the locus admitting solutions with quasi-periodic properties is sufficient to construct the generalized Lamé curve and to allow continuous deformation to the classical Lamé equation; if this locus is empty or the ansatz misses essential solutions for general pole configurations, the parametrization and deformation claims fail.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Lamé curves are built to parametrize quasi-periodic solutions of Lamé equations with multiple singularities on elliptic curves, together with a proof of the Treibich conjecture for up to four extra symmetric pairs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized Lamé curves parametrize quasi-periodic solutions to elliptic equations with multiple poles and prove the Treibich conjecture for up to four symmetric pairs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"709ca166040b5eb4e5d423262269c02743d0cf561872ad8ff18efaa609e2c2a7"},"source":{"id":"2604.21880","kind":"arxiv","version":2},"verdict":{"id":"83d5b245-e035-4393-a539-62898802b5a0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T20:17:29.813787Z","strongest_claim":"We prove the Treibich conjecture stated for r=2 extra symmetric pairs, as well as its generalizations for r ≤ 4. We construct the generalized Lamé curve Y_n(p;τ) which lies in an affine bundle over Sym^n E and parametrizes generalized Hermite-Halphen ansatz solutions, and we prove that the log-free curve V_n(p;τ) is a reduced curve.","one_line_summary":"Generalized Lamé curves are built to parametrize quasi-periodic solutions of Lamé equations with multiple singularities on elliptic curves, together with a proof of the Treibich conjecture for up to four extra symmetric pairs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The restriction to the locus admitting solutions with quasi-periodic properties is sufficient to construct the generalized Lamé curve and to allow continuous deformation to the classical Lamé equation; if this locus is empty or the ansatz misses essential solutions for general pole configurations, the parametrization and deformation claims fail.","pith_extraction_headline":"Generalized Lamé curves parametrize quasi-periodic solutions to elliptic equations with multiple poles and prove the Treibich conjecture for up to four symmetric pairs."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.21880/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T11:39:19.285230Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T00:34:51.445725Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"08fe28f192123ac4b77d631655bebd3c144a59814eb56085c349562c3bc6112c"},"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":"2604.21880","created_at":"2026-06-03T01:05:50.588888+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.21880v2","created_at":"2026-06-03T01:05:50.588888+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.21880","created_at":"2026-06-03T01:05:50.588888+00:00"},{"alias_kind":"pith_short_12","alias_value":"3JAUXWFGDPD6","created_at":"2026-06-03T01:05:50.588888+00:00"},{"alias_kind":"pith_short_16","alias_value":"3JAUXWFGDPD6YXVV","created_at":"2026-06-03T01:05:50.588888+00:00"},{"alias_kind":"pith_short_8","alias_value":"3JAUXWFG","created_at":"2026-06-03T01:05:50.588888+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/3JAUXWFGDPD6YXVVO6QYADBTG3","json":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3.json","graph_json":"https://pith.science/api/pith-number/3JAUXWFGDPD6YXVVO6QYADBTG3/graph.json","events_json":"https://pith.science/api/pith-number/3JAUXWFGDPD6YXVVO6QYADBTG3/events.json","paper":"https://pith.science/paper/3JAUXWFG"},"agent_actions":{"view_html":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3","download_json":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3.json","view_paper":"https://pith.science/paper/3JAUXWFG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.21880&json=true","fetch_graph":"https://pith.science/api/pith-number/3JAUXWFGDPD6YXVVO6QYADBTG3/graph.json","fetch_events":"https://pith.science/api/pith-number/3JAUXWFGDPD6YXVVO6QYADBTG3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3/action/storage_attestation","attest_author":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3/action/author_attestation","sign_citation":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3/action/citation_signature","submit_replication":"https://pith.science/pith/3JAUXWFGDPD6YXVVO6QYADBTG3/action/replication_record"}},"created_at":"2026-06-03T01:05:50.588888+00:00","updated_at":"2026-06-03T01:05:50.588888+00:00"}