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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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","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."}],"snapshot_sha256":"709ca166040b5eb4e5d423262269c02743d0cf561872ad8ff18efaa609e2c2a7"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"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"}],"endpoint":"/pith/2604.21880/integrity.json","findings":[],"snapshot_sha256":"08fe28f192123ac4b77d631655bebd3c144a59814eb56085c349562c3bc6112c","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Chin-Lung Wang, Po-Sheng Wu, You-Cheng Chou","cross_cats":["math.CA"],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-04-23T17:23:20Z","title":"A theory of generalized Lam\\'e curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.21880","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-09T20:17:29.813787Z","id":"83d5b245-e035-4393-a539-62898802b5a0","model_set":{"reader":"grok-4.3"},"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","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.","strongest_claim":"We prove the Treibich conjecture stated for r=2 extra symmetric pairs, as well as its generalizations for r ≤ 4. 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