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For the modular torus this yields #λ_M^{-1}(m) ≤ C (log log(3m))^2."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The normal-turn estimates for McShane-Rivin norm balls hold uniformly enough to convert growth control into a multiplicity bound without additional length-dependent losses."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"On hyperbolic once-punctured tori the multiplicity of any simple geodesic length L is at most C (log L)^2, improving prior logarithmic bounds for Markoff numbers."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L."}],"snapshot_sha256":"b909bb63ca3b6353aa97cd80e419b0069b05a27220094c758332e7e4b24c28ab"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We use normal-turn estimates for McShane--Rivin norm balls to prove that, for every complete finite-area hyperbolic once-punctured torus $X$, the number of simple closed geodesics of length exactly $L\\geq 2$ is at most $C_X(\\log L)^2$. For the modular torus, this gives $$ \\#\\lambda_M^{-1}(m)\\leq C(\\log\\log(3m))^2 $$ for every Markoff number $m$, improving the previous logarithmic Markoff-fiber bounds. 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