{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TTPLCJB725V775Z4IBQRTSN2SR","short_pith_number":"pith:TTPLCJB7","canonical_record":{"source":{"id":"2605.14574","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GT","submitted_at":"2026-05-14T08:45:18Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"63f88bbdaf08e38eb7b8a14097c69ec944fa97fac64aecbda211f058a3b93205","abstract_canon_sha256":"655657a2975d4c837c852cd9ede09737f16f0a0901182948ef98ba5e423ee8b9"},"schema_version":"1.0"},"canonical_sha256":"9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5","source":{"kind":"arxiv","id":"2605.14574","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14574","created_at":"2026-05-17T23:39:05Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14574v1","created_at":"2026-05-17T23:39:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14574","created_at":"2026-05-17T23:39:05Z"},{"alias_kind":"pith_short_12","alias_value":"TTPLCJB725V7","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"TTPLCJB725V775Z4","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"TTPLCJB7","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TTPLCJB725V775Z4IBQRTSN2SR","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14574","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GT","submitted_at":"2026-05-14T08:45:18Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"63f88bbdaf08e38eb7b8a14097c69ec944fa97fac64aecbda211f058a3b93205","abstract_canon_sha256":"655657a2975d4c837c852cd9ede09737f16f0a0901182948ef98ba5e423ee8b9"},"schema_version":"1.0"},"canonical_sha256":"9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:05.436754Z","signature_b64":"6myM3HDItYbAowLHsxEUF9/JEDdf/rOkcnX9+3ddUqNnPr7iOHjRgyNHJepFnS8FrClOJ6otPPCskGSXXQuuBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5","last_reissued_at":"2026-05-17T23:39:05.436047Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:05.436047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14574","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ffkMoHJoQOkqKggJqb1Ji729yiFvYrhc6S+R9uDgoTVTK5204fVrUKDUYhLuy1GXscSniBR8YFW0D9RN+0alAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T17:41:49.423597Z"},"content_sha256":"f3b85e5cf054968df2ffa6c0a9d896fba39bca1464965e7b03228b9b4ada26e5","schema_version":"1.0","event_id":"sha256:f3b85e5cf054968df2ffa6c0a9d896fba39bca1464965e7b03228b9b4ada26e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TTPLCJB725V775Z4IBQRTSN2SR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"McShane-Rivin norm balls and simple-length multiplicities","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.","cross_cats":["math.MG","math.NT"],"primary_cat":"math.GT","authors_text":"Nhat Minh Doan, Van Nguyen, Xiaobin Li","submitted_at":"2026-05-14T08:45:18Z","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. These estimates also give new quantitative information on the local geometry of McShane--Rivin norm balls, including obstructions to infinite-order flatness at certain irrational directions."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every complete finite-area hyperbolic once-punctured torus X, the number of simple closed geodesics of length exactly L ≥ 2 is at most C_X (log L)^2. For the modular torus this yields #λ_M^{-1}(m) ≤ C (log log(3m))^2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b909bb63ca3b6353aa97cd80e419b0069b05a27220094c758332e7e4b24c28ab"},"source":{"id":"2605.14574","kind":"arxiv","version":1},"verdict":{"id":"38f99c47-95c0-4487-baa7-e7756e7c0dd4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:18:59.552058Z","strongest_claim":"For every complete finite-area hyperbolic once-punctured torus X, the number of simple closed geodesics of length exactly L ≥ 2 is at most C_X (log L)^2. For the modular torus this yields #λ_M^{-1}(m) ≤ C (log log(3m))^2.","one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"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.","pith_extraction_headline":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L."},"references":{"count":36,"sample":[{"doi":"","year":2013,"title":"2013 , doi =","work_id":"bd2fe240-a42e-4386-b357-187b71137dd1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Khinchin, Aleksandr Yakovlevich , title =","work_id":"f4ba514a-1ff0-45c7-8a95-9660fdbff278","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Rockett, Andrew M. and Sz. Continued Fractions , publisher =","work_id":"36d8cfe7-700d-4b58-a5ca-0ef8ea8bc791","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Durrett, Rick , title =","work_id":"ed931e10-7526-4a75-b69d-80500362fc1b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Cassels, J. W. S. , title =","work_id":"6b26e16e-5aa1-4b10-bca1-b4a2c888c8d2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":36,"snapshot_sha256":"e4f8c93d56c847cff635043e5a9c169ca5373d492de84b60a1a7a5b669b620c5","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"},"verdict_id":"38f99c47-95c0-4487-baa7-e7756e7c0dd4"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a1i5e1l6BqmY7VYJfvyQnsS4lAGT2eqd+QKqUd6mbGGmlkDzy0eLClt4fqijMymmYXkzGixXZWCFqb5u8cJTDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T17:41:49.424722Z"},"content_sha256":"f96414fc27497922818f0a87891a78940fedfe253ac007cad64cb3b7a5843469","schema_version":"1.0","event_id":"sha256:f96414fc27497922818f0a87891a78940fedfe253ac007cad64cb3b7a5843469"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TTPLCJB725V775Z4IBQRTSN2SR/bundle.json","state_url":"https://pith.science/pith/TTPLCJB725V775Z4IBQRTSN2SR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TTPLCJB725V775Z4IBQRTSN2SR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-20T17:41:49Z","links":{"resolver":"https://pith.science/pith/TTPLCJB725V775Z4IBQRTSN2SR","bundle":"https://pith.science/pith/TTPLCJB725V775Z4IBQRTSN2SR/bundle.json","state":"https://pith.science/pith/TTPLCJB725V775Z4IBQRTSN2SR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TTPLCJB725V775Z4IBQRTSN2SR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TTPLCJB725V775Z4IBQRTSN2SR","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":"655657a2975d4c837c852cd9ede09737f16f0a0901182948ef98ba5e423ee8b9","cross_cats_sorted":["math.MG","math.NT"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GT","submitted_at":"2026-05-14T08:45:18Z","title_canon_sha256":"63f88bbdaf08e38eb7b8a14097c69ec944fa97fac64aecbda211f058a3b93205"},"schema_version":"1.0","source":{"id":"2605.14574","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14574","created_at":"2026-05-17T23:39:05Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14574v1","created_at":"2026-05-17T23:39:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14574","created_at":"2026-05-17T23:39:05Z"},{"alias_kind":"pith_short_12","alias_value":"TTPLCJB725V7","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"TTPLCJB725V775Z4","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"TTPLCJB7","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:f96414fc27497922818f0a87891a78940fedfe253ac007cad64cb3b7a5843469","target":"graph","created_at":"2026-05-17T23:39:05Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"For every complete finite-area hyperbolic once-punctured torus X, the number of simple closed geodesics of length exactly L ≥ 2 is at most C_X (log L)^2. 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. These estimates also give new quantitative information on the local geometry of McShane--Rivin norm balls, including obstructions to infinite-order flatness at certain irrational directions.","authors_text":"Nhat Minh Doan, Van Nguyen, Xiaobin Li","cross_cats":["math.MG","math.NT"],"headline":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GT","submitted_at":"2026-05-14T08:45:18Z","title":"McShane-Rivin norm balls and simple-length multiplicities"},"references":{"count":36,"internal_anchors":0,"resolved_work":36,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"2013 , doi =","work_id":"bd2fe240-a42e-4386-b357-187b71137dd1","year":2013},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Khinchin, Aleksandr Yakovlevich , title =","work_id":"f4ba514a-1ff0-45c7-8a95-9660fdbff278","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Rockett, Andrew M. and Sz. Continued Fractions , publisher =","work_id":"36d8cfe7-700d-4b58-a5ca-0ef8ea8bc791","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Durrett, Rick , title =","work_id":"ed931e10-7526-4a75-b69d-80500362fc1b","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Cassels, J. W. S. , title =","work_id":"6b26e16e-5aa1-4b10-bca1-b4a2c888c8d2","year":null}],"snapshot_sha256":"e4f8c93d56c847cff635043e5a9c169ca5373d492de84b60a1a7a5b669b620c5"},"source":{"id":"2605.14574","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T01:18:59.552058Z","id":"38f99c47-95c0-4487-baa7-e7756e7c0dd4","model_set":{"reader":"grok-4.3"},"one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.","strongest_claim":"For every complete finite-area hyperbolic once-punctured torus X, the number of simple closed geodesics of length exactly L ≥ 2 is at most C_X (log L)^2. For the modular torus this yields #λ_M^{-1}(m) ≤ C (log log(3m))^2.","weakest_assumption":"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."}},"verdict_id":"38f99c47-95c0-4487-baa7-e7756e7c0dd4"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f3b85e5cf054968df2ffa6c0a9d896fba39bca1464965e7b03228b9b4ada26e5","target":"record","created_at":"2026-05-17T23:39:05Z","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":"655657a2975d4c837c852cd9ede09737f16f0a0901182948ef98ba5e423ee8b9","cross_cats_sorted":["math.MG","math.NT"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GT","submitted_at":"2026-05-14T08:45:18Z","title_canon_sha256":"63f88bbdaf08e38eb7b8a14097c69ec944fa97fac64aecbda211f058a3b93205"},"schema_version":"1.0","source":{"id":"2605.14574","kind":"arxiv","version":1}},"canonical_sha256":"9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5","first_computed_at":"2026-05-17T23:39:05.436047Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:05.436047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6myM3HDItYbAowLHsxEUF9/JEDdf/rOkcnX9+3ddUqNnPr7iOHjRgyNHJepFnS8FrClOJ6otPPCskGSXXQuuBA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:05.436754Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14574","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3b85e5cf054968df2ffa6c0a9d896fba39bca1464965e7b03228b9b4ada26e5","sha256:f96414fc27497922818f0a87891a78940fedfe253ac007cad64cb3b7a5843469"],"state_sha256":"86379782368add4a20cc9e69085d0c45ffdd42a534a8d61616ee69f86cfa28f0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5ZU+ylnNAy/kUwyQZo5mRujhA2E0trbg+rwcJm7srIwpn5IlBAj1C4JrEIYOMp9RoF7X/WjRuC8nDwdp28sWAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T17:41:49.429045Z","bundle_sha256":"a54f12e72c283dbc0805580dda5f8e1255b0d28df02cb9ec5ed5bf27ba41d8c4"}}