{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ADFAPXQLSALXGHMPOHC4QV4YNK","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":"c2d24471986313a5be7e9fc55ac64011bf7b6987f8858913910cbfcceb096da7","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-07-02T22:17:15Z","title_canon_sha256":"4f08fd56594300c335bfe5e05dfeb4c3483d8ef8d6e9b6aaf3ad72a1ff6d548e"},"schema_version":"1.0","source":{"id":"1307.0870","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0870","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0870v2","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0870","created_at":"2026-05-18T02:54:46Z"},{"alias_kind":"pith_short_12","alias_value":"ADFAPXQLSALX","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"ADFAPXQLSALXGHMP","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"ADFAPXQL","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:959f59fe55e5e166afeb449e005a518e335c97c56ee29dbe0f6e98f13f7226ba","target":"graph","created_at":"2026-05-18T02:54:46Z","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":"It is shown that $N$ points on a real algebraic curve of degree $n$ in $\\mathbb{R}^d$ always determine $\\gtrsim_{n,d}N^{1+\\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of $N$ points which determine $\\lesssim N$ distinct distances. The method may be applied to other quantities of interest to obtain analogous exponent gaps. An important step in the proof involves understanding the structural rigidity of certain frameworks on curves.","authors_text":"Marcos Charalambides","cross_cats":["cs.CG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-07-02T22:17:15Z","title":"Distinct Distances on Curves via Rigidity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0870","kind":"arxiv","version":2},"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":"sha256:68d9867baa5454507b3c530603ad5a77eb3d3727b11ce2854593f6d473b0d181","target":"record","created_at":"2026-05-18T02:54:46Z","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":"c2d24471986313a5be7e9fc55ac64011bf7b6987f8858913910cbfcceb096da7","cross_cats_sorted":["cs.CG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-07-02T22:17:15Z","title_canon_sha256":"4f08fd56594300c335bfe5e05dfeb4c3483d8ef8d6e9b6aaf3ad72a1ff6d548e"},"schema_version":"1.0","source":{"id":"1307.0870","kind":"arxiv","version":2}},"canonical_sha256":"00ca07de0b9017731d8f71c5c857986a9ff9a3df234294104754ecd6bd6ecdf5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"00ca07de0b9017731d8f71c5c857986a9ff9a3df234294104754ecd6bd6ecdf5","first_computed_at":"2026-05-18T02:54:46.985219Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:46.985219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9hBauQgrwD439FvSOydCNIyxUI6aYT2PgPz+ZEGh19CFTBLfsFrc6l/qRUSQ1uTSW8qz/psQwuxMJ4U0dh3kCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:46.985598Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0870","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68d9867baa5454507b3c530603ad5a77eb3d3727b11ce2854593f6d473b0d181","sha256:959f59fe55e5e166afeb449e005a518e335c97c56ee29dbe0f6e98f13f7226ba"],"state_sha256":"56c7c4ba36c7aa66189763e82324ef969ad015433359f50e4e5f21f8ac930360"}