{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:JZ4DFH6POOEYYZOA5OENHLWUBF","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":"14af3ca4986a2777477d379e2ea6d8bdff18c7ee44da24be13d31e894d4b07ab","cross_cats_sorted":[],"license":"","primary_cat":"math.DS","submitted_at":"2006-10-11T07:33:09Z","title_canon_sha256":"457d5b81358a16afff03efe69707e39e7be0c7d085deada1de8913a38a2f0d24"},"schema_version":"1.0","source":{"id":"math/0610350","kind":"arxiv","version":8}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610350","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610350v8","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610350","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"JZ4DFH6POOEY","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JZ4DFH6POOEYYZOA","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JZ4DFH6P","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:fb0cfd0a15ef88bdf45c2df926775184183b6daf720a998d638aaae12054a00b","target":"graph","created_at":"2026-05-18T04:42:23Z","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":"Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures are living in the (commutative) vector space $\\real^n$, if the toroidal billiard is given on the flat $n$-torus.\n  The billard trajectories, being curves, oftentimes getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e., the billiard flow on the 2-to","authors_text":"Lee M. Goswick, Nandor Simanyi","cross_cats":[],"headline":"","license":"","primary_cat":"math.DS","submitted_at":"2006-10-11T07:33:09Z","title":"Homotopical Rotation Numbers of 2D Billiards"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610350","kind":"arxiv","version":8},"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:a83f9cd9b79f2af38a02ea878785d1f9b5f202aa3cce73c4953098187d54635f","target":"record","created_at":"2026-05-18T04:42:23Z","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":"14af3ca4986a2777477d379e2ea6d8bdff18c7ee44da24be13d31e894d4b07ab","cross_cats_sorted":[],"license":"","primary_cat":"math.DS","submitted_at":"2006-10-11T07:33:09Z","title_canon_sha256":"457d5b81358a16afff03efe69707e39e7be0c7d085deada1de8913a38a2f0d24"},"schema_version":"1.0","source":{"id":"math/0610350","kind":"arxiv","version":8}},"canonical_sha256":"4e78329fcf73898c65c0eb88d3aed4095f9a66e8fc2010ecf31c886c7f64ce98","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e78329fcf73898c65c0eb88d3aed4095f9a66e8fc2010ecf31c886c7f64ce98","first_computed_at":"2026-05-18T04:42:23.290708Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:23.290708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s+8+ee6Mmdnp7508OaXYFvQdP3VklWhjWhAC+bLSPM0KHPRCgcGudUIVZUMm1h0ePeBYhk1CX5rXndXq70bUBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:23.291200Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0610350","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a83f9cd9b79f2af38a02ea878785d1f9b5f202aa3cce73c4953098187d54635f","sha256:fb0cfd0a15ef88bdf45c2df926775184183b6daf720a998d638aaae12054a00b"],"state_sha256":"c5b4834a30322e6213c21b17db7144dc89d72595ee2afae7895f55641d44644f"}