{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:MGMH6MD77BLEZUEPRZ6GDZ7ZIB","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":"2f16ee6b718a15f8cad453e7bf403483449bac3a18ef00a12341517b65f8002d","cross_cats_sorted":["math-ph","math.MP"],"license":"","primary_cat":"math.DS","submitted_at":"2003-01-25T19:28:06Z","title_canon_sha256":"4fd016f74c3697b44a6cdc9fa6e26b762ea61c57526c985482b6b6bb4bdefd70"},"schema_version":"1.0","source":{"id":"math/0301300","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0301300","created_at":"2026-05-18T03:50:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0301300v1","created_at":"2026-05-18T03:50:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0301300","created_at":"2026-05-18T03:50:00Z"},{"alias_kind":"pith_short_12","alias_value":"MGMH6MD77BLE","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"MGMH6MD77BLEZUEP","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"MGMH6MD7","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:9c8f7382193fcc719418095f5b1f7dc01de4fefa3d0b47f9a63036c610e1b6d3","target":"graph","created_at":"2026-05-18T03:50:00Z","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":"In a flat 2-torus with a disk of diameter $r$ removed, let $\\Phi_r(t)$ be the distribution of free-path lengths (the probability that a segment of length larger than $t$ with uniformly distributed origin and direction does not meet the disk).\n  We prove that $\\Phi_r(t/r)$ behaves like $\\frac{2}{\\pi^2 t}$ for each $t>2$ and in the limit as $r\\to 0^+$, in some appropriate sense.\n  We then discuss the implications of this result in the context of kinetic theory.","authors_text":"Emanuele Caglioti, Francois Golse","cross_cats":["math-ph","math.MP"],"headline":"","license":"","primary_cat":"math.DS","submitted_at":"2003-01-25T19:28:06Z","title":"On the distribution of free-path lengths for the periodic Lorentz gas III"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0301300","kind":"arxiv","version":1},"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":"sha256:26b38d9101215bcc042a4fb8d46d36108436abb20b4d69b318e58ed37444a511","target":"record","created_at":"2026-05-18T03:50:00Z","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":"2f16ee6b718a15f8cad453e7bf403483449bac3a18ef00a12341517b65f8002d","cross_cats_sorted":["math-ph","math.MP"],"license":"","primary_cat":"math.DS","submitted_at":"2003-01-25T19:28:06Z","title_canon_sha256":"4fd016f74c3697b44a6cdc9fa6e26b762ea61c57526c985482b6b6bb4bdefd70"},"schema_version":"1.0","source":{"id":"math/0301300","kind":"arxiv","version":1}},"canonical_sha256":"61987f307ff8564cd08f8e7c61e7f9404525db27b23e937bfd477d20238e31f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61987f307ff8564cd08f8e7c61e7f9404525db27b23e937bfd477d20238e31f5","first_computed_at":"2026-05-18T03:50:00.325530Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:00.325530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h2a+IAAo8HUf4P18EnOvbZmaX00lZbf3lzgHOignxRfsUGU4naUYnJN6Qspn6Fje4fu1+tjCXhBZGX+MjFPaDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:00.325997Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0301300","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:26b38d9101215bcc042a4fb8d46d36108436abb20b4d69b318e58ed37444a511","sha256:9c8f7382193fcc719418095f5b1f7dc01de4fefa3d0b47f9a63036c610e1b6d3"],"state_sha256":"5f0d957c9c92160b4b83809d70e4cd3d22b15b31e53c3f6312a4e11eccd2cba8"}