{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:VPLV5BGE47AAS7W4VQMG2DESIT","short_pith_number":"pith:VPLV5BGE","canonical_record":{"source":{"id":"1804.00181","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-31T15:11:48Z","cross_cats_sorted":[],"title_canon_sha256":"da0a3313afc4a646ab0c7279c4043b2fb26024084565b73d6a8a73b5eb42f50f","abstract_canon_sha256":"745b42e371d3b8e9ee0488cfe277db0be84e1b5e1ac25227e0fd227a920c28eb"},"schema_version":"1.0"},"canonical_sha256":"abd75e84c4e7c0097edcac186d0c9244f0007085a8c681b68df951e05bd414a1","source":{"kind":"arxiv","id":"1804.00181","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.00181","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"arxiv_version","alias_value":"1804.00181v4","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.00181","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"pith_short_12","alias_value":"VPLV5BGE47AA","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VPLV5BGE47AAS7W4","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VPLV5BGE","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:VPLV5BGE47AAS7W4VQMG2DESIT","target":"record","payload":{"canonical_record":{"source":{"id":"1804.00181","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-31T15:11:48Z","cross_cats_sorted":[],"title_canon_sha256":"da0a3313afc4a646ab0c7279c4043b2fb26024084565b73d6a8a73b5eb42f50f","abstract_canon_sha256":"745b42e371d3b8e9ee0488cfe277db0be84e1b5e1ac25227e0fd227a920c28eb"},"schema_version":"1.0"},"canonical_sha256":"abd75e84c4e7c0097edcac186d0c9244f0007085a8c681b68df951e05bd414a1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:48.526620Z","signature_b64":"4iRXyZRlPfRhYaEUw4IFHeoRFLVXQ50V1NeSH8+roOM4CvsFUt8FvxFKTn/19AsEIti2dzZnyswE2zwMCL7DBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abd75e84c4e7c0097edcac186d0c9244f0007085a8c681b68df951e05bd414a1","last_reissued_at":"2026-05-17T23:54:48.526084Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:48.526084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.00181","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L0V6rsD9FKBkrvJG5ERaO5SH5Oupmt+I1nUCcsDxqTNVp0MDfBzwcxDVd92wSRFTRsnqx7QVu0GiepWYeEJQBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T19:59:39.101114Z"},"content_sha256":"441e14f7cae1d34895d0a45829903e41147280b6ae24b570ff39ad02bb8b37ef","schema_version":"1.0","event_id":"sha256:441e14f7cae1d34895d0a45829903e41147280b6ae24b570ff39ad02bb8b37ef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:VPLV5BGE47AAS7W4VQMG2DESIT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Olga Paris-Romaskevich, Pascal Hubert","submitted_at":"2018-03-31T15:11:48Z","abstract_excerpt":"Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.00181","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wdF/iNhbHag7ApgjSBUFxjygzwWMF8YxmND0rE76veH5boWypR3nmMpem/OXAN1F//ttHg/8YQ8ls5iEPjl9Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T19:59:39.101474Z"},"content_sha256":"da0c2ae370104565ef7b052c6ae112d0fe2e231d4a6aeb5019514c6dca1e647b","schema_version":"1.0","event_id":"sha256:da0c2ae370104565ef7b052c6ae112d0fe2e231d4a6aeb5019514c6dca1e647b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VPLV5BGE47AAS7W4VQMG2DESIT/bundle.json","state_url":"https://pith.science/pith/VPLV5BGE47AAS7W4VQMG2DESIT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VPLV5BGE47AAS7W4VQMG2DESIT/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":"2026-06-10T19:59:39Z","links":{"resolver":"https://pith.science/pith/VPLV5BGE47AAS7W4VQMG2DESIT","bundle":"https://pith.science/pith/VPLV5BGE47AAS7W4VQMG2DESIT/bundle.json","state":"https://pith.science/pith/VPLV5BGE47AAS7W4VQMG2DESIT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VPLV5BGE47AAS7W4VQMG2DESIT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VPLV5BGE47AAS7W4VQMG2DESIT","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":"745b42e371d3b8e9ee0488cfe277db0be84e1b5e1ac25227e0fd227a920c28eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-31T15:11:48Z","title_canon_sha256":"da0a3313afc4a646ab0c7279c4043b2fb26024084565b73d6a8a73b5eb42f50f"},"schema_version":"1.0","source":{"id":"1804.00181","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.00181","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"arxiv_version","alias_value":"1804.00181v4","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.00181","created_at":"2026-05-17T23:54:48Z"},{"alias_kind":"pith_short_12","alias_value":"VPLV5BGE47AA","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VPLV5BGE47AAS7W4","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VPLV5BGE","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:da0c2ae370104565ef7b052c6ae112d0fe2e231d4a6aeb5019514c6dca1e647b","target":"graph","created_at":"2026-05-17T23:54:48Z","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":"Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : ","authors_text":"Olga Paris-Romaskevich, Pascal Hubert","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-31T15:11:48Z","title":"Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.00181","kind":"arxiv","version":4},"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":"sha256:441e14f7cae1d34895d0a45829903e41147280b6ae24b570ff39ad02bb8b37ef","target":"record","created_at":"2026-05-17T23:54:48Z","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":"745b42e371d3b8e9ee0488cfe277db0be84e1b5e1ac25227e0fd227a920c28eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-31T15:11:48Z","title_canon_sha256":"da0a3313afc4a646ab0c7279c4043b2fb26024084565b73d6a8a73b5eb42f50f"},"schema_version":"1.0","source":{"id":"1804.00181","kind":"arxiv","version":4}},"canonical_sha256":"abd75e84c4e7c0097edcac186d0c9244f0007085a8c681b68df951e05bd414a1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"abd75e84c4e7c0097edcac186d0c9244f0007085a8c681b68df951e05bd414a1","first_computed_at":"2026-05-17T23:54:48.526084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:48.526084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4iRXyZRlPfRhYaEUw4IFHeoRFLVXQ50V1NeSH8+roOM4CvsFUt8FvxFKTn/19AsEIti2dzZnyswE2zwMCL7DBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:48.526620Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.00181","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:441e14f7cae1d34895d0a45829903e41147280b6ae24b570ff39ad02bb8b37ef","sha256:da0c2ae370104565ef7b052c6ae112d0fe2e231d4a6aeb5019514c6dca1e647b"],"state_sha256":"bc9feb57183b62927e7f2804f9ff2e4ebf738d1566b6a22c9f0a3535bebc7149"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"geFJgxWFe9wb/aqTzZq80TxUAoYx1nEyjzth2jAMxaFRZeD6Y793EWAa6d0Fkm4P8zohLLMkT6dZT9UlVZsaCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T19:59:39.103653Z","bundle_sha256":"8685e5d2c512dba6034574900b4a5ab88cd6d2ae35f1633041c5223365663fcd"}}