{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ZPFAZ4BGJGJJLLRPJ7TOOAW3YV","short_pith_number":"pith:ZPFAZ4BG","canonical_record":{"source":{"id":"1403.1953","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-08T09:31:04Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"f9c72654cf1bba4f818fec445682293d725754c0a3b72f75156a67e5dc16f508","abstract_canon_sha256":"faf2280653adf84e5545cdc42f51fc2e8d50964199d945051b156223d89daebb"},"schema_version":"1.0"},"canonical_sha256":"cbca0cf026499295ae2f4fe6e702dbc54179f44d5838b1ed58994bda57427f77","source":{"kind":"arxiv","id":"1403.1953","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.1953","created_at":"2026-05-18T02:56:51Z"},{"alias_kind":"arxiv_version","alias_value":"1403.1953v1","created_at":"2026-05-18T02:56:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.1953","created_at":"2026-05-18T02:56:51Z"},{"alias_kind":"pith_short_12","alias_value":"ZPFAZ4BGJGJJ","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZPFAZ4BGJGJJLLRP","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZPFAZ4BG","created_at":"2026-05-18T12:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ZPFAZ4BGJGJJLLRPJ7TOOAW3YV","target":"record","payload":{"canonical_record":{"source":{"id":"1403.1953","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-08T09:31:04Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"f9c72654cf1bba4f818fec445682293d725754c0a3b72f75156a67e5dc16f508","abstract_canon_sha256":"faf2280653adf84e5545cdc42f51fc2e8d50964199d945051b156223d89daebb"},"schema_version":"1.0"},"canonical_sha256":"cbca0cf026499295ae2f4fe6e702dbc54179f44d5838b1ed58994bda57427f77","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:51.200793Z","signature_b64":"3xNggXeDyBWj9jEli2wQgTq5kEJf1GkT94MPYHnaARStY2I2MGvX+PvIPFtWub9lU5g6HESw7DRoPZfKrPWTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbca0cf026499295ae2f4fe6e702dbc54179f44d5838b1ed58994bda57427f77","last_reissued_at":"2026-05-18T02:56:51.200189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:51.200189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1403.1953","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":"2026-05-18T02:56:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CMe6e/9VlJDq0hK8cajHJlpLqmFdtTC1yNx6GT7hMLOMTbyQwxNvB7vL942HcMsapkb3wQTJtNQMV2z3u1LeAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T16:10:30.306635Z"},"content_sha256":"65d1cb248f9a458cfeeacd7bee70fbfefe6e37b74b8189d51f49854adc51984f","schema_version":"1.0","event_id":"sha256:65d1cb248f9a458cfeeacd7bee70fbfefe6e37b74b8189d51f49854adc51984f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ZPFAZ4BGJGJJLLRPJ7TOOAW3YV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Periodic billiard trajectories and Morse theory on loop spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DS","authors_text":"Kei Irie","submitted_at":"2014-03-08T09:31:04Z","abstract_excerpt":"We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we prove that nonvanishing of relative homology of a certain pair of loop spaces implies the existence of a periodic billiard trajectory. We also prove a parallel result for path spaces. We apply those results to show the existence of short billiard trajectories and short geodesic loops. We also recover two known results on the length of a shortest periodic bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.1953","kind":"arxiv","version":1},"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":"2026-05-18T02:56:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uklBp1hRKCJ64UcUcqBzV03Yko2yVCQN7RVr1oZhoqEJybJm+Hlu7VPsg/bE6UGgSog4yF6/3NwuiRmtguiNCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T16:10:30.306983Z"},"content_sha256":"67c7ee065abc5fa291354c99057663d644e77a75325566771caf2397befa25f3","schema_version":"1.0","event_id":"sha256:67c7ee065abc5fa291354c99057663d644e77a75325566771caf2397befa25f3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV/bundle.json","state_url":"https://pith.science/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV/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":"2026-06-27T16:10:30Z","links":{"resolver":"https://pith.science/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV","bundle":"https://pith.science/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV/bundle.json","state":"https://pith.science/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZPFAZ4BGJGJJLLRPJ7TOOAW3YV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ZPFAZ4BGJGJJLLRPJ7TOOAW3YV","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":"faf2280653adf84e5545cdc42f51fc2e8d50964199d945051b156223d89daebb","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-08T09:31:04Z","title_canon_sha256":"f9c72654cf1bba4f818fec445682293d725754c0a3b72f75156a67e5dc16f508"},"schema_version":"1.0","source":{"id":"1403.1953","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.1953","created_at":"2026-05-18T02:56:51Z"},{"alias_kind":"arxiv_version","alias_value":"1403.1953v1","created_at":"2026-05-18T02:56:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.1953","created_at":"2026-05-18T02:56:51Z"},{"alias_kind":"pith_short_12","alias_value":"ZPFAZ4BGJGJJ","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZPFAZ4BGJGJJLLRP","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZPFAZ4BG","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:67c7ee065abc5fa291354c99057663d644e77a75325566771caf2397befa25f3","target":"graph","created_at":"2026-05-18T02:56:51Z","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":"We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we prove that nonvanishing of relative homology of a certain pair of loop spaces implies the existence of a periodic billiard trajectory. We also prove a parallel result for path spaces. We apply those results to show the existence of short billiard trajectories and short geodesic loops. We also recover two known results on the length of a shortest periodic bi","authors_text":"Kei Irie","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-08T09:31:04Z","title":"Periodic billiard trajectories and Morse theory on loop spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.1953","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":"sha256:65d1cb248f9a458cfeeacd7bee70fbfefe6e37b74b8189d51f49854adc51984f","target":"record","created_at":"2026-05-18T02:56:51Z","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":"faf2280653adf84e5545cdc42f51fc2e8d50964199d945051b156223d89daebb","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-03-08T09:31:04Z","title_canon_sha256":"f9c72654cf1bba4f818fec445682293d725754c0a3b72f75156a67e5dc16f508"},"schema_version":"1.0","source":{"id":"1403.1953","kind":"arxiv","version":1}},"canonical_sha256":"cbca0cf026499295ae2f4fe6e702dbc54179f44d5838b1ed58994bda57427f77","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cbca0cf026499295ae2f4fe6e702dbc54179f44d5838b1ed58994bda57427f77","first_computed_at":"2026-05-18T02:56:51.200189Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:56:51.200189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3xNggXeDyBWj9jEli2wQgTq5kEJf1GkT94MPYHnaARStY2I2MGvX+PvIPFtWub9lU5g6HESw7DRoPZfKrPWTAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:56:51.200793Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.1953","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65d1cb248f9a458cfeeacd7bee70fbfefe6e37b74b8189d51f49854adc51984f","sha256:67c7ee065abc5fa291354c99057663d644e77a75325566771caf2397befa25f3"],"state_sha256":"aa1d56e97d99b052f4bdcdef11984f6293de550c0796ccaa5da813b28a876bf0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"orkHnFlhaow3s3QAKJGGMSd5xHVC2zQpLSgfvGlkm9D+QXQGTdnqt+eGu//PULFx26dUFUIGZYmsRrFVYRMwDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T16:10:30.309019Z","bundle_sha256":"d1c2918b597bf45a6dbcb9a3ee97d6db817582c41a79d8c38a2c93a3eb371996"}}