{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:CUASFSIRXKK5DHQRODMYF67PA2","short_pith_number":"pith:CUASFSIR","canonical_record":{"source":{"id":"1112.0891","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-12-05T11:24:15Z","cross_cats_sorted":["math.DS","math.MP"],"title_canon_sha256":"e0cefe2ecd1af701b990548f1fb19f879534ac4cac21f5487d6bcf86b2acbe40","abstract_canon_sha256":"9f42b48971008a4620396a4d159a5764a385c218036b942c38065b413735b288"},"schema_version":"1.0"},"canonical_sha256":"150122c911ba95d19e1170d982fbef069c9da81e7ffe6fa4e1ddfe28d8808f35","source":{"kind":"arxiv","id":"1112.0891","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0891","created_at":"2026-05-18T01:59:17Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0891v3","created_at":"2026-05-18T01:59:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0891","created_at":"2026-05-18T01:59:17Z"},{"alias_kind":"pith_short_12","alias_value":"CUASFSIRXKK5","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CUASFSIRXKK5DHQR","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CUASFSIR","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:CUASFSIRXKK5DHQRODMYF67PA2","target":"record","payload":{"canonical_record":{"source":{"id":"1112.0891","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-12-05T11:24:15Z","cross_cats_sorted":["math.DS","math.MP"],"title_canon_sha256":"e0cefe2ecd1af701b990548f1fb19f879534ac4cac21f5487d6bcf86b2acbe40","abstract_canon_sha256":"9f42b48971008a4620396a4d159a5764a385c218036b942c38065b413735b288"},"schema_version":"1.0"},"canonical_sha256":"150122c911ba95d19e1170d982fbef069c9da81e7ffe6fa4e1ddfe28d8808f35","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:17.182787Z","signature_b64":"Z7kPVSdBm5nt3I35+G4ZWJjtOOk/mnHMP2+14HkHHsHmfL0Xo0CtgbKfQZuzQtMtaZIZLAIIKgLnn+6bMEKQDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"150122c911ba95d19e1170d982fbef069c9da81e7ffe6fa4e1ddfe28d8808f35","last_reissued_at":"2026-05-18T01:59:17.182042Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:17.182042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.0891","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:59:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d/M3ApH4SR/A7m5Bu8wVnc1OuMcgC7BptPwj8wfnvfo8DUGQ5fGCl8mZQzWOH4BgInVrhHDsvLGdVRXjXIZoBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T01:49:47.783143Z"},"content_sha256":"bde477c7803bcb43a98995a29f8f0139a2bd897c2e2de22ee0495fa56838ab32","schema_version":"1.0","event_id":"sha256:bde477c7803bcb43a98995a29f8f0139a2bd897c2e2de22ee0495fa56838ab32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:CUASFSIRXKK5DHQRODMYF67PA2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Remarks on interior transmission eigenvalues, Weyl formula and branching billiards","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MP"],"primary_cat":"math-ph","authors_text":"Boris Vainberg, Evgeny Lakshtanov","submitted_at":"2011-12-05T11:24:15Z","abstract_excerpt":"The paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter-elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0891","kind":"arxiv","version":3},"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":"2026-05-18T01:59:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PnNWZhrCodYvsaQXWVlTg7EWqfLiI0imsfKdMMF4cWDiJiYZkwv9nwhV/bbZTZ0VTIsnokIM4mwzLG7VSh4FDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T01:49:47.783890Z"},"content_sha256":"92315a157c6b231a00c6871ef5b239b7b72910f7651197b2e291ddf2abf380db","schema_version":"1.0","event_id":"sha256:92315a157c6b231a00c6871ef5b239b7b72910f7651197b2e291ddf2abf380db"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CUASFSIRXKK5DHQRODMYF67PA2/bundle.json","state_url":"https://pith.science/pith/CUASFSIRXKK5DHQRODMYF67PA2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CUASFSIRXKK5DHQRODMYF67PA2/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":"2026-05-31T01:49:47Z","links":{"resolver":"https://pith.science/pith/CUASFSIRXKK5DHQRODMYF67PA2","bundle":"https://pith.science/pith/CUASFSIRXKK5DHQRODMYF67PA2/bundle.json","state":"https://pith.science/pith/CUASFSIRXKK5DHQRODMYF67PA2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CUASFSIRXKK5DHQRODMYF67PA2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:CUASFSIRXKK5DHQRODMYF67PA2","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":"9f42b48971008a4620396a4d159a5764a385c218036b942c38065b413735b288","cross_cats_sorted":["math.DS","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-12-05T11:24:15Z","title_canon_sha256":"e0cefe2ecd1af701b990548f1fb19f879534ac4cac21f5487d6bcf86b2acbe40"},"schema_version":"1.0","source":{"id":"1112.0891","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0891","created_at":"2026-05-18T01:59:17Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0891v3","created_at":"2026-05-18T01:59:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0891","created_at":"2026-05-18T01:59:17Z"},{"alias_kind":"pith_short_12","alias_value":"CUASFSIRXKK5","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CUASFSIRXKK5DHQR","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CUASFSIR","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:92315a157c6b231a00c6871ef5b239b7b72910f7651197b2e291ddf2abf380db","target":"graph","created_at":"2026-05-18T01:59:17Z","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":"The paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter-elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.","authors_text":"Boris Vainberg, Evgeny Lakshtanov","cross_cats":["math.DS","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-12-05T11:24:15Z","title":"Remarks on interior transmission eigenvalues, Weyl formula and branching billiards"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0891","kind":"arxiv","version":3},"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:bde477c7803bcb43a98995a29f8f0139a2bd897c2e2de22ee0495fa56838ab32","target":"record","created_at":"2026-05-18T01:59:17Z","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":"9f42b48971008a4620396a4d159a5764a385c218036b942c38065b413735b288","cross_cats_sorted":["math.DS","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-12-05T11:24:15Z","title_canon_sha256":"e0cefe2ecd1af701b990548f1fb19f879534ac4cac21f5487d6bcf86b2acbe40"},"schema_version":"1.0","source":{"id":"1112.0891","kind":"arxiv","version":3}},"canonical_sha256":"150122c911ba95d19e1170d982fbef069c9da81e7ffe6fa4e1ddfe28d8808f35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"150122c911ba95d19e1170d982fbef069c9da81e7ffe6fa4e1ddfe28d8808f35","first_computed_at":"2026-05-18T01:59:17.182042Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:17.182042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z7kPVSdBm5nt3I35+G4ZWJjtOOk/mnHMP2+14HkHHsHmfL0Xo0CtgbKfQZuzQtMtaZIZLAIIKgLnn+6bMEKQDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:17.182787Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.0891","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bde477c7803bcb43a98995a29f8f0139a2bd897c2e2de22ee0495fa56838ab32","sha256:92315a157c6b231a00c6871ef5b239b7b72910f7651197b2e291ddf2abf380db"],"state_sha256":"023adf87aecbebc28308f3bb134642e66de526005938f4f0e17fbcd292be0b1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A9OV6wHM/CfXRCxo4KUUh9RnyXsWA0pn68J4D5ifwFtbpB8JfvDoyPxSfYIdKNIHfSgx0v1mmRWqJPfJSsGADw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T01:49:47.787266Z","bundle_sha256":"45b70426320f0b91f35764c1860685dd4354d4fd37cba48571ee749fc8157e20"}}