{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HK5BHQ3FPHWDK7H57ZMSS2QSYH","short_pith_number":"pith:HK5BHQ3F","canonical_record":{"source":{"id":"1602.03155","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-02-09T20:22:46Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"6da1be755bc740c420b481493263ba89e65416cd712aa873e6af675c13ac2a06","abstract_canon_sha256":"7883af3939e39a125226bba5201a5428b48a63428652218c5acb15e98c764c1a"},"schema_version":"1.0"},"canonical_sha256":"3aba13c36579ec357cfdfe59296a12c1d805aa53f1f6252dfb42c774f021e3c1","source":{"kind":"arxiv","id":"1602.03155","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03155","created_at":"2026-05-17T23:54:03Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03155v2","created_at":"2026-05-17T23:54:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03155","created_at":"2026-05-17T23:54:03Z"},{"alias_kind":"pith_short_12","alias_value":"HK5BHQ3FPHWD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HK5BHQ3FPHWDK7H5","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HK5BHQ3F","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HK5BHQ3FPHWDK7H57ZMSS2QSYH","target":"record","payload":{"canonical_record":{"source":{"id":"1602.03155","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-02-09T20:22:46Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"6da1be755bc740c420b481493263ba89e65416cd712aa873e6af675c13ac2a06","abstract_canon_sha256":"7883af3939e39a125226bba5201a5428b48a63428652218c5acb15e98c764c1a"},"schema_version":"1.0"},"canonical_sha256":"3aba13c36579ec357cfdfe59296a12c1d805aa53f1f6252dfb42c774f021e3c1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:03.444457Z","signature_b64":"FxK9dtIRmIPcob5Wf/xNA+KsmspSMvXlmdgLjzQ7i5NhyMVlK0qwiquVBTNjBV0f6YRQ7dSTI4gyM8xrIo+QAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3aba13c36579ec357cfdfe59296a12c1d805aa53f1f6252dfb42c774f021e3c1","last_reissued_at":"2026-05-17T23:54:03.443778Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:03.443778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.03155","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vD2CHbG2rW66CXjfAHRr01iVBY+fpeS33vmjudvadYmUDPQjB3MvTRb1DdpScra9B4N4vI39Y0QuA+TCOnVyAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T16:53:04.807193Z"},"content_sha256":"ef42b1f9e9e0d8738fd197b117f607c15aab9c5fdf3fbeb69e9745570a549b33","schema_version":"1.0","event_id":"sha256:ef42b1f9e9e0d8738fd197b117f607c15aab9c5fdf3fbeb69e9745570a549b33"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HK5BHQ3FPHWDK7H57ZMSS2QSYH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.SP","authors_text":"G. Popov, P. Topalov","submitted_at":"2016-02-09T20:22:46Z","abstract_excerpt":"This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03155","kind":"arxiv","version":2},"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":"2026-05-17T23:54:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3AWATIIExcl0nkVCtycHVy/LAGfibB3q2QdDRHaBwRXl6BwZLnbU/8/AZA7Oriay/tTns2fd1mdry9Oo1oMpBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T16:53:04.807887Z"},"content_sha256":"970849de71ac17755661f57ea59b4464398209867c45244b9088a00d06c16d97","schema_version":"1.0","event_id":"sha256:970849de71ac17755661f57ea59b4464398209867c45244b9088a00d06c16d97"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH/bundle.json","state_url":"https://pith.science/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH/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":"2026-06-11T16:53:04Z","links":{"resolver":"https://pith.science/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH","bundle":"https://pith.science/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH/bundle.json","state":"https://pith.science/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HK5BHQ3FPHWDK7H57ZMSS2QSYH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HK5BHQ3FPHWDK7H57ZMSS2QSYH","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":"7883af3939e39a125226bba5201a5428b48a63428652218c5acb15e98c764c1a","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-02-09T20:22:46Z","title_canon_sha256":"6da1be755bc740c420b481493263ba89e65416cd712aa873e6af675c13ac2a06"},"schema_version":"1.0","source":{"id":"1602.03155","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.03155","created_at":"2026-05-17T23:54:03Z"},{"alias_kind":"arxiv_version","alias_value":"1602.03155v2","created_at":"2026-05-17T23:54:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.03155","created_at":"2026-05-17T23:54:03Z"},{"alias_kind":"pith_short_12","alias_value":"HK5BHQ3FPHWD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HK5BHQ3FPHWDK7H5","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HK5BHQ3F","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:970849de71ac17755661f57ea59b4464398209867c45244b9088a00d06c16d97","target":"graph","created_at":"2026-05-17T23:54:03Z","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":"This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the i","authors_text":"G. Popov, P. Topalov","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-02-09T20:22:46Z","title":"From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03155","kind":"arxiv","version":2},"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:ef42b1f9e9e0d8738fd197b117f607c15aab9c5fdf3fbeb69e9745570a549b33","target":"record","created_at":"2026-05-17T23:54:03Z","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":"7883af3939e39a125226bba5201a5428b48a63428652218c5acb15e98c764c1a","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-02-09T20:22:46Z","title_canon_sha256":"6da1be755bc740c420b481493263ba89e65416cd712aa873e6af675c13ac2a06"},"schema_version":"1.0","source":{"id":"1602.03155","kind":"arxiv","version":2}},"canonical_sha256":"3aba13c36579ec357cfdfe59296a12c1d805aa53f1f6252dfb42c774f021e3c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3aba13c36579ec357cfdfe59296a12c1d805aa53f1f6252dfb42c774f021e3c1","first_computed_at":"2026-05-17T23:54:03.443778Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:03.443778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FxK9dtIRmIPcob5Wf/xNA+KsmspSMvXlmdgLjzQ7i5NhyMVlK0qwiquVBTNjBV0f6YRQ7dSTI4gyM8xrIo+QAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:03.444457Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.03155","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef42b1f9e9e0d8738fd197b117f607c15aab9c5fdf3fbeb69e9745570a549b33","sha256:970849de71ac17755661f57ea59b4464398209867c45244b9088a00d06c16d97"],"state_sha256":"24f490efc160be5f99f9011652d48c24ec0cf29a64de6c5206e0a0b8194c036b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"06sLLuL7pyC1W564QDi32nHQzDgZxM7gGcBPKiV3aa7ojw0b696Q7+F4dDs3BKHGFFcnY9L3bOEYNwQZ8DtiAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T16:53:04.811631Z","bundle_sha256":"838b3f8143114f154692942361dccd78cb12e510dd8f91bb0fa4efd2fa695ca9"}}