{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LIGTKSEB4Z5KODFKMMB2SPNOXD","short_pith_number":"pith:LIGTKSEB","canonical_record":{"source":{"id":"1506.00383","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-01T08:30:25Z","cross_cats_sorted":[],"title_canon_sha256":"a5394d3c654703ffe01857bae6baa649531118513df86f577b0d1e8467cbc17f","abstract_canon_sha256":"5e85d79400fd579b34e2e2f1f1a56835e11b42c8c7bf01d1b26481f57fedef54"},"schema_version":"1.0"},"canonical_sha256":"5a0d354881e67aa70caa6303a93daeb8f33af3e70ef235dc5dfc362b6017a23e","source":{"kind":"arxiv","id":"1506.00383","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.00383","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"arxiv_version","alias_value":"1506.00383v1","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.00383","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"pith_short_12","alias_value":"LIGTKSEB4Z5K","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LIGTKSEB4Z5KODFK","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LIGTKSEB","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LIGTKSEB4Z5KODFKMMB2SPNOXD","target":"record","payload":{"canonical_record":{"source":{"id":"1506.00383","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-01T08:30:25Z","cross_cats_sorted":[],"title_canon_sha256":"a5394d3c654703ffe01857bae6baa649531118513df86f577b0d1e8467cbc17f","abstract_canon_sha256":"5e85d79400fd579b34e2e2f1f1a56835e11b42c8c7bf01d1b26481f57fedef54"},"schema_version":"1.0"},"canonical_sha256":"5a0d354881e67aa70caa6303a93daeb8f33af3e70ef235dc5dfc362b6017a23e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:54.706436Z","signature_b64":"Yes6KfJ2xdc9OYZ2jq0YQyRUPkv87wfjSiCszvTzIgkLa6BzjSVaCYaejKd9boeHPPylWz5JCgxiolqowWEfDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a0d354881e67aa70caa6303a93daeb8f33af3e70ef235dc5dfc362b6017a23e","last_reissued_at":"2026-05-18T01:59:54.705888Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:54.705888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.00383","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":"2026-05-18T01:59:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7oMzXdGuSXTwzlqKG0QA/tliE7vXYvnjUmp1NNZY/lbzaXRUzLtdnqkeNb2CLlcsUb6It3n9qyBR1rzhXT+gBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T22:35:08.630533Z"},"content_sha256":"2f8839e7625c99afad71e9e28043fb39de027368583fe9fab72af2fefeef2bea","schema_version":"1.0","event_id":"sha256:2f8839e7625c99afad71e9e28043fb39de027368583fe9fab72af2fefeef2bea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LIGTKSEB4Z5KODFKMMB2SPNOXD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kolmogorov's Theorem for Low-Dimensional Invariant Tori of Hamiltonian Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ivan Kuznetsov, Pavel Plotnikov","submitted_at":"2015-06-01T08:30:25Z","abstract_excerpt":"In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the function $\\Psi$ of two variables defined on semi-cylinder is established. It is proved that if unperturbed Hamiltonian has a saddle point, then under arbitrary perturbations there persists at least one hyperbolic torus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00383","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":"2026-05-18T01:59:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t3y4sNcAYBpBwj3HlLrRgHSqXMNQzGkkrGXNSPujRSphSdts7BglYj4FVh9ei001EFaArQLA7OFk2Orxfm66Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T22:35:08.630876Z"},"content_sha256":"8e503d72dbac663ae070f134810c417f98f6afc4282abbc56aba0fcd3dab88a5","schema_version":"1.0","event_id":"sha256:8e503d72dbac663ae070f134810c417f98f6afc4282abbc56aba0fcd3dab88a5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD/bundle.json","state_url":"https://pith.science/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD/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":"2026-06-26T22:35:08Z","links":{"resolver":"https://pith.science/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD","bundle":"https://pith.science/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD/bundle.json","state":"https://pith.science/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LIGTKSEB4Z5KODFKMMB2SPNOXD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LIGTKSEB4Z5KODFKMMB2SPNOXD","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":"5e85d79400fd579b34e2e2f1f1a56835e11b42c8c7bf01d1b26481f57fedef54","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-01T08:30:25Z","title_canon_sha256":"a5394d3c654703ffe01857bae6baa649531118513df86f577b0d1e8467cbc17f"},"schema_version":"1.0","source":{"id":"1506.00383","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.00383","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"arxiv_version","alias_value":"1506.00383v1","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.00383","created_at":"2026-05-18T01:59:54Z"},{"alias_kind":"pith_short_12","alias_value":"LIGTKSEB4Z5K","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LIGTKSEB4Z5KODFK","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LIGTKSEB","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:8e503d72dbac663ae070f134810c417f98f6afc4282abbc56aba0fcd3dab88a5","target":"graph","created_at":"2026-05-18T01:59:54Z","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 this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the function $\\Psi$ of two variables defined on semi-cylinder is established. It is proved that if unperturbed Hamiltonian has a saddle point, then under arbitrary perturbations there persists at least one hyperbolic torus.","authors_text":"Ivan Kuznetsov, Pavel Plotnikov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-01T08:30:25Z","title":"Kolmogorov's Theorem for Low-Dimensional Invariant Tori of Hamiltonian Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00383","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":"sha256:2f8839e7625c99afad71e9e28043fb39de027368583fe9fab72af2fefeef2bea","target":"record","created_at":"2026-05-18T01:59:54Z","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":"5e85d79400fd579b34e2e2f1f1a56835e11b42c8c7bf01d1b26481f57fedef54","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-01T08:30:25Z","title_canon_sha256":"a5394d3c654703ffe01857bae6baa649531118513df86f577b0d1e8467cbc17f"},"schema_version":"1.0","source":{"id":"1506.00383","kind":"arxiv","version":1}},"canonical_sha256":"5a0d354881e67aa70caa6303a93daeb8f33af3e70ef235dc5dfc362b6017a23e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5a0d354881e67aa70caa6303a93daeb8f33af3e70ef235dc5dfc362b6017a23e","first_computed_at":"2026-05-18T01:59:54.705888Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:54.705888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Yes6KfJ2xdc9OYZ2jq0YQyRUPkv87wfjSiCszvTzIgkLa6BzjSVaCYaejKd9boeHPPylWz5JCgxiolqowWEfDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:54.706436Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.00383","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f8839e7625c99afad71e9e28043fb39de027368583fe9fab72af2fefeef2bea","sha256:8e503d72dbac663ae070f134810c417f98f6afc4282abbc56aba0fcd3dab88a5"],"state_sha256":"00534a8505e0210e08784a309c96562603b521723810f9e0f1be15a417b6fe2a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q2Yo2Z9S+joNhWu3WGxZCcDEnZqAZgIzGQ/tYNt/EhESztUaVBFsAxvEvNMNPfNsY6uKKx1nFHlbDGZyiD79BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T22:35:08.632757Z","bundle_sha256":"e5009b183bad58ebb5f7b12fb621fb0055cc1e40e7c76725eb83b1000feb77a5"}}