{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:LZI2NP6HQNCRW3QZQ3VFNL7HIM","short_pith_number":"pith:LZI2NP6H","canonical_record":{"source":{"id":"1709.02337","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-07T16:22:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2d1a9ee491693836dfe2b16e5123315d91f8eba53c9a65c6cf817b1e49905733","abstract_canon_sha256":"5af4801d3275fa0a462bde04596fa489e02bb105fa1b403e414f77412522819c"},"schema_version":"1.0"},"canonical_sha256":"5e51a6bfc783451b6e1986ea56afe743293891f3d5df82a19b5c1f2e6c2ecca2","source":{"kind":"arxiv","id":"1709.02337","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.02337","created_at":"2026-05-17T23:44:10Z"},{"alias_kind":"arxiv_version","alias_value":"1709.02337v4","created_at":"2026-05-17T23:44:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.02337","created_at":"2026-05-17T23:44:10Z"},{"alias_kind":"pith_short_12","alias_value":"LZI2NP6HQNCR","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"LZI2NP6HQNCRW3QZ","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"LZI2NP6H","created_at":"2026-05-18T12:31:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:LZI2NP6HQNCRW3QZQ3VFNL7HIM","target":"record","payload":{"canonical_record":{"source":{"id":"1709.02337","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-07T16:22:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2d1a9ee491693836dfe2b16e5123315d91f8eba53c9a65c6cf817b1e49905733","abstract_canon_sha256":"5af4801d3275fa0a462bde04596fa489e02bb105fa1b403e414f77412522819c"},"schema_version":"1.0"},"canonical_sha256":"5e51a6bfc783451b6e1986ea56afe743293891f3d5df82a19b5c1f2e6c2ecca2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:10.396556Z","signature_b64":"h3WAThKpVW21nyUWpzonTjN0KPSsd35Bu780nchqHL8/HyM2d1JmT5VN2D58fPDzgYd8zIo7r95wLBNC5QxECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e51a6bfc783451b6e1986ea56afe743293891f3d5df82a19b5c1f2e6c2ecca2","last_reissued_at":"2026-05-17T23:44:10.395861Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:10.395861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.02337","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":"2026-05-17T23:44:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eAX30Id07Af4KVXqkf6Nvh4xdSoYK4K8Yhc2UYyC5+zCNHTy13nphm5nmVsAExXF8kJcSx3d168rfS5/QmRDDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:59:58.847264Z"},"content_sha256":"4854cd98b967d14f0e4e3203c4a4941ca2f9c898d112e64c0697712ee4c0d310","schema_version":"1.0","event_id":"sha256:4854cd98b967d14f0e4e3203c4a4941ca2f9c898d112e64c0697712ee4c0d310"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:LZI2NP6HQNCRW3QZQ3VFNL7HIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Morse theory methods for a class of quasi-linear elliptic systems of higher order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Guangcun Lu","submitted_at":"2017-09-07T16:22:27Z","abstract_excerpt":"We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type result. They are applicable to a wide range of multiple integrals with quasi-linear elliptic Euler equations and systems of higher order."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02337","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":"2026-05-17T23:44:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X5qQ5fUEguD4VDtay4p8P4PEGDCgmV+K9e9jxihhc3JKm3reibFRmoQRFgjlRfPKvCH9bIjH1X1d0yXZC2BFBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:59:58.847636Z"},"content_sha256":"90fed65cef26d38d5ea2b7cdfea0c08111a5d25f79d0345374cd4ee7521414ea","schema_version":"1.0","event_id":"sha256:90fed65cef26d38d5ea2b7cdfea0c08111a5d25f79d0345374cd4ee7521414ea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM/bundle.json","state_url":"https://pith.science/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM/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":"2026-06-22T09:59:58Z","links":{"resolver":"https://pith.science/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM","bundle":"https://pith.science/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM/bundle.json","state":"https://pith.science/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LZI2NP6HQNCRW3QZQ3VFNL7HIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:LZI2NP6HQNCRW3QZQ3VFNL7HIM","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":"5af4801d3275fa0a462bde04596fa489e02bb105fa1b403e414f77412522819c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-07T16:22:27Z","title_canon_sha256":"2d1a9ee491693836dfe2b16e5123315d91f8eba53c9a65c6cf817b1e49905733"},"schema_version":"1.0","source":{"id":"1709.02337","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.02337","created_at":"2026-05-17T23:44:10Z"},{"alias_kind":"arxiv_version","alias_value":"1709.02337v4","created_at":"2026-05-17T23:44:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.02337","created_at":"2026-05-17T23:44:10Z"},{"alias_kind":"pith_short_12","alias_value":"LZI2NP6HQNCR","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"LZI2NP6HQNCRW3QZ","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"LZI2NP6H","created_at":"2026-05-18T12:31:28Z"}],"graph_snapshots":[{"event_id":"sha256:90fed65cef26d38d5ea2b7cdfea0c08111a5d25f79d0345374cd4ee7521414ea","target":"graph","created_at":"2026-05-17T23:44:10Z","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 develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type result. They are applicable to a wide range of multiple integrals with quasi-linear elliptic Euler equations and systems of higher order.","authors_text":"Guangcun Lu","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-07T16:22:27Z","title":"Morse theory methods for a class of quasi-linear elliptic systems of higher order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02337","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":"sha256:4854cd98b967d14f0e4e3203c4a4941ca2f9c898d112e64c0697712ee4c0d310","target":"record","created_at":"2026-05-17T23:44:10Z","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":"5af4801d3275fa0a462bde04596fa489e02bb105fa1b403e414f77412522819c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-07T16:22:27Z","title_canon_sha256":"2d1a9ee491693836dfe2b16e5123315d91f8eba53c9a65c6cf817b1e49905733"},"schema_version":"1.0","source":{"id":"1709.02337","kind":"arxiv","version":4}},"canonical_sha256":"5e51a6bfc783451b6e1986ea56afe743293891f3d5df82a19b5c1f2e6c2ecca2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5e51a6bfc783451b6e1986ea56afe743293891f3d5df82a19b5c1f2e6c2ecca2","first_computed_at":"2026-05-17T23:44:10.395861Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:10.395861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h3WAThKpVW21nyUWpzonTjN0KPSsd35Bu780nchqHL8/HyM2d1JmT5VN2D58fPDzgYd8zIo7r95wLBNC5QxECQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:10.396556Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.02337","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4854cd98b967d14f0e4e3203c4a4941ca2f9c898d112e64c0697712ee4c0d310","sha256:90fed65cef26d38d5ea2b7cdfea0c08111a5d25f79d0345374cd4ee7521414ea"],"state_sha256":"0481c28a8e7fe2e128da00f85136996d9b227cb10c8ccd13a593bb9535fd29bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ruFQm+9t9EmEA3zaS9P2ikQApmFefA4lYJJL3fgaKYGsYBigWgskKTZceqzAn+is3zzqEqx2Zss+WCRUGpbEAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T09:59:58.849541Z","bundle_sha256":"65e0f388d1e4cee007811067465042f79edca8562eb4ccd90e965498d5e68815"}}