{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:BI473CDB53FKCMITFZZ6Z7CQVH","short_pith_number":"pith:BI473CDB","canonical_record":{"source":{"id":"1211.2127","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-06T12:26:17Z","cross_cats_sorted":["math.AP","math.GT"],"title_canon_sha256":"ff66db433a8dd006d8aa611e6a40858ef6734a6b6bd1c035b6c55e1f87f89645","abstract_canon_sha256":"0972f6032b382eb0c90639e1aeaa6da11f0ecc0ccad1046ca4daefb63bf23c76"},"schema_version":"1.0"},"canonical_sha256":"0a39fd8861eecaa131132e73ecfc50a9e95efe378c648d3176f11e53eb0ba4eb","source":{"kind":"arxiv","id":"1211.2127","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2127","created_at":"2026-05-18T02:50:01Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2127v2","created_at":"2026-05-18T02:50:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2127","created_at":"2026-05-18T02:50:01Z"},{"alias_kind":"pith_short_12","alias_value":"BI473CDB53FK","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"BI473CDB53FKCMIT","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"BI473CDB","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:BI473CDB53FKCMITFZZ6Z7CQVH","target":"record","payload":{"canonical_record":{"source":{"id":"1211.2127","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-06T12:26:17Z","cross_cats_sorted":["math.AP","math.GT"],"title_canon_sha256":"ff66db433a8dd006d8aa611e6a40858ef6734a6b6bd1c035b6c55e1f87f89645","abstract_canon_sha256":"0972f6032b382eb0c90639e1aeaa6da11f0ecc0ccad1046ca4daefb63bf23c76"},"schema_version":"1.0"},"canonical_sha256":"0a39fd8861eecaa131132e73ecfc50a9e95efe378c648d3176f11e53eb0ba4eb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:01.019203Z","signature_b64":"4tuYH0h1N7gLSbOMBrg9EJ5FHIdgpxtTFVQ1rAZD2qwUVyeD/sSZZGW84qLskkRQNunE1IbAwH/3omBbNjb1Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a39fd8861eecaa131132e73ecfc50a9e95efe378c648d3176f11e53eb0ba4eb","last_reissued_at":"2026-05-18T02:50:01.018721Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:01.018721Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.2127","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":"2026-05-18T02:50:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"31rMCrSZMw+fWhAyPkHfOe8DGIl17tuH1WirH/jXNUDRQf7iEyiLvYL4hFqfkrdgTlrFqJr8B6n6MWB8y3FgCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:30:21.086388Z"},"content_sha256":"1317df34e3827fa480ae3e9cb6dbaabffa7daa5567056d465a5acce25a56d1bd","schema_version":"1.0","event_id":"sha256:1317df34e3827fa480ae3e9cb6dbaabffa7daa5567056d465a5acce25a56d1bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:BI473CDB53FKCMITFZZ6Z7CQVH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The splitting lemmas for nonsmooth functionals on Hilbert spaces I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.GT"],"primary_cat":"math.FA","authors_text":"Guangcun Lu","submitted_at":"2012-11-06T12:26:17Z","abstract_excerpt":"The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\\int_\\Omega f(x, u,..., D^mu)dx$ as in (\\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2127","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":"2026-05-18T02:50:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"deuBeKCh0RkwbJYraEhzZ2mpRNifRCiMvpZGXnxk5Wo1hncdR4yxwWQor+ZUpyU607jupis6nZ/omf3PloQEDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:30:21.086746Z"},"content_sha256":"fdb25ca47e3326e32db5b0917b8bfbed3ed61bffa1a584e5a27aa28427cf389f","schema_version":"1.0","event_id":"sha256:fdb25ca47e3326e32db5b0917b8bfbed3ed61bffa1a584e5a27aa28427cf389f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BI473CDB53FKCMITFZZ6Z7CQVH/bundle.json","state_url":"https://pith.science/pith/BI473CDB53FKCMITFZZ6Z7CQVH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BI473CDB53FKCMITFZZ6Z7CQVH/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":"2026-06-22T09:30:21Z","links":{"resolver":"https://pith.science/pith/BI473CDB53FKCMITFZZ6Z7CQVH","bundle":"https://pith.science/pith/BI473CDB53FKCMITFZZ6Z7CQVH/bundle.json","state":"https://pith.science/pith/BI473CDB53FKCMITFZZ6Z7CQVH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BI473CDB53FKCMITFZZ6Z7CQVH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:BI473CDB53FKCMITFZZ6Z7CQVH","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":"0972f6032b382eb0c90639e1aeaa6da11f0ecc0ccad1046ca4daefb63bf23c76","cross_cats_sorted":["math.AP","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-06T12:26:17Z","title_canon_sha256":"ff66db433a8dd006d8aa611e6a40858ef6734a6b6bd1c035b6c55e1f87f89645"},"schema_version":"1.0","source":{"id":"1211.2127","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2127","created_at":"2026-05-18T02:50:01Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2127v2","created_at":"2026-05-18T02:50:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2127","created_at":"2026-05-18T02:50:01Z"},{"alias_kind":"pith_short_12","alias_value":"BI473CDB53FK","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"BI473CDB53FKCMIT","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"BI473CDB","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:fdb25ca47e3326e32db5b0917b8bfbed3ed61bffa1a584e5a27aa28427cf389f","target":"graph","created_at":"2026-05-18T02:50:01Z","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 Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\\int_\\Omega f(x, u,..., D^mu)dx$ as in (\\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but","authors_text":"Guangcun Lu","cross_cats":["math.AP","math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-06T12:26:17Z","title":"The splitting lemmas for nonsmooth functionals on Hilbert spaces I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2127","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":"sha256:1317df34e3827fa480ae3e9cb6dbaabffa7daa5567056d465a5acce25a56d1bd","target":"record","created_at":"2026-05-18T02:50:01Z","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":"0972f6032b382eb0c90639e1aeaa6da11f0ecc0ccad1046ca4daefb63bf23c76","cross_cats_sorted":["math.AP","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-06T12:26:17Z","title_canon_sha256":"ff66db433a8dd006d8aa611e6a40858ef6734a6b6bd1c035b6c55e1f87f89645"},"schema_version":"1.0","source":{"id":"1211.2127","kind":"arxiv","version":2}},"canonical_sha256":"0a39fd8861eecaa131132e73ecfc50a9e95efe378c648d3176f11e53eb0ba4eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a39fd8861eecaa131132e73ecfc50a9e95efe378c648d3176f11e53eb0ba4eb","first_computed_at":"2026-05-18T02:50:01.018721Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:01.018721Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4tuYH0h1N7gLSbOMBrg9EJ5FHIdgpxtTFVQ1rAZD2qwUVyeD/sSZZGW84qLskkRQNunE1IbAwH/3omBbNjb1Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:01.019203Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.2127","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1317df34e3827fa480ae3e9cb6dbaabffa7daa5567056d465a5acce25a56d1bd","sha256:fdb25ca47e3326e32db5b0917b8bfbed3ed61bffa1a584e5a27aa28427cf389f"],"state_sha256":"f7245101937ddc78386b7ea13dea48780001600ce573968a40cf2b2040b83dcf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R/Sy5Nz34x1bkoiGHFXNaKyKlgu+lD7Tw9V5YTsKsjcsPh8JyUjJgF8LUZRa3vrs1XPm1fxY0tyazUKAhazgBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T09:30:21.088804Z","bundle_sha256":"57844d5665be0245cfa8ce62b852bf6d9e4ab827573e861baaacc1f12f5b8d9e"}}