{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:O4WTURIWU4CGGUJQ7ZZRP43ASG","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":"ea28cc473b3d24948197540e7a5846da5b37c141f29bb70c7ea0e160afe7e062","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-08-05T13:54:33Z","title_canon_sha256":"6008931a3f34664adecf67b24456057c38846b4290cac5458a1863d4803d0fde"},"schema_version":"1.0","source":{"id":"2608.04862","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.04862","created_at":"2026-08-06T01:47:25Z"},{"alias_kind":"arxiv_version","alias_value":"2608.04862v1","created_at":"2026-08-06T01:47:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.04862","created_at":"2026-08-06T01:47:25Z"},{"alias_kind":"pith_short_12","alias_value":"O4WTURIWU4CG","created_at":"2026-08-06T01:47:25Z"},{"alias_kind":"pith_short_16","alias_value":"O4WTURIWU4CGGUJQ","created_at":"2026-08-06T01:47:25Z"},{"alias_kind":"pith_short_8","alias_value":"O4WTURIW","created_at":"2026-08-06T01:47:25Z"}],"graph_snapshots":[{"event_id":"sha256:8ca18e0442eda36aafe8513dfe204f99bc5d9aac0ebe1217d0b3339b11a8d0a2","target":"graph","created_at":"2026-08-06T01:47:25Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.04862/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The classical normally hyperbolic invariant manifold theorem asserts that a \\(C^1\\) normally hyperbolic invariant manifold persists under \\(C^1\\) small perturbations. For a family of standard-like dissipative twist maps, we show that the threshold \\((1-\\sqrt{\\lambda})^2\\) for the \\(C^1\\)-norm of the perturbation is sharp: there exists a $C^\\infty$ perturbation \\(\\phi\\) with \\(\\|\\phi\\|_{C^1} = (1-\\sqrt{\\lambda})^2\\) such that the map preserves a unique invariant graph, but this graph possesses non-differentiable points. On the other hand, whenever \\(\\|\\phi\\|_{C^1} < (1-\\sqrt{\\lambda})^2\\), the ","authors_text":"Junhao Li, Lin Wang, Yujie Huang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-08-05T13:54:33Z","title":"On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.04862","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:c8549a1e50f5e3ea858c2e166069924d0860523e83197cf5b098f629c89cfc20","target":"record","created_at":"2026-08-06T01:47:25Z","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":"ea28cc473b3d24948197540e7a5846da5b37c141f29bb70c7ea0e160afe7e062","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-08-05T13:54:33Z","title_canon_sha256":"6008931a3f34664adecf67b24456057c38846b4290cac5458a1863d4803d0fde"},"schema_version":"1.0","source":{"id":"2608.04862","kind":"arxiv","version":1}},"canonical_sha256":"772d3a4516a704635130fe7317f36091a80e11c70f901a908df3fcbf1d4a2fec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"772d3a4516a704635130fe7317f36091a80e11c70f901a908df3fcbf1d4a2fec","first_computed_at":"2026-08-06T01:47:25.155585Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-06T01:47:25.155585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l+tYOyMii70kP8c//H9tOvGqPAlvNbmrkMuf6+Sd5K407aJnG+HI9SJkOH7a5N19ys3oiTZ9I3DrpsBjHpTlBg==","signature_status":"signed_v1","signed_at":"2026-08-06T01:47:25.157081Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.04862","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8549a1e50f5e3ea858c2e166069924d0860523e83197cf5b098f629c89cfc20","sha256:8ca18e0442eda36aafe8513dfe204f99bc5d9aac0ebe1217d0b3339b11a8d0a2"],"state_sha256":"64448442121cb59efa2d034260b2241fc4709c2f9a2c83cf489c223b41c25d4f"}