{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:3IFGJZFJXI55BVSUYB75XQHRBJ","short_pith_number":"pith:3IFGJZFJ","canonical_record":{"source":{"id":"1004.5237","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-29T09:39:52Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"a880ac37056e09a05ac73f41a4d5bf4c8d1095e6cf46042e7ca231b765f3a1e9","abstract_canon_sha256":"5c5e4edb0826b5813888406ab9e44bfcf384e6a55690452a65ae39441700ceb7"},"schema_version":"1.0"},"canonical_sha256":"da0a64e4a9ba3bd0d654c07fdbc0f10a464600c77628228db60c6c69a0e890c2","source":{"kind":"arxiv","id":"1004.5237","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5237","created_at":"2026-05-18T02:07:45Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5237v2","created_at":"2026-05-18T02:07:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5237","created_at":"2026-05-18T02:07:45Z"},{"alias_kind":"pith_short_12","alias_value":"3IFGJZFJXI55","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3IFGJZFJXI55BVSU","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3IFGJZFJ","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:3IFGJZFJXI55BVSUYB75XQHRBJ","target":"record","payload":{"canonical_record":{"source":{"id":"1004.5237","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-29T09:39:52Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"a880ac37056e09a05ac73f41a4d5bf4c8d1095e6cf46042e7ca231b765f3a1e9","abstract_canon_sha256":"5c5e4edb0826b5813888406ab9e44bfcf384e6a55690452a65ae39441700ceb7"},"schema_version":"1.0"},"canonical_sha256":"da0a64e4a9ba3bd0d654c07fdbc0f10a464600c77628228db60c6c69a0e890c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:07:45.217336Z","signature_b64":"WUXj4DJy5ttarp+m8BP00HuO81KQyvzvGtYAQytEeKauXskx/R2yQvMgC2fkh2b1Vbncfp9PVe11/jeArNYwAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da0a64e4a9ba3bd0d654c07fdbc0f10a464600c77628228db60c6c69a0e890c2","last_reissued_at":"2026-05-18T02:07:45.216703Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:07:45.216703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.5237","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":"2026-05-18T02:07:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4W9sOrCRL5wdD7p6i59pdqtuLGLm/vtLyJ14djbRIEBTe9HchETbozJRrEcs7TXfWiz9xOHgVu25uHNZTb7WBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T11:52:46.044172Z"},"content_sha256":"a738d538d17417a34c7e7999def8c8473fdcf767cb7d6908d1be0572eafbdf1b","schema_version":"1.0","event_id":"sha256:a738d538d17417a34c7e7999def8c8473fdcf767cb7d6908d1be0572eafbdf1b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:3IFGJZFJXI55BVSUYB75XQHRBJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Steady water waves with multiple critical layers: interior dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Gabriele Villari, Joachim Escher, Mats Ehrnstr\\\"om","submitted_at":"2010-04-29T09:39:52Z","abstract_excerpt":"We study small-amplitude steady water waves with multiple critical layers. Those are rotational two-dimensional gravity-waves propagating over a perfect fluid of finite depth. It is found that arbitrarily many critical layers with cat's-eye vortices are possible, with different structure at different levels within the fluid. The corresponding vorticity depends linearly on the stream function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5237","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":"2026-05-18T02:07:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mNEDfV+oOvKrRbvNaoS6hUqryveaYouxyT77g9MRPV3+f9vXFkbMA80xmhbdZpsM991LyS4kkRnRKLrhNz2xAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T11:52:46.044567Z"},"content_sha256":"0eb080b5f44ddb0af3c916e3c65b5032104ed0fcc0bd63eab267dea8467b4901","schema_version":"1.0","event_id":"sha256:0eb080b5f44ddb0af3c916e3c65b5032104ed0fcc0bd63eab267dea8467b4901"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3IFGJZFJXI55BVSUYB75XQHRBJ/bundle.json","state_url":"https://pith.science/pith/3IFGJZFJXI55BVSUYB75XQHRBJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3IFGJZFJXI55BVSUYB75XQHRBJ/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":"2026-06-05T11:52:46Z","links":{"resolver":"https://pith.science/pith/3IFGJZFJXI55BVSUYB75XQHRBJ","bundle":"https://pith.science/pith/3IFGJZFJXI55BVSUYB75XQHRBJ/bundle.json","state":"https://pith.science/pith/3IFGJZFJXI55BVSUYB75XQHRBJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3IFGJZFJXI55BVSUYB75XQHRBJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:3IFGJZFJXI55BVSUYB75XQHRBJ","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":"5c5e4edb0826b5813888406ab9e44bfcf384e6a55690452a65ae39441700ceb7","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-29T09:39:52Z","title_canon_sha256":"a880ac37056e09a05ac73f41a4d5bf4c8d1095e6cf46042e7ca231b765f3a1e9"},"schema_version":"1.0","source":{"id":"1004.5237","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5237","created_at":"2026-05-18T02:07:45Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5237v2","created_at":"2026-05-18T02:07:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5237","created_at":"2026-05-18T02:07:45Z"},{"alias_kind":"pith_short_12","alias_value":"3IFGJZFJXI55","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3IFGJZFJXI55BVSU","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3IFGJZFJ","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:0eb080b5f44ddb0af3c916e3c65b5032104ed0fcc0bd63eab267dea8467b4901","target":"graph","created_at":"2026-05-18T02:07:45Z","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 study small-amplitude steady water waves with multiple critical layers. Those are rotational two-dimensional gravity-waves propagating over a perfect fluid of finite depth. It is found that arbitrarily many critical layers with cat's-eye vortices are possible, with different structure at different levels within the fluid. The corresponding vorticity depends linearly on the stream function.","authors_text":"Gabriele Villari, Joachim Escher, Mats Ehrnstr\\\"om","cross_cats":["math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-29T09:39:52Z","title":"Steady water waves with multiple critical layers: interior dynamics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5237","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:a738d538d17417a34c7e7999def8c8473fdcf767cb7d6908d1be0572eafbdf1b","target":"record","created_at":"2026-05-18T02:07:45Z","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":"5c5e4edb0826b5813888406ab9e44bfcf384e6a55690452a65ae39441700ceb7","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-29T09:39:52Z","title_canon_sha256":"a880ac37056e09a05ac73f41a4d5bf4c8d1095e6cf46042e7ca231b765f3a1e9"},"schema_version":"1.0","source":{"id":"1004.5237","kind":"arxiv","version":2}},"canonical_sha256":"da0a64e4a9ba3bd0d654c07fdbc0f10a464600c77628228db60c6c69a0e890c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da0a64e4a9ba3bd0d654c07fdbc0f10a464600c77628228db60c6c69a0e890c2","first_computed_at":"2026-05-18T02:07:45.216703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:07:45.216703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WUXj4DJy5ttarp+m8BP00HuO81KQyvzvGtYAQytEeKauXskx/R2yQvMgC2fkh2b1Vbncfp9PVe11/jeArNYwAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:07:45.217336Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.5237","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a738d538d17417a34c7e7999def8c8473fdcf767cb7d6908d1be0572eafbdf1b","sha256:0eb080b5f44ddb0af3c916e3c65b5032104ed0fcc0bd63eab267dea8467b4901"],"state_sha256":"d9e9495003987771188a7fd3e3b024ce168471d18e1ea368131aa7f9f2105748"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tOfG6JfLXHDG8xgm5vMz1wmIjWZqz+qmLbgzoVvnzNCeuMu+ap39p3CwInR/zX2LB0p61+sBw3VTv6zzI4xvBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T11:52:46.046555Z","bundle_sha256":"bbe0613f94a2aeb99000fac00ae4dfd6872ca1eacc2aeab25804d4f32acdbed1"}}