{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:X4H7UQC6MT3SRQN4W6LWS4WMCJ","short_pith_number":"pith:X4H7UQC6","canonical_record":{"source":{"id":"2410.06175","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CV","submitted_at":"2024-10-08T16:21:07Z","cross_cats_sorted":["math.AP","math.DG"],"title_canon_sha256":"1dd0a7c82c15d3defa8f23da521426a46bd172a64e94a7ea6526ffa270e70bee","abstract_canon_sha256":"60c2b651ab5f62b90c80825a86aba82f20f888cd359fb5ab3c777838f6e07c35"},"schema_version":"1.0"},"canonical_sha256":"bf0ffa405e64f728c1bcb7976972cc1275220a4de27fdd6558cafd574f50672f","source":{"kind":"arxiv","id":"2410.06175","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.06175","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"arxiv_version","alias_value":"2410.06175v2","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.06175","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"pith_short_12","alias_value":"X4H7UQC6MT3S","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"pith_short_16","alias_value":"X4H7UQC6MT3SRQN4","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"pith_short_8","alias_value":"X4H7UQC6","created_at":"2026-07-05T11:33:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:X4H7UQC6MT3SRQN4W6LWS4WMCJ","target":"record","payload":{"canonical_record":{"source":{"id":"2410.06175","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CV","submitted_at":"2024-10-08T16:21:07Z","cross_cats_sorted":["math.AP","math.DG"],"title_canon_sha256":"1dd0a7c82c15d3defa8f23da521426a46bd172a64e94a7ea6526ffa270e70bee","abstract_canon_sha256":"60c2b651ab5f62b90c80825a86aba82f20f888cd359fb5ab3c777838f6e07c35"},"schema_version":"1.0"},"canonical_sha256":"bf0ffa405e64f728c1bcb7976972cc1275220a4de27fdd6558cafd574f50672f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:33:16.473978Z","signature_b64":"aDCioND/67tdcMHa/QmP/tpOQ9ihp3Eftgqw59e3ze0QEa1LK3JbmNhPiyGg/5dt/kKyB78TZD30GZa0hR3BCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf0ffa405e64f728c1bcb7976972cc1275220a4de27fdd6558cafd574f50672f","last_reissued_at":"2026-07-05T11:33:16.473568Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:33:16.473568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.06175","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":"2026-07-05T11:33:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YIlH3Yilo5na1XLl/fzlocnUfyKdo20go5SmsgyK7869EBCoc+E2jhDmPMOvqlVFEthChf1S2I+HevyjCyCfBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T01:46:08.966624Z"},"content_sha256":"dd3716e973b2405dd8472183914bc6f22ccbd703b8b16637722c2262d37c17cb","schema_version":"1.0","event_id":"sha256:dd3716e973b2405dd8472183914bc6f22ccbd703b8b16637722c2262d37c17cb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:X4H7UQC6MT3SRQN4W6LWS4WMCJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Holomorphic dependence for the Beltrami equation in Sobolev spaces","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AP","math.DG"],"primary_cat":"math.CV","authors_text":"Christian El Emam, Nathaniel Sagman","submitted_at":"2024-10-08T16:21:07Z","abstract_excerpt":"We prove that, given a path of Beltrami differentials on $\\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\\infty}_{loc}(\\Omega)$ of an open subset $\\Omega\\subset \\mathbb C$, the canonical solutions to the Beltrami equation vary holomorphically in $W^{l+1,p}_{loc}(\\Omega)$ for admissible $p > 2$. This extends a foundational result of Ahlfors and Bers (the case $l = 0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06175","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.06175/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:33:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zuZSCglDYVxu6LJHx0Jq2vlJ2H6MqiLz1uahPy+aVWClmZUFl+Q/UmmL9ApzJhY+TqaBT596VzrRqQc0/bTDAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T01:46:08.967421Z"},"content_sha256":"1085a847ef4bf8da81e261478bc7cb62bcc2eaed804f322d945dd48ad7e1a994","schema_version":"1.0","event_id":"sha256:1085a847ef4bf8da81e261478bc7cb62bcc2eaed804f322d945dd48ad7e1a994"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ/bundle.json","state_url":"https://pith.science/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ/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":"2026-08-10T01:46:08Z","links":{"resolver":"https://pith.science/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ","bundle":"https://pith.science/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ/bundle.json","state":"https://pith.science/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X4H7UQC6MT3SRQN4W6LWS4WMCJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:X4H7UQC6MT3SRQN4W6LWS4WMCJ","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":"60c2b651ab5f62b90c80825a86aba82f20f888cd359fb5ab3c777838f6e07c35","cross_cats_sorted":["math.AP","math.DG"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CV","submitted_at":"2024-10-08T16:21:07Z","title_canon_sha256":"1dd0a7c82c15d3defa8f23da521426a46bd172a64e94a7ea6526ffa270e70bee"},"schema_version":"1.0","source":{"id":"2410.06175","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.06175","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"arxiv_version","alias_value":"2410.06175v2","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.06175","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"pith_short_12","alias_value":"X4H7UQC6MT3S","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"pith_short_16","alias_value":"X4H7UQC6MT3SRQN4","created_at":"2026-07-05T11:33:16Z"},{"alias_kind":"pith_short_8","alias_value":"X4H7UQC6","created_at":"2026-07-05T11:33:16Z"}],"graph_snapshots":[{"event_id":"sha256:1085a847ef4bf8da81e261478bc7cb62bcc2eaed804f322d945dd48ad7e1a994","target":"graph","created_at":"2026-07-05T11:33:16Z","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/2410.06175/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that, given a path of Beltrami differentials on $\\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\\infty}_{loc}(\\Omega)$ of an open subset $\\Omega\\subset \\mathbb C$, the canonical solutions to the Beltrami equation vary holomorphically in $W^{l+1,p}_{loc}(\\Omega)$ for admissible $p > 2$. This extends a foundational result of Ahlfors and Bers (the case $l = 0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data.","authors_text":"Christian El Emam, Nathaniel Sagman","cross_cats":["math.AP","math.DG"],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CV","submitted_at":"2024-10-08T16:21:07Z","title":"Holomorphic dependence for the Beltrami equation in Sobolev spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06175","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:dd3716e973b2405dd8472183914bc6f22ccbd703b8b16637722c2262d37c17cb","target":"record","created_at":"2026-07-05T11:33:16Z","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":"60c2b651ab5f62b90c80825a86aba82f20f888cd359fb5ab3c777838f6e07c35","cross_cats_sorted":["math.AP","math.DG"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CV","submitted_at":"2024-10-08T16:21:07Z","title_canon_sha256":"1dd0a7c82c15d3defa8f23da521426a46bd172a64e94a7ea6526ffa270e70bee"},"schema_version":"1.0","source":{"id":"2410.06175","kind":"arxiv","version":2}},"canonical_sha256":"bf0ffa405e64f728c1bcb7976972cc1275220a4de27fdd6558cafd574f50672f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf0ffa405e64f728c1bcb7976972cc1275220a4de27fdd6558cafd574f50672f","first_computed_at":"2026-07-05T11:33:16.473568Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:33:16.473568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aDCioND/67tdcMHa/QmP/tpOQ9ihp3Eftgqw59e3ze0QEa1LK3JbmNhPiyGg/5dt/kKyB78TZD30GZa0hR3BCA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:33:16.473978Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.06175","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd3716e973b2405dd8472183914bc6f22ccbd703b8b16637722c2262d37c17cb","sha256:1085a847ef4bf8da81e261478bc7cb62bcc2eaed804f322d945dd48ad7e1a994"],"state_sha256":"622addad2ab0053a489bb4191a057bdd62080fabe1073b264620888fe2242ddf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1n3ea4C7rVJyCc7T4pto/h0bntdtX87xuyvqYTi9Cx1UvQX9fSq+X6OTMICDh0iLWrRsW93UXDZ3RD8VibVHDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T01:46:08.974076Z","bundle_sha256":"238fd5462a40020c46440c5724112fb3af6c60132011e23b23af74954f15381a"}}