{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QDRVFRQPTCLG6IPCHMOOAODBTE","short_pith_number":"pith:QDRVFRQP","canonical_record":{"source":{"id":"1208.4659","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-08-23T02:39:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8f0da3446c2f4381c9adc2d5fa7233d163fea1d979fbc5cc99d73f4777852f27","abstract_canon_sha256":"762f76023a4cffd4e7bf0a382287f41f1e41fde352f21e4afe449ca38d7950c0"},"schema_version":"1.0"},"canonical_sha256":"80e352c60f98966f21e23b1ce038619900b07e1e32a55b01aa03212a12fa1809","source":{"kind":"arxiv","id":"1208.4659","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4659","created_at":"2026-05-18T02:58:46Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4659v3","created_at":"2026-05-18T02:58:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4659","created_at":"2026-05-18T02:58:46Z"},{"alias_kind":"pith_short_12","alias_value":"QDRVFRQPTCLG","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QDRVFRQPTCLG6IPC","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QDRVFRQP","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QDRVFRQPTCLG6IPCHMOOAODBTE","target":"record","payload":{"canonical_record":{"source":{"id":"1208.4659","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-08-23T02:39:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8f0da3446c2f4381c9adc2d5fa7233d163fea1d979fbc5cc99d73f4777852f27","abstract_canon_sha256":"762f76023a4cffd4e7bf0a382287f41f1e41fde352f21e4afe449ca38d7950c0"},"schema_version":"1.0"},"canonical_sha256":"80e352c60f98966f21e23b1ce038619900b07e1e32a55b01aa03212a12fa1809","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:46.572736Z","signature_b64":"4bqniah30fjEmbZMf9j5dxEOpzkwH4A21KHNEZlQeplXSXznLYwjtq/PHN6oKgPeik3CjEzg/LPFTKKQhEuiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80e352c60f98966f21e23b1ce038619900b07e1e32a55b01aa03212a12fa1809","last_reissued_at":"2026-05-18T02:58:46.571974Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:46.571974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.4659","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d/rAiMRUfqnG4OZTcuEzDuTt61Mx6XpQuj4J9d8c7NXh/iPmViZDeKU5RaEBxo5oc9P6IxThm2LiyLgDHLDsAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:16:26.847671Z"},"content_sha256":"f21c30ec757fc87cc2b5c70158564ef2259f50818c47d14765409349e48a47b1","schema_version":"1.0","event_id":"sha256:f21c30ec757fc87cc2b5c70158564ef2259f50818c47d14765409349e48a47b1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QDRVFRQPTCLG6IPCHMOOAODBTE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CV","authors_text":"Andrew Lorent","submitted_at":"2012-08-23T02:39:27Z","abstract_excerpt":"In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of studies proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4659","kind":"arxiv","version":3},"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":"2026-05-18T02:58:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wWnXFUXBc1HKoUlJ0l0MINrbNOS3GRUKio1968ptKJGkoBj7aGk2Cf3XFDoHuYaakk2t0JeWDV5VKwO5xMqZBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:16:26.848392Z"},"content_sha256":"794fe6496dc9257b1141d8384641976ac636841912d89aff81237838ae416593","schema_version":"1.0","event_id":"sha256:794fe6496dc9257b1141d8384641976ac636841912d89aff81237838ae416593"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QDRVFRQPTCLG6IPCHMOOAODBTE/bundle.json","state_url":"https://pith.science/pith/QDRVFRQPTCLG6IPCHMOOAODBTE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QDRVFRQPTCLG6IPCHMOOAODBTE/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":"2026-06-09T07:16:26Z","links":{"resolver":"https://pith.science/pith/QDRVFRQPTCLG6IPCHMOOAODBTE","bundle":"https://pith.science/pith/QDRVFRQPTCLG6IPCHMOOAODBTE/bundle.json","state":"https://pith.science/pith/QDRVFRQPTCLG6IPCHMOOAODBTE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QDRVFRQPTCLG6IPCHMOOAODBTE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QDRVFRQPTCLG6IPCHMOOAODBTE","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":"762f76023a4cffd4e7bf0a382287f41f1e41fde352f21e4afe449ca38d7950c0","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-08-23T02:39:27Z","title_canon_sha256":"8f0da3446c2f4381c9adc2d5fa7233d163fea1d979fbc5cc99d73f4777852f27"},"schema_version":"1.0","source":{"id":"1208.4659","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4659","created_at":"2026-05-18T02:58:46Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4659v3","created_at":"2026-05-18T02:58:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4659","created_at":"2026-05-18T02:58:46Z"},{"alias_kind":"pith_short_12","alias_value":"QDRVFRQPTCLG","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QDRVFRQPTCLG6IPC","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QDRVFRQP","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:794fe6496dc9257b1141d8384641976ac636841912d89aff81237838ae416593","target":"graph","created_at":"2026-05-18T02:58:46Z","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":"In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of studies proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide","authors_text":"Andrew Lorent","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-08-23T02:39:27Z","title":"Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4659","kind":"arxiv","version":3},"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:f21c30ec757fc87cc2b5c70158564ef2259f50818c47d14765409349e48a47b1","target":"record","created_at":"2026-05-18T02:58:46Z","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":"762f76023a4cffd4e7bf0a382287f41f1e41fde352f21e4afe449ca38d7950c0","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2012-08-23T02:39:27Z","title_canon_sha256":"8f0da3446c2f4381c9adc2d5fa7233d163fea1d979fbc5cc99d73f4777852f27"},"schema_version":"1.0","source":{"id":"1208.4659","kind":"arxiv","version":3}},"canonical_sha256":"80e352c60f98966f21e23b1ce038619900b07e1e32a55b01aa03212a12fa1809","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80e352c60f98966f21e23b1ce038619900b07e1e32a55b01aa03212a12fa1809","first_computed_at":"2026-05-18T02:58:46.571974Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:46.571974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4bqniah30fjEmbZMf9j5dxEOpzkwH4A21KHNEZlQeplXSXznLYwjtq/PHN6oKgPeik3CjEzg/LPFTKKQhEuiBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:46.572736Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.4659","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f21c30ec757fc87cc2b5c70158564ef2259f50818c47d14765409349e48a47b1","sha256:794fe6496dc9257b1141d8384641976ac636841912d89aff81237838ae416593"],"state_sha256":"d5b5bf35a08015ef40efb2ee7e9d0cf84d063402b3007a428c9c1b6a42a6bbec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QtmvE94kJhEDLyKJqfktzcfpe8wCKpSUdn1T3qwB/N4UFoRWAELxNBmlagdJsK3P2deR/IMbRFcZC73FP78zCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T07:16:26.852417Z","bundle_sha256":"2e9dc8214dbcd98d3adc4e1b14da545a33f3ce5d42990ddcd259ff482cb9edd6"}}