{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:YT5O7UYR3D6LZMF3RMGX43NS7N","short_pith_number":"pith:YT5O7UYR","canonical_record":{"source":{"id":"2606.24461","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-23T11:55:15Z","cross_cats_sorted":[],"title_canon_sha256":"c6562bf3e3b86ad7e41215f071f650ec9432b74b39598e013cd8fb2ec6594001","abstract_canon_sha256":"5935091640200b79b2cd456666a2cab19af80eb5565d3898325f0d9a11778850"},"schema_version":"1.0"},"canonical_sha256":"c4faefd311d8fcbcb0bb8b0d7e6db2fb59481ee9326c31b0620247b5a996d957","source":{"kind":"arxiv","id":"2606.24461","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24461","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24461v1","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24461","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"pith_short_12","alias_value":"YT5O7UYR3D6L","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"pith_short_16","alias_value":"YT5O7UYR3D6LZMF3","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"pith_short_8","alias_value":"YT5O7UYR","created_at":"2026-06-24T01:15:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:YT5O7UYR3D6LZMF3RMGX43NS7N","target":"record","payload":{"canonical_record":{"source":{"id":"2606.24461","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-23T11:55:15Z","cross_cats_sorted":[],"title_canon_sha256":"c6562bf3e3b86ad7e41215f071f650ec9432b74b39598e013cd8fb2ec6594001","abstract_canon_sha256":"5935091640200b79b2cd456666a2cab19af80eb5565d3898325f0d9a11778850"},"schema_version":"1.0"},"canonical_sha256":"c4faefd311d8fcbcb0bb8b0d7e6db2fb59481ee9326c31b0620247b5a996d957","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:15:31.035229Z","signature_b64":"jmiFBLjHmH5l3uNuMPp6u1mmLVSYIbwQJx8VGZSIek6coBAy7+oPZ+FhYmE5KYc6SVsIaUzXnwA/gYFXeB0CDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4faefd311d8fcbcb0bb8b0d7e6db2fb59481ee9326c31b0620247b5a996d957","last_reissued_at":"2026-06-24T01:15:31.034843Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:15:31.034843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.24461","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-24T01:15:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fKVCx2PqNPqhZNEFLDsR8MxZM0RIaSEkawpDs/RRSbBq0eWrO/q54joyE2sVRIDBRgtbfYdFqeA+CufOchWCDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T10:22:15.467937Z"},"content_sha256":"73b752a12521a1297f0bebc2608b51386ab762e0736df69b612b2a0c003a8e38","schema_version":"1.0","event_id":"sha256:73b752a12521a1297f0bebc2608b51386ab762e0736df69b612b2a0c003a8e38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:YT5O7UYR3D6LZMF3RMGX43NS7N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Usual Square Function on Weakly Flat Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Benjamin Jaye, Tobias Wang","submitted_at":"2026-06-23T11:55:15Z","abstract_excerpt":"We study the usual square function estimate associated with the Cauchy single-layer kernel in the plane, without assuming Ahlfors-David regularity. We prove that a finite Radon measure with positive and finite upper density is rectifiable if it satisfies the usual square function estimate and a weak flatness condition. We also prove that, under the same finiteness and density hypotheses, the weak flatness condition follows when the support is contained in a locally two-sided NTA curve. As a corollary, rectifiability follows when the support is contained in a quasicircle."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24461","kind":"arxiv","version":1},"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/2606.24461/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":"2026-06-24T01:15:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DeKTkEKMH24FXU652TnR/d3wNL+bTCSes1KxJZ/EoWo5D6v1O4bel8SQnNvQQ9k+fmkWKG/UFOvXW4RFtT97CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T10:22:15.468300Z"},"content_sha256":"e83404689cd8dc315978567471750657d5c76defeded3547945093f1decf168e","schema_version":"1.0","event_id":"sha256:e83404689cd8dc315978567471750657d5c76defeded3547945093f1decf168e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YT5O7UYR3D6LZMF3RMGX43NS7N/bundle.json","state_url":"https://pith.science/pith/YT5O7UYR3D6LZMF3RMGX43NS7N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YT5O7UYR3D6LZMF3RMGX43NS7N/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":"2026-06-29T10:22:15Z","links":{"resolver":"https://pith.science/pith/YT5O7UYR3D6LZMF3RMGX43NS7N","bundle":"https://pith.science/pith/YT5O7UYR3D6LZMF3RMGX43NS7N/bundle.json","state":"https://pith.science/pith/YT5O7UYR3D6LZMF3RMGX43NS7N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YT5O7UYR3D6LZMF3RMGX43NS7N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:YT5O7UYR3D6LZMF3RMGX43NS7N","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":"5935091640200b79b2cd456666a2cab19af80eb5565d3898325f0d9a11778850","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-23T11:55:15Z","title_canon_sha256":"c6562bf3e3b86ad7e41215f071f650ec9432b74b39598e013cd8fb2ec6594001"},"schema_version":"1.0","source":{"id":"2606.24461","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24461","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24461v1","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24461","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"pith_short_12","alias_value":"YT5O7UYR3D6L","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"pith_short_16","alias_value":"YT5O7UYR3D6LZMF3","created_at":"2026-06-24T01:15:31Z"},{"alias_kind":"pith_short_8","alias_value":"YT5O7UYR","created_at":"2026-06-24T01:15:31Z"}],"graph_snapshots":[{"event_id":"sha256:e83404689cd8dc315978567471750657d5c76defeded3547945093f1decf168e","target":"graph","created_at":"2026-06-24T01:15:31Z","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/2606.24461/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the usual square function estimate associated with the Cauchy single-layer kernel in the plane, without assuming Ahlfors-David regularity. We prove that a finite Radon measure with positive and finite upper density is rectifiable if it satisfies the usual square function estimate and a weak flatness condition. We also prove that, under the same finiteness and density hypotheses, the weak flatness condition follows when the support is contained in a locally two-sided NTA curve. As a corollary, rectifiability follows when the support is contained in a quasicircle.","authors_text":"Benjamin Jaye, Tobias Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-23T11:55:15Z","title":"The Usual Square Function on Weakly Flat Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24461","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":"sha256:73b752a12521a1297f0bebc2608b51386ab762e0736df69b612b2a0c003a8e38","target":"record","created_at":"2026-06-24T01:15:31Z","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":"5935091640200b79b2cd456666a2cab19af80eb5565d3898325f0d9a11778850","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-23T11:55:15Z","title_canon_sha256":"c6562bf3e3b86ad7e41215f071f650ec9432b74b39598e013cd8fb2ec6594001"},"schema_version":"1.0","source":{"id":"2606.24461","kind":"arxiv","version":1}},"canonical_sha256":"c4faefd311d8fcbcb0bb8b0d7e6db2fb59481ee9326c31b0620247b5a996d957","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4faefd311d8fcbcb0bb8b0d7e6db2fb59481ee9326c31b0620247b5a996d957","first_computed_at":"2026-06-24T01:15:31.034843Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:15:31.034843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jmiFBLjHmH5l3uNuMPp6u1mmLVSYIbwQJx8VGZSIek6coBAy7+oPZ+FhYmE5KYc6SVsIaUzXnwA/gYFXeB0CDA==","signature_status":"signed_v1","signed_at":"2026-06-24T01:15:31.035229Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.24461","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73b752a12521a1297f0bebc2608b51386ab762e0736df69b612b2a0c003a8e38","sha256:e83404689cd8dc315978567471750657d5c76defeded3547945093f1decf168e"],"state_sha256":"30b845f0c181b281a3a5f528f58e69c0444dc2122236e5102766cd1ac9dc49d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6DifjQmqwDZzCdGKNOCltJpjL9cpaxEukG1Y0u5SUNlug+oBaZ3JCWmQIkyYNaj8wCzoH6QSpweb3Byxz3J4DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T10:22:15.470205Z","bundle_sha256":"15ee3efa4d0f827e2bd8fc2270a735a2952dd6d8c8b4f8e8345598dfeaa74a0d"}}