{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:BI3MZBRTPE3FPZSDRYMU5Y5DRJ","short_pith_number":"pith:BI3MZBRT","canonical_record":{"source":{"id":"2605.04394","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-06T01:34:44Z","cross_cats_sorted":[],"title_canon_sha256":"44f4e2d3e536623dd46544a48b98aef1e46a237eb3c03e68a7810ed5d77ee208","abstract_canon_sha256":"3f9968ee95dd7b45f43536c32a71205e08a26dc4851cf0504ee7683d7f3536c5"},"schema_version":"1.0"},"canonical_sha256":"0a36cc8633793657e6438e194ee3a38a79b69c5ab914160d3b68496a3b68d36d","source":{"kind":"arxiv","id":"2605.04394","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.04394","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"arxiv_version","alias_value":"2605.04394v2","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.04394","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"pith_short_12","alias_value":"BI3MZBRTPE3F","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"pith_short_16","alias_value":"BI3MZBRTPE3FPZSD","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"pith_short_8","alias_value":"BI3MZBRT","created_at":"2026-05-26T01:02:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:BI3MZBRTPE3FPZSDRYMU5Y5DRJ","target":"record","payload":{"canonical_record":{"source":{"id":"2605.04394","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-06T01:34:44Z","cross_cats_sorted":[],"title_canon_sha256":"44f4e2d3e536623dd46544a48b98aef1e46a237eb3c03e68a7810ed5d77ee208","abstract_canon_sha256":"3f9968ee95dd7b45f43536c32a71205e08a26dc4851cf0504ee7683d7f3536c5"},"schema_version":"1.0"},"canonical_sha256":"0a36cc8633793657e6438e194ee3a38a79b69c5ab914160d3b68496a3b68d36d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:02:34.889645Z","signature_b64":"CK0CU22JUp+OcyKbpFQh25+WJ+Im1Xfa0rVShx5GXDJBwSG4WYknwHpebt44imu/ZMMqpACRO24O0eQOBjyUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a36cc8633793657e6438e194ee3a38a79b69c5ab914160d3b68496a3b68d36d","last_reissued_at":"2026-05-26T01:02:34.888869Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:02:34.888869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.04394","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":"2026-05-26T01:02:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UfzuDDZR/npJBOqCjfld0oA7+4/dNOPn5eCS9Sgqgekjiy8vgMXMwsAt+5ugofuWdulmTMeHrDctbQQpJGywDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T14:53:16.331985Z"},"content_sha256":"97255fc019779508d7673d5b8b204f975d864fc448ea22a993048924392f310e","schema_version":"1.0","event_id":"sha256:97255fc019779508d7673d5b8b204f975d864fc448ea22a993048924392f310e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:BI3MZBRTPE3FPZSDRYMU5Y5DRJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A unified geometric perspective on Zygmund's conjecture for maximal functions associated with vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields.","cross_cats":[],"primary_cat":"math.CA","authors_text":"Lingxiao Zhang","submitted_at":"2026-05-06T01:34:44Z","abstract_excerpt":"The Zygmund vector field maximal function conjecture is a long-standing open problem. This paper establishes a new boundedness criterion that significantly weakens the existing conditions in the literature. Specifically, the required decay condition is relaxed from the power-type decay of Bourgain for Zygmund's conjecture and the exponential-logarithmic decay of Lacey and Li for Stein's conjecture, to a logarithmic polynomial decay. Unlike the traditional framework that separates finite-type and non-finite-type operators, this paper offers a unified geometric view of both settings. The new cri"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The refinement assumes the vector fields satisfy the (unspecified in abstract) weaker condition identified by the refined argument; without the full paper the exact form and verification of this condition cannot be assessed.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Refines Bourgain's method to establish boundedness of maximal functions under a weaker condition than previously known, strengthening an implicit Lacey-Li result.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3b6268bbcb06efe9d1b814fac6f8c4f8779dd00fc774fa45b896ef3b6fe8b604"},"source":{"id":"2605.04394","kind":"arxiv","version":2},"verdict":{"id":"2be98dcf-f7cb-4ebe-b50f-8425709b8264","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T16:55:35.178926Z","strongest_claim":"By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known.","one_line_summary":"Refines Bourgain's method to establish boundedness of maximal functions under a weaker condition than previously known, strengthening an implicit Lacey-Li result.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The refinement assumes the vector fields satisfy the (unspecified in abstract) weaker condition identified by the refined argument; without the full paper the exact form and verification of this condition cannot be assessed.","pith_extraction_headline":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04394/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:44:00.960619Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:20.011448Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:29:10.181808Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e015d598fff024546c44347ba1d8006dbc948019434d7e4cf89b307d11297c04"},"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":"2be98dcf-f7cb-4ebe-b50f-8425709b8264"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-26T01:02:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cbkJzG3lYflyWrKev3FpalMxohM5NTqMpl8afiDkuP9zGVTKVNfXWj6WNomXNct/azV+6Ag+4+ajoNvSWFo8Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T14:53:16.333007Z"},"content_sha256":"95bc2fabc76570653d3e58bdbb65fc49bc4a7274d408f252966cb7ce7fc4fc70","schema_version":"1.0","event_id":"sha256:95bc2fabc76570653d3e58bdbb65fc49bc4a7274d408f252966cb7ce7fc4fc70"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ/bundle.json","state_url":"https://pith.science/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ/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":"2026-06-10T14:53:16Z","links":{"resolver":"https://pith.science/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ","bundle":"https://pith.science/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ/bundle.json","state":"https://pith.science/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BI3MZBRTPE3FPZSDRYMU5Y5DRJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BI3MZBRTPE3FPZSDRYMU5Y5DRJ","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":"3f9968ee95dd7b45f43536c32a71205e08a26dc4851cf0504ee7683d7f3536c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-06T01:34:44Z","title_canon_sha256":"44f4e2d3e536623dd46544a48b98aef1e46a237eb3c03e68a7810ed5d77ee208"},"schema_version":"1.0","source":{"id":"2605.04394","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.04394","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"arxiv_version","alias_value":"2605.04394v2","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.04394","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"pith_short_12","alias_value":"BI3MZBRTPE3F","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"pith_short_16","alias_value":"BI3MZBRTPE3FPZSD","created_at":"2026-05-26T01:02:34Z"},{"alias_kind":"pith_short_8","alias_value":"BI3MZBRT","created_at":"2026-05-26T01:02:34Z"}],"graph_snapshots":[{"event_id":"sha256:95bc2fabc76570653d3e58bdbb65fc49bc4a7274d408f252966cb7ce7fc4fc70","target":"graph","created_at":"2026-05-26T01:02:34Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The refinement assumes the vector fields satisfy the (unspecified in abstract) weaker condition identified by the refined argument; without the full paper the exact form and verification of this condition cannot be assessed."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Refines Bourgain's method to establish boundedness of maximal functions under a weaker condition than previously known, strengthening an implicit Lacey-Li result."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields."}],"snapshot_sha256":"3b6268bbcb06efe9d1b814fac6f8c4f8779dd00fc774fa45b896ef3b6fe8b604"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-20T11:44:00.960619Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:20.011448Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T14:29:10.181808Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.04394/integrity.json","findings":[],"snapshot_sha256":"e015d598fff024546c44347ba1d8006dbc948019434d7e4cf89b307d11297c04","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Zygmund vector field maximal function conjecture is a long-standing open problem. This paper establishes a new boundedness criterion that significantly weakens the existing conditions in the literature. Specifically, the required decay condition is relaxed from the power-type decay of Bourgain for Zygmund's conjecture and the exponential-logarithmic decay of Lacey and Li for Stein's conjecture, to a logarithmic polynomial decay. Unlike the traditional framework that separates finite-type and non-finite-type operators, this paper offers a unified geometric view of both settings. The new cri","authors_text":"Lingxiao Zhang","cross_cats":[],"headline":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-06T01:34:44Z","title":"A unified geometric perspective on Zygmund's conjecture for maximal functions associated with vector fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.04394","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T16:55:35.178926Z","id":"2be98dcf-f7cb-4ebe-b50f-8425709b8264","model_set":{"reader":"grok-4.3"},"one_line_summary":"Refines Bourgain's method to establish boundedness of maximal functions under a weaker condition than previously known, strengthening an implicit Lacey-Li result.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Refining Bourgain's argument identifies a weaker condition for boundedness of maximal functions associated with planar vector fields.","strongest_claim":"By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known.","weakest_assumption":"The refinement assumes the vector fields satisfy the (unspecified in abstract) weaker condition identified by the refined argument; without the full paper the exact form and verification of this condition cannot be assessed."}},"verdict_id":"2be98dcf-f7cb-4ebe-b50f-8425709b8264"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:97255fc019779508d7673d5b8b204f975d864fc448ea22a993048924392f310e","target":"record","created_at":"2026-05-26T01:02:34Z","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":"3f9968ee95dd7b45f43536c32a71205e08a26dc4851cf0504ee7683d7f3536c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-06T01:34:44Z","title_canon_sha256":"44f4e2d3e536623dd46544a48b98aef1e46a237eb3c03e68a7810ed5d77ee208"},"schema_version":"1.0","source":{"id":"2605.04394","kind":"arxiv","version":2}},"canonical_sha256":"0a36cc8633793657e6438e194ee3a38a79b69c5ab914160d3b68496a3b68d36d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a36cc8633793657e6438e194ee3a38a79b69c5ab914160d3b68496a3b68d36d","first_computed_at":"2026-05-26T01:02:34.888869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T01:02:34.888869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CK0CU22JUp+OcyKbpFQh25+WJ+Im1Xfa0rVShx5GXDJBwSG4WYknwHpebt44imu/ZMMqpACRO24O0eQOBjyUAw==","signature_status":"signed_v1","signed_at":"2026-05-26T01:02:34.889645Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.04394","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97255fc019779508d7673d5b8b204f975d864fc448ea22a993048924392f310e","sha256:95bc2fabc76570653d3e58bdbb65fc49bc4a7274d408f252966cb7ce7fc4fc70"],"state_sha256":"d2e095526b8ff4fad06816f72ebfea90a79fd0d30af34e0bb95612a9896ca6db"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ajdAy3ynV/DbTycVIxgXitWbtjTztosIoMm97LeocvV2mRFtAYOCudGDJ7hHacWj5n85OtTsUoiEkke/sVQxBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T14:53:16.337039Z","bundle_sha256":"e024ded4a920de76e9760dbd436f4b2829c617c46391d8e41bc0cecb053aaaa8"}}