{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:RUADPOXQAF2UDUH65MSMDMAL67","short_pith_number":"pith:RUADPOXQ","schema_version":"1.0","canonical_sha256":"8d0037baf0017541d0feeb24c1b00bf7f896b595e2e9f3ded565fc793513c0a5","source":{"kind":"arxiv","id":"2508.21214","version":1},"attestation_state":"computed","paper":{"title":"Propagation of smallness near codimension two for gradients of harmonic functions","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Benjamin Foster, Josep Gallegos","submitted_at":"2025-08-28T21:09:06Z","abstract_excerpt":"Let $u$ be a harmonic function in the unit ball $B_1 \\subset \\mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $\\epsilon$-small in size on a set $E\\subset B_{1/2}$ with positive $(n-2+\\delta)$-dimensional Hausdorff content for some $\\delta>0$, then $\\sup_{B_{1/2}} |\\nabla u| \\leq C \\epsilon^\\alpha$ with $C,\\alpha>0$ depending only on $n,\\delta$ and the $(n-2+\\delta)$-Hausdorff content of $E$. This is an improvement over a similar result of Logunov and Malinnikova that required $\\delta>1-c_n$ for a small dimensional co"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2508.21214","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-08-28T21:09:06Z","cross_cats_sorted":[],"title_canon_sha256":"8f73fad54774fb5a25125963495d3a84c9621678b8f15350d21b59c387e55eeb","abstract_canon_sha256":"cdbcd079be5d61b6336d59392f29a00d5fc64067f31a68cefd3faa9163d71c82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:01:30.172495Z","signature_b64":"QekPYhEAw58RNWy6tssKSYCE4pM1WgtB6N6GcOG8tNdlXlBvXDdHma4ysutyU05kh6mu9ByRr68SoHdwoMSrDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d0037baf0017541d0feeb24c1b00bf7f896b595e2e9f3ded565fc793513c0a5","last_reissued_at":"2026-07-05T12:01:30.171935Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:01:30.171935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Propagation of smallness near codimension two for gradients of harmonic functions","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Benjamin Foster, Josep Gallegos","submitted_at":"2025-08-28T21:09:06Z","abstract_excerpt":"Let $u$ be a harmonic function in the unit ball $B_1 \\subset \\mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $\\epsilon$-small in size on a set $E\\subset B_{1/2}$ with positive $(n-2+\\delta)$-dimensional Hausdorff content for some $\\delta>0$, then $\\sup_{B_{1/2}} |\\nabla u| \\leq C \\epsilon^\\alpha$ with $C,\\alpha>0$ depending only on $n,\\delta$ and the $(n-2+\\delta)$-Hausdorff content of $E$. This is an improvement over a similar result of Logunov and Malinnikova that required $\\delta>1-c_n$ for a small dimensional co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.21214","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/2508.21214/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2508.21214","created_at":"2026-07-05T12:01:30.171997+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.21214v1","created_at":"2026-07-05T12:01:30.171997+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.21214","created_at":"2026-07-05T12:01:30.171997+00:00"},{"alias_kind":"pith_short_12","alias_value":"RUADPOXQAF2U","created_at":"2026-07-05T12:01:30.171997+00:00"},{"alias_kind":"pith_short_16","alias_value":"RUADPOXQAF2UDUH6","created_at":"2026-07-05T12:01:30.171997+00:00"},{"alias_kind":"pith_short_8","alias_value":"RUADPOXQ","created_at":"2026-07-05T12:01:30.171997+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67","json":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67.json","graph_json":"https://pith.science/api/pith-number/RUADPOXQAF2UDUH65MSMDMAL67/graph.json","events_json":"https://pith.science/api/pith-number/RUADPOXQAF2UDUH65MSMDMAL67/events.json","paper":"https://pith.science/paper/RUADPOXQ"},"agent_actions":{"view_html":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67","download_json":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67.json","view_paper":"https://pith.science/paper/RUADPOXQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.21214&json=true","fetch_graph":"https://pith.science/api/pith-number/RUADPOXQAF2UDUH65MSMDMAL67/graph.json","fetch_events":"https://pith.science/api/pith-number/RUADPOXQAF2UDUH65MSMDMAL67/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/action/storage_attestation","attest_author":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/action/author_attestation","sign_citation":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/action/citation_signature","submit_replication":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/action/replication_record"}},"created_at":"2026-07-05T12:01:30.171997+00:00","updated_at":"2026-07-05T12:01:30.171997+00:00"}