{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:RUADPOXQAF2UDUH65MSMDMAL67","short_pith_number":"pith:RUADPOXQ","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"},"canonical_sha256":"8d0037baf0017541d0feeb24c1b00bf7f896b595e2e9f3ded565fc793513c0a5","source":{"kind":"arxiv","id":"2508.21214","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.21214","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"2508.21214v1","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.21214","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"RUADPOXQAF2U","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"pith_short_16","alias_value":"RUADPOXQAF2UDUH6","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"pith_short_8","alias_value":"RUADPOXQ","created_at":"2026-07-05T12:01:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:RUADPOXQAF2UDUH65MSMDMAL67","target":"record","payload":{"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"},"canonical_sha256":"8d0037baf0017541d0feeb24c1b00bf7f896b595e2e9f3ded565fc793513c0a5","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"},"source_kind":"arxiv","source_id":"2508.21214","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":"2026-07-05T12:01:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GSxXwXvjSJvT8rQFGvdawlwDeACqExa244cP6HQflpJ/aXNUteDmxDXjPq6zyoA3+fyK64zQZ2NzVoK8wOe6Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T10:03:38.766675Z"},"content_sha256":"fb20b90d844ee179591e2375fb6af51eb18cd8f771ba2c70530a03d7e41f4cf4","schema_version":"1.0","event_id":"sha256:fb20b90d844ee179591e2375fb6af51eb18cd8f771ba2c70530a03d7e41f4cf4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:RUADPOXQAF2UDUH65MSMDMAL67","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:01:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lP8rkEXYueLzWPjazKYQx918fYcIc7Bo7C0LJe2wVWNnspxVqzVW77Eayj7QBKNxL2wqiz9Mn2q0/GX44dLhDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T10:03:38.767275Z"},"content_sha256":"27fbd78dd6262ee6bfa5db4045de5b25fae188bd0a417422e5674a5ad938aa0c","schema_version":"1.0","event_id":"sha256:27fbd78dd6262ee6bfa5db4045de5b25fae188bd0a417422e5674a5ad938aa0c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/bundle.json","state_url":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RUADPOXQAF2UDUH65MSMDMAL67/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":"2026-08-18T10:03:38Z","links":{"resolver":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67","bundle":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/bundle.json","state":"https://pith.science/pith/RUADPOXQAF2UDUH65MSMDMAL67/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RUADPOXQAF2UDUH65MSMDMAL67/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:RUADPOXQAF2UDUH65MSMDMAL67","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":"cdbcd079be5d61b6336d59392f29a00d5fc64067f31a68cefd3faa9163d71c82","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-08-28T21:09:06Z","title_canon_sha256":"8f73fad54774fb5a25125963495d3a84c9621678b8f15350d21b59c387e55eeb"},"schema_version":"1.0","source":{"id":"2508.21214","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.21214","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"2508.21214v1","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.21214","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"RUADPOXQAF2U","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"pith_short_16","alias_value":"RUADPOXQAF2UDUH6","created_at":"2026-07-05T12:01:30Z"},{"alias_kind":"pith_short_8","alias_value":"RUADPOXQ","created_at":"2026-07-05T12:01:30Z"}],"graph_snapshots":[{"event_id":"sha256:27fbd78dd6262ee6bfa5db4045de5b25fae188bd0a417422e5674a5ad938aa0c","target":"graph","created_at":"2026-07-05T12:01:30Z","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/2508.21214/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Benjamin Foster, Josep Gallegos","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-08-28T21:09:06Z","title":"Propagation of smallness near codimension two for gradients of harmonic functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.21214","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:fb20b90d844ee179591e2375fb6af51eb18cd8f771ba2c70530a03d7e41f4cf4","target":"record","created_at":"2026-07-05T12:01:30Z","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":"cdbcd079be5d61b6336d59392f29a00d5fc64067f31a68cefd3faa9163d71c82","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-08-28T21:09:06Z","title_canon_sha256":"8f73fad54774fb5a25125963495d3a84c9621678b8f15350d21b59c387e55eeb"},"schema_version":"1.0","source":{"id":"2508.21214","kind":"arxiv","version":1}},"canonical_sha256":"8d0037baf0017541d0feeb24c1b00bf7f896b595e2e9f3ded565fc793513c0a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d0037baf0017541d0feeb24c1b00bf7f896b595e2e9f3ded565fc793513c0a5","first_computed_at":"2026-07-05T12:01:30.171935Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:01:30.171935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QekPYhEAw58RNWy6tssKSYCE4pM1WgtB6N6GcOG8tNdlXlBvXDdHma4ysutyU05kh6mu9ByRr68SoHdwoMSrDg==","signature_status":"signed_v1","signed_at":"2026-07-05T12:01:30.172495Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.21214","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb20b90d844ee179591e2375fb6af51eb18cd8f771ba2c70530a03d7e41f4cf4","sha256:27fbd78dd6262ee6bfa5db4045de5b25fae188bd0a417422e5674a5ad938aa0c"],"state_sha256":"d63a902d8e14bbc008c6a78e59a372611f4a5286c938a3fa1ef3765a49b24aa5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rMD1Y2bbmmww+L640uxlGfKJrSMs/vRZdlAY6WB2he5yvrvHL2LoO2T+NF7JJsR8p+6h4k4Vax0BQGRbQikiBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T10:03:38.773229Z","bundle_sha256":"bbcd9166d3147f7f54f93a5248d6a6beb01881b44e98a388cc13fd469a6bc49f"}}