{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:ITVQAJEBUFKWBCQJAKIJSE7PPL","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":"11e8ba1475372807e158611f5bc2d18870653f704baad3cc9deaa04004f40d00","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-01-06T11:45:12Z","title_canon_sha256":"e093d0c8c16db413d52b97cee24bd773250be7a711d1843255fa191e30625e09"},"schema_version":"1.0","source":{"id":"2501.02948","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.02948","created_at":"2026-07-05T09:57:27Z"},{"alias_kind":"arxiv_version","alias_value":"2501.02948v1","created_at":"2026-07-05T09:57:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.02948","created_at":"2026-07-05T09:57:27Z"},{"alias_kind":"pith_short_12","alias_value":"ITVQAJEBUFKW","created_at":"2026-07-05T09:57:27Z"},{"alias_kind":"pith_short_16","alias_value":"ITVQAJEBUFKWBCQJ","created_at":"2026-07-05T09:57:27Z"},{"alias_kind":"pith_short_8","alias_value":"ITVQAJEB","created_at":"2026-07-05T09:57:27Z"}],"graph_snapshots":[{"event_id":"sha256:1fe1513ee30a64460f70d40132bd907d52e95128570f2a4f2e84031131b9f576","target":"graph","created_at":"2026-07-05T09:57:27Z","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/2501.02948/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a sufficient condition for a Borel subset $E\\subset X$ of a complete metric space with $\\mathcal{H}^n(E)<\\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti representation. Precisely, we show that if $\\mathcal{H}^n|_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss.\n  An important step of ou","authors_text":"David Bate, Julian Weigt","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-01-06T11:45:12Z","title":"Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.02948","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":"sha256:89da53d0d2ed844de001bae30ff1736b87a158a939be610cae11fdc4d0e7e2cd","target":"record","created_at":"2026-07-05T09:57:27Z","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":"11e8ba1475372807e158611f5bc2d18870653f704baad3cc9deaa04004f40d00","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-01-06T11:45:12Z","title_canon_sha256":"e093d0c8c16db413d52b97cee24bd773250be7a711d1843255fa191e30625e09"},"schema_version":"1.0","source":{"id":"2501.02948","kind":"arxiv","version":1}},"canonical_sha256":"44eb002481a155608a0902909913ef7ae33f63e4a58c4c50490be5d0eb017907","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"44eb002481a155608a0902909913ef7ae33f63e4a58c4c50490be5d0eb017907","first_computed_at":"2026-07-05T09:57:27.879855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:57:27.879855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fPy9zJSSxhBW1AP7uQT60DPFiQL4XlJTVSmDV9Ewa9xt0lLneS4Ydqfo7/Pecwf8nm4hvagMvuDlFGrfb9PTDg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:57:27.880250Z","signed_message":"canonical_sha256_bytes"},"source_id":"2501.02948","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89da53d0d2ed844de001bae30ff1736b87a158a939be610cae11fdc4d0e7e2cd","sha256:1fe1513ee30a64460f70d40132bd907d52e95128570f2a4f2e84031131b9f576"],"state_sha256":"cda293b984febcaba65e40d5b4bcd71de32e56ea99a66597b6453043fd841ae0"}