{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1999:RY6IZVDVBRJ4NLFWW7ECQJBPYU","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":"b05a91f675002c44cd080d218f6838450e7effaa1057fde0f55c27060a84fe5f","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"math.CA","submitted_at":"1999-07-01T00:00:00Z","title_canon_sha256":"12faca6e4348af006637edd07724ff42bda36db29837d7d6274b7b0c0466fb87"},"schema_version":"1.0","source":{"id":"math/9907209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9907209","created_at":"2026-05-18T01:05:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/9907209v1","created_at":"2026-05-18T01:05:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9907209","created_at":"2026-05-18T01:05:32Z"},{"alias_kind":"pith_short_12","alias_value":"RY6IZVDVBRJ4","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_16","alias_value":"RY6IZVDVBRJ4NLFW","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_8","alias_value":"RY6IZVDV","created_at":"2026-05-18T12:25:49Z"}],"graph_snapshots":[{"event_id":"sha256:bcf8562b7e856c4cb3a07e6d594145f283c07693059e1759357882306bf02a8e","target":"graph","created_at":"2026-05-18T01:05:32Z","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"},"paper":{"abstract_excerpt":"We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable.","authors_text":"Brian White","cross_cats":["math.DG"],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"1999-07-01T00:00:00Z","title":"Rectifiability of flat chains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9907209","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":"sha256:9ac91b47e73c4e39bfd15946424bfe9903c67a87d7c6530871ab2febd108be65","target":"record","created_at":"2026-05-18T01:05:32Z","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":"b05a91f675002c44cd080d218f6838450e7effaa1057fde0f55c27060a84fe5f","cross_cats_sorted":["math.DG"],"license":"","primary_cat":"math.CA","submitted_at":"1999-07-01T00:00:00Z","title_canon_sha256":"12faca6e4348af006637edd07724ff42bda36db29837d7d6274b7b0c0466fb87"},"schema_version":"1.0","source":{"id":"math/9907209","kind":"arxiv","version":1}},"canonical_sha256":"8e3c8cd4750c53c6acb6b7c828242fc5391cd4d1364962735b5a017a941d9a1e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e3c8cd4750c53c6acb6b7c828242fc5391cd4d1364962735b5a017a941d9a1e","first_computed_at":"2026-05-18T01:05:32.678305Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:32.678305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vxU2NhJ9RY3avUnTwYB6aj4ywCm4NA4rYPRN1K/wwCa5MdH6g+IzbcBrAkNbfEPXhfNkkx6h2k7iq8HhGbmAAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:32.679033Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9907209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ac91b47e73c4e39bfd15946424bfe9903c67a87d7c6530871ab2febd108be65","sha256:bcf8562b7e856c4cb3a07e6d594145f283c07693059e1759357882306bf02a8e"],"state_sha256":"85f00b563fd24830e51cca3a1a936c0ab31397a78873c15f2f407f567b520228"}