{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:Z4VIMEFMJTXADFFCWLF7NBGTBE","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":"2ee12115296bd657f3dbf741f86ca9a551e7960dbf81c00ef4b6b88f93e5ac3f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-08-10T21:37:01Z","title_canon_sha256":"e5da5770633fb32c3cfc4cadc123a69ac8535fdad5fbd85ff23a9b6ec141da56"},"schema_version":"1.0","source":{"id":"2608.10253","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.10253","created_at":"2026-08-12T00:23:11Z"},{"alias_kind":"arxiv_version","alias_value":"2608.10253v1","created_at":"2026-08-12T00:23:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.10253","created_at":"2026-08-12T00:23:11Z"},{"alias_kind":"pith_short_12","alias_value":"Z4VIMEFMJTXA","created_at":"2026-08-12T00:23:11Z"},{"alias_kind":"pith_short_16","alias_value":"Z4VIMEFMJTXADFFC","created_at":"2026-08-12T00:23:11Z"},{"alias_kind":"pith_short_8","alias_value":"Z4VIMEFM","created_at":"2026-08-12T00:23:11Z"}],"graph_snapshots":[{"event_id":"sha256:8bbc55a24f0e041622e99744ccaefd85e6fb5fa699f834a41b9d10f331342250","target":"graph","created_at":"2026-08-12T00:23:11Z","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/2608.10253/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $\\mathbb{R}^d$, a classical theorem of Federer shows that the 1-dimensional Hausdorff measure of such sets is controlled by the multiplicity-weighted lengths of finitely many linearly independent projections. We develop a framework for extending Federer's result into a diverse set of nonlinear problems. This technique yields a unified approach for studying sets through their lower-dimensional nonlinear images, as well","authors_text":"Caleb Marshall, Krystal Taylor, Paige Bright, Rosemarie Bongers","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-08-10T21:37:01Z","title":"Applications of Nonlinear Projections to Rectifiable 1-sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.10253","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:f0567968ca09e00090c2a79f1977f3f1f38ac74ae80281887a8e201144d0eec6","target":"record","created_at":"2026-08-12T00:23:11Z","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":"2ee12115296bd657f3dbf741f86ca9a551e7960dbf81c00ef4b6b88f93e5ac3f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-08-10T21:37:01Z","title_canon_sha256":"e5da5770633fb32c3cfc4cadc123a69ac8535fdad5fbd85ff23a9b6ec141da56"},"schema_version":"1.0","source":{"id":"2608.10253","kind":"arxiv","version":1}},"canonical_sha256":"cf2a8610ac4cee0194a2b2cbf684d30939e5e31610fc29012f1e476ce65ef9e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf2a8610ac4cee0194a2b2cbf684d30939e5e31610fc29012f1e476ce65ef9e6","first_computed_at":"2026-08-12T00:23:11.377933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-12T00:23:11.377933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OJfJlrvl26COrFSrrdlzmn5UqNQruHzjUwLL8BsG0TCHZjxMMmeLUxQJ6WxGpVcW2KG10f59ZGzRbbWJJc50Cw==","signature_status":"signed_v1","signed_at":"2026-08-12T00:23:11.379853Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.10253","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0567968ca09e00090c2a79f1977f3f1f38ac74ae80281887a8e201144d0eec6","sha256:8bbc55a24f0e041622e99744ccaefd85e6fb5fa699f834a41b9d10f331342250"],"state_sha256":"867e03c79dfef2921ae11dfc532569b83c0d2298549a3c347b5221c9c2a94b84"}