{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:P3RIX6O2VXDABK5SBIHYV7ACS4","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":"5bfc9dd8bf0825ebf4972eeadbf29ddf56242f55b13b277a12d24ea41add6a6f","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"2007-01-31T14:57:24Z","title_canon_sha256":"d72b56d77ddb154424a1608076c5cae4e483b6819cba0961869a2a834ab49bcc"},"schema_version":"1.0","source":{"id":"math/0701926","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0701926","created_at":"2026-05-18T01:09:15Z"},{"alias_kind":"arxiv_version","alias_value":"math/0701926v1","created_at":"2026-05-18T01:09:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701926","created_at":"2026-05-18T01:09:15Z"},{"alias_kind":"pith_short_12","alias_value":"P3RIX6O2VXDA","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"P3RIX6O2VXDABK5S","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"P3RIX6O2","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:f827e1b07dbeee2ce292b36b75c54d9a8ffb7f9ef9f4a972da6d2d917f3db6db","target":"graph","created_at":"2026-05-18T01:09:15Z","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":"Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space, and properties of generated $\\sigma$-ideals are studied. These $\\sigma$-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.","authors_text":"Lud\\v{e}k Zaj\\'i\\v{c}ek","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2007-01-31T14:57:24Z","title":"On Lipschitz and d.c. surfaces of finite codimension in a Banach space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701926","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:ba4d1a329d64ec290235aacc34d2412d788d1d63b1eef055647e02791c516582","target":"record","created_at":"2026-05-18T01:09:15Z","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":"5bfc9dd8bf0825ebf4972eeadbf29ddf56242f55b13b277a12d24ea41add6a6f","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"2007-01-31T14:57:24Z","title_canon_sha256":"d72b56d77ddb154424a1608076c5cae4e483b6819cba0961869a2a834ab49bcc"},"schema_version":"1.0","source":{"id":"math/0701926","kind":"arxiv","version":1}},"canonical_sha256":"7ee28bf9daadc600abb20a0f8afc029700b0a404f6daf726128db693d2ab62ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ee28bf9daadc600abb20a0f8afc029700b0a404f6daf726128db693d2ab62ce","first_computed_at":"2026-05-18T01:09:15.468670Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:15.468670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AKeZkVTPRbuaS65gP0h3eAptEs43afopwL+sqkQ7ZqeciyfspS3Pi7Lc8yCRtfHEbwXV9LFRchXpcrS8FJ9mCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:15.469309Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0701926","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba4d1a329d64ec290235aacc34d2412d788d1d63b1eef055647e02791c516582","sha256:f827e1b07dbeee2ce292b36b75c54d9a8ffb7f9ef9f4a972da6d2d917f3db6db"],"state_sha256":"201eaae5373d25d570a2c8d39ec7f6a24f04f6f5976bb0d8c09ce37e2dbf77d9"}