{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:OZ3XOW4VYHVCZ5SFK2YFTGU7IV","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":"1e9ad631ba1371ce849a1878aea80b5d9600069d4e6ea670f9d8dd218f066675","cross_cats_sorted":["math.CA","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-20T11:06:06Z","title_canon_sha256":"72b87d9e6a16c211ebe5e5edb8927cc4b37dd826e99093514d902c3991e4bc70"},"schema_version":"1.0","source":{"id":"1209.4491","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4491","created_at":"2026-05-18T02:44:07Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4491v1","created_at":"2026-05-18T02:44:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4491","created_at":"2026-05-18T02:44:07Z"},{"alias_kind":"pith_short_12","alias_value":"OZ3XOW4VYHVC","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"OZ3XOW4VYHVCZ5SF","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"OZ3XOW4V","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:6aff18c1f3fa3df3f3fa3c682a08347c670875c704e1b45587f81cb12cfca094","target":"graph","created_at":"2026-05-18T02:44:07Z","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 investigate differences between upper and lower porosity. In finite dimensional Banach spaces every upper porous set is directionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in the plane which is not even a countable union of directionally lower porous sets.","authors_text":"Gareth Speight","cross_cats":["math.CA","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-20T11:06:06Z","title":"Directional Lower Porosity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4491","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":"sha256:483a0f500c557c10fb1cd3d6dcf37344b08db36a3b1267a7e9f9e38055e66dfa","target":"record","created_at":"2026-05-18T02:44:07Z","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":"1e9ad631ba1371ce849a1878aea80b5d9600069d4e6ea670f9d8dd218f066675","cross_cats_sorted":["math.CA","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-20T11:06:06Z","title_canon_sha256":"72b87d9e6a16c211ebe5e5edb8927cc4b37dd826e99093514d902c3991e4bc70"},"schema_version":"1.0","source":{"id":"1209.4491","kind":"arxiv","version":1}},"canonical_sha256":"7677775b95c1ea2cf64556b0599a9f45742deb1d8a2ab2ea790503803c66a6d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7677775b95c1ea2cf64556b0599a9f45742deb1d8a2ab2ea790503803c66a6d8","first_computed_at":"2026-05-18T02:44:07.124562Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:07.124562Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zs0xrneU1MowTevq8Qgj0Otnj5YgpwlRKx58fKSe9xtgC9KhMiOYhRmQe3CSA9XaxGPhsDID3IYHpnLyH9umDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:07.124977Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4491","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:483a0f500c557c10fb1cd3d6dcf37344b08db36a3b1267a7e9f9e38055e66dfa","sha256:6aff18c1f3fa3df3f3fa3c682a08347c670875c704e1b45587f81cb12cfca094"],"state_sha256":"440c992cc09c77953623cad41fd09a4e3d8bd8e686480623cfd323a318619312"}