{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ZSX5VTFY6V4ZV6JP56J2DQQEIX","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":"715b571c6945ad9b053478eac090c053bf0193702f718c08910d51df3be680fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-29T21:01:46Z","title_canon_sha256":"71192917c52f1ad06fbc33ec37ae15af25d0f8aab8e43d6fd927551d77a3c7b2"},"schema_version":"1.0","source":{"id":"1412.8481","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8481","created_at":"2026-05-18T01:11:08Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8481v4","created_at":"2026-05-18T01:11:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8481","created_at":"2026-05-18T01:11:08Z"},{"alias_kind":"pith_short_12","alias_value":"ZSX5VTFY6V4Z","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZSX5VTFY6V4ZV6JP","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZSX5VTFY","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:09b054595b43004cbe206f552cc544e24a60f221f331fbefc591d98208711a03","target":"graph","created_at":"2026-05-18T01:11:08Z","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":"Let $\\mathcal{M}(\\Omega, \\mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(\\Omega, \\mu)$. Let $B \\subset \\mathcal{M}(\\Omega, \\mu)$ be a set of finitely supported measurable functions such that the essential range of each $f \\in B$ is a subset of $\\{ 0,1 \\}$. The main result of this paper shows that for any $p \\in (0, \\infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(\\Omega, \\mu)$ if and only if $B$ is an affinely independent subset of $\\mathcal{M}(\\Omega, \\mu)$ (when $\\mathcal{M}(\\Omega, \\mu)$ is considered as a real vec","authors_text":"Anthony Weston","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-29T21:01:46Z","title":"The geometry of two-valued subsets of $L_{p}$-spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8481","kind":"arxiv","version":4},"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":"sha256:6ed97f1fd50c88e2dd2cc0f52e3e77566cc132eeaed2143d2528313f8a4de1bc","target":"record","created_at":"2026-05-18T01:11:08Z","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":"715b571c6945ad9b053478eac090c053bf0193702f718c08910d51df3be680fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-29T21:01:46Z","title_canon_sha256":"71192917c52f1ad06fbc33ec37ae15af25d0f8aab8e43d6fd927551d77a3c7b2"},"schema_version":"1.0","source":{"id":"1412.8481","kind":"arxiv","version":4}},"canonical_sha256":"ccafdaccb8f5799af92fef93a1c20445cfb36c9ae5e4c56dec37f3c9035fe5e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ccafdaccb8f5799af92fef93a1c20445cfb36c9ae5e4c56dec37f3c9035fe5e8","first_computed_at":"2026-05-18T01:11:08.672001Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:08.672001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o+2OTLs7IwaBiSAnK5Cha7AObU1tujdNWm3BuF4scuz1MyjOCdJXt+Zsmecy5XZ72VUsChuahaUtoirDeC6RCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:08.672460Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.8481","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ed97f1fd50c88e2dd2cc0f52e3e77566cc132eeaed2143d2528313f8a4de1bc","sha256:09b054595b43004cbe206f552cc544e24a60f221f331fbefc591d98208711a03"],"state_sha256":"d2baca654f36cf2323530a1fe7dd348e1a2faa9d6e718942aa1eacf888da8696"}