{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:GLQEYYGW7MJ5UHJ5QGYD7MI35X","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":"1a7b957e85b565bbc33e7b45c7ae2803ed8a861a0127b7c1858251a1733f9371","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-04-02T17:20:12Z","title_canon_sha256":"6177797adee9df5b23b2baab65ecace4bd1726f5ae8ab802237c42c6e4a0f81c"},"schema_version":"1.0","source":{"id":"2404.02116","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.02116","created_at":"2026-06-19T16:12:12Z"},{"alias_kind":"arxiv_version","alias_value":"2404.02116v4","created_at":"2026-06-19T16:12:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.02116","created_at":"2026-06-19T16:12:12Z"},{"alias_kind":"pith_short_12","alias_value":"GLQEYYGW7MJ5","created_at":"2026-06-19T16:12:12Z"},{"alias_kind":"pith_short_16","alias_value":"GLQEYYGW7MJ5UHJ5","created_at":"2026-06-19T16:12:12Z"},{"alias_kind":"pith_short_8","alias_value":"GLQEYYGW","created_at":"2026-06-19T16:12:12Z"}],"graph_snapshots":[{"event_id":"sha256:b63f92ddaf9e2389d817c8113a81785ffc16f7b86c658bcfdf145e02bfb755d1","target":"graph","created_at":"2026-06-19T16:12:12Z","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/2404.02116/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is well-known that the Sobolev spaces $W^{k,p}(\\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \\in \\{0,1\\}$, but not if $k \\ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\\mathbb R^d)$ is a vector lattice in this case.\n  We also prove a related abstract result: if $(T(t))_{t \\in [0,\\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results ob","authors_text":"Felix L. Schwenninger, Jochen Gl\\\"uck, Sahiba Arora","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-04-02T17:20:12Z","title":"The lattice structure of negative Sobolev and extrapolation spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.02116","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":"sha256:3312d715c7bcd44ef361606827fbd4b702b40fdc75e8fc79141b49f5b3365008","target":"record","created_at":"2026-06-19T16:12:12Z","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":"1a7b957e85b565bbc33e7b45c7ae2803ed8a861a0127b7c1858251a1733f9371","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-04-02T17:20:12Z","title_canon_sha256":"6177797adee9df5b23b2baab65ecace4bd1726f5ae8ab802237c42c6e4a0f81c"},"schema_version":"1.0","source":{"id":"2404.02116","kind":"arxiv","version":4}},"canonical_sha256":"32e04c60d6fb13da1d3d81b03fb11bede1e1605948baa5fdeb1de1a36f9d725d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32e04c60d6fb13da1d3d81b03fb11bede1e1605948baa5fdeb1de1a36f9d725d","first_computed_at":"2026-06-19T16:12:12.576201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:12.576201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TagZUYpsNl6rAdXptQpXbEzvLEZq52zeQOFjgD4Rnqy27YhNae7qXqFIjReNBwzcIpboExzfDgYqH+jOHyyBAQ==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:12.576597Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.02116","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3312d715c7bcd44ef361606827fbd4b702b40fdc75e8fc79141b49f5b3365008","sha256:b63f92ddaf9e2389d817c8113a81785ffc16f7b86c658bcfdf145e02bfb755d1"],"state_sha256":"1f3aad76051a0b9385dccd8698f37e42cf12b0eaf99f37d16512453265ed9fe7"}