{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:A4LH65ZDKGXMBRWUTVJSTH3VCF","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":"822fd036b12aeb3c5ca178b4016a6484e26f387251e667570a7b8148d18e871a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-11-14T15:40:56Z","title_canon_sha256":"e10f22ff46d27aea2357aba76c26884f81cb0a0f16023b57f468548ee5a621e7"},"schema_version":"1.0","source":{"id":"1711.05149","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.05149","created_at":"2026-07-05T01:08:20Z"},{"alias_kind":"arxiv_version","alias_value":"1711.05149v3","created_at":"2026-07-05T01:08:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.05149","created_at":"2026-07-05T01:08:20Z"},{"alias_kind":"pith_short_12","alias_value":"A4LH65ZDKGXM","created_at":"2026-07-05T01:08:20Z"},{"alias_kind":"pith_short_16","alias_value":"A4LH65ZDKGXMBRWU","created_at":"2026-07-05T01:08:20Z"},{"alias_kind":"pith_short_8","alias_value":"A4LH65ZD","created_at":"2026-07-05T01:08:20Z"}],"graph_snapshots":[{"event_id":"sha256:da33cce3d3e2f26ec9a7a5002d8d909aed126ea7e2c495d633700a9dd00402fe","target":"graph","created_at":"2026-07-05T01:08:20Z","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/1711.05149/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\\|x\\pm y\\| > 1$ for distinct elements $x,y\\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set $A$ may be chosen in a way that $\\|x\\pm y\\| \\geqslant 1+\\varepsilon$ for some $\\varepsilon > 0$ and distinct $x,y\\in A$. Under additional structural properties of $X$, such as no","authors_text":"Petr H\\'ajek, Tomasz Kania, Tommaso Russo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-11-14T15:40:56Z","title":"Symmetrically separated sequences in the unit sphere of a Banach space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.05149","kind":"arxiv","version":3},"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":"sha256:90973acb5081776367818280683803bf5fef56ad34b014f9bc602f75b450b106","target":"record","created_at":"2026-07-05T01:08:20Z","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":"822fd036b12aeb3c5ca178b4016a6484e26f387251e667570a7b8148d18e871a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-11-14T15:40:56Z","title_canon_sha256":"e10f22ff46d27aea2357aba76c26884f81cb0a0f16023b57f468548ee5a621e7"},"schema_version":"1.0","source":{"id":"1711.05149","kind":"arxiv","version":3}},"canonical_sha256":"07167f772351aec0c6d49d53299f75117060fa99f8f27e5b6f6d79969ac446da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"07167f772351aec0c6d49d53299f75117060fa99f8f27e5b6f6d79969ac446da","first_computed_at":"2026-07-05T01:08:20.846151Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:08:20.846151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bNJsN9BwXqEiKjfnagGtCIw79eHyzQdspGmsxePUwYfXxdpakPcnii5hap0kcN2U+MmY6aZlay/sIEDixa0tBg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:08:20.846607Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.05149","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90973acb5081776367818280683803bf5fef56ad34b014f9bc602f75b450b106","sha256:da33cce3d3e2f26ec9a7a5002d8d909aed126ea7e2c495d633700a9dd00402fe"],"state_sha256":"bc4d3290cba4ea09dc995762a6fadaa4cfbed1f47f9e491152f8a5d57ea04d23"}