{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:WCR23N7WE2WV6CD3UWKP24CXPW","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":"fe81b5373aaa862c43661405b73ad8629804348fd9ef3d0e506cc378a1bbb5dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-08-13T14:01:37Z","title_canon_sha256":"b9babcf9bb57c5a4540bc1635e69f1fd547d37001854720a8ab199c59b9bddd2"},"schema_version":"1.0","source":{"id":"2608.13264","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.13264","created_at":"2026-08-14T01:00:48Z"},{"alias_kind":"arxiv_version","alias_value":"2608.13264v1","created_at":"2026-08-14T01:00:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.13264","created_at":"2026-08-14T01:00:48Z"},{"alias_kind":"pith_short_12","alias_value":"WCR23N7WE2WV","created_at":"2026-08-14T01:00:48Z"},{"alias_kind":"pith_short_16","alias_value":"WCR23N7WE2WV6CD3","created_at":"2026-08-14T01:00:48Z"},{"alias_kind":"pith_short_8","alias_value":"WCR23N7W","created_at":"2026-08-14T01:00:48Z"}],"graph_snapshots":[{"event_id":"sha256:4424ec054264c6171cdb2eba9e710ad24cda4f3edce5543bcafc26a197d17c8f","target":"graph","created_at":"2026-08-14T01:00:48Z","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/2608.13264/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $\\zeta_n:=\\arccos(-\\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\\mathbb{S}^n$.\n  We prove that $$ d_{\\mathrm{GH}}(\\mathbb{S}^n,\\mathbb{S}^{n+1})=\\frac{\\zeta_n}{2} \\qquad(n\\geq1), $$ resolving a conjecture of Lim, M\\'emoli, and Smith. All cases $n\\geq4$ were previously open.\n  This equality is established by explicitly constructing a family of correspondences $\\mathcal R_n\\subseteq \\mathbb{S","authors_text":"Donghan Kim, Facundo Memoli, Sunhyuk Lim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-08-13T14:01:37Z","title":"The Gromov-Hausdorff Distance Between Consecutive Spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13264","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":"sha256:05b501b721c840cc0202f573898802c83436956127618b6eb33e28dd63f58657","target":"record","created_at":"2026-08-14T01:00:48Z","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":"fe81b5373aaa862c43661405b73ad8629804348fd9ef3d0e506cc378a1bbb5dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-08-13T14:01:37Z","title_canon_sha256":"b9babcf9bb57c5a4540bc1635e69f1fd547d37001854720a8ab199c59b9bddd2"},"schema_version":"1.0","source":{"id":"2608.13264","kind":"arxiv","version":1}},"canonical_sha256":"b0a3adb7f626ad5f087ba594fd70577d8001c1c5d605b54c6d95ae8ad97ac6df","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0a3adb7f626ad5f087ba594fd70577d8001c1c5d605b54c6d95ae8ad97ac6df","first_computed_at":"2026-08-14T01:00:48.621521Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-14T01:00:48.621521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MjOQaInvIo5xgdpZMwuj4Zn5VXjIWSbaOynP/Zu03ySaRrFsHPNCp0gkp6tKMXjF9947FtbXZ7NLDPQncGbNAQ==","signature_status":"signed_v1","signed_at":"2026-08-14T01:00:48.623536Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.13264","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05b501b721c840cc0202f573898802c83436956127618b6eb33e28dd63f58657","sha256:4424ec054264c6171cdb2eba9e710ad24cda4f3edce5543bcafc26a197d17c8f"],"state_sha256":"cececf8dd2573e9bbfa0237d7ac3a8a383b3e53765e3334500a8d0e6b34b2140"}