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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2608.13264","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-08-13T14:01:37Z","cross_cats_sorted":[],"title_canon_sha256":"b9babcf9bb57c5a4540bc1635e69f1fd547d37001854720a8ab199c59b9bddd2","abstract_canon_sha256":"fe81b5373aaa862c43661405b73ad8629804348fd9ef3d0e506cc378a1bbb5dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-14T01:00:48.623536Z","signature_b64":"MjOQaInvIo5xgdpZMwuj4Zn5VXjIWSbaOynP/Zu03ySaRrFsHPNCp0gkp6tKMXjF9947FtbXZ7NLDPQncGbNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0a3adb7f626ad5f087ba594fd70577d8001c1c5d605b54c6d95ae8ad97ac6df","last_reissued_at":"2026-08-14T01:00:48.621521Z","signature_status":"signed_v1","first_computed_at":"2026-08-14T01:00:48.621521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Gromov-Hausdorff Distance Between Consecutive Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Donghan Kim, Facundo Memoli, Sunhyuk Lim","submitted_at":"2026-08-13T14:01:37Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13264","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.13264/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.13264","created_at":"2026-08-14T01:00:48.622536+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.13264v1","created_at":"2026-08-14T01:00:48.622536+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.13264","created_at":"2026-08-14T01:00:48.622536+00:00"},{"alias_kind":"pith_short_12","alias_value":"WCR23N7WE2WV","created_at":"2026-08-14T01:00:48.622536+00:00"},{"alias_kind":"pith_short_16","alias_value":"WCR23N7WE2WV6CD3","created_at":"2026-08-14T01:00:48.622536+00:00"},{"alias_kind":"pith_short_8","alias_value":"WCR23N7W","created_at":"2026-08-14T01:00:48.622536+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW","json":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW.json","graph_json":"https://pith.science/api/pith-number/WCR23N7WE2WV6CD3UWKP24CXPW/graph.json","events_json":"https://pith.science/api/pith-number/WCR23N7WE2WV6CD3UWKP24CXPW/events.json","paper":"https://pith.science/paper/WCR23N7W"},"agent_actions":{"view_html":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW","download_json":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW.json","view_paper":"https://pith.science/paper/WCR23N7W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.13264&json=true","fetch_graph":"https://pith.science/api/pith-number/WCR23N7WE2WV6CD3UWKP24CXPW/graph.json","fetch_events":"https://pith.science/api/pith-number/WCR23N7WE2WV6CD3UWKP24CXPW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW/action/storage_attestation","attest_author":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW/action/author_attestation","sign_citation":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW/action/citation_signature","submit_replication":"https://pith.science/pith/WCR23N7WE2WV6CD3UWKP24CXPW/action/replication_record"}},"created_at":"2026-08-14T01:00:48.622536+00:00","updated_at":"2026-08-14T01:00:48.622536+00:00"}