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Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\\ge \\Lambda_0>0$. Then there exist four distinct embedded minimal two-spheres $\\Sigma_1,\\ldots,\\Sigma_4\\subset (S^3,g)$ such that $\\operatorname{area}_{g}(\\Sigma_i)\\le 12\\pi(i+1)/\\Lambda_0$ for every $i=1,\\l"},"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":"2605.21607","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-05-20T18:15:16Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"9853488791927efa5ee992db5eca0c9a5808898876a5d58c44838771554d1852","abstract_canon_sha256":"86918d0a1ebbe999e582be94d6691ac5c93db5caf502ecee119473d654978d2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:25.156378Z","signature_b64":"uOhDKJgDSivdYA4UUgwtLSkK4pqbZR8/yeZp6DonI/U8P7M7JNu6M9xCSsvK5Ic2q6ztf1UjTIWQF4FDbgTFAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0139ac19ff470eca391d10e710a8951a74e75bfc3e30c2065740f86fd523756e","last_reissued_at":"2026-05-22T01:03:25.155508Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:25.155508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal spheres and scalar curvature","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Talant Talipov","submitted_at":"2026-05-20T18:15:16Z","abstract_excerpt":"In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\\ge \\Lambda_0>0$. 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