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The proof is inspired by the recent work of Frank and Ivanisvili~\\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their idea of cubic-majorant reduction, but replace thei"},"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":"2606.02847","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-01T20:11:29Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"ca24f8328a99bd8962251e433f42a1326447513537a62743c964f8c21d4402cf","abstract_canon_sha256":"14ebccdad85fdbb4d054f02603f89f06aec9165a5690f27983e5aecd6ce00a01"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:24.475252Z","signature_b64":"61kdLRMBrkYPA3dHPZaKYJp0B3YmsM8q0nNqsT1H4ALdfvgT00p77IToePNuA2ubOsMiWKU+YSRlWhjT2cBjCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15c2e2ea14eb6339c643ab12d7052daadd47b77c26d9bb13f195dd1962e440f7","last_reissued_at":"2026-06-03T01:05:24.474890Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:24.474890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Log-Sobolev Inequalities for Finite Cyclic Groups with Word-Length","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.CA","authors_text":"Haonan Zhang, Xinyuan Xie","submitted_at":"2026-06-01T20:11:29Z","abstract_excerpt":"Let $\\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\\pi$, and let $-A_{\\psi_n}$ be the Laplacian with respect to the word length $\n  \\psi_n(k) = \\min(k,n-k). $ We prove the sharp log-Sobolev inequality $$\n  \\operatorname{Ent}_{\\pi}(f^2)\n  \\le 2\\pi\\bigl(f A_{\\psi_n} f\\bigr),\n  \\qquad f:\\mathbb Z_n \\to \\mathbb C, $$ for every $n \\ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. 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