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Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \\in [-T,0], the pointed entropy, with W_{x}(s) <= 0, and W_{x}(s) = 0 iff (M,g(t)) is isometric to the trivial flow on R^n. Our main theorem asserts the following: There exists an \\epsilon>0, depending only on T and on lower scalar curvature and \\mu-entropy bounds for (M,g(-T)), such that W_{x_0}("},"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":"1205.0380","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-05-02T11:07:56Z","cross_cats_sorted":[],"title_canon_sha256":"d0600e35f4ef67997a09b2214e4ea26710aaa602442295da85b62bb2734c0800","abstract_canon_sha256":"aeee7a9f13672c082cc16ff698321a352ad33f4de5f38369bfb23e08d2ad20af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:34.687656Z","signature_b64":"6dgwhL3YSxJ5Wezmnc3r9tSXd8CxX4aVK2F0qwq+0rFlFPuQ4qsPD7h5lgmByr795iG9M+sd4+LBdPnNngqaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39f7cb86050a332f8d38e2c82e32f807896a8da9ed773b2c8273233cc54a5c89","last_reissued_at":"2026-05-18T03:56:34.687126Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:34.687126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New logarithmic Sobolev inequalities and an \\epsilon-regularity theorem for the Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Hans-Joachim Hein","submitted_at":"2012-05-02T11:07:56Z","abstract_excerpt":"In this note we prove a new \\epsilon-regularity theorem for the Ricci flow. 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