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We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \\emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \\emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$.\n  More generally, we prove a similar result for maps $f:X \\to Y$ for any compact Rieman"},"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":"1611.03513","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-11-10T21:05:51Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"5c4eedc7c4c358bd631819a4744e167dc16420ecd0871f0a5c4d6246c3e9e92b","abstract_canon_sha256":"26fa91d17c0f61b2c559dfc63a9e69db379cfe7bde2f1159531c798dd4634e76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:13:50.451472Z","signature_b64":"lQFQ0q4oPy6CX0b1hxbDoLZOA5tFglM5RsehyAm5qWNjMfsdkx7DkFmyq/YuLbNUJeH87PqtELBz5wRoQET2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"471a8230cf53cb696d7653ed79d28c03d22682d5209f62b7d2d65c1950b5217d","last_reissued_at":"2026-07-05T01:13:50.451016Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:13:50.451016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantitative nullhomotopy and rational homotopy type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Fedor Manin, Gregory R. 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