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Using the contraction property of the SO(3) left Jacobian, we show that\n  ||exp(theta)||_op <= 1 + ||theta||_F\n  for all theta in se(3). We then derive a self-contained O(R^2) upper bound for the gradient Lipschitz constant, with explicit constant 4.02, and construct an objective J* satisfying\n  L_J*(R; se(3)) >= 0.0505 R^2\n  for R >= 2.\n  These results place se(3) between compact Lie algebra"},"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":"2605.31076","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-29T09:44:52Z","cross_cats_sorted":[],"title_canon_sha256":"a788db45ea0abd81264e090e84ad6b980270e4d0110d5b41343cc7a66d938919","abstract_canon_sha256":"7f00118affec79886950302f0a4d699cadd51bada7464b23bfaf489466fa833b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:03:34.801980Z","signature_b64":"2cUNVe9MR7OB3+obliL3IArmsjL+DfmuUXAomlvsuHsypyWk8wYAsvu7E/Pg3f8CTX/3lPdTc7dyAnqEQHDDDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48ec12e95b40c26ee49ed9e265974ee0226cae2a862e3182bd5f4a210f03eea0","last_reissued_at":"2026-06-01T01:03:34.801056Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:03:34.801056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Operator-norm bounds and a quadratic lower-growth example for the special Euclidean algebra se(3)","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Sooraj K.C, Vivek Mishra","submitted_at":"2026-05-29T09:44:52Z","abstract_excerpt":"We prove operator-norm and gradient Lipschitz bounds for exponential-map parameterizations on the special Euclidean algebra se(3), providing an explicit example of intermediate polynomial growth behavior. 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