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In such cases, there is a filtration of the singular set, $S_0\\subset S_1\\cdots S_{n-1}:= S$, where $S^k:= \\{x\\in X:\\text{ no tangent cone at $x$ is }(k+1)\\text{-symmetric}\\}$; equivalently no tangent cone splits off a Euclidean factor $\\mathbb{R}^{k+1}$ isometrically. Moreover, by \\cite{ChCoI}, $\\dim S"},"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":"1805.07988","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-05-21T11:09:55Z","cross_cats_sorted":[],"title_canon_sha256":"565f9286de8127ab7bd7d0a8fc74794faf3156aae08dd983d10cf0189887a328","abstract_canon_sha256":"50c7140a0cbc25f3b63444703e6e86bc75ef216bcd502eeb7708be714f8d0cb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:31.957638Z","signature_b64":"aKF+jbF4DjvpGiv64sYFZd3SqU2mBFVLXRQes46w8D9lXcYwfS7tFDnNGzMsUQ244jsxb6OYLLvdHvtEQx6wAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"465b15861c9aff0ed94f82650cc1c4e7b451ca6f9528c472061060be416b7b03","last_reissued_at":"2026-05-18T00:15:31.957115Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:31.957115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Jeff Cheeger, Wenshuai Jiang","submitted_at":"2018-05-21T11:09:55Z","abstract_excerpt":"This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\\stackrel{d_{GH}}{\\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $Rc_{M^n_i}\\geq -(n-1)$ as well as the noncollapsing assumption $Vol(B_1(p_i))>v>0$. In such cases, there is a filtration of the singular set, $S_0\\subset S_1\\cdots S_{n-1}:= S$, where $S^k:= \\{x\\in X:\\text{ no tangent cone at $x$ is }(k+1)\\text{-symmetric}\\}$; equivalently no tangent cone splits off a Euclidean factor $\\mathbb{R}^{k+1}$ isometrically. 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