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Here, we define quantitative singular strata $\\cS^j_{\\eta,r}$ satisfying $\\cup_{\\eta>0}\\cap_{0<r} \\cS^j_{\\eta,r}=\\cS^j$. Sharpening the known parabolic Hausdorff dimension bound $\\dim \\cS^j\\leq j$, we prove the effective Minkowski estimates that the volume of $r$-tubular neighborhoods of $\\cS^j_{\\eta,r}$ satisfies $\\Vol (T_r(\\cS^j_{\\eta,r})\\cap B_1)\\leq Cr^{N+2-j-\\varep"},"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":"1207.3619","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-07-16T10:14:32Z","cross_cats_sorted":[],"title_canon_sha256":"bfcc4c67cb232c4e3b4a6d3712f05147d5c6ccf3b01b95098d2a300b783737c1","abstract_canon_sha256":"9ea282d7044596f980a2c6af46800e09d54d4cafe5a86d6dd981ff35c994e973"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:58.484824Z","signature_b64":"jPKlqo+rkC4ijUWGM9odeYsElSzFLF2n8KkcclwJ/KQmbWAmijGLT+dLWE9jp4Mv3ektCD6jsZt/3PYH9TuiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"279b0e10dce59e30c8afa7d18ed569a4bbab309458cc8e3b94b26bae07d965e0","last_reissued_at":"2026-05-18T03:41:58.484170Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:58.484170Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantitative Stratification and the Regularity of Mean Curvature Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Jeff Cheeger, Robert Haslhofer","submitted_at":"2012-07-16T10:14:32Z","abstract_excerpt":"Let $\\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. 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Sharpening the known parabolic Hausdorff dimension bound $\\dim \\cS^j\\leq j$, we prove the effective Minkowski estimates that the volume of $r$-tubular neighborhoods of $\\cS^j_{\\eta,r}$ satisfies $\\Vol (T_r(\\cS^j_{\\eta,r})\\cap B_1)\\leq Cr^{N+2-j-\\varep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3619","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1207.3619","created_at":"2026-05-18T03:41:58.484268+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.3619v2","created_at":"2026-05-18T03:41:58.484268+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3619","created_at":"2026-05-18T03:41:58.484268+00:00"},{"alias_kind":"pith_short_12","alias_value":"E6NQ4EG44WPD","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"E6NQ4EG44WPDBSFP","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"E6NQ4EG4","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS","json":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS.json","graph_json":"https://pith.science/api/pith-number/E6NQ4EG44WPDBSFPU7IY5VLJUS/graph.json","events_json":"https://pith.science/api/pith-number/E6NQ4EG44WPDBSFPU7IY5VLJUS/events.json","paper":"https://pith.science/paper/E6NQ4EG4"},"agent_actions":{"view_html":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS","download_json":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS.json","view_paper":"https://pith.science/paper/E6NQ4EG4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.3619&json=true","fetch_graph":"https://pith.science/api/pith-number/E6NQ4EG44WPDBSFPU7IY5VLJUS/graph.json","fetch_events":"https://pith.science/api/pith-number/E6NQ4EG44WPDBSFPU7IY5VLJUS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS/action/storage_attestation","attest_author":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS/action/author_attestation","sign_citation":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS/action/citation_signature","submit_replication":"https://pith.science/pith/E6NQ4EG44WPDBSFPU7IY5VLJUS/action/replication_record"}},"created_at":"2026-05-18T03:41:58.484268+00:00","updated_at":"2026-05-18T03:41:58.484268+00:00"}