{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:STTFJWYHBHMN3YIGFECJ4N3RCE","short_pith_number":"pith:STTFJWYH","schema_version":"1.0","canonical_sha256":"94e654db0709d8dde10629049e3771113ec099eca45ebd8416cd526cef860ae9","source":{"kind":"arxiv","id":"2606.08887","version":1},"attestation_state":"computed","paper":{"title":"Bounding Curvature Measure on Manifolds with Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Nan Li","submitted_at":"2026-06-08T00:11:16Z","abstract_excerpt":"Let $X$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$, and let $\\eta > 0$. Define $\\mathcal{S}^{k}_\\eta(X)$ as the set of $(k,\\eta)$-singular points in $X$ whose tangent cones are $\\eta$-away from splitting off $\\mathbb{R}^{k+1}$ isometrically. For a point $p \\in X$, assume that $M = B_2(p) \\setminus (\\mathcal{S}^{n-2}_\\eta(X) \\cup \\partial X)$ is a smooth manifold equipped with the Riemannian metric induced by $X$. We prove that the integral of the scalar curvature of $M$ over $B_1(p)$ is bounded from above by a constant depending only on $n$ and $\\eta$. As a special case, th"},"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":"2606.08887","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-08T00:11:16Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"a5df28d5aca09fd4e2b4e6e45faa211f5915d906729de16c4084c41134d7f75c","abstract_canon_sha256":"4c0e22c30cb87b552b2b3559551145086ed8afd0820f548a5c6d58f75f48b2b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:45.395932Z","signature_b64":"f+gpydh8O0A+R6XRSsbIvl9Nze0TStamp177Q0egV+tz1DYgpIGtemiP4Jf1X569l1Jqs4TdC/Lxl29JbGUDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94e654db0709d8dde10629049e3771113ec099eca45ebd8416cd526cef860ae9","last_reissued_at":"2026-06-09T02:07:45.395073Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:45.395073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding Curvature Measure on Manifolds with Singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Nan Li","submitted_at":"2026-06-08T00:11:16Z","abstract_excerpt":"Let $X$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$, and let $\\eta > 0$. Define $\\mathcal{S}^{k}_\\eta(X)$ as the set of $(k,\\eta)$-singular points in $X$ whose tangent cones are $\\eta$-away from splitting off $\\mathbb{R}^{k+1}$ isometrically. For a point $p \\in X$, assume that $M = B_2(p) \\setminus (\\mathcal{S}^{n-2}_\\eta(X) \\cup \\partial X)$ is a smooth manifold equipped with the Riemannian metric induced by $X$. We prove that the integral of the scalar curvature of $M$ over $B_1(p)$ is bounded from above by a constant depending only on $n$ and $\\eta$. As a special case, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08887","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08887/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.08887","created_at":"2026-06-09T02:07:45.395223+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.08887v1","created_at":"2026-06-09T02:07:45.395223+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08887","created_at":"2026-06-09T02:07:45.395223+00:00"},{"alias_kind":"pith_short_12","alias_value":"STTFJWYHBHMN","created_at":"2026-06-09T02:07:45.395223+00:00"},{"alias_kind":"pith_short_16","alias_value":"STTFJWYHBHMN3YIG","created_at":"2026-06-09T02:07:45.395223+00:00"},{"alias_kind":"pith_short_8","alias_value":"STTFJWYH","created_at":"2026-06-09T02:07:45.395223+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/STTFJWYHBHMN3YIGFECJ4N3RCE","json":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE.json","graph_json":"https://pith.science/api/pith-number/STTFJWYHBHMN3YIGFECJ4N3RCE/graph.json","events_json":"https://pith.science/api/pith-number/STTFJWYHBHMN3YIGFECJ4N3RCE/events.json","paper":"https://pith.science/paper/STTFJWYH"},"agent_actions":{"view_html":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE","download_json":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE.json","view_paper":"https://pith.science/paper/STTFJWYH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.08887&json=true","fetch_graph":"https://pith.science/api/pith-number/STTFJWYHBHMN3YIGFECJ4N3RCE/graph.json","fetch_events":"https://pith.science/api/pith-number/STTFJWYHBHMN3YIGFECJ4N3RCE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE/action/storage_attestation","attest_author":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE/action/author_attestation","sign_citation":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE/action/citation_signature","submit_replication":"https://pith.science/pith/STTFJWYHBHMN3YIGFECJ4N3RCE/action/replication_record"}},"created_at":"2026-06-09T02:07:45.395223+00:00","updated_at":"2026-06-09T02:07:45.395223+00:00"}