{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:3C4QZMPB3ETS3QYWLREXRYOQR7","short_pith_number":"pith:3C4QZMPB","canonical_record":{"source":{"id":"1603.03647","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-11T14:44:35Z","cross_cats_sorted":["math.DG","math.SP"],"title_canon_sha256":"f808cc9243f241bdc94dcb1b49351687e7549932722b9fee7f33cc1d2d40d2fe","abstract_canon_sha256":"bb7d06666445da6a833ff9b92ffcceebea853afa4e7bc95bbec92c1f73890514"},"schema_version":"1.0"},"canonical_sha256":"d8b90cb1e1d9272dc3165c4978e1d08ff3497ef69969240c7b776a43862f6f39","source":{"kind":"arxiv","id":"1603.03647","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.03647","created_at":"2026-05-17T23:41:37Z"},{"alias_kind":"arxiv_version","alias_value":"1603.03647v2","created_at":"2026-05-17T23:41:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.03647","created_at":"2026-05-17T23:41:37Z"},{"alias_kind":"pith_short_12","alias_value":"3C4QZMPB3ETS","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3C4QZMPB3ETS3QYW","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3C4QZMPB","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:3C4QZMPB3ETS3QYWLREXRYOQR7","target":"record","payload":{"canonical_record":{"source":{"id":"1603.03647","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-11T14:44:35Z","cross_cats_sorted":["math.DG","math.SP"],"title_canon_sha256":"f808cc9243f241bdc94dcb1b49351687e7549932722b9fee7f33cc1d2d40d2fe","abstract_canon_sha256":"bb7d06666445da6a833ff9b92ffcceebea853afa4e7bc95bbec92c1f73890514"},"schema_version":"1.0"},"canonical_sha256":"d8b90cb1e1d9272dc3165c4978e1d08ff3497ef69969240c7b776a43862f6f39","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:37.579284Z","signature_b64":"0AYU7MCpfqrkC66ZpuTJrmEntrlCr9tEA7/qbmTZalJUP0eXG5hmNXoKD+xvQMGpI76JrOItCTwxdXe+lTmrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8b90cb1e1d9272dc3165c4978e1d08ff3497ef69969240c7b776a43862f6f39","last_reissued_at":"2026-05-17T23:41:37.578689Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:37.578689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.03647","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZRCA8I3XlfiQGKxuwPQ8YX3lwHehvb4ZhrKSA5E4JkRlst6OJkTPHPsOEbIinQdTZeaQWmhLSvTOXBvwCI/wBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T19:20:42.931045Z"},"content_sha256":"44b13af9782c6425408b43ef774d19e18fb5316a11db20cd4d6f88ae1061ae38","schema_version":"1.0","event_id":"sha256:44b13af9782c6425408b43ef774d19e18fb5316a11db20cd4d6f88ae1061ae38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:3C4QZMPB3ETS3QYWLREXRYOQR7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.SP"],"primary_cat":"math.AP","authors_text":"Alan McIntosh, Andreas Ros\\'en, Lashi Bandara","submitted_at":"2016-03-11T14:44:35Z","abstract_excerpt":"We prove that the Atiyah-Singer Dirac operator ${\\mathrm D}_{\\mathrm g}$ in ${\\mathrm L}^2$ depends Riesz continuously on ${\\mathrm L}^{\\infty}$ perturbations of complete metrics ${\\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\\mathrm g} \\to {\\mathrm D}_{\\mathrm g}(1 + {\\mathrm D}_{\\mathrm g}^2)^{-\\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\\'on's first commutator and the Kato square root problem. We also show perturbation results f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03647","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bKT5AsizWsG9uz8yg6+uNMhYpwkd85Q45XIUEtPzrWkyI+vnnP250N3Q2FEHeWbgMGYm+JSTxXHzoGxeBA9BDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T19:20:42.931746Z"},"content_sha256":"947fdff7c27c423cfe9d847aa91071e3ee6cb74918868a1000699de38e6ee691","schema_version":"1.0","event_id":"sha256:947fdff7c27c423cfe9d847aa91071e3ee6cb74918868a1000699de38e6ee691"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3C4QZMPB3ETS3QYWLREXRYOQR7/bundle.json","state_url":"https://pith.science/pith/3C4QZMPB3ETS3QYWLREXRYOQR7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3C4QZMPB3ETS3QYWLREXRYOQR7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T19:20:42Z","links":{"resolver":"https://pith.science/pith/3C4QZMPB3ETS3QYWLREXRYOQR7","bundle":"https://pith.science/pith/3C4QZMPB3ETS3QYWLREXRYOQR7/bundle.json","state":"https://pith.science/pith/3C4QZMPB3ETS3QYWLREXRYOQR7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3C4QZMPB3ETS3QYWLREXRYOQR7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3C4QZMPB3ETS3QYWLREXRYOQR7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bb7d06666445da6a833ff9b92ffcceebea853afa4e7bc95bbec92c1f73890514","cross_cats_sorted":["math.DG","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-11T14:44:35Z","title_canon_sha256":"f808cc9243f241bdc94dcb1b49351687e7549932722b9fee7f33cc1d2d40d2fe"},"schema_version":"1.0","source":{"id":"1603.03647","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.03647","created_at":"2026-05-17T23:41:37Z"},{"alias_kind":"arxiv_version","alias_value":"1603.03647v2","created_at":"2026-05-17T23:41:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.03647","created_at":"2026-05-17T23:41:37Z"},{"alias_kind":"pith_short_12","alias_value":"3C4QZMPB3ETS","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3C4QZMPB3ETS3QYW","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3C4QZMPB","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:947fdff7c27c423cfe9d847aa91071e3ee6cb74918868a1000699de38e6ee691","target":"graph","created_at":"2026-05-17T23:41:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that the Atiyah-Singer Dirac operator ${\\mathrm D}_{\\mathrm g}$ in ${\\mathrm L}^2$ depends Riesz continuously on ${\\mathrm L}^{\\infty}$ perturbations of complete metrics ${\\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\\mathrm g} \\to {\\mathrm D}_{\\mathrm g}(1 + {\\mathrm D}_{\\mathrm g}^2)^{-\\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\\'on's first commutator and the Kato square root problem. We also show perturbation results f","authors_text":"Alan McIntosh, Andreas Ros\\'en, Lashi Bandara","cross_cats":["math.DG","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-11T14:44:35Z","title":"Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03647","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:44b13af9782c6425408b43ef774d19e18fb5316a11db20cd4d6f88ae1061ae38","target":"record","created_at":"2026-05-17T23:41:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bb7d06666445da6a833ff9b92ffcceebea853afa4e7bc95bbec92c1f73890514","cross_cats_sorted":["math.DG","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-11T14:44:35Z","title_canon_sha256":"f808cc9243f241bdc94dcb1b49351687e7549932722b9fee7f33cc1d2d40d2fe"},"schema_version":"1.0","source":{"id":"1603.03647","kind":"arxiv","version":2}},"canonical_sha256":"d8b90cb1e1d9272dc3165c4978e1d08ff3497ef69969240c7b776a43862f6f39","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8b90cb1e1d9272dc3165c4978e1d08ff3497ef69969240c7b776a43862f6f39","first_computed_at":"2026-05-17T23:41:37.578689Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:37.578689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0AYU7MCpfqrkC66ZpuTJrmEntrlCr9tEA7/qbmTZalJUP0eXG5hmNXoKD+xvQMGpI76JrOItCTwxdXe+lTmrCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:37.579284Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.03647","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44b13af9782c6425408b43ef774d19e18fb5316a11db20cd4d6f88ae1061ae38","sha256:947fdff7c27c423cfe9d847aa91071e3ee6cb74918868a1000699de38e6ee691"],"state_sha256":"2596c9b504ec1e3e80080dcc521f08777536f94b176fb1f5dfce9879b2090b2e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A0jwSIQcwEFJYwOpbQO2+D14zgYGsgenyi5QSG/k0HuvNfBM/Rvy3xBi565yjF4hLuXOg5d24aCZdJnSHYlZAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T19:20:42.935445Z","bundle_sha256":"e8394db3d6e840e6d083784762a82133d612f0a24dc4f56a1a6d4c8916413f7a"}}