{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KLXAEU6QF66FQQ3FFGX5P3WOBU","short_pith_number":"pith:KLXAEU6Q","canonical_record":{"source":{"id":"1504.02043","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-08T17:32:34Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"660472f9548fa3dce96545d308acd2359d38ce944763b621d9d4e336591b2051","abstract_canon_sha256":"7d59be93f3ee82b2f64b9c6afd9f832bf9997a43fe6ef9afcb9a5511d051d3fe"},"schema_version":"1.0"},"canonical_sha256":"52ee0253d02fbc58436529afd7eece0d158b1390ff80e53bfeef1f99696597f0","source":{"kind":"arxiv","id":"1504.02043","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02043","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02043v5","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02043","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"pith_short_12","alias_value":"KLXAEU6QF66F","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KLXAEU6QF66FQQ3F","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KLXAEU6Q","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KLXAEU6QF66FQQ3FFGX5P3WOBU","target":"record","payload":{"canonical_record":{"source":{"id":"1504.02043","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-08T17:32:34Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"660472f9548fa3dce96545d308acd2359d38ce944763b621d9d4e336591b2051","abstract_canon_sha256":"7d59be93f3ee82b2f64b9c6afd9f832bf9997a43fe6ef9afcb9a5511d051d3fe"},"schema_version":"1.0"},"canonical_sha256":"52ee0253d02fbc58436529afd7eece0d158b1390ff80e53bfeef1f99696597f0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:48.623834Z","signature_b64":"2gcW5GIh3ZDGA09MlEtkDHe4HopVyEk1hrEWRQw5ErkWeSAi7m/cXI5wsDUm71vLBRX5A3nMenP3uxsvZkzcAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52ee0253d02fbc58436529afd7eece0d158b1390ff80e53bfeef1f99696597f0","last_reissued_at":"2026-05-18T00:13:48.623173Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:48.623173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.02043","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:13:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YBmhEzo4R+C7md3kdKLsN2HRhBJvkWtUz6VoXkupKgEblM3qIpDYmJ7HVOvoqaLCJNhRdE7eRAtZvzwwY/m+DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:23:02.227450Z"},"content_sha256":"0a4bb854b428c0b10260503bcb722da51af1eda6ca020e7cdfd414d7768b5cef","schema_version":"1.0","event_id":"sha256:0a4bb854b428c0b10260503bcb722da51af1eda6ca020e7cdfd414d7768b5cef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KLXAEU6QF66FQQ3FFGX5P3WOBU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Daniele Valtorta","submitted_at":"2015-04-08T17:32:34Z","abstract_excerpt":"In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\\subseteq M\\to N$ between Riemannian manifolds. If $S^k(f)\\equiv\\{x\\in M: \\text{ no tangent map at $x$ is }k+1\\text{-symmetric}\\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\\dim S^k\\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable.\n  In the case of minimizing harmonic maps we go further, and prove that the singular set $S(f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02043","kind":"arxiv","version":5},"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":"2026-05-18T00:13:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O1pSxZgCy0gTrQqDNmxfidKMawUleMe2G1RPaIGM3j9dyN7uoMF7bMpkKUsIhYN354ym1vRUwse7brLC1fLIDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:23:02.227928Z"},"content_sha256":"0d4acc28588f186b8fd9dc1a5e64e621dcd3e569f30827c16436904a9bc93690","schema_version":"1.0","event_id":"sha256:0d4acc28588f186b8fd9dc1a5e64e621dcd3e569f30827c16436904a9bc93690"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU/bundle.json","state_url":"https://pith.science/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU/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":"2026-05-27T23:23:02Z","links":{"resolver":"https://pith.science/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU","bundle":"https://pith.science/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU/bundle.json","state":"https://pith.science/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KLXAEU6QF66FQQ3FFGX5P3WOBU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KLXAEU6QF66FQQ3FFGX5P3WOBU","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":"7d59be93f3ee82b2f64b9c6afd9f832bf9997a43fe6ef9afcb9a5511d051d3fe","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-08T17:32:34Z","title_canon_sha256":"660472f9548fa3dce96545d308acd2359d38ce944763b621d9d4e336591b2051"},"schema_version":"1.0","source":{"id":"1504.02043","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02043","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02043v5","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02043","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"pith_short_12","alias_value":"KLXAEU6QF66F","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"KLXAEU6QF66FQQ3F","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"KLXAEU6Q","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:0d4acc28588f186b8fd9dc1a5e64e621dcd3e569f30827c16436904a9bc93690","target":"graph","created_at":"2026-05-18T00:13:48Z","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":"In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\\subseteq M\\to N$ between Riemannian manifolds. If $S^k(f)\\equiv\\{x\\in M: \\text{ no tangent map at $x$ is }k+1\\text{-symmetric}\\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\\dim S^k\\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable.\n  In the case of minimizing harmonic maps we go further, and prove that the singular set $S(f","authors_text":"Aaron Naber, Daniele Valtorta","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-08T17:32:34Z","title":"Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02043","kind":"arxiv","version":5},"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":"sha256:0a4bb854b428c0b10260503bcb722da51af1eda6ca020e7cdfd414d7768b5cef","target":"record","created_at":"2026-05-18T00:13:48Z","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":"7d59be93f3ee82b2f64b9c6afd9f832bf9997a43fe6ef9afcb9a5511d051d3fe","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-08T17:32:34Z","title_canon_sha256":"660472f9548fa3dce96545d308acd2359d38ce944763b621d9d4e336591b2051"},"schema_version":"1.0","source":{"id":"1504.02043","kind":"arxiv","version":5}},"canonical_sha256":"52ee0253d02fbc58436529afd7eece0d158b1390ff80e53bfeef1f99696597f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"52ee0253d02fbc58436529afd7eece0d158b1390ff80e53bfeef1f99696597f0","first_computed_at":"2026-05-18T00:13:48.623173Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:48.623173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2gcW5GIh3ZDGA09MlEtkDHe4HopVyEk1hrEWRQw5ErkWeSAi7m/cXI5wsDUm71vLBRX5A3nMenP3uxsvZkzcAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:48.623834Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.02043","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a4bb854b428c0b10260503bcb722da51af1eda6ca020e7cdfd414d7768b5cef","sha256:0d4acc28588f186b8fd9dc1a5e64e621dcd3e569f30827c16436904a9bc93690"],"state_sha256":"df468de1764c4db612868695edfefaa063fbb274146d0271fd39f2bfdccec3d5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U950HgG1zF35pZ72fuWoBvYDEJwAVqITUML8mgyWaxQn2ySj601AN2X2ewP+NUtHtJ1aGEj8TPT/m6SMBCOwBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:23:02.230722Z","bundle_sha256":"0d13c2dea8019f0b1ff7b861df01a1f9b2584d6f902c180e02d298d1d3906185"}}