{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FNHFO7HXVZSAFHXEXQV6C4V7W3","short_pith_number":"pith:FNHFO7HX","canonical_record":{"source":{"id":"1509.03211","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-10T16:12:44Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"155bc3d449c07e2dcd6eca2c13b1a5a222140f213c4e23477a2c206ffff46868","abstract_canon_sha256":"f47f88442bc02948a50b5a74f191cc07193878119890cf74ec5671378be500d9"},"schema_version":"1.0"},"canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","source":{"kind":"arxiv","id":"1509.03211","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03211","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03211v2","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03211","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"FNHFO7HXVZSA","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FNHFO7HXVZSAFHXE","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FNHFO7HX","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FNHFO7HXVZSAFHXEXQV6C4V7W3","target":"record","payload":{"canonical_record":{"source":{"id":"1509.03211","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-10T16:12:44Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"155bc3d449c07e2dcd6eca2c13b1a5a222140f213c4e23477a2c206ffff46868","abstract_canon_sha256":"f47f88442bc02948a50b5a74f191cc07193878119890cf74ec5671378be500d9"},"schema_version":"1.0"},"canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:00.238610Z","signature_b64":"94FBEOnOjNqJkw1sToskFOjc7c5NsyyGVh8CAo7zGE4CGAIVzd3eHqqGUGFRaov0QNy7FQu0HzNXYoU3KoxeDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","last_reissued_at":"2026-05-18T00:21:00.238098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:00.238098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.03211","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":"2026-05-18T00:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ltnh0vKgUnV3+4Y3FvcHpIi5/lbQ2LdiQeh25Q+Ae6HowBZNxYAakpY/cmDAtfZE5WHZ1jEArMwGXHZMz7zXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:23:11.961489Z"},"content_sha256":"5782a327f9a1639a47ce2f665874b3d9c635a4e94bd10c40f21292bef1cf9edb","schema_version":"1.0","event_id":"sha256:5782a327f9a1639a47ce2f665874b3d9c635a4e94bd10c40f21292bef1cf9edb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FNHFO7HXVZSAFHXEXQV6C4V7W3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structure of sets which are well approximated by zero sets of harmonic polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Matthew Badger, Max Engelstein, Tatiana Toro","submitted_at":"2015-09-10T16:12:44Z","abstract_excerpt":"The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how \"degree $k$ points\" sit inside zero sets of harmonic polynomials in $\\mathbb R^n$ of degree $d$ (for all $n\\geq 2$ and $1\\leq k\\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03211","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":"2026-05-18T00:21:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7pKVLWc4ZUtzlF68w9CVTv5DsWVLFSdDlLOl9E+3QOCaORG4ZKUudBTIpt2cpI0yHulWgNskzV1qL15X9TZ5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:23:11.962139Z"},"content_sha256":"26f94e361a01528aca70268bef10b5ebf22a17ac9d11f3109cb17b6ffe52ed23","schema_version":"1.0","event_id":"sha256:26f94e361a01528aca70268bef10b5ebf22a17ac9d11f3109cb17b6ffe52ed23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/bundle.json","state_url":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/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":"2026-05-28T17:23:11Z","links":{"resolver":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3","bundle":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/bundle.json","state":"https://pith.science/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FNHFO7HXVZSAFHXEXQV6C4V7W3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FNHFO7HXVZSAFHXEXQV6C4V7W3","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":"f47f88442bc02948a50b5a74f191cc07193878119890cf74ec5671378be500d9","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-10T16:12:44Z","title_canon_sha256":"155bc3d449c07e2dcd6eca2c13b1a5a222140f213c4e23477a2c206ffff46868"},"schema_version":"1.0","source":{"id":"1509.03211","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03211","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03211v2","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03211","created_at":"2026-05-18T00:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"FNHFO7HXVZSA","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FNHFO7HXVZSAFHXE","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FNHFO7HX","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:26f94e361a01528aca70268bef10b5ebf22a17ac9d11f3109cb17b6ffe52ed23","target":"graph","created_at":"2026-05-18T00:21:00Z","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":"The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how \"degree $k$ points\" sit inside zero sets of harmonic polynomials in $\\mathbb R^n$ of degree $d$ (for all $n\\geq 2$ and $1\\leq k\\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of s","authors_text":"Matthew Badger, Max Engelstein, Tatiana Toro","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-10T16:12:44Z","title":"Structure of sets which are well approximated by zero sets of harmonic polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03211","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":"sha256:5782a327f9a1639a47ce2f665874b3d9c635a4e94bd10c40f21292bef1cf9edb","target":"record","created_at":"2026-05-18T00:21:00Z","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":"f47f88442bc02948a50b5a74f191cc07193878119890cf74ec5671378be500d9","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-09-10T16:12:44Z","title_canon_sha256":"155bc3d449c07e2dcd6eca2c13b1a5a222140f213c4e23477a2c206ffff46868"},"schema_version":"1.0","source":{"id":"1509.03211","kind":"arxiv","version":2}},"canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b4e577cf7ae64029ee4bc2be172bfb6ecadc735c32d984ffd4e8c5f9c532284","first_computed_at":"2026-05-18T00:21:00.238098Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:00.238098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"94FBEOnOjNqJkw1sToskFOjc7c5NsyyGVh8CAo7zGE4CGAIVzd3eHqqGUGFRaov0QNy7FQu0HzNXYoU3KoxeDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:00.238610Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.03211","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5782a327f9a1639a47ce2f665874b3d9c635a4e94bd10c40f21292bef1cf9edb","sha256:26f94e361a01528aca70268bef10b5ebf22a17ac9d11f3109cb17b6ffe52ed23"],"state_sha256":"922a9393a5c3721d638d53b33dff468fda7a2c9650b020f5a626f8dfd3f09c99"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"61RfPN+P6HuHFjWHDxfJa982FD6OBXB23OnzHoIHZwKIvgxzdxwc1Lb0F7wH3uUARb6MryyUwwZEzstwrBVFBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T17:23:11.965246Z","bundle_sha256":"ad0a337b4286754ae07c2254d5bc67d9416531382932196914b912d2d84c061b"}}