{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:E2SZVMNPPDEPSHLLYON6SGKRQZ","short_pith_number":"pith:E2SZVMNP","canonical_record":{"source":{"id":"2507.09587","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-13T11:39:21Z","cross_cats_sorted":[],"title_canon_sha256":"c44ab6c3e54eac7ab05ead58b533524987c321515ab9e634369fdeaa6939f028","abstract_canon_sha256":"4c61fc3080c4de3e85baa2cdea71fb76d61233d14098905d26b64bbbce8c75b4"},"schema_version":"1.0"},"canonical_sha256":"26a59ab1af78c8f91d6bc39be919518665512251f8edcb8ee3cff694c7d4ce88","source":{"kind":"arxiv","id":"2507.09587","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.09587","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"arxiv_version","alias_value":"2507.09587v2","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.09587","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"pith_short_12","alias_value":"E2SZVMNPPDEP","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"pith_short_16","alias_value":"E2SZVMNPPDEPSHLL","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"pith_short_8","alias_value":"E2SZVMNP","created_at":"2026-07-05T11:37:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:E2SZVMNPPDEPSHLLYON6SGKRQZ","target":"record","payload":{"canonical_record":{"source":{"id":"2507.09587","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-13T11:39:21Z","cross_cats_sorted":[],"title_canon_sha256":"c44ab6c3e54eac7ab05ead58b533524987c321515ab9e634369fdeaa6939f028","abstract_canon_sha256":"4c61fc3080c4de3e85baa2cdea71fb76d61233d14098905d26b64bbbce8c75b4"},"schema_version":"1.0"},"canonical_sha256":"26a59ab1af78c8f91d6bc39be919518665512251f8edcb8ee3cff694c7d4ce88","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:37:26.318269Z","signature_b64":"AzXCaUXf1GAG8HT33MCZYvGtuFXb7CvGh+S8TFhO+HsaB6h5I+wBDO8581U2E+XJC77d1xPUDSb8DLvcWW9DBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26a59ab1af78c8f91d6bc39be919518665512251f8edcb8ee3cff694c7d4ce88","last_reissued_at":"2026-07-05T11:37:26.317746Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:37:26.317746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.09587","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":"2026-07-05T11:37:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j7+CulFjkY3waf2guCMflij+tL+ukQjqmeVFHReWX47HJXexa3uQhJxWjXuTIJbTr9TDqHseqR/LbgRuLeckBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:17:36.603987Z"},"content_sha256":"7ece25374996d35101a624ffcc511aaad56da0fcac1e93e5fcb4761d6ad34f16","schema_version":"1.0","event_id":"sha256:7ece25374996d35101a624ffcc511aaad56da0fcac1e93e5fcb4761d6ad34f16"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:E2SZVMNPPDEPSHLLYON6SGKRQZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Liouville theorem for singular solutions to nonlocal equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Minhyun Kim, Se-Chan Lee","submitted_at":"2025-07-13T11:39:21Z","abstract_excerpt":"We study singular solutions to the fractional Laplace equation and, more generally, to nonlocal linear equations with measurable kernels. We establish B\\^ocher type results that characterize the behavior of singular solutions near the singular point. In addition, we prove Liouville theorems for singular solutions. To this end, we construct fundamental solutions for nonlocal linear operators and establish a localized comparison principle."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.09587","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.09587/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:37:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hUPPeq8PDhzOJ21UtSvERFblZkK9c3nLPlFqkGqVB2RnDYOcdxoeOgHV7BohzoN851jxCgn9Wn0rggoJh/xOAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:17:36.604890Z"},"content_sha256":"696fcb0f7a88b6115fd861876009d6905b18cd5d4dd39831700f77c72df5791e","schema_version":"1.0","event_id":"sha256:696fcb0f7a88b6115fd861876009d6905b18cd5d4dd39831700f77c72df5791e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ/bundle.json","state_url":"https://pith.science/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ/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":"2026-08-12T18:17:36Z","links":{"resolver":"https://pith.science/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ","bundle":"https://pith.science/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ/bundle.json","state":"https://pith.science/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E2SZVMNPPDEPSHLLYON6SGKRQZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:E2SZVMNPPDEPSHLLYON6SGKRQZ","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":"4c61fc3080c4de3e85baa2cdea71fb76d61233d14098905d26b64bbbce8c75b4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-13T11:39:21Z","title_canon_sha256":"c44ab6c3e54eac7ab05ead58b533524987c321515ab9e634369fdeaa6939f028"},"schema_version":"1.0","source":{"id":"2507.09587","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.09587","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"arxiv_version","alias_value":"2507.09587v2","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.09587","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"pith_short_12","alias_value":"E2SZVMNPPDEP","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"pith_short_16","alias_value":"E2SZVMNPPDEPSHLL","created_at":"2026-07-05T11:37:26Z"},{"alias_kind":"pith_short_8","alias_value":"E2SZVMNP","created_at":"2026-07-05T11:37:26Z"}],"graph_snapshots":[{"event_id":"sha256:696fcb0f7a88b6115fd861876009d6905b18cd5d4dd39831700f77c72df5791e","target":"graph","created_at":"2026-07-05T11:37:26Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.09587/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study singular solutions to the fractional Laplace equation and, more generally, to nonlocal linear equations with measurable kernels. We establish B\\^ocher type results that characterize the behavior of singular solutions near the singular point. In addition, we prove Liouville theorems for singular solutions. To this end, we construct fundamental solutions for nonlocal linear operators and establish a localized comparison principle.","authors_text":"Minhyun Kim, Se-Chan Lee","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-13T11:39:21Z","title":"Liouville theorem for singular solutions to nonlocal equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.09587","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":"sha256:7ece25374996d35101a624ffcc511aaad56da0fcac1e93e5fcb4761d6ad34f16","target":"record","created_at":"2026-07-05T11:37:26Z","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":"4c61fc3080c4de3e85baa2cdea71fb76d61233d14098905d26b64bbbce8c75b4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-13T11:39:21Z","title_canon_sha256":"c44ab6c3e54eac7ab05ead58b533524987c321515ab9e634369fdeaa6939f028"},"schema_version":"1.0","source":{"id":"2507.09587","kind":"arxiv","version":2}},"canonical_sha256":"26a59ab1af78c8f91d6bc39be919518665512251f8edcb8ee3cff694c7d4ce88","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26a59ab1af78c8f91d6bc39be919518665512251f8edcb8ee3cff694c7d4ce88","first_computed_at":"2026-07-05T11:37:26.317746Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:37:26.317746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AzXCaUXf1GAG8HT33MCZYvGtuFXb7CvGh+S8TFhO+HsaB6h5I+wBDO8581U2E+XJC77d1xPUDSb8DLvcWW9DBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:37:26.318269Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.09587","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ece25374996d35101a624ffcc511aaad56da0fcac1e93e5fcb4761d6ad34f16","sha256:696fcb0f7a88b6115fd861876009d6905b18cd5d4dd39831700f77c72df5791e"],"state_sha256":"2fff7f748ba91d24e8c629887b5ed465c1baa5ca384ae20528aa4729db86a7aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RYvfBqv9jj0WqHBm4oCY2NJt0YjccV5Vh1Uk1lUDTWSut5WCB7OwdYGhZIj/23IOnCWUH1DWXdreIRtBCIjTDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T18:17:36.610460Z","bundle_sha256":"d37834950dbd451ab00d0162db7fe8fd2962298fea16de9a9d1a8d26ae3027fa"}}