{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:DF45RMUMNRRLTFPYFMBWYOTA7O","short_pith_number":"pith:DF45RMUM","canonical_record":{"source":{"id":"1401.7402","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-29T03:23:53Z","cross_cats_sorted":[],"title_canon_sha256":"a4f8dc77bfa653b3625d45fd81733ccd176b379688a27036b43679d9469f471a","abstract_canon_sha256":"72b07e8046843a1d1eeb639d450b0b0924c047114de8c1c33d54bae4bbf71c74"},"schema_version":"1.0"},"canonical_sha256":"1979d8b28c6c62b995f82b036c3a60fb85d9566c82066b660e3efe45a0bbf6fe","source":{"kind":"arxiv","id":"1401.7402","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.7402","created_at":"2026-05-18T03:00:48Z"},{"alias_kind":"arxiv_version","alias_value":"1401.7402v1","created_at":"2026-05-18T03:00:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.7402","created_at":"2026-05-18T03:00:48Z"},{"alias_kind":"pith_short_12","alias_value":"DF45RMUMNRRL","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DF45RMUMNRRLTFPY","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DF45RMUM","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:DF45RMUMNRRLTFPYFMBWYOTA7O","target":"record","payload":{"canonical_record":{"source":{"id":"1401.7402","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-29T03:23:53Z","cross_cats_sorted":[],"title_canon_sha256":"a4f8dc77bfa653b3625d45fd81733ccd176b379688a27036b43679d9469f471a","abstract_canon_sha256":"72b07e8046843a1d1eeb639d450b0b0924c047114de8c1c33d54bae4bbf71c74"},"schema_version":"1.0"},"canonical_sha256":"1979d8b28c6c62b995f82b036c3a60fb85d9566c82066b660e3efe45a0bbf6fe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:48.601193Z","signature_b64":"hN0QIXo1ukPizrzOTLHvn9QPX/JLlMHpgKJKMpNJDFlS15Zd2SjYs1vo0bPscrhrWoYG64Knh6lmiW26C8j+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1979d8b28c6c62b995f82b036c3a60fb85d9566c82066b660e3efe45a0bbf6fe","last_reissued_at":"2026-05-18T03:00:48.600272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:48.600272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.7402","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:00:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2krdZWMIPEdpDJBvsA1QiDusJ+id238+6VMtH3KD91BW/28a95S/H7znXFaKieroHIUXMFvkPUgyvZKsJ8w/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T15:49:06.783566Z"},"content_sha256":"9f88f91f64924a4525cc6dab8eb193e8b2010d411606703ecb7f3ee04dd61c8f","schema_version":"1.0","event_id":"sha256:9f88f91f64924a4525cc6dab8eb193e8b2010d411606703ecb7f3ee04dd61c8f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:DF45RMUMNRRLTFPYFMBWYOTA7O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Liouville Theorem for the Fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan","submitted_at":"2014-01-29T03:23:53Z","abstract_excerpt":"We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove\n  Every $\\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7402","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":""},"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":"2026-05-18T03:00:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bT0XWxQeJdnb7KiL+Bv0TuRYnXILR0/c1O/MgxXjA4KuDf2Ufv16fkyKP/v6lvlr4GiO/hS554PFIAhOKgT7Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T15:49:06.783894Z"},"content_sha256":"d881ed96438efb806f71fc1d367656261b50f1bed9f74e5d89ca681b5d5b44a0","schema_version":"1.0","event_id":"sha256:d881ed96438efb806f71fc1d367656261b50f1bed9f74e5d89ca681b5d5b44a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DF45RMUMNRRLTFPYFMBWYOTA7O/bundle.json","state_url":"https://pith.science/pith/DF45RMUMNRRLTFPYFMBWYOTA7O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DF45RMUMNRRLTFPYFMBWYOTA7O/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":"2026-07-06T15:49:06Z","links":{"resolver":"https://pith.science/pith/DF45RMUMNRRLTFPYFMBWYOTA7O","bundle":"https://pith.science/pith/DF45RMUMNRRLTFPYFMBWYOTA7O/bundle.json","state":"https://pith.science/pith/DF45RMUMNRRLTFPYFMBWYOTA7O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DF45RMUMNRRLTFPYFMBWYOTA7O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:DF45RMUMNRRLTFPYFMBWYOTA7O","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":"72b07e8046843a1d1eeb639d450b0b0924c047114de8c1c33d54bae4bbf71c74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-29T03:23:53Z","title_canon_sha256":"a4f8dc77bfa653b3625d45fd81733ccd176b379688a27036b43679d9469f471a"},"schema_version":"1.0","source":{"id":"1401.7402","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.7402","created_at":"2026-05-18T03:00:48Z"},{"alias_kind":"arxiv_version","alias_value":"1401.7402v1","created_at":"2026-05-18T03:00:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.7402","created_at":"2026-05-18T03:00:48Z"},{"alias_kind":"pith_short_12","alias_value":"DF45RMUMNRRL","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"DF45RMUMNRRLTFPY","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"DF45RMUM","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:d881ed96438efb806f71fc1d367656261b50f1bed9f74e5d89ca681b5d5b44a0","target":"graph","created_at":"2026-05-18T03:00:48Z","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":"We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove\n  Every $\\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant.","authors_text":"Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-29T03:23:53Z","title":"A Liouville Theorem for the Fractional Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7402","kind":"arxiv","version":1},"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":"sha256:9f88f91f64924a4525cc6dab8eb193e8b2010d411606703ecb7f3ee04dd61c8f","target":"record","created_at":"2026-05-18T03:00:48Z","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":"72b07e8046843a1d1eeb639d450b0b0924c047114de8c1c33d54bae4bbf71c74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-29T03:23:53Z","title_canon_sha256":"a4f8dc77bfa653b3625d45fd81733ccd176b379688a27036b43679d9469f471a"},"schema_version":"1.0","source":{"id":"1401.7402","kind":"arxiv","version":1}},"canonical_sha256":"1979d8b28c6c62b995f82b036c3a60fb85d9566c82066b660e3efe45a0bbf6fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1979d8b28c6c62b995f82b036c3a60fb85d9566c82066b660e3efe45a0bbf6fe","first_computed_at":"2026-05-18T03:00:48.600272Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:00:48.600272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hN0QIXo1ukPizrzOTLHvn9QPX/JLlMHpgKJKMpNJDFlS15Zd2SjYs1vo0bPscrhrWoYG64Knh6lmiW26C8j+Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:00:48.601193Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.7402","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f88f91f64924a4525cc6dab8eb193e8b2010d411606703ecb7f3ee04dd61c8f","sha256:d881ed96438efb806f71fc1d367656261b50f1bed9f74e5d89ca681b5d5b44a0"],"state_sha256":"e5a48491a0ca0ae08171c17acb737b73efc2f71e457b9dcabdd6cbaabf9f7db9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Yd+086AlcNhfpwbyWRbYh+zLmy8WgbnOjfXYzwjbYgMqLmvwIJSjpt0EaC8HDyVIUmrepKNmGT1a0Lw/FsrCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T15:49:06.785753Z","bundle_sha256":"60187465ec53bb09b3df6203fcc34fac85d03f85b19e4523b7ae1a8c04336128"}}