{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:FKR7H6VTATAIXRCXZKGTSFVOUF","short_pith_number":"pith:FKR7H6VT","canonical_record":{"source":{"id":"2506.20121","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-25T04:21:31Z","cross_cats_sorted":[],"title_canon_sha256":"65ec880e84d643b9b8d7bee40c0f0030d7407a48dc94278c72cc35aa16a7224d","abstract_canon_sha256":"ba5c3405e48182411138bb4fb67da1dfa20f5afa79937621ea03085cf9ecc370"},"schema_version":"1.0"},"canonical_sha256":"2aa3f3fab304c08bc457ca8d3916aea160f6d4c3ef2053b96ba0f17cd26426d8","source":{"kind":"arxiv","id":"2506.20121","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.20121","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"arxiv_version","alias_value":"2506.20121v1","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.20121","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"pith_short_12","alias_value":"FKR7H6VTATAI","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"pith_short_16","alias_value":"FKR7H6VTATAIXRCX","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"pith_short_8","alias_value":"FKR7H6VT","created_at":"2026-07-05T11:27:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:FKR7H6VTATAIXRCXZKGTSFVOUF","target":"record","payload":{"canonical_record":{"source":{"id":"2506.20121","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-25T04:21:31Z","cross_cats_sorted":[],"title_canon_sha256":"65ec880e84d643b9b8d7bee40c0f0030d7407a48dc94278c72cc35aa16a7224d","abstract_canon_sha256":"ba5c3405e48182411138bb4fb67da1dfa20f5afa79937621ea03085cf9ecc370"},"schema_version":"1.0"},"canonical_sha256":"2aa3f3fab304c08bc457ca8d3916aea160f6d4c3ef2053b96ba0f17cd26426d8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:27:00.645033Z","signature_b64":"7lwvwCc9qjP98xDG5lnGk1t5qAbUPVnQW9NI8fAfktCypP78qKu6/tnCC717DJ+YWFmSE55ua84y8nAx6Pt3CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2aa3f3fab304c08bc457ca8d3916aea160f6d4c3ef2053b96ba0f17cd26426d8","last_reissued_at":"2026-07-05T11:27:00.644600Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:27:00.644600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.20121","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":"2026-07-05T11:27:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZN3RziCZapp+OJQVWc374oRfOD90/m81giKftP0QEGK0MOyXLe/wQh8fSSWGXj6EAf1DEcqjQfu0ODNMgwvRDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T19:14:04.223855Z"},"content_sha256":"f38bb425211683b16736c58cc8cda550b09c04ba35375cbb31fbc05469b3b10d","schema_version":"1.0","event_id":"sha256:f38bb425211683b16736c58cc8cda550b09c04ba35375cbb31fbc05469b3b10d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:FKR7H6VTATAIXRCXZKGTSFVOUF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Lee","submitted_at":"2025-06-25T04:21:31Z","abstract_excerpt":"Existence of the fundamental solution of the logarithmic Laplacian (in dimensions $d \\geq 3$) was established by Huyuan Chen and Laurent V\\'eron (2024). In this note, we present an alternative approach, based on a modification on the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange and Ehrenpreis. Moreover, we give a variant of the Liouville theorem for the logarithmic Laplacian and give some further clarification regarding a conjecture posed by Chen and V\\'eron regarding the behavior of solutions in dimensions 1 and 2."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.20121","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.20121/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":"2026-07-05T11:27:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qItz8OiNvYk1R7cgicSwmr3daMqZm54EPnpeA4iNRw6D6Ce/qfAuWSZQR8RmMM69pTFyEwpc7evWfgbzWOjwDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T19:14:04.224482Z"},"content_sha256":"f5840c901162a711fabd02847e694b498d6909f30ef654c4af0f71fbb23b67eb","schema_version":"1.0","event_id":"sha256:f5840c901162a711fabd02847e694b498d6909f30ef654c4af0f71fbb23b67eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FKR7H6VTATAIXRCXZKGTSFVOUF/bundle.json","state_url":"https://pith.science/pith/FKR7H6VTATAIXRCXZKGTSFVOUF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FKR7H6VTATAIXRCXZKGTSFVOUF/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":"2026-08-08T19:14:04Z","links":{"resolver":"https://pith.science/pith/FKR7H6VTATAIXRCXZKGTSFVOUF","bundle":"https://pith.science/pith/FKR7H6VTATAIXRCXZKGTSFVOUF/bundle.json","state":"https://pith.science/pith/FKR7H6VTATAIXRCXZKGTSFVOUF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FKR7H6VTATAIXRCXZKGTSFVOUF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:FKR7H6VTATAIXRCXZKGTSFVOUF","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":"ba5c3405e48182411138bb4fb67da1dfa20f5afa79937621ea03085cf9ecc370","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-25T04:21:31Z","title_canon_sha256":"65ec880e84d643b9b8d7bee40c0f0030d7407a48dc94278c72cc35aa16a7224d"},"schema_version":"1.0","source":{"id":"2506.20121","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.20121","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"arxiv_version","alias_value":"2506.20121v1","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.20121","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"pith_short_12","alias_value":"FKR7H6VTATAI","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"pith_short_16","alias_value":"FKR7H6VTATAIXRCX","created_at":"2026-07-05T11:27:00Z"},{"alias_kind":"pith_short_8","alias_value":"FKR7H6VT","created_at":"2026-07-05T11:27:00Z"}],"graph_snapshots":[{"event_id":"sha256:f5840c901162a711fabd02847e694b498d6909f30ef654c4af0f71fbb23b67eb","target":"graph","created_at":"2026-07-05T11:27:00Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.20121/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Existence of the fundamental solution of the logarithmic Laplacian (in dimensions $d \\geq 3$) was established by Huyuan Chen and Laurent V\\'eron (2024). In this note, we present an alternative approach, based on a modification on the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange and Ehrenpreis. Moreover, we give a variant of the Liouville theorem for the logarithmic Laplacian and give some further clarification regarding a conjecture posed by Chen and V\\'eron regarding the behavior of solutions in dimensions 1 and 2.","authors_text":"David Lee","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-25T04:21:31Z","title":"Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.20121","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:f38bb425211683b16736c58cc8cda550b09c04ba35375cbb31fbc05469b3b10d","target":"record","created_at":"2026-07-05T11:27:00Z","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":"ba5c3405e48182411138bb4fb67da1dfa20f5afa79937621ea03085cf9ecc370","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-25T04:21:31Z","title_canon_sha256":"65ec880e84d643b9b8d7bee40c0f0030d7407a48dc94278c72cc35aa16a7224d"},"schema_version":"1.0","source":{"id":"2506.20121","kind":"arxiv","version":1}},"canonical_sha256":"2aa3f3fab304c08bc457ca8d3916aea160f6d4c3ef2053b96ba0f17cd26426d8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2aa3f3fab304c08bc457ca8d3916aea160f6d4c3ef2053b96ba0f17cd26426d8","first_computed_at":"2026-07-05T11:27:00.644600Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:27:00.644600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7lwvwCc9qjP98xDG5lnGk1t5qAbUPVnQW9NI8fAfktCypP78qKu6/tnCC717DJ+YWFmSE55ua84y8nAx6Pt3CQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:27:00.645033Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.20121","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f38bb425211683b16736c58cc8cda550b09c04ba35375cbb31fbc05469b3b10d","sha256:f5840c901162a711fabd02847e694b498d6909f30ef654c4af0f71fbb23b67eb"],"state_sha256":"5d9800ff14e4f3f73fcc05abc212543544d3cd36a83ab54d9a4a49abda67cdf0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7Ox8vwEwesoRhREqCWWvambXQU5DZ/01zS/+ELzPPrGJMNMXycVhGWr6TSv9PY/uLPldKTiV8v3xGnFx+M3aBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T19:14:04.229920Z","bundle_sha256":"403b9170f579dccaca399f62287228cb07ea2e6a7536d07b0c0a7def7dbb8955"}}