{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:OA2CIUJG75EWRR57KIOCXRWXVE","short_pith_number":"pith:OA2CIUJG","schema_version":"1.0","canonical_sha256":"7034245126ff4968c7bf521c2bc6d7a901036785e169db2cd68af7585c8ac8b7","source":{"kind":"arxiv","id":"2604.11436","version":2},"attestation_state":"computed","paper":{"title":"Fourier-based potential theory without an explicit Green's function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.AP","authors_text":"Fredrik Fryklund","submitted_at":"2026-04-13T13:20:09Z","abstract_excerpt":"Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regul"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2604.11436","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-04-13T13:20:09Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"780e6461e00d1922438e4dbae43c280d08a84f0113f27524adbce5ff196bae2b","abstract_canon_sha256":"6d8584d4c6a63c80290bef354dba7eabe7f26d1f1991ae5875ec7f84cfd77808"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:14:28.742256Z","signature_b64":"9apNhrEMcStdNl6J/QmjlKR8HDg8OJMIubsaleGeG/24/AQN+57eOYOl9ce6o9HAEC3BC9gNRHQNQ7wvq157Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7034245126ff4968c7bf521c2bc6d7a901036785e169db2cd68af7585c8ac8b7","last_reissued_at":"2026-06-23T03:14:28.741808Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:14:28.741808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fourier-based potential theory without an explicit Green's function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.AP","authors_text":"Fredrik Fryklund","submitted_at":"2026-04-13T13:20:09Z","abstract_excerpt":"Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regul"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The parabolic regularization produces a valid decomposition into nonlocal and localized components whose asymptotic expansions remain accurate for small ε, under the assumption that the operator belongs to the class of strongly elliptic systems.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Fourier-symbol-based potential theory with parabolic regularization decomposes solutions and provides asymptotic expansions for volume, single-layer, and double-layer potentials without explicit Green's functions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e455fd53d59ab65552d61f2e38f7494dda6758bb8c815f6c251c0a109a9f3d70"},"source":{"id":"2604.11436","kind":"arxiv","version":2},"verdict":{"id":"4565baef-a8ea-4ffa-8e10-8f1b0f81b267","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:40:56.700720Z","strongest_claim":"we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual.","one_line_summary":"A Fourier-symbol-based potential theory with parabolic regularization decomposes solutions and provides asymptotic expansions for volume, single-layer, and double-layer potentials without explicit Green's functions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The parabolic regularization produces a valid decomposition into nonlocal and localized components whose asymptotic expansions remain accurate for small ε, under the assumption that the operator belongs to the class of strongly elliptic systems.","pith_extraction_headline":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11436/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2604.11436","created_at":"2026-06-23T03:14:28.741865+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.11436v2","created_at":"2026-06-23T03:14:28.741865+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11436","created_at":"2026-06-23T03:14:28.741865+00:00"},{"alias_kind":"pith_short_12","alias_value":"OA2CIUJG75EW","created_at":"2026-06-23T03:14:28.741865+00:00"},{"alias_kind":"pith_short_16","alias_value":"OA2CIUJG75EWRR57","created_at":"2026-06-23T03:14:28.741865+00:00"},{"alias_kind":"pith_short_8","alias_value":"OA2CIUJG","created_at":"2026-06-23T03:14:28.741865+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE","json":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE.json","graph_json":"https://pith.science/api/pith-number/OA2CIUJG75EWRR57KIOCXRWXVE/graph.json","events_json":"https://pith.science/api/pith-number/OA2CIUJG75EWRR57KIOCXRWXVE/events.json","paper":"https://pith.science/paper/OA2CIUJG"},"agent_actions":{"view_html":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE","download_json":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE.json","view_paper":"https://pith.science/paper/OA2CIUJG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.11436&json=true","fetch_graph":"https://pith.science/api/pith-number/OA2CIUJG75EWRR57KIOCXRWXVE/graph.json","fetch_events":"https://pith.science/api/pith-number/OA2CIUJG75EWRR57KIOCXRWXVE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE/action/storage_attestation","attest_author":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE/action/author_attestation","sign_citation":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE/action/citation_signature","submit_replication":"https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE/action/replication_record"}},"created_at":"2026-06-23T03:14:28.741865+00:00","updated_at":"2026-06-23T03:14:28.741865+00:00"}