{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WP342Q76CWVFS6OBXIPTDKJTCP","short_pith_number":"pith:WP342Q76","schema_version":"1.0","canonical_sha256":"b3f7cd43fe15aa5979c1ba1f31a93313c9f66bf7e25e715901a3aa6b04188526","source":{"kind":"arxiv","id":"2605.17645","version":1},"attestation_state":"computed","paper":{"title":"A Degree-Two Hilbert--P\\'olya Realisation by Causal Riemann-Surface Operators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"J-self-adjoint analytic pencils on square-root Riemann surfaces recover the local Euler factors of an elliptic L-function at Langlands degree two.","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.NT","authors_text":"Kejun Liu","submitted_at":"2026-05-17T20:46:30Z","abstract_excerpt":"We study a local Euler-factor version of the Hilbert--Polya idea at Langlands degree two. The operators are J-self-adjoint analytic pencils whose spectral parameter lives on the Riemann surface of the square-root function. The fractional kernel is the Laplace transform of a causal response, so KK analyticity forces the branch-point cover on which the RSCO pencil lives. For the N=2 canonical case, the spectral curve is the elliptic curve E_0: y^2 = x^3 + 8x (LMFDB 256b2, conductor 256) and the local Euler factors of L(E_0,s) are recovered by an explicit off-shell basepoint in the resolvent. The"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.17645","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-17T20:46:30Z","cross_cats_sorted":["math-ph","math.MP","math.SP"],"title_canon_sha256":"b89c37b10f27a2a61747aa2ffc22ae7d4a103b83598917784cba50f2701b9bb7","abstract_canon_sha256":"7bb45c017cd7b15f98781fac358c02b2c699ce4ef7c3df10fae4a5ae126ac3ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:04:50.343549Z","signature_b64":"qntczbO+7o8EN91YWf4rjEnFC+/N49VHjT1mREwplUZ5Bj5fXQOLEdC33R0lUXjADvvoPYhky8AELgzgFYYVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3f7cd43fe15aa5979c1ba1f31a93313c9f66bf7e25e715901a3aa6b04188526","last_reissued_at":"2026-05-20T00:04:50.342705Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:04:50.342705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Degree-Two Hilbert--P\\'olya Realisation by Causal Riemann-Surface Operators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"J-self-adjoint analytic pencils on square-root Riemann surfaces recover the local Euler factors of an elliptic L-function at Langlands degree two.","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.NT","authors_text":"Kejun Liu","submitted_at":"2026-05-17T20:46:30Z","abstract_excerpt":"We study a local Euler-factor version of the Hilbert--Polya idea at Langlands degree two. The operators are J-self-adjoint analytic pencils whose spectral parameter lives on the Riemann surface of the square-root function. The fractional kernel is the Laplace transform of a causal response, so KK analyticity forces the branch-point cover on which the RSCO pencil lives. For the N=2 canonical case, the spectral curve is the elliptic curve E_0: y^2 = x^3 + 8x (LMFDB 256b2, conductor 256) and the local Euler factors of L(E_0,s) are recovered by an explicit off-shell basepoint in the resolvent. The"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For the N=2 canonical case, the spectral curve is the elliptic curve E_0: y^2 = x^3 + 8x (LMFDB 256b2, conductor 256) and the local Euler factors of L(E_0,s) are recovered by an explicit off-shell basepoint in the resolvent. The basepoint is real because the Hasse-Weil bound supplies the needed inequality.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the fractional kernel being the Laplace transform of a causal response, together with the J-self-adjoint analytic pencil condition, forces the branch-point cover on which the RSCO pencil lives and permits recovery of the local Euler factors via the resolvent basepoint (with Hasse-Weil supplying reality).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper gives a local Hilbert-Pólya realization for elliptic L-functions at degree two via causal Riemann-surface operators, recovering local Euler factors for the curve y² = x³ + 8x and extending to a family through quadratic matching.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"J-self-adjoint analytic pencils on square-root Riemann surfaces recover the local Euler factors of an elliptic L-function at Langlands degree two.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"247011304e02d8bd1a5f29f1877f644f4d1f29183987d43d1875d6a9a5489464"},"source":{"id":"2605.17645","kind":"arxiv","version":1},"verdict":{"id":"48af78eb-5b70-45ed-9827-f505e430c7c4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:29:13.891823Z","strongest_claim":"For the N=2 canonical case, the spectral curve is the elliptic curve E_0: y^2 = x^3 + 8x (LMFDB 256b2, conductor 256) and the local Euler factors of L(E_0,s) are recovered by an explicit off-shell basepoint in the resolvent. The basepoint is real because the Hasse-Weil bound supplies the needed inequality.","one_line_summary":"The paper gives a local Hilbert-Pólya realization for elliptic L-functions at degree two via causal Riemann-surface operators, recovering local Euler factors for the curve y² = x³ + 8x and extending to a family through quadratic matching.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the fractional kernel being the Laplace transform of a causal response, together with the J-self-adjoint analytic pencil condition, forces the branch-point cover on which the RSCO pencil lives and permits recovery of the local Euler factors via the resolvent basepoint (with Hasse-Weil supplying reality).","pith_extraction_headline":"J-self-adjoint analytic pencils on square-root Riemann surfaces recover the local Euler factors of an elliptic L-function at Langlands degree two."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17645/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.415850Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:40:53.957356Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T22:22:36.783963Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.550113Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.470414Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5b1c31945a0a13191212f46be60e58ad9dfb6f95c61df986cc6aa718776cc157"},"references":{"count":38,"sample":[{"doi":"","year":null,"title":"Symmetry , volume =","work_id":"08758fe9-3c80-4180-bbee-40fd6e4e152e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Yakaboylu, Enderalp , title =. J. Phys. A: Math. Theor. , volume =","work_id":"71b73d0f-01cd-4fa5-bfe4-9a50de02f628","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Connes, Alain and Moscovici, Henri , title =. Proc. Natl. Acad. Sci. USA , volume =","work_id":"f7793941-fad1-4596-855c-e2b3d7efc018","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"and Snaith, Nina C","work_id":"f4cdcff8-81c4-4326-833b-690546230099","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"and Sarnak, Peter , title =","work_id":"b4373638-e2e5-4852-af83-576c1e3db16a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":38,"snapshot_sha256":"0c7eefdafd690b1019fd642a2e3b07ac4782e0933d4993579fb6285a1b898c09","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"fb6f3b44434355c073d46eae9c1c0bdcf82ea5598f03dbb580307304abe2eb20"},"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":"2605.17645","created_at":"2026-05-20T00:04:50.342851+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.17645v1","created_at":"2026-05-20T00:04:50.342851+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17645","created_at":"2026-05-20T00:04:50.342851+00:00"},{"alias_kind":"pith_short_12","alias_value":"WP342Q76CWVF","created_at":"2026-05-20T00:04:50.342851+00:00"},{"alias_kind":"pith_short_16","alias_value":"WP342Q76CWVFS6OB","created_at":"2026-05-20T00:04:50.342851+00:00"},{"alias_kind":"pith_short_8","alias_value":"WP342Q76","created_at":"2026-05-20T00:04:50.342851+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP","json":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP.json","graph_json":"https://pith.science/api/pith-number/WP342Q76CWVFS6OBXIPTDKJTCP/graph.json","events_json":"https://pith.science/api/pith-number/WP342Q76CWVFS6OBXIPTDKJTCP/events.json","paper":"https://pith.science/paper/WP342Q76"},"agent_actions":{"view_html":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP","download_json":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP.json","view_paper":"https://pith.science/paper/WP342Q76","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.17645&json=true","fetch_graph":"https://pith.science/api/pith-number/WP342Q76CWVFS6OBXIPTDKJTCP/graph.json","fetch_events":"https://pith.science/api/pith-number/WP342Q76CWVFS6OBXIPTDKJTCP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP/action/storage_attestation","attest_author":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP/action/author_attestation","sign_citation":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP/action/citation_signature","submit_replication":"https://pith.science/pith/WP342Q76CWVFS6OBXIPTDKJTCP/action/replication_record"}},"created_at":"2026-05-20T00:04:50.342851+00:00","updated_at":"2026-05-20T00:04:50.342851+00:00"}