{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LAMMMGM52CQEWXELGAO4PEMQOV","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":"f591061b8f09fc7848a46c5cb420d6a5e02acdbb5cacb891bbdc56841be90612","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T17:24:36Z","title_canon_sha256":"eb210e76ba2c013a9f47bfc19970cce8bc602d505642930de3aba064d1dd96fc"},"schema_version":"1.0","source":{"id":"2605.15107","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.15107","created_at":"2026-05-17T21:18:33Z"},{"alias_kind":"arxiv_version","alias_value":"2605.15107v1","created_at":"2026-05-17T21:18:33Z"},{"alias_kind":"pith_short_12","alias_value":"LAMMMGM52CQE","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"LAMMMGM52CQEWXEL","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"LAMMMGM5","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:50c04ec57316b3b24d634c5a59394cb569c267e7b4c87715c6ed662ce079ffff","target":"graph","created_at":"2026-05-17T21:57:19Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We prove a two-variable refinement with an additional parameter a. Our proof relies entirely on q-series combined with the Bailey pairs. The original even identity and the odd identity then follow as corollaries by letting a=1."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the Bailey pair technique applies directly to this specific two-color series and its even/odd parts without requiring modular completions or additional verification steps."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Andrews and El Bachraoui prove a two-variable generalization of the Hecke sum identity for S(q) via Bailey pairs, recovering the even and odd cases as corollaries when a=1."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A two-variable refinement with parameter a of the Hecke sum for two-color partitions is proved using only q-series and Bailey pairs."}],"snapshot_sha256":"e9af26003876451b78ad2e4187dc07a5a71cf6478e4f749d29d794ee8986163e"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"223b73e92b6ccdcbcacd129a338c6c332d6e979b8e9c5c7b3455fd3639592933"},"paper":{"abstract_excerpt":"The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$.\n  Our proof relies entirely on $q$-series combined with the Bailey pairs The original","authors_text":"George E. Andrews, Mohamed El Bachraoui","cross_cats":[],"headline":"A two-variable refinement with parameter a of the Hecke sum for two-color partitions is proved using only q-series and Bailey pairs.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T17:24:36Z","title":"Solutions for Hecke Sum Questions of Banerjee and Bringmann"},"references":{"count":7,"internal_anchors":1,"resolved_work":7,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"G. E. Andrews,The Theory of Partitions, Addison-Wesley, Reading, MA, 1976; reprinted by Cambridge Uni- versity Press, Cambridge, 1998","work_id":"5786e449-ace0-40f6-a77b-ee57e59c8417","year":1976},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"G. E. Andrews,q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, vol. 66, American Ma","work_id":"e43466ef-b44f-4940-9553-d1900aadb7f0","year":1986},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"G. E. Andrews and M. El Bachraoui,Congruences for two-color partitions with odd smallest part, arXiv:2410.14190","work_id":"71faf992-d8f5-45a7-a99c-2eda0ba9eb06","year":null},{"cited_arxiv_id":"2605.10300","doi":"","is_internal_anchor":true,"ref_index":4,"title":"Proof of a conjecture of Andrews and Bachraoui on a Hecke sum","work_id":"b8f55b65-5e02-47b7-9b26-34a7161fdd51","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"G. Gasper and M. Rahman,Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Appli- cations, vol. 96, Cambridge University Press, Cambridge, 2004","work_id":"2d742963-570f-48d3-9644-db9382563a5f","year":2004}],"snapshot_sha256":"98676fb8f4e00d7ce8fa66e9fd5bc29a0d034aa01af73f0d0e0f0901931e0528"},"source":{"id":"2605.15107","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T03:09:16.578618Z","id":"edf704da-c6dd-4696-b726-5b0bca953836","model_set":{"reader":"grok-4.3"},"one_line_summary":"Andrews and El Bachraoui prove a two-variable generalization of the Hecke sum identity for S(q) via Bailey pairs, recovering the even and odd cases as corollaries when a=1.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A two-variable refinement with parameter a of the Hecke sum for two-color partitions is proved using only q-series and Bailey pairs.","strongest_claim":"We prove a two-variable refinement with an additional parameter a. Our proof relies entirely on q-series combined with the Bailey pairs. The original even identity and the odd identity then follow as corollaries by letting a=1.","weakest_assumption":"That the Bailey pair technique applies directly to this specific two-color series and its even/odd parts without requiring modular completions or additional verification steps."}},"verdict_id":"edf704da-c6dd-4696-b726-5b0bca953836"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4d93e7a019eb3382e2bf3aa60dab08ae7f362703de969efd82ab2aab19985449","target":"record","created_at":"2026-05-17T21:18:33Z","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":"f591061b8f09fc7848a46c5cb420d6a5e02acdbb5cacb891bbdc56841be90612","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T17:24:36Z","title_canon_sha256":"eb210e76ba2c013a9f47bfc19970cce8bc602d505642930de3aba064d1dd96fc"},"schema_version":"1.0","source":{"id":"2605.15107","kind":"arxiv","version":1}},"canonical_sha256":"5818c6199dd0a04b5c8b301dc79190757c7eb5741f589403a551e3280297de77","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5818c6199dd0a04b5c8b301dc79190757c7eb5741f589403a551e3280297de77","first_computed_at":"2026-05-17T21:40:25.776754Z","kind":"pith_receipt","last_reissued_at":"2026-05-17T21:57:19.108916Z","receipt_version":"0.2","signature_status":"unsigned_v0"},"source_id":"2605.15107","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d93e7a019eb3382e2bf3aa60dab08ae7f362703de969efd82ab2aab19985449","sha256:50c04ec57316b3b24d634c5a59394cb569c267e7b4c87715c6ed662ce079ffff"],"state_sha256":"bbe98cfadc7187a051c0a19b9dd379911049a21594309754a95f577437c429b4"}