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If any nonnegative integer can be written in the form $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$, where $a,b,c,d$ are positive integers, then we call $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$ a universal quaternary sum. In this paper, we determine the universality of many quaternary sums when $r,s,t,u \\in \\{3,4,5,8\\}$, using the theory of Ramanujan's theta function identities"},"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":"2507.13645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-07-18T04:15:40Z","cross_cats_sorted":[],"title_canon_sha256":"ba96dcaa4f1c2ada52641423394be7226700dab9ed9f6cc1a112a971c50f03df","abstract_canon_sha256":"a04cf362b3ade1460d0966067717dcb4d987178d4a4b200305f0aef0a2194418"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:39:20.613732Z","signature_b64":"JVID7+fVKv3y/ahnCyUCAFgNij13cjGCqG4P9/wkwfq+xaUuWDyRBX+oNdHuuiq2ONmzOJzeUQ+weLMaWaKoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a075aec29d6af55497a9459ce544f7559f701c732e52e546f3e56db46cf63ce6","last_reissued_at":"2026-07-05T11:39:20.613294Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:39:20.613294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"A. 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