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We prove that $$ \\sqrt{s^+(G)}\\le\\left(1-\\frac{1}{\\omega(G)}\\right)n. $$ This strengthens Wilf's classical spectral Tur\\'{a}n theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin--Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro--Wei greedy algorithm on the complement"},"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":"2607.18044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-20T15:12:53Z","cross_cats_sorted":[],"title_canon_sha256":"4c7732a59f109c4f4f131b98c898897c1a0a86d541551ca331092da0ea64267f","abstract_canon_sha256":"e2129b4845520fdf7e6e06282485269af003c9923a8750fd6936f5cec5b778ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T02:22:13.149386Z","signature_b64":"PUgi0Ll3mbIWBU7MUUC+1x8/iNVvq7JWWIqcexJ1+RW8RpTez/3NG9eSDBD/gRr/AkVUfn6/Pa7U71nlAkK4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab2444171b326dcfe27e2fd2f61bb4fb48b735a2ac38227da248d8135b29aee2","last_reissued_at":"2026-07-21T02:22:13.148512Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T02:22:13.148512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A positive square-energy strengthening of Tur\\'an's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Quanyu Tang, Shengtong Zhang, Yinchen Liu","submitted_at":"2026-07-20T15:12:53Z","abstract_excerpt":"Let $G$ be an $n$-vertex graph with clique number $\\omega(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. 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