A positive square-energy strengthening of Turán's theorem

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. We prove that $$\sqrt{s^{+}(G)}\le (1-\frac{1}{\omega (G)})n\mathrm{.}$$ This strengthens Wilf’s classical spectral Turán 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.