Correction terms and the non-orientable slice genus

By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound to the non-orientable slice genus ${\gamma }_4(K)$ in terms of the signature $\sigma (K)$ and the concordance invariants $V_i(\bar{K})$, which strengthens a previous bound given by Batson, and which coincides with Ozsv'ath-Stipsicz-Szab'o’s bound in terms of their $\upsilon$ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable non-orientable genus is sometimes better than the one on ${\gamma }_4(K)$.