Counting $K_4$-saturating non-edges in Nosal graphs

A non-edge of a $K_4$-free graph is $K_4$-saturating if adding it creates a copy of $K_4$, and $f(G)$ counts these non-edges. Erdős and Tuza showed that the minimum of $f$ over $n$-vertex $K_4$-free graphs jumps from zero to a quadratic quantity once the number of edges exceeds $\lfloor n^2/4\rfloor$. Balogh and Liu [J. Combin. Theory Ser. B 109 (2014)] determined the asymptotic size of this jump. In this paper, we study the jump phenomenon at Nosal's spectral threshold $λ(G)>\sqrt m$, where $λ(G)$ is the spectral radius and $m$ is the number of edges. This spectral threshold does not involve the order, and its boundary contains unbalanced graphs far below the Mantel threshold. We prove that every $K_4$-free graph $G$ with $λ(G)>\sqrt m$ satisfies $f(G)+ \frac{1}{2}q(G)> \frac{1}{9} λ(G)^2$, where $q(G)$ counts the vertices in triangles. The constant $\frac{1}{9}$ is best possible, and the triangle-free case of this inequality reduces to Nosal's theorem. Consequently, the minimum of $f$ over $K_4$-free graphs with $m$ edges jumps from $0$ to $\frac19m+O(\sqrt m\,)$ at the Nosal threshold, and the error order $\sqrt m$ is best possible. For large $λ(G)$, we also prove $f(G)>\frac19λ(G)^2-\frac13λ(G)$, and graphs on the boundary $λ(G)^2=m$ attain this value. In the extremal graphs, the three edges of every triangle have booksizes in the ratio $4:1:1$, so these graphs are different from those of Balogh and Liu. We also show that the spectral Mantel condition $λ(G)>λ(T_{n,2})$ forces asymptotically fewer $K_4$-saturating non-edges than the condition $e(G)>\lfloor n^2/4\rfloor$. Our proof transfers the Balogh-Liu theorem through a weighted blow-up and replaces the choice of an extremal edge by a zero-sum comparison over all edges of a neighborhood.