A Buchsbaum theory for Frobenius closure

We give a partial characterization for when the difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ is independent of the choice of parameter ideal $\mathfrak{q}\subseteq R$ in an excellent equidimensional local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$. Here, $\mathfrak{q}^F$ is the Frobenius closure of $\mathfrak{q}$ and $e(\mathfrak{q})$ denotes the Hilbert--Samuel multiplicity of $\mathfrak{q}$. In addition to ideal-theoretic equivalences, our characterization involves the derived category and is motivated by Schenzel's criterion of the Buchsbaum property as well as similar results of Ma-Quy in the setting of tight closure.