Excess decay for minimizing hypercurrents mod 2Q

We consider codimension 1 area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\Sigma$ of Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt}\left(T\right)\setminus {\mathrm{spt}}^p\left(\partial T\right)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in Minter and Wickramasekera (2024) as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $\Sigma$. This improvement is in fact crucial in De Lellis et al., (2022) to prove that the singular set of $T$ can be decomposed into a $C^{1,\alpha }$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.