Homological $k$-systole in $n$-manifolds with positive intermediate curvature

In this paper, we prove optimal $k$-systolic inequalities and characterize the case of equality on closed $n$-dimensional Riemannian manifolds with positive intermediate curvature for $3\leq n\leq 7$. This unifies prior works of Bray-Brendle-Neves \cite{BrayBrenleNevesrigidity} and Chu-Lee-Zhu \cite{chuleezhu_n_systole}, and extends them to higher codimensions. The proof is inspired by our recent work on splitting theorems under intermediate curvature \cite{chenhong2026}.