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}.