Stable systolic inequalities via mod n covering

We introduce a mod $n$ covering based approach to stable systolic inequalities. The idea is to prescribe a cohomology class mod $n$ which forces the desired cup product or index to be nonzero, and then find a short integral lift of that class. The method is especially effective in rank two as we can compute the covering constant. As a curvature free application, we improve the stable two systolic bound for $S^2\times S^2$ to $2$. The same bound holds for every oriented four manifold with $b_2=2$. Under a positive scalar curvature lower bound, the mod $n$ covering method combined with a sharp cowaist inequality for line bundles gives stable two systolic bounds. This gives the sharp stable two systolic inequality for odd complex projective spaces and an $O(m\log m)$ bound for $(S^2)^m$ when scalar curvature is at least $2m$. For $S^2\times S^2$ one gets that every metric with scalar curvature at least $4$ has stable two systole at most $8\pi$.