K-area bounds for smooth sphere bundles

Let $E\to B$ be a smooth $S^n$ bundle with oriented Riemannian total space $(E,g)$, where $n\ge 2$. We prove K-area$(E,g)\le 4\pi n(n-1)\mathrm{/}\sigma$ whenever every fibre has intrinsic scalar curvature at least $\sigma >0$, independently of the dimension of $B$. In particular, unit round fibres give K-area$(E,g)\le 4\pi$. For arbitrary metrics on two-sphere fibres, K-area is bounded by the supremum of their areas. We also obtain stronger bounds from fibrewise scalar spectra.