Power-law violations of Yau's nodal upper bound

We construct a smooth Riemannian metric on $S^4\times S^1$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than ${\lambda }^{1/2+{ϵ}_0}$ for one fixed ${ϵ}_0>0$. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.