A uniqueness theorem for warped $N>16$ Minkowski backgrounds with fluxes

We demonstrate that warped Minkowski space backgrounds, ${\mathbb{R}}^{n-1,1}{\times }_w{M}^{d-n}$, $n\ge 3$, that preserve strictly more than 16 supersymmetries in $d=11$ and type II $d=10$ supergravities and with fields which may not be smooth everywhere are locally isometric to the ${\mathbb{R}}^{d-1,1}$ Minkowski vacuum. In particular, all such flux compactification vacua of these theories have the same local geometry as the maximally supersymmetric vacuum ${\mathbb{R}}^{n-1,1}\times T^{d-n}$.