  We study translation invariant stochastic processes on $\mathbb{R}^d$ or
$\mathbb{Z}^d$ whose diffraction spectrum or structure function $S(k)$, i.e.
the Fourier transform of the truncated total pair correlation function,
vanishes on an open set $U$ in the wave space. A key family of such processes
are stealthy hyperuniform point processes, for which the origin $k=0$ is in
$U$; these are of much current physical interest. We show that all such
processes exhibit the following remarkable maximal rigidity : namely, the
configuration outside a bounded region determines, with probability 1, the
exact value (or the exact locations of the points) of the process inside the
region. In particular, such processes are completely determined by their tail.
In the 1D discrete setting (i.e. $\mathbb{Z}$-valued processes on
$\mathbb{Z}$), this can also be seen as a consequence of a recent theorem of
Borichev, Sodin and Weiss; in higher dimensions or in the continuum, such a
phenomenon seems novel. For stealthy hyperuniform point processes, we prove the
Zhang-Stillinger-Torquato conjecture that such processes have bounded holes
(empty regions), with a universal bound that depends inversely on the size of
$U$.
