  We prove the long-standing conjecture regarding the existence of infinitely
many high-frequency modulational instability ``isolas" for a Stokes wave in
arbitrary depth $ \mathtt{h} > 0 $, subject to longitudinal perturbations. We
completely describe the spectral bands with non-zero real part away from the
origin of the $L^2(\mathbb{R})$-spectrum of the water waves system linearized
at a Stokes waves of small amplitude $ \epsilon > 0 $. The unstable spectrum is
the union of isolas of elliptical shape, parameterized by integers $
\mathtt{p}\geq 2 $, with semiaxis of size $ |\beta_1^{(\mathtt{p})}
(\mathtt{h})| \epsilon^\mathtt{p}+ O(\epsilon^{\mathtt{p}+1} )$ where
$\beta_1^{( \mathtt{p})} (\mathtt{h})$ is a nonzero analytic function of the
depth $ \mathtt{h} $ that depends on the Taylor coefficients of the Stokes
waves up to order $\mathtt{p}$.
