  This study establishes the existence of inertial manifolds for the
hyperviscous Navier-Stokes equations (HNSE) on a 2D periodic domain:
\begin{equation*} \partial_t u+ \nu(-\Delta) ^{\beta}u+(u\cdot \nabla )u+\nabla
p=f, \;\; \text{on} \;\; \mathbb{T}^2, \end{equation*} with $\nabla \cdot u=0$,
for any $\beta > \frac{17}{12} $. The exponent $\beta = \frac{3}{2}$ is
identified as the "critical" value for the inertial manifold problem in 2D
HNSE, below which the spectral gap condition is not satisfied. A breakthrough
in this work is that it extends the theory to "supercritical" regimes where
$\beta < \frac{3}{2}$. An important aspect of our argument involves a refined
analysis on the sparse distribution of lattice points in annular regions.
