  We prove that all Galerkin truncations of the 2d stochastic Navier-Stokes
equations in vorticity form on any rectangular torus subjected to hypoelliptic,
additive stochastic forcing are chaotic at sufficiently small viscosity,
provided the frequency truncation satisfies $N\geq 392$. By "chaotic" we mean
having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic
exponential growth of the derivative with respect to generic initial
conditions. A sufficient condition for such results was derived in previous
joint work with Alex Blumenthal which reduces the question to the
non-degeneracy of a matrix Lie algebra implying H\"ormander's condition for the
Markov process lifted to the sphere bundle (projective hypoellipticity). The
purpose of this work is to reformulate this condition to be more amenable for
Galerkin truncations of PDEs and then to verify this condition using a) a
reduction to genericity properties of a diagonal sub-algebra inspired by the
root space decomposition of semi-simple Lie algebras and b) computational
algebraic geometry executed by Maple in exact rational arithmetic. Note that
even though we use a computer assisted proof, the result is valid for all
aspect ratios and all sufficiently high dimensional truncations; in fact,
certain steps simplify in the formal infinite dimensional limit.
