  We prove a conditional theorem on the positivity of the Lyapunov exponent for
a Schr\"odinger cocycle over a skew shift base with a cosine potential and the
golden ratio as frequency. For coupling below 1, which is the threshold for
Herman's subharmonicity trick, we formulate three conditions on the Lyapunov
exponent in a finite but large volume and on the associated large deviation
estimates at that scale. Our main results demonstrate that these finite-size
conditions imply the positivity of the infinite volume Lyapunov exponent. This
paper shows that it is possible to make the techniques developed for the study
of Schr\"odinger operators with deterministic potentials, based on large
deviation estimates and the avalanche principle, effective.
