  The Restricted Planar Circular 3-Body Problem models the motion of a body of
negligible mass under the gravitational influence of two massive bodies, called
the primaries, which perform circular orbits coplanar with that of the massless
body. In rotating coordinates, it can be modelled by a two degrees of freedom
Hamiltonian system, which has five critical points called the Lagrange points.
Among them, the point L3 is a saddle-center which is collinear with the
primaries and beyond the largest of the two. The papers arXiv:2107.09942 and
arXiv:2107.09941 provide an asymptotic formula for the distance between the one
dimensional stable and unstable manifolds of L3 in a transverse section for
small values of the mass ratio $\mu$. This distance is exponentially small with
respect to $\mu$ and its first order depends on what is usually called a Stokes
constant. The non-vanishing of this constant implies that the distance between
the invariant manifolds at the section is not zero. In this paper, we prove
that the Stokes constant is non-zero. The proof is computer assisted.
