** Authors: **

Maxime Murray and J.D. Mireles James

** Abstract:**We develop computer assisted arguments for proving the existence of transverse
homoclinic connecting orbits, and apply these arguments for a number of non-perturbative parameter
and energy values in the spatial equilateral circular restricted four body problem. The
idea is to formulate the desired connecting orbits as solutions of certain two point boundary
value problems for orbit segments which originate and terminate on the local stable/unstable
manifolds attached to a periodic orbit. These boundary value problems are studied via a
Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of
rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the
problem is controlling the boundary conditions, which must lie on the local stable/unstable
manifolds of the periodic orbit. For this portion of the problem we use a parameterization
method to develop Fourier-Taylor approximations equipped with a-posteriori error bounds.
This requires validated computation of a finite number of Fourier-Taylor coefficients via
Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences
of Fourier coefficients, followed by a fixed point argument to bound the tail terms
of the Taylor expansion. Transversality follows as a consequence of the Newton-Kantorovich
argument.

(Submitted).