  The dynamics of many important high-dimensional dynamical systems are both
chaotic and complex, meaning that strong reducing hypotheses are required to
understand the dynamics. The highly influential chaotic hypothesis of
Gallavotti and Cohen states that the large-scale dynamics of high-dimensional
systems are effectively uniformly hyperbolic, which implies many felicitous
statistical properties. We obtain direct and reliable numerical evidence,
contrary to the chaotic hypothesis, of the existence of non-hyperbolic
large-scale dynamical structures in a mean-field coupled system. To do this we
reduce the system to its thermodynamic limit, which we approximate numerically
with a Chebyshev basis transfer operator discretisation. This enables us to
obtain a high precision estimate of a homoclinic tangency, implying a failure
of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under
perturbation. As a result, the chaotic hypothesis should not be {\it a priori}
assumed to hold in all systems, and a better understanding of the domain of its
validity is required.
