  Poincar\'e's work more than one century ago, or Laskar's numerical
simulations from the 1990's on, have irrevocably impaired the long-held belief
that the Solar System should be stable. But mathematical mechanisms explaining
this instability have remained mysterious. In 1968, Arnold conjectured the
existence of "Arnold diffusion" in celestial mechanics. We prove Arnold's
conjecture in the planetary spatial $4$-body problem as well as in the
corresponding hierarchical problem (where the bodies are increasingly
separated), and show that this diffusion leads, on a long time interval, to
some large-scale instability. Along the diffusive orbits, the mutual
inclination of the two inner planets is close to $\pi/2$, which hints at why
even marginal stability in planetary systems may exist only when inner planets
are not inclined.
  More precisely, consider the normalised angular momentum of the second
planet, obtained by rescaling the angular momentum by the square root of its
semimajor axis and by an adequate mass factor (its direction and norm give the
plane of revolution and the eccentricity of the second planet). It is a vector
of the unit $3$-ball. We show that any finite sequence in this ball may be
realised, up to an arbitrary precision, as a sequence of values of the
normalised angular momentum in the $4$-body problem. For example, the second
planet may flip from prograde nearly horizontal revolutions to retrograde ones.
As a consequence of the proof, the non-recurrent set of any finite-order
secular normal form accumulates on circular motions -- a weak form of a
celebrated conjecture of Herman.

Dynamical Systems