  Families of breathers have been found for certain integrable PDEs but are
believed to be rare in non-integrable ones such as nonlinear Klein-Gordon
equations. In this paper we consider small breathers for semilinear
Klein-Gordon equations with analytic odd nonlinearities. A breather with small
amplitude exists only when its temporal frequency is close to be resonant with
the linear Klein-Gordon dispersion relation. Our main result is that, for such
frequencies, we rigorously identify the leading order term in the exponentially
small (with respect to the small amplitude) obstruction to the existence of
small breathers in terms of the so-called Stokes constant which depends on the
nonlinearity analytically, but is independent of the frequency. As a corollary
it proves that, for generic analytic odd nonlinearities, there does not exist
any {\it small} breather of {\it any} temporal frequency, even though this had
been intuitive for any single given frequency due to the dimension counting. In
particular, this gives a rigorous justification of a formal asymptotic argument
by Kruskal and Segur \cite{KS87} in the analysis of small breathers.
  We rely on the spatial dynamics approach where breathers can be seen as
homoclinic orbits. The birth of such small homoclinics is analyzed via a
singular perturbation setting where a Bogdanov-Takens type bifurcation is
coupled to infinitely many rapidly oscillatory directions. The leading order
term of the exponentially small splitting between the stable/unstable invariant
manifolds is obtained through a careful analysis of the analytic continuation
of their parameterizations. This requires the study of another limit equation
in the complexified evolution variable, the so-called \emph{inner equation}.
