  Correlation functions of adiabatic modes in cosmology are constrained by an
infinite number of consistency relations, which relate N+1-point correlation
functions with a soft-momentum scalar or tensor mode to a symmetry
transformation on N-point correlation functions of hard-momentum modes. They
constrain, at each order n, the q^n behavior of the soft limits. In this paper
we show that all consistency relations derive from a single, master identity,
which follows from the Slavnov-Taylor identity for spatial diffeomorphisms.
This master identity is valid at any value of q and therefore goes beyond the
soft limit. By differentiating it n times with respect to the soft momentum, we
recover the consistency relations at each q order. Our approach underscores the
role of spatial diffeomorphism invariance at the root of cosmological
consistency relations. It also offers new insights on the necessary conditions
for their validity: a physical contribution to the vertex functional must
satisfy certain analyticity properties in the soft limit in order for the
consistency relations to hold. For standard inflationary models, this is
equivalent to requiring that mode functions have constant growing-mode
solutions. For more exotic models in which modes do not "freeze" in the usual
sense, the analyticity requirement offers an unambiguous criterion.
