  We use the amplitude soft bootstrap method to explore the space of effective
field theories (EFT) of massless vectors and scalars. It is known that
demanding vanishing soft limits fixes uniquely a special class of EFTs:
non-linear sigma model, scalar Galileon and Born-Infeld theories. Based on the
amplitudes analysis, we conjecture no-go theorems for higher-derivative vector
theories and theories with coupled vectors and scalars. We then allow for more
general soft theorems where the non-trivial part of the soft limit of the
(n+1)-pt amplitude is equal to a linear combination of n-pt amplitudes. We
derive the form of these soft theorems for general power-counting and spins of
particles and use it as an input into the soft bootstrap method in the case of
Galileon power-counting and coupled scalar-vector theories. We show that this
unifies the description of existing Galileon theories and leads us to the
discovery of a new exceptional theory: Special scalar-vector Galileon.
