  Any non-trivial scattering with massless fields in four spacetime dimensions
will generically produce an out-state with memory. Scattering with any massless
fields violates the standard assumption of asymptotic completeness -- that all
"in" and "out" states lie in the standard (zero memory) Fock space -- and
therefore leads to infrared (IR) divergences in the standard $S$-matrix
amplitudes. In this paper we define an infrared finite scattering theory which
assumes only (1) the existence of in/out algebras and (2) that Heisenberg
evolution is an automorphism of these algebras. The resulting "superscattering"
map $\$$ allows for transitions between different in/out memory states and
agrees with the standard $S$-matrix when it is defined. We construct
$\$$-amplitudes by defining (3) a "generalized asymptotic completeness" which
accommodates states with memory in the space of asymptotic states and (4) a
complete basis of improper states which generalize the usual $n$-particle
momentum basis to account for states with memory. Using only general properties
of $\$$, we prove an analog of the Weinberg soft theorems in quantum gravity
and QED which imply that all $\$$-amplitudes are well-defined in the infrared.
We comment on how one must generalize this framework to consider
$\$$-amplitudes for theories with collinear divergences (e.g., massless QED and
Yang-Mills theories).
