  We recently proved threshold saturation for spatially coupled sparse
superposition codes on the additive white Gaussian noise channel. Here we
generalize our analysis to a much broader setting. We show for any memoryless
channel that spatial coupling allows generalized approximate message-passing
(GAMP) decoding to reach the potential (or Bayes optimal) threshold of the code
ensemble. Moreover in the large input alphabet size limit: i) the GAMP
algorithmic threshold of the underlying (or uncoupled) code ensemble is simply
expressed as a Fisher information; ii) the potential threshold tends to
Shannon's capacity. Although we focus on coding for sake of coherence with our
previous results, the framework and methods are very general and hold for a
wide class of generalized estimation problems with random linear mixing.
