  For a boundary CFT to give a good approximation to the bulk flat-space
S-matrix, a number of conditions need to be satisfied: some of those are
investigated here. In particular, one would like to identify an appropriate set
of approximate asymptotic scattering states, constructed purely via boundary
data. We overview, elaborate, and simplify obstacles encountered with existing
proposals for these. Those corresponding to normalizable wavefunctions undergo
multiple interactions; we contrast this situation with that needed for a
flat-space LSZ treatment. Non-normalizable wavefunctions can have spurious
interactions, due either to power-law tails of wavepackets or to their
non-normalizable behavior, which obscure S-matrix amplitudes we wish to
extract; although in the latter case we show that such gravitational
interactions can be finite, as a result of gravitational red shift. We outline
an illustrative construction of arbitrary normalizable wavepackets from
boundary data, that also yields such spurious interactions. Another set of
non-trivial questions regard the form of unitarity relations for the bulk
S-matrix, and in particular its normalization and multi-particle cuts. These
combined constraints, together with those found earlier on boundary singularity
structure needed for bulk momentum conservation and other physical/analytic
properties, are a non-trivial collection of obstacles to surmount if a
fine-grained S-matrix, as opposed to a coarse-grained construction, is to be
defined purely from boundary data.
