  We study the constraints on HST models of AdS space-time. The causal diamonds
of HST along time-like geodesics of AdS space-time, fit nicely into the FRW
patch of AdS space. The coordinate singularity of the FRW patch is identified
with the proper time at which the Hilbert space of the causal diamond becomes
infinite dimensional. For diamonds much smaller than the AdS radius, RAdS, the
time dependent Hamiltonians of HST are the same as those used to describe
similar diamonds in Minkowski space. In particular, they are invariant under
the fuzzy analog of volume preserving diffeomorphisms of the holographic
screen, which leads to fast scrambling of perturbations on the horizon of a
black hole of size smaller than RAdS. We argue that, in order to take a limit
of this system which converges to a CFT, one must choose Hamiltonians, in a
range of proper times of order RAdS, which break this invariance, and become
local in a particular choice of basis for the variables. We show that,
beginning with flat, sub-RAdS, patches of dimension D, the resulting CFT,
constructed from the variables of HST, is inconsistent with the entropy of
large black holes, unless one has at least two compact dimensions, whose size
is of order RAdS. The argument is connected to a new observation about the
scrambling rate of information localized on the compact dimensions. Our
construction explains why large AdS black holes do not have the fast scrambling
property. Our present approach cannot deal with models where string theory is
weakly coupled and RAdS is of order the string scale, because the relationship
between area and entropy is non-universal in such models. On spatial length
scales longer than RAdS, our mapping of HST variables into CFT shares much with
the Tensor Network Renormalization Group (TNRG)[1] and is a sort of evolving
error correcting code[2].
