  We consider effective field theory around classical background geometries
with a gauge theory dual, in the class of LLM geometries. These are dual to
half-BPS states of $\cal{N}=$ 4 SYM. We find that the language of code
subspaces is natural for discussing the set of nearby states, which are built
by acting with effective fields on these backgrounds. This work extends our
previous work by going beyond the strict infinite $N$ limit. We further discuss
how one can extract the topology of the state beyond $N\rightarrow\infty$ and
find that uncertainty and entanglement entropy calculations still provide a
useful tool to do so. Finally, we discuss obstructions to writing down a
globally defined metric operator. We find that the answer depends on the choice
of reference state that one starts with. Therefore there is ambiguity in trying
to write an operator that describes the metric globally.
