  I derive a family of Ryu--Takayanagi formulae that are valid in the large $N$
limit of holographic quantum error-correcting codes, and parameterized by a
choice of UV cutoff in the bulk. The bulk entropy terms are matched with a
family of von Neumann factors nested inside the large $N$ von Neumann algebra
describing the bulk effective field theory. These factors are mapped onto one
another by a family of conditional expectations, which are interpreted as a
renormalization group flow for the code subspace. Under this flow, I show that
the renormalizations of the area term and the bulk entropy term exactly
compensate each other. This result provides a concrete realization of the
ER=EPR paradigm, as well as an explicit proof of a conjecture due to Susskind
and Uglum.
