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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 achoice of UV cutoff in the bulk. The bulk entropy terms are matched with afamily of von Neumann factors nested inside the large $N$ von Neumann algebradescribing the bulk effective field theory. These factors are mapped onto oneanother by a family of conditional expectations, which are interpreted as arenormalization group flow for the code subspace. Under this flow, I show thatthe renormalizations of the area term and the bulk entropy term exactlycompensate each other. This result provides a concrete realization of theER=EPR paradigm, as well as an explicit proof of a conjecture due to Susskindand Uglum.

Large $N$ von Neumann algebras and the renormalization of Newton's constant