Remark: The exact criterion is not a geometric dictionary

Theorem thm:exact concerns one specified isometry $V:C\to R\otimes B$ and recovery of its entire logical matrix algebra. It is not an area law, a Ryu–Takayanagi relation, an entanglement-first-law identity, an island formula, or a bulk–boundary code.

No spacetime dimension, metric, horizon area, Newton constant, boundary conformal field theory, or gravitational equation appears among its hypotheses. Naming $R$ “boundary” and $B$ “bulk” does not add those data.