Geometry requires an additional certificate

There is no universal implication

Proposition prop:dephasing already shows that an exactly available observable can coexist with failure of full-code recovery. Nor does perfect quantum recovery establish classical concentration: identity recovery of a qubit in $|+⟩$ has zero channel error, while the variance of $Z$ is one.

Without Definition def:geometry, the finite and modular data impose no mathematical equations identifying a metric. This is a relative nonidentification statement, not a theorem that arbitrary geometries realize those data.