Characterization of measurements failing the lower-bound condition

Characterize the repeatable measurements in nomic toy theory that fail the inclusion condition β({0} ⊕ V) ⊆ I_pre, particularly by providing a useful characterization of those for which the additional predicted information is not absorbed into the predictively inaccessible subspace.

Background

For a repeatable measurement, the paper constructs a map from the predicted-information quotient to the retrodicted-information quotient. When β({0} ⊕ V) is contained in I_pre, this map is injective, so the two quotient descriptions are isomorphic.

The authors note that the inclusion holds for many of their principal measurement constructions but fails for the predictive interaction M^pred. They do not provide a useful characterization of the exceptional measurements.