  The relationship according to which one physical theory encompasses the
domain of empirical validity of another is widely known as "reduction." Here it
is argued that one popular methodology for showing that one theory reduces to
another, associated with the so-called "Bronstein cube" of physical theories,
rests on an over-simplified characterization of the type of mathematical
relationship between theories that typically underpins reduction. An
alternative methodology, based on a certain simple geometrical relationship
between dis- tinct state space models of the same physical system, is then
described and illustrated with examples. Within this approach, it is shown how
and under what conditions inter-model reductions involving distinct model pairs
can be composed or chained together to yield a direct reduction between
theoretically remote descriptions of the same system. Building on this
analysis, we consider cases in which a single reduction between two models may
be effected via distinct composite reductions differing in their intermediate
layer of description, and motivate a set of formal consistency requirements on
the mappings between model state spaces and on the subsets of the model state
spaces that characterize such reductions. These constraints are explicitly
shown to hold in the reduction of a non-relativistic classical model to a model
of relativistic quantum mechanics, which may be effected via distinct composite
reductions in which the intermediate layer of description is either a model of
non-relativistic quantum mechanics or of relativistic classical mechanics. Some
brief speculations are offered as to whether and how this sort of consistency
requirement between distinct composite reductions might serve to constrain the
relationship that any unification of the Standard Model with general relativity
must bear to these theories.
