  In the framework of ontological models, the inherently nonclassical features
of quantum theory always seem to involve properties that are fine tuned, i.e.
properties that hold at the operational level but break at the ontological
level. Their appearance at the operational level is due to unexplained special
choices of the ontological parameters, which is what we mean by a fine tuning.
Famous examples of such features are contextuality and nonlocality. In this
article, we develop a theory-independent mathematical framework for
characterizing operational fine tunings. These are distinct from causal fine
tunings -- already introduced by Wood and Spekkens in [NJP,17 033002(2015)] --
as the definition of an operational fine tuning does not involve any
assumptions about the underlying causal structure. We show how known examples
of operational fine tunings, such as Spekkens' generalized contextuality,
violation of parameter independence in Bell experiment, and ontological time
asymmetry, fit into our framework. We discuss the possibility of finding new
fine tunings and we use the framework to shed new light on the relation between
nonlocality and generalized contextuality. Although nonlocality has often been
argued to be a form of contextuality, this is only true when nonlocality
consists of a violation of parameter independence. We formulate our framework
also in the language of category theory using the concept of functors.
