Kochen-Specker and ontology

Classically, you have objects o₁, o₂, .... and each object o_i has properties p_ij. Each property has a value (or at least a magnitude). In quantum mechanics, the situation is different.

If you have an "object" like an electron, it doesn't have a value for a would-be property (Kochen-Specker theorem), even if that property's value is measurable/observable. But you can't have objects that have no properties. Nevertheless, something (at least related to the electron) must exist.

Happily, that something does, in fact, have at least the "property" of obeying Schroedinger's equation (if I interpret Auyang, How is Quantum Field Theory Possible correctly). Apparently, this property is a property 'only up to probability' or 'modulo probability' or whatever you want to call it. I don't know if all such properties are.

I would add that Auyang's property is of a vector in a Hilbert Space H. H is given in the terms/coordinates of a particular measuring apparatus. Thus a third quantum system w (for Wigner) must be able to represent the electron+apparatus system (which generally has more properties than the properties of the electron union the properties of the apparatus) in its terms/coordinates. This is because there is no preferred system, ontologically, among the various quantum systems.

What's the group(-oid?)(??) of transformations specifiable-up-to-probability among the various systems? What kind of network do they form?