[tel]

Didn't read in enough detail to pick out the use of K.

My understanding is that the separation comes from an internal/external dichotomy. Internally we only have our observations as support for the metric, externally (and for a nicer mathematical construction) we need more structure including explicit construction of the manifold. This feels plausible externally, but involves some amount of arbitrary choice which is what is flexed to introduce the paradox.

To be honest, I'm not much of a physicist, but this idea shows up all over in mathematics.