Thanks for writing this.
On the finiteness point, I conjecture that “finite dimensional” (|B| is finite) is sufficient for all of my results so far, although some of my proofs actually use “finite” (|S| is finite). The example with real numbers is still finite dimensional, so I don’t expect any problems.
Seems right. I still think it’s funky that X_1 and X_2 are conditionally non-orthogonal even when the range of the variables is unbounded.
Yeah, this is the point that orthogonality is a stronger notion than just all values being mutually compatible. Any x1 and x2 values are mutually compatible, but we don’t call them orthogonal. This is similar to how we don’t want to say that x1 and (the level sets of) x1+x2 are compatible.
The coordinate system has a collection of surgeries, you can take a point and change the x1 value without changing the other values. When you condition on E, that surgery is no longer well defined. However the surgery of only changing the x4 value is still well defined, and the surgery of changing x1 x2 and x3 simultaneously is still well defined (provided you change them to something compatible with E).
We could define a surgery that says that when you increase x1, you decrease x2 by the same amount, but that is a new surgery that we invented, not one that comes from the original coordinate system.