Constant,

I agree with you that systems which are not totally constrained will show a variety of outcomes and that the relative frequencies of the outcomes are a function of the physics of the system. I'm not sure I'd agree that the relative frequencies can be derived solely from the geometry of the system in the same way as distance, etc. The critical factor missing from your exposition is the measure on the relative frequencies of the initial conditions.

In the case of the coin toss, we can say that if we positively, absolutely know that the measure on the relative frequencies of the initial conditions is insufficiently sharp, then thanks to the geometry of the system, we can make some reasonable approximations which imply that the relative frequency of the outcomes will be very close to equal.

The prediction of equal frequencies is critically founded on a *state of information*, not a state of the world. It's objective only in the sense that anyone with the same state of information must make the same prediction.

Relative frequency really is a different type of thing than probability, and it's unfortunate that people want to use the same name for these two different things just because they both happen to obey Kolmogorov's axioms.