I think we're talking on slightly different terms. I was thinking of the Hubble radius, which in the limit equates to Open/Flat/Closed iff there is no cosmological constant (Dark energy). This does not seem to be the case. With a cosmological constant, the Hubble radius is relevant because of results on black hole entropy, which would limit the entropy content of a patch of the universe which had a finitely bounded Hubble radius. I was referring to the regression of the boundary as the "expansion of the universe". The two work roughly similarly in cases where there is a cosmological constant.

I have no formal training in cosmology. In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation. In a flat universe, your causal boundary recedes at a constant c, and thus peak entropy in the patch containing your past light cone goes as t^2. It is not clear that there is a finite bound on the whole of a flat spacetime. I agree entirely on your closed/open comments.

Omega could alternatively assert that the majority of the universe is open with a negative cosmological constant, which would be both stable and have the energy in your cosmological horizon unbounded by any constant.

As to attacking the premises; I entirely agree.