...on the rock and roll...
No, matter time is not infinite, the matter time universe is finite and closed...and simple. My matter time universe is a butt simple time pulse of matter with a Fourier transform that is a matter spectrum with a peak for every matter interaction of every object in the universe. There is a bunch of phase stuff as you might imagine as both time and matter have both amplitude and phase and all action is driven by the Schrödinger equation. Action, once again, is simply a change in matter with time for an object.
Since space is a projection of matter time action, whatever your space in space time works just fine in matter time as well. But do realize that when you say the universe is flat, you infer that there is a space and yet we know that GR's model for gravity has deep flaws.
The speed of light has a really simple definition in matter time. It is just the rate of collapse of the universe at this time. The increasing red shift of light from galaxies is not due to red Doppler shift, it is because the universe pulse began with a universe that had all force at zero and projected to 3,000 Mpc Now we are about 81% of that size. In fact, the collapsing universe, these red spectra are actually blue shifted by the velocity of collapse.
Note that the CMB velocity is 99.96% of the speed of light and so our universe is seemingly tied to the speed of light close to its boundary, the CMB. In space time, this is a coincidence and in a billion years or so, the CMB will disappear over the event horizon. In matter time, the rate of change of the universe is the speed of light, collapsing, not expanding.
Given the universe pulse in time, all force increases in sync with its collapse, and so all of the right constants shift together. The fine structure splitting, for example, appears the same in early galaxies as it is today even though force was much smaller in early galaxies.
The scaling between gravity and charge forces is related to the size the universe relative to the size of the atom, but they both due to matter decay.
There is an interesting topology for matter time in that matter seems to have a centered or horn torus shape. Since all force is due to matter exchange, that means that the projection of force in space is centered on action, centers of mass, and not on particle centers.
You mentioned an omega parameter...sounds like de Chardin? Singular axioms like the omega point just don't seem consistent with the universe, where there seems to be a minimum of three axioms for a universe, so I call it a trimal universe. Although you can build a universe out of space, time, and action, I find that matter, time, and action makes a very nice simple set of axioms that then result in space as a convenient construct of our consciousness.