1998 Mitsubishi Eclipse Alternator

My wife was driving our car three days ago when all of the sudden it just shut off with no warning. We got the car back home by charging the battery a few times and just drove on the battery power. Today I replaced the battery and did some tests to make sure everything was working. When I hooked the v.O.M. (Voltage ohm meter) to the battery I only got a reading of 12.22 with the car running. I then tested on the back of the alternator and got the same exact reading.
The fuse link looks good visually, and all fuses under the hood and inside the car test good, but it is still not charging. There is less than half an inch total play in the belt, but it does squeal for a few seconds when you first start the car. I do have a new belt for the alternator, but it is zero degrees here right now and I have no heated garage. The alternator was replaced in march and has a lifetime warranty, but it is a pain to remove.
The car stays running when you remove the positive cable from the battery. I cannot figure out what else there could be wrong with it, unless there is a break in the charge wire that goes back to the battery, or there is another fusible link somewhere in the charge loop? Please help me figure this out, and thank you for your time.

Would not the dummy light on the dash come on if the voltage regulator went out, or could it just be that it is under charging that the light has not come on? These cars do not have a computer controlled voltage regulator do they? I am also wondering if maybe a relay is supposed to have four poles coming out of the bottom, because the one in my car has three, the other one is pushed up inside right next to a rectangular ceramic white piece.

We got everything working again, and come to find out that the whole time all it needed was to have the belt replaced. Even though it visually looked like it was moving as fast as it should be, the belt was slipping just enough to make the alternator undercharge. But it's fixed now

thank you for all your help

Electrical problem
1997 Mitsubishi Eclipse 4 cyl Front Wheel Drive Automatic 146000 miles

1997 Mitsubishi Eclipse Spyder GST 2.0 turbo, auto, 146000 mi,
Alternator will only start charging if you rev it high enough for the turbo to kick in over 4k RPM, I have searched and found posts and replies regarding this problem but none seem to have much in answers, I have replaced the battery, alternator (used ebay), ignition switch, and fan belt. I put a volt meter on the battery and before starting it's about 12.5 volts after it starts then it drops to 10.5 volts and will stay there at this point the charging sys & brake dash lights come on dimly and the engine starts running rough, then I accelerate to over 4k RPM and the alternator starts charging all the lights will brighten the lights on the dash go back to normal and the engine start running good, the volt meter still only show about 12.5 volts, I also tested output direct from the alternator and a common ground which reads lower than 2 volts then it bumps up to 12.5 volts when charging. I removed the alternator took it to an auto parts house they ran a 3 point check which it passed. This car has been to an auto electric shop and that's why it has replaced battery and alternator, they said that it may be the computer but not for sure so they gave me the car back saying how much more you willing to pay for us to solve this and for me to give it a shot at all the basics they know me and that I am mechanically capable. So my question does anyone have any fixes that have worked so I can give it a try.