The receiver does make the calculations to line up the 4 spheres. Like I said the accuracy of my brothers receiver is 3 meters. Thats what it is. Its not 3 meters plus other error. All error is accounted for.
You seem to be confusing 'theoretical land' with the real world. Theoretically, if you had a perfectly clean, noiseless signal, which contained perfectly clean, noiseless data, obtained from noiseless sensors, theoretically, the GPS unit could calculate your position to three meters due to it's internal calculation and clock errors. However, in the real world, the signal is not clean. You can't just wave your hand and say, "all error is taken into account, it's just three meters." There are at least 12 meters of uncertainty contained within the signal itself.