Jim...
As you so astutely pointed out on pages 2 and 3 of your treatise on bandgap design, by choosing the multiplier between R1 and R2 to be (epsilon) the math is made terribly simple. And, by choosing R1 appropriately, you can make the current arbitrarily low through the mechanism.
However (and I admit that I've only modeled it in Electronic Workbench for the time being)...
If you use 10K for R1, this makes R2 27K (using 5% values) and R3 1.2K. All very straightforward. The bandgap voltage comes out 1.320 volts (at room temperature).
Doing a temperature sweep from -15 to +40C, the curve is nearly a straight line from 1.332 volts (-15) to 1.317 volts (+40) giving a total delta of 15 millivolts over the range, or about 270 uV per degree C. Doing a monte carlo with 5% parts says that the bandgap will be somewhere between 1.25 and
1.4 volts under worst case conditions. Certainly we can adjust this small deviation out quite easily.
HOWEVER...
If we start diddling (excuse me, heuristically engineering) with the resistor values, just to see what happens, we find that the temperature curve may be made parabolic by an appropriate choice of R2 and R3, without much change in the percentage error of the monte carlo sweep.
For example, leaving R1 at 10K, making R2 56K and R3 2.7K, the bandgap voltage is 1.696 volts, the monte carlo gives us values from 1.6 to 1.8 volts worst-case, but the temperature sweep is a distorted parabola with the end temperatures (high and low) both being roughly equivalent at 1.680 volts, but a peak (inflection point) of the parabola at 1.7 volts at 22°C.
As a matter of fact, I can put the peak of the parabola at any temperature I choose by an appropriate choice for R3. Values of 2K and 3K will put the inflection point at -15 or +40°C respectively.
I can't explain the parabolic function. I've looked your equations over until I could probably recite them from memory and I can't find anything that looks like the equation of a parabola.
What am I missing?
Jim