Thank you for all of your responses. All of this is very helpful. It seems like if raising the temperature about 120 degrees above room temperature in hot process could reduce cure time from four weeks to about an hour, then raising the temperature about thirty degrees could maybe reduce cure time from four weeks to two weeks or three weeks.
I think you are confusing 'cure' with 'saponification'. They are two different processes.
In saponification (which simply means to turn into soap), the oils and lye go through their chemical dance to become tongue-neutral/zapless soap. As they do their dance, heat is generated. Sometimes the heat is enough to make the batter go into the gel-stage, which will speed up the process of the lye and oils turning into soap, but sometimes not enough heat is generated and the saponification process will proceed slower. Adding heat, such as applying heat in CPOP or during HP, forces the soap batter to go into the gel-stage, which will cause the lye and oils to become tongue-neutral/zapless soap all the more quicker.
The main point of covering molds or CPOPing or HPing is to ensure that the soap goes fully through the gel stage to complete the saponification process (instead of going through partial gel, which causes unevenly colored soap). It has nothing to do with 'cure'.
Once you have tongue-neutral/zapless soap, saponification is complete- in other words, the lye has done its job of converting the oils into soap.
Now, while you can certainly use your soap the moment it becomes tongue-neutral, it will not last as long or be as good as it will be weeks later. It is still in it's infancy stage and has not reached the best it can possibly be in terms of hardness, longevity, mildness and lathering abilities. It needs time to bring those qualities to full maturity....
That's where 'cure' comes in. Heat will help to evaporate water off, sure, but it will not help your soap to become as mild or as bubbly as what you formulated it to be. Only time and exposure to air will bring such things to fruition. As much as we'd like our soap to become adults quickly, the process just doesn't work that way, or else we'd all have dehydrators. lol
Anyway, I want to shorten the time before I can use my soaps; there is so much I want to learn, and I don't want it to take so long to learn it. I know patience is a virtue but gee whiz... four weeks, six weeks, eight weeks is a long time to me. If small increases in temperature don't cause the curing process to become more even or improve it in some way, then why do recipes usually say to put a towel around the soap for the first twenty-four hours?
Although I can empathize (waiting is hard for all of us!), it looks like you may have picked the wrong hobby then!
If you want soap that's ready in a day, melt and pour may be more to your liking.