Motorman,
The smallest spiral screw extractor I could find--and purchase--has a .065 diameter at the tip, on a .127 diameter shank. I don't give that much chance of working in a .112-40 or similar size screw, whether it is Phillips head or hex recess cap. Not enough head depth to drill and engage. No wall left on the screw shank when you go deeper. Even if centered perfectly, if the screw was truly stuck, it is going to snap off. Which isn't really a bad thing. A guy can give it a try and the results are the same as what I included in the earlier list of suggestions: drill the head.
If someone has a source for a smaller spiral extractor, I'd like to buy one. Screws break in the middle of the night when stores are closed. And, the night before you need something to work. At least in Dave's world....
Dave
PS--I've heard of using a reverse twist drill bit on small screws. The combination of the drilling torque plus the vibration, plus the heat might do it. I have not tried this, and I don't think I have any reverse bits, so this is simply speculation.
PPS--I checked on the small screw extractor that I bought. It is an Irwin/Hanson EX-1 (53401) used with a 5/64 drill bit. Irwin also sells the reverse bits in various sizes. Made in the US of A.