I agree that if A & B are different, P & Q must be different as well, yes.
Yes, I agree. But only partially. If we included a function of the software that did the following, it might still work:
"I'm sorry to report, but your key has resulted in 12 collisions.... wait.... accessing the Basekey information .... unique file found from key."
Now, although there were 12 collisions (where other files started and ended at the exact same place, had the same number of 1's in their files structure, and otherwise fit all the criteria, 11 of them had at least 1 bit different in the initial 64K Bit Key included in the 4K file, so the program was successfully able to weed out the other 11. But let's say instead of that, we get this message:
"From unique key sampling, there are still 2 collisions, what would you like to do?" You access: "save both to disk." Now you have two files on your desktop. You try the first one in your videoplayer and it doesn't work. You try the 2nd, and your movie is playing right before your eyes. You take about an hour to try and see what the 2nd file is ... you try it as PDF, Doc file, Zip file, audio, everything, finally you discover it's a 3DS Max scene file from some guy in Florida who does 3D work for some studio there. Amazing, it's exactly the same size as your file down to the last bit, but it's an actual working file that someone else made. Now you have to wonder: who made the file? That man, or Nature? Nature came first. And it's numbers don't change. Maybe the man discovered the art he is making through accessing Nature? Creepy.
But perhaps this is how we solve the overlapping argument once and for all.
The problem with your "Partial Agree" is that there will be not 12 collisions, there will be infinite number of collisions.