Dude best thing to do for a PSP 1001 on 6.31 OFW, install 6.31 HEN (to enable homebrew) and then use Hellcat's Recovery Flasher 1.65 (do this on only 1000s and fully hackable 2000s) and install 5.50 GEN-D3 CFW Pack (don't forget to rename the 5.50 EBOOT to 550.PBP and place it on the root of your memory stick) Then just run the 5.50 Prometheus patch. Put all folders in PSP/GAME. And who says a PSP 1001 insn't hackable get your facts right.
Got my first PSP (a 1001) last week, it was on 6.31 OFW, followed this guide and had 5.50 Prom-4 with zero problems. thanks!