Unfortunately, downgrading a PSP from firmware 6.37 to 6.31 without using a Pandora Battery or Magic Memory Stick is not possible. Here's why:
Attempting to downgrade without these tools can brick your device (render it permanently unusable). Avoid any methods not thoroughly tested and documented by the PSP hacking community.
Always back up your PSP data before attempting any firmware modifications. Downgrading is challenging—proceed with caution and thorough research.