Downgrading a PSP from 4.05 carries a high risk of permanent bricking and often involves tools violating Sony's terms. Here's why you should install Custom Firmware (CFW) instead.
No downgrade needed. Firmware 4.05 is an excellent base for modern CFW:
Update to 6.61 (latest official) via System Update → Network Update. You'll lose homebrew but gain maximum stability and PSN access (though PSP PSN store is closed).
Bottom line: Install CFW on 4.05 instead of downgrading. It's safer, reversible, and unlocks far more functionality.