Hi I was just to ask something real quick? Is it allowed to speed run the Japanese version on ps1 ?

The US Playstation version is considered the "standard" version. Faster original versions (PC, Saturn) will have loads converted to match. Slower versions (Japanese PS1, PS3) will not, but will still be listed.

I see well I guess I can do the Japanese version even if it’s is the slower version anyways.

Doing on the emulator like bizhawk is not good for me to do :(

Yeah the JP versions are totally fine to run. The main difference is all the bosses have voiced dialog in that version which loses like 30 seconds over the US version (https://megamanrta.fandom.com/wiki/Mega_Man_X4#Mega_Man_X4_vs._Rockman_X4). Also, if Bizhawk won't work for you for whatever reason, a really good alternative is DuckStation if you still want to try to emulate.

Well I tried the bizhawk emulation with using livesplit and it not good. The audio sound quality was messed up while I was using the OBS program. I mean it fine that you use emulation while recording on OBS but using with livesplit with emulate to run the game is not okay due to audio issues.

I don’t know if can try out duckstation if it could run or not, I’m not so sure .

I will say that I was able to use Bizhawk and Livesplit with an autosplitter with no issues. Not sure what was causing your audio issues but do what you've got to do to play the game. Good luck with your runs!