The battery light in my car comes on when I hit 4000 rpms and I'm not really sure why its happening just replaced alternator and battery and it was fine then this just started happening
Check the fuse and connections for the PGM-FI and the voltage regulator plug on the back of the alternator. the control module varies the draw from the alternator based on total vehicle speed and rpm. So anything related to the understanding of these two variables could cause an error.
I would bet the belts just not quite tight enough, try tightening it a bit more and try it. High RPM belts slipping a bit but might not sqeel. Seen it before .
Also 13v is ok....It will range from 13-15 depends on loads battery condition all that.