Ok, so if i understand this correctly, the neutral light only works if the headlight switch is on? Correct?
If thats the case this is easy... The neutral light has its ground/negative lead switched by the neutral switch in the trans. Most likely what happened is someone took it apart up at the handlebars and plugged the wire in the wrong spot. Looking at the manual and the 85 125M i have access to, depending your year you will have a Light Green/Red wire that goes to the neutral switch in the trans and either a Yellow/Red or a Black/Brown for the positive. Its very possible as i have seen it before that they plugged the positive wire for the light into the wrong connector, make sure the colors are matching.
The Neutral indicator does not share anything with the headlight switch, the negative side goes to the switch in the trans and the positive side goes to the output from the ignition switch. Its most likely that someone rewired it or hooked up wrong.
Oh and the neutral light will work with the light switch off and no battery, just have to have the ignition switch turned on and the engine running in that case for it to work.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....