The white wire supplies power from the regulator to the field coil, which is grounded on the other end. With a stock regulator, with the key on and the motor not running, you will see very nearly the same voltage on the white wire and the black wire. A solid state regulator will have some voltage drop, so with 12.6 volts on the black wire, it may read 12.0 volts, or maybe even a little lower, but certainly not down into the Milli-volt range.