This is what the BCM looks like on the Cruze, has seven connectors points on it labeled X1 through X7, on ebay for as cheap as 50 bucks, but may have to pay your dealer a hundred bucks to program this. Don't get this, not a darn thing to do with emissions, should be a free download, but just another way to get robbed. Guys on this board had to lay out a hundred bucks just to get their fog lamps to work. This is a darn right dirty trick, shame on you GM.
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Not sure which socket is X1, but B+ or 12 volts enters pin 3 of X2, B+ is an old term referring to vacuum tubes, Vcc is used for low voltage solid state. With parking lights on, pin 6 of X4 should show voltage. Doesn't appear to be very easy to measure.
Circuit diagram shows a point contact relay contacts, now this is a good question, is it a point contact relay or a power MOSFET transistor. Only way to find out is to open it and hand trace the lands. Typically a transistor with an overload will short out so the lights will be on all the time. Where one tiny spec of dirt can keep the contacts from making contact.
So at this point, your guess is as good as mine, but for me, worth looking at before buying a new one, also found cold solder joints on those many connector pins, ha, saved me 1,300 bucks on one vehicles ECU. And more than once.
An old saying is the the best connector is no connector, offers some convenience, but sure introduces new problems, Made In China doesn't help either.
Curious as to how you are making out with your dead tail lamps. If it happened to you, can happen to the rest of us. Can be something as simple as those fuse tabs are spread apart, you never want to rock any of these to pull them out, have to come out straight. Talk about cheap, and not only hardware problems but software problems as well, only need one corrupt byte.
Ha, have a total of 18 marker and parking lamps on my motorhome, the shear volume was the major problem, but for getting 12V, only had to pull out the head lamp switch out one notch. Law doesn't help either, one bulb not working is reason to get stopped and even searched, and you have to be very very nice.
Ha, talk about fighting for freedom, don't have to go the middle east, we should be doing this right here. Called this political electronics.