On the e53 the temp indicator on the dash is not a thermostat or even close.

If it's at noon is supposedly at closed loop but bad programming will allow noon indication and open loop. (no idea how much fuel I wasted driving open loop for about 4-5 months with stuck open tstat).

If the needles didn't point up you're open loop.

It's possible for a wire to tstat to be a sensor but I think all e53 use map tstat.

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)