URGENT tire pressure question please!!

[crosvs]

So as some of you know, I recently acquired a 2005 645 cabrio (with sports package standard 19" rims) (Pix coming soon!!)

It needed new tires though, and I opted out of the expensive and not-so-good OEM Bridgestone run-flats. Instead, I purchased a set of Goodyear Eagle F1 tires.

Thing is though, neither I nor the tire installer were sure what tire pressures to use!! The door jamb and owner's manual specify the ludicrously high PSI settings for run-flats which obviously are inapplicable to the GY Eagles, while the GY Eagles recommend something around 29/31 F/R which is what the tire shop said they set them to.

HOWEVER: just now after driving around for a bit I decided to check the pressures with my cool sharper image color-coded digital tire gague lol. Strangely, I couldn't get it to read the pressure from any tires at first -- couldn't hear any air at all and thus no reading. Finally, however, one of the rear tires gave an instantaneous reading of ... ready for it? 55 PSI!!!!!

The max tire pressure indicated on the sidewall is 51. What's going on here??

And more importantly, how am I to figure out what the optimal tire pressure is for these GY Eagle F1 tires when all the published information for the car pertains solely to the run-flats??

Thanks!!