Proof that BF3 exists?
Boron trifluoride, as described in Wikipedia is proved to exist the way everything in chemistry is ultimately proved, by experiment.  The octet rule is really only a model that describes the electronic structure of some but not all molecules.  Boron trifluoride is one of those (many, actually) molecules for which the octet rule is not really correct.  The actual electronic structure is not just a single Lewis structure but a combination of multiple contributing structures, some of which do satisfy the octet rule and some of which don't.