Sounds like he wants to use this 'proof' as a way to exert power over you. Like a sword of Damocles that he can drop on you when you're not behaving like
his girlfriend should. Bleh!
Where do you stand in this? Are you going to put up with your bf basically not trusting you? Will you allow yourself to be hold hostage over some intangible proof? Would you stay with him if he never revealed his evidence or admitted that it is false or doesn't exist?
(My guess is that all your answers to these questions are 'No!' - and rightly so)