Thanks for your question.
I can certainly understand your frustration, especially since you have been completely honest and have done your best to prove this to him. You've obviously done nothing wrong, and while I respect your efforts to correct this misunderstanding, the problem seems to be his own inability to trust you despite your best efforts. You may have handled everything properly, but if he has trust issues, it's going to continue to be an uphill battle until he starts to recognize that.
Often times a person's insecurities will cause them to behave like this, and if he has been known to lie in the past, it may be his assumption that you would do the same thing, even if that is not the case. While it's understandable that you would make some efforts to try and prove your loyalty, after a certain point, the responsibility is on him to put some trust in you and let this situation go.
There may be a lot of great things about him, but it does sound like trust is going to be an ongoing issue until he is willing to give you the benefit of the doubt. If there was something specific that he could verbalize that would help him to trust you, that would give you a place to start. Otherwise, it's possible that even if you solve this particular issue, that these trust issues will continue manifest itself in a different way.
I definitely wish you the best with all of this, and if there's anything else I can do to help please let me know.