However, I didn't give up. I was eventually able to convince them to give me proof by being extremely annoying and repeatedly e-mailing the CEO and others in the company until they relented.
The "proof" they had was one of the most ridiculous misunderstandings of how the web works that I've ever seen; it's as if their click fraud department was run by someone who had no computer skills beyond using Excel. I wrote a comprehensive document explaining what happened in the scenario they saw, why it happened, how to prevent it, and admonished them for stonewalling me on something that was so clearly not my fault and not an instance of click fraud.
As a result, they reinstated me and paid my earnings, but I feel terrible for what I'm sure were hundreds of other people who were wrongly terminated and weren't able to get a response from them like I did.