Agreed.
(And you are permitted to infer that some form of documentation exists).
Most certainly an acknowledgement of an allegation can exist without any documentation of said allegation, so no, you cannot infer that documentation exists based solely on its acknowledgement, unless you are, in this case, considering the reporting of that acknowledgement by the media as the documentation, but I do not believe that is what you were trying to say.
So yes, you can . . . especially when the payments occurred over the space of thirteen years.
And even if Cain left no paper trail, the existence of documentation is irrelevant because he ADMITS doing it. They are no longer alleged payments. That was the comment that brought me to this thread.
That, and the tortured definition of alms-giving.