>>jeremy+(OP)
> Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
Does this mean that the modern mathematician must be able to assemble all the digits of the number's representation (say, in decimal)? Given some formalization of this requirement, there would be a fairly set upper bound on the acceptable integers.
>>baddox+Wf
Determining something exactly does not mean providing its digits. It means uniquely identifying the semantic meaning of an expression. As an analogy, you can know what a word means when it's spoken to you even if you can't spell it. Or you can understand what a program written down on paper computes (and prove that it computes that thing) without knowing the explicit output of a particular input.
>>j2kun+gv
There must be more to it than that. For one thing, it would be necessary to determine which of the submitted numbers is larger, which according to my interpretation requires more than "uniquely identifying the semantic meaning of an expression."