>>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
It occurs to me that you could formalize the requirement by saying that there must be a provably terminating algorithm that produces precisely the decimal representation of the integer. That way, numbers like Graham's number can be submitted despite there being no physical way to represent the number in decimal (in the observable Universe).