How does strong NP-completeness agree with encoding complexity?
I figured it out after looking at this paper. My error in thinking came from generally assuming some direct (i.e. linear) relationship between the numbers in the source problem and those in the target problem to which the reduction is mapping. However, producing unary output in the reduction in polytime requires that such a connection must only exist in those cases where the source problem is a number problem which has its numerical values polynomially bound in its binary input length. If, on the other hand, the source problem is allowed to have exponentially large numbers, the #bits required to encode the numbers in the target problem may only scale logarithmically with the #bits required for the numbers in the source problem, ensuring polynomial boundedness of the maximum value and enabling changing from binary to unary encoding in polytime.