 Solution

Use an abstract data type with two total count functions, $I,O:U\to \{0,\dots ,M\}$, where $U=\{1,\dots ,99999999\}$ and $M=2^{31}-1$. $I(u)$ counts incoming payments and $O(u)$ outgoing payments. Initially $I(u)=O(u)=0$ for every valid user. A user without a stored record is therefore observable as two zero counts; no separate registration operation is necessary.

The explicit bound $M$ matches the source's Int record fields. Increment operations require space in the relevant count, so no integer overflow is permitted. An unbounded mathematical counting abstract data type would instead require an arbitrary-precision representation or an explicit overflow policy. The large user domain does not require allocating a record for every user: a sparse map with a default value can realize these total functions.