[NiceCatch]

Ok, I'm going to step through a couple of examples. I've found one combination that works already, which I'll review below.

Say m+n=11 (11 is not a sum of primes)
(m,n)=(5,6),(4,7),(3,8), or (2,9)
Say (m,n)=(5,6) ---> m*n = 30, so (m,n) could equal (5,6), (15,2), (3,10) in X's eyes. However, 3+10=13=11+2 (two primes), so that gets thrown out the window. As does m+n=11, because in such a case, X could believe (m,n) is EITHER (5,6) OR (15,2). Since X KNOWS what the combo is, m*n can't possibly equal 30, therefore (m,n) can't be (5,6).

If you go through all the possibilities for m+n=11, you'll find that both (3,8) and (4,7) have products whose factors only add up to a "k" number (not a sum of two primes) in only one way. This means that Y would see TWO possible pairs that would fit. So m+n therefore cannot equal 11.

Let's try the next smallest "k" number (not a sum of two primes), 17. (m,n) = (2,15), (3,14), (4,13), (5,12), (6,11), (7,10), or (8,9).

Doing the same process, you find that six of the pairs named have products whose factors can add up to more than one "k" number. This obviously discounts them.

Interestingly, the pair (4,13) does not have a product whose factors can add up to more than one "k" number.

4*13=52
52=(4*13) or (2*26), 2+26=28=23+5 (two prime numbers), therefore (m,n) can be (4,13).

So that is one solution. This does not discount the possibility of other solutions. Basically there are two stipulations: m*n must be a number such that there must be a unique set of two of its factors that adds up to the summation of two non-primes. Additionally, m+n must equal some z where the following is true: given all pairs of numbers between 2 and 100 that add up to z, there must be a unique set of two numbers such that any two factors of the product of the two numbers can only add up to the summation of two non-prime numbers in a single way.

Sorry, I know there is some better way to word that. I hope the gist of it comes through, though.