This is an extremely elementary question but I want to make sure I understand this right.

If someone adds two numbers in their mind and tells us the sum, it is impossible to prove mathematically what two numbers they chose (in most cases). The number of distinct ways to add two numbers to form say, an integer $n$, is something like $ceil((N+1)/2)$ ([here](https://math.stackexchange.com/questions/3034937/how-many-ways-can-a-number-be-written-as-a-sum-of-two-non-negative-integers)). This shows us that there is more than one way, for any number from 2 and beyond.

I think I am thinking about a function mapping each pair of elements to its sum, and observing that it is not injective.

I am just trying to understand why. As integers are sort of the conceptual basis for the concept of a group (I think), is the fact that different elements can combine to form the same element a result of "closure", in that, when we look at the Cayley table of a group, we will see each element of the group exactly once? So is it a necessary condition on any group that the function (x, y) -> (x + y) can not be bijective? Or is there any relationship between groups cycles and elements having unique composition?

Maybe I am completely off-track here but I am trying to ask the fundamental question of why addition is not reversible and what kind of related structure does on the other hand have that property. What deep difference in structure or properties do they have, allowing this?
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If you want 4, 1+3 and 2+2 to be separate elements of your collection, then you should also be able to involve these elements in sums separately. so 5+4, 5+(1+3), and 5+(2+2) should all be different elements of your structure as well.

Consider the set of expressions built up out of base elements via binary sums, with the trivial operation of addition being "add parentheses and stick a + sign between". So addition takes 5 and 4 as inputs and outputs 5+4 and addition takes 5 and 1+3 as inputs and outputs 5+(1+3).

This is a reasonable operation, but it's not associative (we view 5+(1+3) and (5+1)+3 as different expressions) and it doesn't have inverses, so it's not a group.

If you want a group, the operation on the one element group is bijective.
Multiplication of prime numbers is reversible, in the sense that any positive integer is expressible as a product of primes in just one way (not counting changing order...).

(The computational difficulty of reversing the operation, that is, of *factoring* versus multiplying, is the basis of the RSA cryptosystem and other things...)