... no need to worry about absolute values for this purpose.

Yes, that's the point I was getting at. I also wanted to hint that you don't even need to be concerned with the numbers themselves, but only with the least-significant bit of each number (in 2s'-complement binary representation). The LS bit determines the odd/even status of a number, so adding up all these bits mod 2 (i.e., masking with 1) gives the answer.

Please forgive me if I have not expressed this in a correct formal way ...

No worries. I doubt if I would recognize a correct formal expression of this proposition if you hit me in the face with it.