My mistake with respect to the sum being over all time, thank you for clarifying.
No. If a=b, then a+γb=b+γa. The ordering between identical utilities won’t matter for the total sum, and the individual that is currently behind will be prioritised.
While the ordering between identical utilities does not affect the total sum, it does affect the individual valuation. a can be prioritized over b just by the ordering, even though they have identical utility. Unless I am missing something obvious.
I have realized that I am coming off like I don’t understand algebra, which is a result of my failure to communicate. As unlikely as I am making it sound, I understand what you are saying and already knew it.
What I mean is this:
Despite a = b, it could “look like” a < b or b > a if you didn’t have access to the world but only to the (expanded) sum. If you can ask for the difference between the total sum and the sum ignoring a, but not for the actual value of a.
I can’t think of a non-pathological case where this would actually matter, but it seems like a desirable desideratum that a = b will always “look like” a = b regardless of what kind of (sufficiently fine-grained) information that you have.
EDIT : After reading your above comment about willingness to sacrifice elegance, I kind of wish I hadn’t said anything at all, considering my comments are all in the interest of what I would consider elegance. To be sure, I think elegance is a legitimate practical concern, but I wouldn’t have engaged with you initially had I known your view.