Either I've misunderstood your response, or you've misunderstood the challenge. The challenge is that the same bound variable need never be repeated. Sure for any given single sentence containing n different variables you can define an equivalent single predicate with n arguments, but you can't do this for each of the infinitely many sentences in a language.
But you _can_! That's how you'd build up sentences if the language's mechanism for composing concepts involves uniting them into larger predicates.
That said, I have internally equated what you desire to a well-studied notion in computer science (which would need adaptation before being moved to natural language). I may have misunderstood, but I think that your notion of eliminating repeated references is equivalent to eliminating explicit references altogether.
Other things you may want to look at: combinatory logic,
concatenative programming (or stack-based programming), point-free
style (or tacit programming), and the J programming language.
Thanks for taking the trouble to make the suggestions, but I don't have the requisite intellectual skills to scan outlines of those areas in the hope of descrying a notation that doesn't require variable repetition. If you think one of them provides a notation that doesn't require variable repetition, it would be very precious to meif you were to explain it to me.
Combinatory logic provides a notation based on S and K (or another one based on B, C, K, and W) for expressing predicates without explicit reference.
Concatenative programming uses the idea of a stack to avoid explicit reference.
The J Programming Language uses ideas from the above to be very terse and linear without explicit variables.
--and.