Solving equations from a logic puzzle about sumproduct

Question

I have a question that I tried to solve, and I think I partially did it. Now I want to know if I did everything correctly and, if yes, is there any proof regarding this?

Answer 1

As long as the function $f$ allows for arrays with non-integer elements, computing $a$ along with $c_1 = (1,...,1)$ and $c_2 = (\sqrt{2}, \sqrt{3}, \sqrt{5},...)$ should work.

To get to another solution $b$ that satisfies $f(a,c_1) = f(b,c_1)$ and $f(a,c_2) = f(b,c_2)$, we can start with $a$ then repeatedly subtract $1$ from one component and add $1$ to another component. But all of the pairwise differences in $c_2$ are "linearly independent" up to integer scalar multiples, so any sequence of these moves would necessarily change $f(b, c_2)$.