is this range proof i made sound?
If you're using the syntax $&lt;a, b&gt;$ to mean dot-product, then the assumption that you make:

a binary vector is the only vector where $&lt;a,a&gt;=&lt;a,1&gt;$

is incorrect.  It is correct in the ring $\mathbb{Z}$, however we're not in the integers, we're in a finite field.
Counterexamples in finite fields are easy to find; for example, in $GF(11)$ (picked solely because it's large enough to be nontrivial, but small enough for the computation to be easy), we find that for $a = \{ 2, 5 \}$, we have:
$$&lt;a, a&gt; = 2 \cdot 2 + 5 \cdot 5 = 4 + 3 = 7$$
$$&lt;a, 1&gt; = 2 \cdot 1 + 5 \cdot 1 = 2 + 5 = 7$$