Quartic distinct-distance bound on a maximally circular curve

There are absolute positive integers $T,C_0$ with the following property.

Let $G$ be an irreducible and absolutely irreducible real plane polynomial of degree $d$, with real zero set $C$. Assume that $G$ is not a circle factor and is maximally circular: its local order at each of the two complex circular points at infinity is $d/2$. For a finite sample $S\subset C$, write $N=|S|$ and

Then

Both constants are independent of the curve and sample. The conclusion gives the lower bound $D(S)\ge C_0^{-1/3}d^{-4/3}N^{4/3}$ under the stated threshold.