Balanced intrinsic cell covers on a maximally circular curve

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 $d/2$. Let $S\subset C$ be a finite sample with $N=|S|\ge 2$.

There is one disjoint split $S=P_1\sqcup P_2$ into nonempty sets, satisfying

Both point sets lie on $C$. The incidence configuration uses $C$ as both carrier curves, each represented by the same irreducible degree-$d$ polynomial $G$. The real curve is neither a line nor a circle, so the line- and circle-specific normalization conditions impose no additional restriction on the split. This split is chosen independently of the scale below.

For every integer $j$ with $24d\le j$ and $jd<N$, there are a finite set of cells $\Omega$, target sets $Q_\omega\subset P_2\times P_2$, label sets $L_\omega\subset P_1\times P_1$, and an exceptional set $B\subset P_2\times P_2$ with the following properties. A label $(a,b)$ is incident with a target $(p,q)$ when $\|a-p\|^2=\|b-q\|^2$.

1. $|B|\le 4d^2N$.
2. Every target outside $B$ belongs to some $Q_\omega$ with $\omega\in\Omega$.
3. Every label incident with a target in $Q_\omega$ belongs to $L_\omega$.
4. $|\Omega|\le 2d^2j^2$.
5. $|Q_\omega|\le 4N^2/(d^2j^2)$ for every $\omega\in\Omega$.
6. $\sum_{\omega\in\Omega}|L_\omega|\le 20d^2jN^2$.