Auxiliary incidence bound at every scale

There is an absolute positive integer $K$ 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 $d/2$. Let $S\subset C$ be finite and put $N=|S|$.

Take any incidence configuration whose two carrier sets are both $C$, each represented by an irreducible real polynomial of degree at most $d$. Its finite point sets $P_1,P_2$ are subsets of $C$ and form a disjoint split $S=P_1\sqcup P_2$. The carrier is neither a line nor a circle: in particular neither copy is a vertical line, the pair is neither parallel lines, perpendicular lines, nor concentric circles, and the conditional restrictions on line or circle carriers hold automatically. Assume

Let $I$ count ordered pairs of a label $(a,b)\in P_1\times P_1$ and a target $(p,q)\in P_2\times P_2$ satisfying $\|a-p\|^2=\|b-q\|^2$. For every integer $j\ge24d$,

The constant $K$ is independent of the polynomial, sample, incidence configuration, and scale. The balanced cell-cover construction supplies such a configuration for every sample with $N\ge2$.