Szemerédi–Trotter theorem for unit circles ($I(P,\mathcal{C})\lesssim |P{|}^{2/3}|\mathcal{C}{|}^{2/3}+|P|+|\mathcal{C}|$)

For distinct equal-radius circles and a finite point set in the real plane, incidences between points and curves satisfy $I(P,\mathcal{C})=O(|P{|}^{2/3}|\mathcal{C}{|}^{2/3}+|P|+|\mathcal{C}|)$. Scaling the plane reduces any common positive radius to one. The constant is universal. The fixed-radius condition bounds the number of circles through two prescribed distinct points by two.