Inverse Theorems for Point-Sphere Incidences over Finite Fields

Abstract

Let Q be a nondegenerate quadratic form on $\mathbb F_q^d$, with q odd & $d\ge2$. We prove a two-scale inverse theory for point-sphere incidences. At the maximal-incidence endpoint, put $μ=1-I(P,S)/(|P||S|)$. If $μ<(d+1)/(d+2)^2$, then deleting at most a proportion $\sqrt{(d+1)μ}$ from each side places the retained configuration in one explicit lifted-linear model: the sphere parameters $Φ(S_Q(c,r))=(2c,r-Q(c))$ lie in an affine space L of dimension at most d, & the retained points lie in its common affine-quadric base. The natural deletion distance satisfies $μ\le dist_{lin}(P,S) \le2\sqrt{(d+1)μ}$ and the square-root exponent is optimal. Under codimension-two nonconcentration, & provided at least two spheres survive, the model reduces to one coaxal pencil. This classification has a sharp lower-scale boundary. A polynomial-root chart on one lifted rational normal curve realizes every admissible d-uniform hypergraph; consequently any list of individual point-set templates approximating all resulting constant-K systems within $o(q^d)$ edits has size $\exp(Ω_d(q^d))$. Exact fixed-nonzero-radius centered direct sums occur even when $|S|\ge4q/K^2$, & a prescribed-radius twisted-cubic system in dimension three has a complete rich-pair graph but neither a heavy pair section nor a quantitatively nontrivial centered block decomposition. At the fixed-radius deviation scale, what remains universally true is sharp adaptive extraction. If $σ(P,S)\ge Kq^{(d-1)/2}\sqrt{|P||S|}$ & $K^2q^{d-1}|S|\ge4|P|$, then at least $K^2|S|/4$ ordered pairs have a common P-section of size at least $K^2q^{d-1}/(4|S|)$; for $d\ge3$ the witness count improves to $K^2q|S|/8$. We prove mixed-radius profile stability for both signs, a log-free refinement, a cap-free positive-surplus decomposition, & applications to affine hyperplanes, pinned distances, dot products.