Inverse Theorems for Point-Sphere Incidences over Finite Fields

We prove the first inverse theorem for point–sphere incidence bounds over finite fields in dimensions $d\ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the structural characterization of configurations that nearly saturate these bounds has remained completely open. Specifically, if a configuration of points $P\subset {\mathbb{F}}_q^d$ and spheres $\mathcal{S}$ exceeds the random incidence baseline by a factor $K$ in the moderate-sphere regime, then there exists a subset $P^{\mathrm{\prime }}\subset P$ of size $$\mathrm{\mid }{P}^{\mathrm{\prime }}\mathrm{\mid }\gtrsim K{q}^{(d-1)\mathrm{/}2}$$ contained in the zero set of a polynomial $F$ of degree at most $C{K}^C$. This yields a one-sided result: we identify necessary algebraic obstructions to extremality, without asserting sufficiency. The proof introduces a new rigidity mechanism for finite-field incidence geometry. Near-extremality manifests as persistent overlap among bisector hyperplanes. We prove that such persistent coincidence cannot occur without forcing the emergence of bounded-complexity algebraic certificates. The argument proceeds by isolating high-overlap layers via energy stratification, followed by a projective polynomial dichotomy applied to the set of normal directions. As applications, we obtain the first inverse-type results for pinned distance and dot-product problems over finite fields, resolving structural questions inaccessible to standard polynomial or Fourier-analytic methods.