Parity-Sensitive Fourier Uncertainty and Zero-Set Rigidity on the Finite Parabola

Let $p\ge5$ be prime, let $R\subset\mathbb F_p$ satisfy $1\le |R|\le p-1$, and set \[ F(a,b)=\sum_{x\in R}c_xω^{ax^2+bx}, \qquad (a,b)\in\mathbb F_p^2, \qquad c_x\ne0, \] where $ω=e^{2πi/p}$. We prove a parity-sensitive uncertainty principle for complex Fourier spectra supported on the finite