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

Let $p\ge 5$ be prime, let $R\subset {\mathbb{F}}_p$ satisfy $1\le \mathrm{\mid }R\mathrm{\mid }\le p-1$, and set $$F(a,b)=\sum _{x\in R}{c}_x{\omega }^{a{x}^2+bx},(a,b)\in {\mathbb{F}}_p^2,c_x\mathrm{\ne }0,$$ where $\omega =e^{2\pi i\mathrm{/}p}$. We prove a parity-sensitive uncertainty principle for complex Fourier spectra supported on the finite parabola. If $\mathrm{\mid }R\mathrm{\mid }=2r$, then $$\mathrm{\mid }Z(F)\mathrm{\mid }\le p+2r-2,$$ and the bound is sharp for every even support size $2\le \mathrm{\mid }R\mathrm{\mid }\le p-1$, including the endpoint $\mathrm{\mid }R\mathrm{\mid }=p-1$. If $\mathrm{\mid }R\mathrm{\mid }=2r+1$, then $$\mathrm{\mid }Z(F)\mathrm{\mid }\le \min \{p+2r-2,r(r+1)\}\mathrm{.}$$ Thus fixed odd spectral sparsity forces a number of zeros bounded independently of $p$, whereas sufficiently large even zero sets are rigid: for even support $\mathrm{\mid }R\mathrm{\mid }=2r$, if $\mathrm{\mid }Z(F)\mathrm{\mid }>r(r+1)$, then $F$ vanishes identically on a nonvertical affine line and has at most $r(r-1)$ further zeros. The proof reduces arbitrary complex coefficients to cyclotomic data and uses a truncated $(1-\omega )$-adic expansion. The first nonzero finite-field jets satisfy $${\mathrm{\partial }}_b^2{Q}_j={\mathrm{\partial }}_a{Q}_{j-1},$$ turning the two-dimensional zero problem into a multiplicity and component-persistence problem for algebraic curves. As an application, we solve the total-support uncertainty problem for the standard complete set of $p+1$ mutually unbiased bases in ${\mathbb{C}}^p$: $$\underset{0\mathrm{\ne }\psi \in {\mathbb{C}}^p}{\min }\sum _j\mathrm{\mid }{\mathrm{supp}}_{{\mathcal{B}}_j}(\psi )\mathrm{\mid }=p^2-p+2.$$ We also classify all extremizing projective states and give an exact enumeration formula.