Polynomial Unit-Distance Lower Bounds over Prime Fields

For each a in {1,3}, we prove that infinitely many primes p congruent to a modulo 4 admit a set of exactly p points in F_p^2 with at least p1.0058 unordered pairs at quadratic distance one. We transfer Sawin's number-field unit-distance construction to prime fields using a field-uniform quantitative Chebotarev estimate of Zaman. A relative ideal-norm bound prevents collisions without requiring a bounded global denominator. Reduction at split or inert primes gives the required quadratic form, and disjoint translates supply exactly p points. The exponent is certified by rational interval arithmetic. This is a lower-bound result for AIM-COMBINATORICS-0257, not a determination of the optimal exponent or a result for every sufficiently large prime. The AI-assisted preprint is self-audited and unrefereed; novelty and absolute priority are not certified.