A near-quadratic lower bound for sets with no unique sums

Let $m(p)$ be the least size of a subset of ${\text{F}}_p$ with at least two elements for which every sum has two distinct representations as unordered pairs, allowing repetition. We prove that, for every prime $p\ge 64$, $$m(p)\ge 2^{-80}{\left(\frac{\log p}{\log \log p}\right)}^2\mathrm{.}$$ The argument compresses the full integer collision lattice by unit-pivot elimination. A shared random sample and forests of bounded diameter give $O(\sqrt{n}+n\mathrm{/}\log p)$ surviving coordinates of polynomial height for a minimal set of size $n$. A nonzero minor divisible by $p$ then gives the lower bound. We also construct weakly ternary-balanced seeds yielding $$m(p)\le \frac{(\log p{)}^2}{2(\log 3{)}^2}+(\frac{1}{4\log 3}+o(1))\frac{(\log p{)}^2}{\log \log p}\mathrm{.}$$ Consequently $m(p)=(\log p{)}^{2+o(1)}$ as $p$ tends to infinity through the primes. The constant-factor order of $m(p)$ remains undetermined.