Critical Restricted Sumsets at the Boundary $\mid A\mid +\mid B\mid =p$

Let $p$ be an odd prime, and write $A\text{RS}B=\{a+b:a\in A,b\in B,a\mathrm{\ne }b\}$. We classify all pairs $A,B\subseteq \text{Fp}$ satisfying $\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }=p$, $\mathrm{\mid }A\mathrm{\mid }>\mathrm{\mid }B\mathrm{\mid }=\mathrm{\ell }\ge 1$, and $\mathrm{\mid }A\text{RS}B\mathrm{\mid }=p-2$. After normalizing the two missing sums to $\{0,1\}$, we obtain exactly $\mathrm{\ell }+1$ explicit models. The classification forces $B\subseteq A$ and yields, for fixed $p,\mathrm{\ell }$, exactly $\left(\genfrac{}{}{0px}{}{p}{2}\right)(\mathrm{\ell }+1)$ ordered pairs and $1+\lfloor \mathrm{\ell }\mathrm{/}2\rfloor$ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For $\mathrm{\ell }\ge 3$ and $p\ge 4\mathrm{\ell }-5$, the cardinality $\mathrm{\mid }B-B\mathrm{\mid }$ determines the equivalence class at fixed $p,\mathrm{\ell }$. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in ${\mathbb{F}}_{13}$, with size gap three and $\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }=p-2$, that contradicts a proposed inverse statement below the boundary.