A signed-threshold reduction and three exact finite cases

For real $a<b$ and every $n\ge2$, the maximum spread over real symmetric matrices with entries in $[a,b]$ equals the maximum over the $2^{2n-1}$ signed-threshold endpoint matrices $\frac{a+b}2\mathbf 1\mathbf 1^T+\frac{b-a}2S$ with $S_{ij}=d_id_j\varepsilon_{\min(i,j)}$. The proof perturbs the linear functional $xx^T-yy^T$ given by extreme eigenvectors of a maximizer, orders coordinate ratios, and passes to a limit. Enumerating this family, computing exact integer characteristic polynomials, and checking rational upper root bounds for every distinct polynomial, without a floating-point eigensolver, gives maximum spreads $\sqrt{73}$, $\sqrt{133}$ and $\sqrt{161}$ on $[a,1]$ for $(n,a)=(8,1/2),(10,0),(11,0)$. Rank-two block matrices with entries in $\{a,1\}$ attain each value. The reduction alone does not show that a maximizing member of the family has rank two.