Three Regimes in the Sharp Quadratic Stability Problem for Dittert's Functional

We study the sharp global quadratic stability constant for Dittert's functional on the simplex of nonnegative (n\times n) matrices with total entry sum (n). Writing (U_n=J_n/n), we consider [\kappa_n^*=\inf_{A\ne U_n}\frac{\Phi(U_n)-\Phi(A)}{|A-U_n|_F^2}.] Exact witnesses and reproducible numerical searches reveal three distinct low-dimensional regimes. For (n=3,4,5), the best observed finite-distance witnesses are cycle matrices of the form ((P+Q)/2). For (n=6,7), they are replaced by highly symmetric block minimizers. From (n=8) onward, the best observed branch is a nearly uniform matrix with a single zero entry. Two general results are proved for every (n\ge3). First, the local Hessian scale (L_n) is never sharp: (\kappa_n^<L_n). Second, an explicit one-dimensional one-zero family yields the strict upper bound[\kappa_n^\le \widetilde Z_n<Z_n,]where (Z_n) is the doubly stochastic one-zero value. We derive the asymptotic expansion (Z_n/\gamma_n\to1/2) from below and quantify the exponentially small departure of the optimized branch from double stochasticity. The computational archive provides exact rational upper bounds for (n=6,\ldots,9), KKT and Hessian diagnostics, and comparisons of seven prescribed-zero boundary patterns through (n=12). Global optimality beyond the rigorously established (n=3) case remains open.