Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma

Dittert’s conjecture asserts that, among nonnegative $n\times n$ matrices whose entries sum to $n$, the functional $\phi (A)={\prod }_{i=1}^n{r}_i+{\prod }_{j=1}^n{c}_j-\mathrm{per}(A)$ is uniquely maximized by the uniform matrix $J_n\mathrm{/}n$. This paper proves the conjecture for $n=16$. The key observation is that, for a near-maximizer, the deficits of the row-sum and column-sum products satisfy a single joint constraint rather than two independent bounds. Combining this joint-deficit estimate with a Pinsker-type subset-sum bound yields a sharper scalar dilation to a doubly superstochastic matrix. The Knopp-Sinkhorn boundary lower bound for permanents then excludes maximizers with a zero entry, and Hwang’s positive-support theorem identifies the unique maximizer. Together with Pang’s result for $n\ge 17$ (arXiv:2606.01531), this establishes Dittert’s conjecture for every $n\ge 16$.