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

Abstract

Dittert's conjecture asserts that, among nonnegative $n\times n$ matrices whose entries sum to $n$, the functional $φ(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\operatorname{per}(A)$ is uniquely maximized by the uniform matrix $J_n/n$. 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 $n\ge 16$.

Disclosure

“superstochastic matrix. The proof strategy, central lemma, and most of the initial exposition were produced by OpenAI’s GPT-5.6 Sol through ChatGPT under the direction of the author. See the disclosure at the end of the paper. 1”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-23T19:00:03.243245Z

Structural counts