We study correlated rounding on the hypersimplex, the base polytope of the uniform matroid. For each point $x$ in the hypersimplex, the goal is to sample a $k$-subset $A(x)$ with marginals $x$, while coupling the samples for all choices of $x$ so that nearby inputs produce nearby sets. We give a constant-stretch scheme. Our scheme samples the maximum-entropy $k$-subset distribution with prescribed marginals using a common random ordering and common uniform thresholds. For every $x,y\in[0,1]^n$ with $\sum_i x_i=\sum_i y_i=k$, it satisfies $\mathbb{E}[|A(x)\triangle A(y)|]\le 6\|x-y\|_1$. This improves the previous $O(\log k)$ bound for hypersimplex correlated rounding and answers an open question raised by Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc. By adding dummy coordinates, the same result gives stretch at most $12$ for the at-most-$k$ polytope. The proof was found in a GPT 5.5 Pro Extended conversation prompted by the authors, and Codex was used to help assemble the manuscript under the authors' supervision.

翻译：我们研究了超单纯形（均匀拟阵的基多面体）上的相关舍入问题。对于超单纯形中的每个点 $x$，目标是在边缘分布为 $x$ 的条件下采样一个 $k$-子集 $A(x)$，同时对所有 $x$ 的选择进行耦合，使得邻近的输入产生邻近的集合。我们给出了一种常数拉伸方案。该方案使用一个共同的随机排序和共同的均匀阈值，对具有指定边缘分布的最大熵 $k$-子集分布进行采样。对于每个 $x,y\in[0,1]^n$ 且 $\sum_i x_i=\sum_i y_i=k$，它满足 $\mathbb{E}[|A(x)\triangle A(y)|]\le 6\|x-y\|_1$。这改进了此前超单纯形相关舍入的 $O(\log k)$ 界，并回答了 Naor、Raju、Shetty、Srinivasan、Valieva 和 Wajc 提出的一个开放问题。通过添加虚拟坐标，相同的结果对至多 $k$ 多面体给出了不超过 $12$ 的拉伸。该证明是在作者提示下通过 GPT 5.5 Pro Extended 对话获得的，并在作者监督下使用 Codex 协助汇编了手稿。