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\begin{document}
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\npptitlepage{Nearly Minimax\\ Variance Estimation\\ Under Rough Random Design}{P.\ M.\ Aronow\\ Patrick Lopatto}
\begin{authorsnote}
This paper was written by the authors with the assistance of large language models, which included suggesting arguments, contributing to drafting and revision, and writing code for computational checks. It represents an update to a prior version of this paper (arXiv:2607.13170v3), with significant improvements to the bounds previously reported. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. However, the resulting document remains lacking in exposition and overall coherence. We are nonetheless releasing this manuscript on Hexagon to disseminate the result as quickly as possible, because we believe it may be of interest to the statistics community. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

P.L.\ was partially supported by NSF
grant DMS-2450004.
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