What was already known
Rogers gave an O(n log n) translative upper bound. Earlier Ω(n log n) examples concerned single-lattice coverings, which do not establish an unrestricted translative lower bound.
AI involvement
OpenAI is credited as author of the model-produced lower-bound manuscript. The release does not identify a human reviewer or establish autonomy for this proof; the single-lattice companion is a separate result.
OpenAI is the credited author of the two manuscripts and the repository describes model-produced results. The specific human direction of this pair is not disclosed.
Importance
Answers the linear-bound question and determines the worst-case unrestricted translative order.
Limitations
The rational-polytope version is a further corollary. Lean source states the lower theorem, but the reviewed release did not establish a successful build or comparator result for it. The companion single-lattice upper theorem has separate scope and credit.
Technical verification
The repository contains Lean source declaring a translative lower theorem, a single-lattice upper theorem and their combined order theorem, plus Comparator challenge statements. The formalization catalogue explicitly lists the single-lattice upper theorem, but a completed build or Comparator result for the translative lower theorem, the claim that answers this entry's question, was not established in the reviewed sources. A formal-looking source file or challenge configuration alone does not show a completed machine check of that lower theorem; no human check is documented.
Human understanding
No named human explanation of the model-produced random-slab construction, localization and all-pattern exclusion is documented. Few is the evidence default, not a report that researchers cannot understand it.
Correction and publication history
If correct, answers the linear-bound question and determines the worst-case unrestricted translative order.
Read this published revision