OpenAI · 092-01 · 2026-09-23 · Convex and metric geometry
A single-lattice covering bound of order n log n
preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026
Family 092. The optimal order of convex-body covering density.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Determines the optimal worst-case covering density as , for both lattice and unrestricted translative coverings. Every convex body in , , admits a lattice covering of density at most ; centrally symmetric examples in every sufficiently large dimension require at least even without the lattice restriction, for absolute .
Every convex body in ℝn, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most , for an absolute constant C. No symmetry or boundary regularity is assumed.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 092: The optimal order of convex-body covering density. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.