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Math-2646-A2 · revision r1 · on Math-2646

OpenAI claim summary for Math-2646

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 092; 2026-09-23).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Single-lattice covering density of order n log n
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Single-lattice covering density of order n log n

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 092; 2026-09-23), the asserted result is that every convex body in dimension at least two admits a single-lattice covering of density at most an absolute multiple of $n\log n$.

Mathematical statement of the claimed result

Manuscript 1.
An absolute constant $C>0$ is claimed such that for every dimension $n\ge2$ and every convex body $K\subset\mathbb R^n$ there is one full-rank lattice $\Lambda$ with \[ K+\Lambda=\mathbb R^n,\qquad |K|/\det\Lambda\le Cn\log n. \] The ratio is lattice covering density. Neither central symmetry nor smooth boundary is needed, and the covering uses one lattice rather than a union of independently translated lattices.

Outline of the argument

For manuscript 1 (family 092; 2026-09-23), a hierarchy of branching lattice blocks supplies usable anchors. Folded Gaussian sampling and simultaneous shears cover a finite collection of horizontal patterns, after which a final small-hole removal produces an exact covering by one lattice.

Formal verification

For manuscript 1 (family 092; 2026-09-23), OpenAI lists Lean statements for this assertion: every convex body in dimension at least two admits a single-lattice covering of density at most an absolute multiple of $n\log n$.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are CoveringDensity.lean, SingleLatticeCovering.lean.

Their named declarations include OAI.CoveringOrder.optimal_order, OAI.SingleLatticeCovering.single_lattice_covering.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources