OAI.SingleLatticeCovering.Assembly.eventual_single_lattice_covering: intermediate lemma from OpenAI, The optimal order of convex-body covering density

Proved

OAI.SingleLatticeCovering.Assembly.eventual_single_lattice_covering

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.CoveringOrder.optimal_order, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The optimal order of convex-body covering density, OpenAI Math Release, September 23, 2026, result family 92, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SingleLatticeCovering.Assembly.eventual_single_lattice_covering of lean/OAI/Geometry/LatticeCovering/Assembly.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SingleLatticeCovering.Assembly, with open MeasureTheory, open Filter Topology, open scoped BigOperators Pointwise ENNReal):

theorem eventual_single_lattice_covering :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ K : Set (Fin n → ℝ),
      IsCompact K → Convex ℝ K → (interior K).Nonempty →
      ∃ (Λ : Submodule ℤ (Fin n → ℝ)) (_ : DiscreteTopology Λ), IsZLattice ℝ Λ ∧
        K+(Λ : Set (Fin n → ℝ))=Set.univ ∧
        (volume K).toReal/ZLattice.covolume Λ volume ≤ C*(n : ℝ)*Real.log (n : ℝ)

Used by. The proof of OAI.CoveringOrder.optimal_order of this split imports it.

Preamble

import Mathlib

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

namespace SingleLatticeCovering.Assembly
open MeasureTheory
open Filter Topology
open scoped BigOperators Pointwise ENNReal

theorem eventual_single_lattice_covering :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ K : Set (Fin n → ℝ),
      IsCompact K → Convex ℝ K → (interior K).Nonempty →
      ∃ (Λ : Submodule ℤ (Fin n → ℝ)) (_ : DiscreteTopology Λ), IsZLattice ℝ Λ ∧
        K+(Λ : Set (Fin n → ℝ))=Set.univ ∧
        (volume K).toReal/ZLattice.covolume Λ volume ≤ C*(n : ℝ)*Real.log (n : ℝ) := by
  sorry

end SingleLatticeCovering.Assembly

end

end OAI
end

Source

OpenAI, The optimal order of convex-body covering density, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 92; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/LatticeCovering/Assembly.lean (declaration OAI.SingleLatticeCovering.Assembly.eventual_single_lattice_covering)

View graph