OAI.SingleLatticeCovering.GaussianProjection.actual_smoothed_lower: intermediate lemma from OpenAI, A single-lattice covering bound of order n log n

Proved

OAI.SingleLatticeCovering.GaussianProjection.actual_smoothed_lower

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.SingleLatticeCovering.single_lattice_covering, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, A single-lattice covering bound of order n log n, 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.GaussianProjection.actual_smoothed_lower of lean/OAI/Geometry/LatticeCovering/ProjectionBounds.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SingleLatticeCovering.GaussianProjection, with open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity Prekopa Radial, open scoped ENNReal Topology RealInnerProductSpace BigOperators):

theorem actual_smoothed_lower {n k D : ℕ} (hn : 0 < n) (hk : 2 ≤ k)
    {K : Set (Isotropization.E n)} (hK : IsCompact K) (hc : Convex ℝ K)
    (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ]
    (hdef : μ=(volume K)⁻¹ • volume.restrict K) (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ)
    {s r T ε : ℝ} (hs : s^2*(n:ℝ)=1) (hr : 1 ≤ r) (hT : 0 < T)
    (hε : 0 < ε) (hε1 : ε ≤ 1/10) (hDε : (D:ℝ)*ε ≤ 1/48)
    (hloss : shellLoss n k s r T ε < 1/(4*fourthMomentConstant))
    (hloss8 : shellLoss n k s r T ε ≤ 1/8) :
    ∃ u : ℝ, 1/4 ≤ u ∧ u ≤ 4 ∧ ∃ g : MatrixSpace n D, ∀ y : Isotropization.E D,
      (3/4)*density (1/(2*u)) y-projectionError n D s r ≤ normalizedProjection μ s r g y

Definitions. The statement is written with OAI.SingleLatticeCovering.GaussianDensity.density, OAI.SingleLatticeCovering.GaussianProjection.MatrixSpace, OAI.SingleLatticeCovering.GaussianProjection.normalizedProjection, OAI.SingleLatticeCovering.GaussianProjection.projectionError, OAI.SingleLatticeCovering.GaussianProjection.shellLoss, OAI.SingleLatticeCovering.Isotropization.E, OAI.SingleLatticeCovering.Isotropization.IsIsotropic and OAI.SingleLatticeCovering.Prekopa.fourthMomentConstant (definitions bundle OAIB3Sl000 of this split).

Used by. The proof of OAI.SingleLatticeCovering.single_lattice_covering of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIB3Sl000

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

section
namespace SingleLatticeCovering.GaussianProjection
open MeasureTheory ProbabilityTheory Set Filter Isotropization GaussianDensity Prekopa Radial
open scoped ENNReal Topology RealInnerProductSpace BigOperators

theorem actual_smoothed_lower {n k D : ℕ} (hn : 0 < n) (hk : 2 ≤ k)
    {K : Set (Isotropization.E n)} (hK : IsCompact K) (hc : Convex ℝ K)
    (μ : Measure (Isotropization.E n)) [IsProbabilityMeasure μ]
    (hdef : μ=(volume K)⁻¹ • volume.restrict K) (hμ : MemLp id 2 μ) (hiso : IsIsotropic μ)
    {s r T ε : ℝ} (hs : s^2*(n:ℝ)=1) (hr : 1 ≤ r) (hT : 0 < T)
    (hε : 0 < ε) (hε1 : ε ≤ 1/10) (hDε : (D:ℝ)*ε ≤ 1/48)
    (hloss : shellLoss n k s r T ε < 1/(4*fourthMomentConstant))
    (hloss8 : shellLoss n k s r T ε ≤ 1/8) :
    ∃ u : ℝ, 1/4 ≤ u ∧ u ≤ 4 ∧ ∃ g : MatrixSpace n D, ∀ y : Isotropization.E D,
      (3/4)*density (1/(2*u)) y-projectionError n D s r ≤ normalizedProjection μ s r g y := by
  sorry

end SingleLatticeCovering.GaussianProjection

end

end OAI
end

Source

OpenAI, A single-lattice covering bound of order n log n, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 92; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-single-lattice-covering-bound-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/ProjectionBounds.lean (declaration OAI.SingleLatticeCovering.GaussianProjection.actual_smoothed_lower)

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