OAI.Laughlin.Certificate.scaled_four_body_certificates: intermediate lemma from OpenAI, Uniform stability of the spherical Laughlin gap (result family 269)

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OAI.Laughlin.Certificate.scaled_four_body_certificates

by dbenbenn · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Laughlin.mainTarget_proved, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Uniform stability of the spherical Laughlin gap (result family 269), OpenAI Math Release, October 5, 2026, result family 269, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Laughlin.Certificate.scaled_four_body_certificates of lean/OAI/Analysis/Laughlin/FourBody/ScaledAll.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Laughlin.Certificate):

theorem scaled_four_body_certificates (D : ℕ) (hD₁ : 1 ≤ D) (hD₂ : D ≤ 23) :
    (scaledGram D * (certificateDiagonal D-scaledError D) * scaledGram D).PosSemidef

Definitions. The statement is written with OAI.Laughlin.Certificate.certificateDiagonal, OAI.Laughlin.Certificate.scaledError and OAI.Laughlin.Certificate.scaledGram (definitions bundle OAIQ047L000 of this split).

Used by. The proof of OAI.Laughlin.Fock.source_fourBody_limitpositive of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ047L000

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

namespace Laughlin.Certificate

theorem scaled_four_body_certificates (D : ℕ) (hD₁ : 1 ≤ D) (hD₂ : D ≤ 23) :
    (scaledGram D * (certificateDiagonal D-scaledError D) * scaledGram D).PosSemidef := by
  sorry

end Laughlin.Certificate

end OAI
end

Source

OpenAI, Uniform stability of the spherical Laughlin gap (result family 269), OpenAI Math Release, October 5, 2026, https://github.com/openai/math (result family 269; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Laughlin/FourBody/ScaledAll.lean (declaration OAI.Laughlin.Certificate.scaled_four_body_certificates)

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