OAI.DimensionTen.compositeChoi_kernel_rank: intermediate lemma from OpenAI, Entanglement without distillable secret key in local dimension ten (result family 272)

Proved

OAI.DimensionTen.compositeChoi_kernel_rank

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.DimensionTen.main_pair, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Entanglement without distillable secret key in local dimension ten (result family 272), OpenAI Math Release, September 27, 2026, result family 272, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.DimensionTen.compositeChoi_kernel_rank of lean/OAI/Analysis/Quantum/DimensionTen/Quantum/DimensionTen/Main.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.DimensionTen, with open scoped Matrix Kronecker ComplexOrder, open Matrix):

lemma compositeChoi_kernel_rank {x : Fin 4 → ℂ} (hx : (pencil x).rank < 4) :
    compositeChoi *ᵥ productVector (veronese x) (veronese x) = 0

Definitions. The statement is written with OAI.DimensionTen.pencil and OAI.DimensionTen.veronese (definitions bundle OAIQ033D009 of this split); OAI.DimensionTen.compositeChoi and OAI.DimensionTen.productVector (the published bundle Def_DimensionTenPair).

Used by. The proof of OAI.DimensionTen.main_pair of this split imports it.

Preamble

import Mathlib
import Definitions.Def_DimensionTenPair
import Definitions.Def_OAIQ033D009

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
noncomputable section
open scoped Matrix Kronecker ComplexOrder
open Matrix
namespace DimensionTen

lemma compositeChoi_kernel_rank {x : Fin 4 → ℂ} (hx : (pencil x).rank < 4) :
    compositeChoi *ᵥ productVector (veronese x) (veronese x) = 0 := by
  sorry

end DimensionTen

end
end

end OAI
end

Source

OpenAI, Entanglement without distillable secret key in local dimension ten (result family 272), OpenAI Math Release, September 27, 2026, https://github.com/openai/math (result family 272; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Quantum/DimensionTen/Quantum/DimensionTen/Main.lean (declaration OAI.DimensionTen.compositeChoi_kernel_rank)

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