OAI.DimensionTen.Border.minors_row_12: intermediate lemma from OpenAI, Entanglement without distillable secret key in local dimension ten (result family 272)

Proved

OAI.DimensionTen.Border.minors_row_12

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.Border.minors_row_12 of lean/OAI/Analysis/Quantum/DimensionTen/Quantum/DimensionTen/MinorData.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.DimensionTen.Border, with open Matrix):

lemma minors_row_12 {R : Type*} [CommRing R] (x : Fin 4 → R) :
    minor x 12 = ((D.map (Int.castRingHom R)) *ᵥ homogeneousLow x +
      (H.map (Int.castRingHom R)) *ᵥ high (fun j => x j.succ)) 12

Definitions. The statement is written with OAI.DimensionTen.Border.D (definitions bundle OAIQ033D000 of this split); OAI.DimensionTen.Border.high and OAI.DimensionTen.Border.homogeneousLow (definitions bundle OAIQ033D003 of this split); OAI.DimensionTen.Border.H and OAI.DimensionTen.Border.minor (definitions bundle OAIQ033D012 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ033D012

Formal statement

section

attribute [-instance] _root_.OAI.DimensionTen.Mod41.instFactPrimeOfNatNat_solutions
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 Matrix
namespace DimensionTen.Border

lemma minors_row_12 {R : Type*} [CommRing R] (x : Fin 4 → R) :
    minor x 12 = ((D.map (Int.castRingHom R)) *ᵥ homogeneousLow x +
      (H.map (Int.castRingHom R)) *ᵥ high (fun j => x j.succ)) 12 := by
  sorry

end DimensionTen.Border

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/MinorData.lean (declaration OAI.DimensionTen.Border.minors_row_12)

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