OAI.DimensionTen.Mod139.iterate6: intermediate lemma from OpenAI, Entanglement without distillable secret key in local dimension ten (result family 272)

Proved

OAI.DimensionTen.Mod139.iterate6

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.Mod139.iterate6 of lean/OAI/Analysis/Quantum/DimensionTen/Quantum/DimensionTen/Mod139Powers03.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.DimensionTen.Mod139, with open Polynomial):

theorem iterate6 {R : Type*} [CommRing R] [CharP R 139]
    (z : R) (hz : f z = 0) : z ^ (139 ^ 6) = u6 z

Definitions. The statement is written with OAI.DimensionTen.Mod139.f (definitions bundle OAIQ033D006 of this split); OAI.DimensionTen.Mod139.u6 (definitions bundle OAIQ033D013 of this split).

Used by. The proof of OAI.DimensionTen.Mod139.iterate12 of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ033D013

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

noncomputable section
open Polynomial
namespace DimensionTen.Mod139

theorem iterate6 {R : Type*} [CommRing R] [CharP R 139]
    (z : R) (hz : f z = 0) : z ^ (139 ^ 6) = u6 z := by
  sorry

end DimensionTen.Mod139

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/Mod139Powers03.lean (declaration OAI.DimensionTen.Mod139.iterate6)

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