OAI.DimensionTen.exists_swap_of_mod131: intermediate lemma from OpenAI, Entanglement without distillable secret key in local dimension ten (result family 272)

Proved

OAI.DimensionTen.exists_swap_of_mod131

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.exists_swap_of_mod131 of lean/OAI/Analysis/Quantum/DimensionTen/Quantum/DimensionTen/GaloisCertificate.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

lemma exists_swap_of_mod131
    (h131 : f.map (Int.castRingHom (ZMod 131)) = Mod131.f X) :
    ∃ σ : Gal(K/ℚ), (MulAction.toPerm (σ ^ 17) : Equiv.Perm (f.rootSet K)).IsSwap

Definitions. The statement is written with OAI.DimensionTen.Mod131.f (definitions bundle OAIQ033D003 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ033D003

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 NumberField Polynomial
namespace DimensionTen

section Roots
variable {K : Type*} [Field K] [NumberField K] [IsGalois ℚ K] [DecidableEq K]
variable (f : ℤ[X]) (hm : f.Monic) (hs : (f.map (algebraMap ℤ K)).Separable)
    (hp : (f.map (algebraMap ℤ K)).Splits)
    (hc : Fintype.card (f.rootSet K) = 20)

include hm hs hp hc in
lemma exists_swap_of_mod131
    (h131 : f.map (Int.castRingHom (ZMod 131)) = Mod131.f X) :
    ∃ σ : Gal(K/ℚ), (MulAction.toPerm (σ ^ 17) : Equiv.Perm (f.rootSet K)).IsSwap := by
  sorry

end Roots
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/GaloisCertificate.lean (declaration OAI.DimensionTen.exists_swap_of_mod131)

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