OAI.Kaplansky.mainTarget_of_actualDefectQuasitrace: intermediate lemma from OpenAI, A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

Proved

OAI.Kaplansky.mainTarget_of_actualDefectQuasitrace

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Kaplansky.kaplansky_quasitrace_counterexample, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness, OpenAI Math Release, September 23, 2026, result family 294, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Kaplansky.mainTarget_of_actualDefectQuasitrace of lean/OAI/Analysis/Kaplansky/Kaplansky/ActualAlgebra.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Kaplansky, with open scoped BigOperators):

theorem mainTarget_of_actualDefectQuasitrace
    (hq : ∃ τ : OneQuasitrace ActualDefectAlgebra, τ.Normalized ∧ τ.IsTwo) : MainTarget

Definitions. The statement is written with OAI.Kaplansky.ActualDefectAlgebra (definitions bundle OAIQ4Kq009 of this split); OAI.Kaplansky.actualDefectAlgebra_cstar (definitions bundle OAIQ4Kq011 of this split); OAI.Kaplansky.MainTarget, OAI.Kaplansky.OneQuasitrace, OAI.Kaplansky.OneQuasitrace.IsTwo and OAI.Kaplansky.OneQuasitrace.Normalized (the published bundle Def_KaplanskyQuasitrace).

Used by. The proof of OAI.Kaplansky.kaplansky_quasitrace_counterexample of this split imports it.

Preamble

import Mathlib
import Lean.Elab.Tactic.Omega
import Definitions.Def_KaplanskyQuasitrace
import Definitions.Def_OAIQ4Kq011

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

noncomputable section

namespace Kaplansky

section
open scoped BigOperators
attribute [local instance 2000] CStarAlgebra.spectralOrder CStarAlgebra.spectralOrderedRing

theorem mainTarget_of_actualDefectQuasitrace
    (hq : ∃ τ : OneQuasitrace ActualDefectAlgebra, τ.Normalized ∧ τ.IsTwo) : MainTarget := by
  sorry

end

end Kaplansky

end

end OAI
end

Source

OpenAI, A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 294; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Kaplansky/Kaplansky/ActualAlgebra.lean (declaration OAI.Kaplansky.mainTarget_of_actualDefectQuasitrace)

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