OAI.Kadison.FinitePatching.exists_countable_unitary_tests: intermediate lemma from OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288)

Proved

OAI.Kadison.FinitePatching.exists_countable_unitary_tests

by dbenbenn · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Kadison.uniform_commutator, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, result family 288, github.com/openai/math, Apache License 2.0), ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Kadison.FinitePatching, with open scoped ComplexOrder Topology, open Filter, open Words):

theorem exists_countable_unitary_tests
    (hnormal : NormalGNS.BoundedWOTContinuous M τ)
    (hfaith : ∀ a : M,τ (star a*a)=0 → a=0) (hone : τ 1=1)
    (hirr : ∀ b : M,(∀ n (a : Matrix (ι n) (ι n) ℂ),stage n a*b=b*stage n a) →
      ∃ c : ℂ,b=algebraMap ℂ M c)
    (hgrow : Tendsto (fun n => (Fintype.card (ι n):ℝ)) atTop atTop)
    (hstage : ∀ n a,(stage n a : H →L[ℂ] H)∈R)
    (hdiffuse : DiffuseProjection.HasNoMinimalProjections R)
    (x : ℕ → M) (k : ℕ → ℤ) (hx : ∀ j,‖x j‖ ≤ 1)
    (hcenter : ∀ j,τ (x j)=0) (hk : ∀ i,k i≠0) :
    ∃ u : ℕ → unitary M, (∀ n, ((u n:M) : H →L[ℂ] H)∈R) ∧
      (∀ i,Tendsto (fun n => τ ((PartialUnitary.ofUnitary (u n)).power (k i))) atTop (𝓝 0)) ∧
      (∀ l : List (Letter ℕ ℕ), l≠[] → Valid l →
        Tendsto (fun n => τ (word x k (PartialUnitary.ofUnitary (u n)) l)) atTop (𝓝 0))

Definitions. The statement is written with OAI.Kadison.NormalGNS.BoundedWOTContinuous, OAI.Kadison.VonNeumannCStar.isClosed and OAI.Kadison.VonNeumannCStar.smulMemClass (definitions bundle OAIQ039C002 of this split); OAI.Kadison.DiffuseProjection.HasNoMinimalProjections (definitions bundle OAIQ039C008 of this split); OAI.Kadison.FinitePatching.PartialUnitary.power (definitions bundle OAIQ039C009 of this split); OAI.Kadison.VonNeumannCStar.cstarAlgebra (definitions bundle OAIQ039C022 of this split); OAI.Kadison.FinitePatching.Words.Letter and OAI.Kadison.FinitePatching.Words.Valid (definitions bundle OAIQ039C025 of this split); OAI.Kadison.FinitePatching.PartialUnitary.ofUnitary and OAI.Kadison.FinitePatching.Words.word (definitions bundle OAIQ039C035 of this split).

Used by. The proof of OAI.Kadison.FinitePatching.exists_all_unitary_tests of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C035

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

section

noncomputable section

open scoped ComplexOrder Topology

open Filter

namespace Kadison.FinitePatching

open Words

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

variable (M R : VonNeumannAlgebra H) (hRM : R ≤ M)

variable (τ : M →ₚ[ℂ] ℂ) (hτ : ∀ a b : M,τ (a*b)=τ (b*a))

variable {ι : ℕ → Type*} [∀ n,Fintype (ι n)] [∀ n,LinearOrder (ι n)]
  [∀ n,Nonempty (ι n)]

variable (stage : ∀ n,Matrix (ι n) (ι n) ℂ →⋆ₐ[ℂ] M)

variable (hmono : ∀ {n m : ℕ},n ≤ m → (stage n).range ≤ (stage m).range)

include hRM hτ hmono in
theorem exists_countable_unitary_tests
    (hnormal : NormalGNS.BoundedWOTContinuous M τ)
    (hfaith : ∀ a : M,τ (star a*a)=0 → a=0) (hone : τ 1=1)
    (hirr : ∀ b : M,(∀ n (a : Matrix (ι n) (ι n) ℂ),stage n a*b=b*stage n a) →
      ∃ c : ℂ,b=algebraMap ℂ M c)
    (hgrow : Tendsto (fun n => (Fintype.card (ι n):ℝ)) atTop atTop)
    (hstage : ∀ n a,(stage n a : H →L[ℂ] H)∈R)
    (hdiffuse : DiffuseProjection.HasNoMinimalProjections R)
    (x : ℕ → M) (k : ℕ → ℤ) (hx : ∀ j,‖x j‖ ≤ 1)
    (hcenter : ∀ j,τ (x j)=0) (hk : ∀ i,k i≠0) :
    ∃ u : ℕ → unitary M, (∀ n, ((u n:M) : H →L[ℂ] H)∈R) ∧
      (∀ i,Tendsto (fun n => τ ((PartialUnitary.ofUnitary (u n)).power (k i))) atTop (𝓝 0)) ∧
      (∀ l : List (Letter ℕ ℕ), l≠[] → Valid l →
        Tendsto (fun n => τ (word x k (PartialUnitary.ofUnitary (u n)) l)) atTop (𝓝 0)) := by
  sorry

end Kadison.FinitePatching

end

end
end

end OAI
end

Source

OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 288; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf; Apache License 2.0)

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