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

Proved

OAI.Kadison.LocalQuantization.exists_small_projection

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.LocalQuantization, with open scoped InnerProductSpace ComplexOrder Topology BigOperators, open Filter, open TracialGNS):

theorem exists_small_projection
    (hgrow : Tendsto (fun n => (Fintype.card (ι n):ℝ)) atTop atTop)
    (x : J → A)
    (hx : ∀ j, Tendsto (fun n => algebraVector τ (MatrixExpectation.expectation (stage n) (x j)))
      atTop (𝓝 0))
    {ε η : ℝ} (hε : 0<ε) (hη : 0<η) :
    ∃ (n : ℕ) (q : A), q∈(stage n).range ∧ IsStarProjection q ∧
      0<(τ q).re ∧ (τ q).re<η ∧
      (∑ j, ‖algebraVector τ (q*x j*q)‖^2)<ε^2*(τ q).re

Definitions. The statement is written with OAI.Kadison.TracialGNS.algebraVector (definitions bundle OAIQ039C003 of this split); OAI.Kadison.MatrixExpectation.expectation (definitions bundle OAIQ039C012 of this split).

Used by. The proof of OAI.Kadison.LocalQuantization.exists_centered_compression of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C003
import Definitions.Def_OAIQ039C012

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

section

noncomputable section

open scoped InnerProductSpace ComplexOrder Topology BigOperators

open Filter

namespace Kadison.LocalQuantization

open TracialGNS

section FiniteTests

variable {A J : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
  [Fintype J]

variable (τ : A →ₚ[ℂ] ℂ) (hτ : ∀ a b : A, τ (a*b)=τ (b*a)) (hone : τ 1=1)

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] A)

include hτ hone in
theorem exists_small_projection
    (hgrow : Tendsto (fun n => (Fintype.card (ι n):ℝ)) atTop atTop)
    (x : J → A)
    (hx : ∀ j, Tendsto (fun n => algebraVector τ (MatrixExpectation.expectation (stage n) (x j)))
      atTop (𝓝 0))
    {ε η : ℝ} (hε : 0<ε) (hη : 0<η) :
    ∃ (n : ℕ) (q : A), q∈(stage n).range ∧ IsStarProjection q ∧
      0<(τ q).re ∧ (τ q).re<η ∧
      (∑ j, ‖algebraVector τ (q*x j*q)‖^2)<ε^2*(τ q).re := by
  sorry

end FiniteTests

end Kadison.LocalQuantization

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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