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

Proved

OAI.Kadison.FinitePatching.trace_centered_compression_bound

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, open TracialL2):

lemma trace_centered_compression_bound {p q b a c w z : A}
    (hp : IsStarProjection p) (hq : IsStarProjection q) (hqp : q ≤ p)
    (hw : w*q=w) (hz : q*z=z)
    (ha : ‖a‖ ≤ 1) (hc : ‖c‖ ≤ 1) (hwn : ‖w‖ ≤ 1) (hzn : ‖z‖ ≤ 1)
    {δ : ℝ} (hδ : 0 ≤ δ)
    (hsmall : norm τ (q*cornerCenter τ p b*q)^2 ≤ δ^2*(τ q).re) :
    ‖τ (a*w*b*z*c)‖ ≤ (δ+‖cornerScalar τ p b‖)*(τ q).re

Definitions. The statement is written with OAI.Kadison.FinitePatching.cornerCenter and OAI.Kadison.FinitePatching.cornerScalar (definitions bundle OAIQ039C010 of this split); OAI.Kadison.TracialL2.norm (definitions bundle OAIQ039C017 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ039C010
import Definitions.Def_OAIQ039C017

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 ComplexOrder

namespace Kadison.FinitePatching

open TracialL2

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

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

include hτ in
lemma trace_centered_compression_bound {p q b a c w z : A}
    (hp : IsStarProjection p) (hq : IsStarProjection q) (hqp : q ≤ p)
    (hw : w*q=w) (hz : q*z=z)
    (ha : ‖a‖ ≤ 1) (hc : ‖c‖ ≤ 1) (hwn : ‖w‖ ≤ 1) (hzn : ‖z‖ ≤ 1)
    {δ : ℝ} (hδ : 0 ≤ δ)
    (hsmall : norm τ (q*cornerCenter τ p b*q)^2 ≤ δ^2*(τ q).re) :
    ‖τ (a*w*b*z*c)‖ ≤ (δ+‖cornerScalar τ p b‖)*(τ q).re := 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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