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

Proved

OAI.Kadison.FinitePatching.trace_word_second_order_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 BigOperators):

lemma trace_word_second_order_bound (τ : A →ₗ[ℂ] ℂ) (old new : I → A)
    (l : List I) {S D : ℝ} (hS : 0 ≤ S) (hD : 0 ≤ D)
    (hsingle : ∀ s i t, l=s++i::t →
      ‖τ (wordProduct old s*new i*wordProduct old t)‖ ≤ S)
    (hdouble : ∀ s i t j r, l=s++i::(t++j::r) →
      ‖τ (wordProduct old s*new i*wordProduct old t*new j*
        wordProduct (fun k => old k+new k) r)‖ ≤ D) :
    ‖τ (wordProduct (fun i => old i+new i) l)-τ (wordProduct old l)‖ ≤
      (l.length:ℝ)*S+(l.length:ℝ)^2*D

Definitions. The statement is written with OAI.Kadison.FinitePatching.wordProduct (definitions bundle OAIQ039C010 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

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 BigOperators

namespace Kadison.FinitePatching

variable {A I : Type*} [NormedRing A]

variable [NormedAlgebra ℂ A]

lemma trace_word_second_order_bound (τ : A →ₗ[ℂ] ℂ) (old new : I → A)
    (l : List I) {S D : ℝ} (hS : 0 ≤ S) (hD : 0 ≤ D)
    (hsingle : ∀ s i t, l=s++i::t →
      ‖τ (wordProduct old s*new i*wordProduct old t)‖ ≤ S)
    (hdouble : ∀ s i t j r, l=s++i::(t++j::r) →
      ‖τ (wordProduct old s*new i*wordProduct old t*new j*
        wordProduct (fun k => old k+new k) r)‖ ≤ D) :
    ‖τ (wordProduct (fun i => old i+new i) l)-τ (wordProduct old l)‖ ≤
      (l.length:ℝ)*S+(l.length:ℝ)^2*D := 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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