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

Proved

OAI.Kadison.FreeBlocks.free_fejer_norm

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.FreeBlocks, with open scoped ComplexOrder, open FreeMoments FreeWordTensor DihedralCorner):

theorem free_fejer_norm (e : Corner hp) (heproj : IsStarProjection e)
    (he : e∈cornerFactor hp (S false) hpD) (m : ℕ) (hm : 0 < m) :
    ‖(e:A)*((fejerCorner hp (d*s) m:Corner hp):A)*(e:A)‖≤
      4*Real.sqrt 2*Real.sqrt (halfCornerTrace hp t.functional.toLinearMap e).re*
        Real.sqrt ((m:ℝ)⁻¹)+2*(halfCornerTrace hp t.functional.toLinearMap e).re

Definitions. The statement is written with OAI.Kadison.Corner, OAI.Kadison.cornerSubalgebra, OAI.Kadison.instAlgebraComplexCorner, OAI.Kadison.instModuleComplexCorner, OAI.Kadison.instRingCorner, OAI.Kadison.instStarModuleComplexCorner and OAI.Kadison.instStarRingCorner (definitions bundle OAIQ039C004 of this split); OAI.Kadison.FreeMoments.Free and OAI.Kadison.FreeWordTensor.FaithfulState (definitions bundle OAIQ039C010 of this split); OAI.Kadison.halfCornerTrace (definitions bundle OAIQ039C054 of this split); OAI.Kadison.cornerFactor and OAI.Kadison.DihedralCorner.fejerCorner (definitions bundle OAIQ039C059 of this split).

Used by. The proof of OAI.Kadison.FreeBlocks.free_hankel_bound of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C059

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier _root_.OAI.Kadison.CyclicColumns.instDecidableEq_solutions _root_.OAI.Kadison.UnitaryDualAction.action _root_.OAI.Kadison.VonNeumannCStar.cstarAlgebra _root_.OAI.Kadison.VonNeumannCStar.isClosed _root_.OAI.Kadison.VonNeumannCStar.smulMemClass _root_.OAI.Kadison.VonNeumannCStar.starOrderedRing
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.FreeBlocks

open FreeMoments FreeWordTensor DihedralCorner

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

variable (t : FaithfulState A) (S : Bool → StarSubalgebra ℂ A)
  (hfree : Free S t.functional.toLinearMap)

variable (htr : ∀ a b : A, t.functional (a*b)=t.functional (b*a))

variable (d s : unitary A) (hd : (d:A)∈S false) (hs : (s:A)∈S true)
  (hd0 : t.functional (d:A)=0) (hs0 : t.functional (s:A)=0)
  (hr : MulAut.conj d (d*s)=(d*s)⁻¹)
  (hd2 : (d:A)*(d:A)=1)

variable {p : A} (hp : IsStarProjection p)
  (hpform : p=(1/2:ℂ) • (1+(d:A))) (hpd : p*(d:A)=p) (hdp : (d:A)*p=p)
  (hpD : p∈S false)

include hfree htr hd hs hd0 hs0 hr hd2 hpform hpd hdp in
theorem free_fejer_norm (e : Corner hp) (heproj : IsStarProjection e)
    (he : e∈cornerFactor hp (S false) hpD) (m : ℕ) (hm : 0 < m) :
    ‖(e:A)*((fejerCorner hp (d*s) m:Corner hp):A)*(e:A)‖≤
      4*Real.sqrt 2*Real.sqrt (halfCornerTrace hp t.functional.toLinearMap e).re*
        Real.sqrt ((m:ℝ)⁻¹)+2*(halfCornerTrace hp t.functional.toLinearMap e).re := by
  sorry

end Kadison.FreeBlocks

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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