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

Proved

OAI.Kadison.FreeBlocks.base_block_compression

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 UniformSpace, open scoped InnerProductSpace ComplexOrder, open ContinuousLinearMap, open FreeWordTensor FreeMoments DihedralCorner):

lemma base_block_compression
    (G L : t.functional.GNS →L[ℂ] t.functional.GNS)
    (B : ∀ k, multiplicityLevel t S true k →L[ℂ] multiplicityLevel t S true k)
    (hb : ∀ k a (x : letterSpace t S false) v,
      G (repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k v)=
        repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k (B k v))
    (hb' : ∀ k a (x : letterSpace t S false) v,
      G.adjoint (repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k v)=
        repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k ((B k).adjoint v))
    (sl : letterSpace t S true) (sn : ‖sl‖=1) :
    (ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
      (baseEmbedding t S hfree).toContinuousLinearMap).comp
      ((G*L-L*G).comp (baseEmbedding t S hfree).toContinuousLinearMap)=
    B 0*((ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
      (baseEmbedding t S hfree).toContinuousLinearMap).comp
      (L.comp (baseEmbedding t S hfree).toContinuousLinearMap))-
      ((ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
      (baseEmbedding t S hfree).toContinuousLinearMap).comp
      (L.comp (baseEmbedding t S hfree).toContinuousLinearMap))*B 0

Definitions. The statement is written with OAI.Kadison.FreeMoments.Free, OAI.Kadison.FreeWordTensor.FaithfulState, OAI.Kadison.FreeWordTensor.faithfulNormed, OAI.Kadison.FreeWordTensor.letterSpace, OAI.Kadison.FreeWordTensor.preSpace and OAI.Kadison.FreeWordTensor.Space (definitions bundle OAIQ039C010 of this split); OAI.Kadison.FreeBlocks.factorSpace (definitions bundle OAIQ039C025 of this split); OAI.Kadison.FreeBlocks.multiplicityLevel and OAI.Kadison.FreeBlocks.multiplicityTensor (definitions bundle OAIQ039C036 of this split); OAI.Kadison.FreeBlocks.baseEmbedding (definitions bundle OAIQ039C055 of this split); OAI.Kadison.FreeBlocks.repeatedOutput (definitions bundle OAIQ039C059 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ039C059

Formal statement

section

attribute [-instance] _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 UniformSpace

open scoped InnerProductSpace ComplexOrder

open ContinuousLinearMap

namespace Kadison.FreeBlocks

open FreeWordTensor FreeMoments DihedralCorner

section Base

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

variable (t : FaithfulState A) (S : Bool → StarSubalgebra ℂ A)
  (f : D →⋆ₐ[ℂ] A) (hf : Function.Injective f) (hS : S false=f.range)
  (hfree : Free S t.functional.toLinearMap)

lemma base_block_compression
    (G L : t.functional.GNS →L[ℂ] t.functional.GNS)
    (B : ∀ k, multiplicityLevel t S true k →L[ℂ] multiplicityLevel t S true k)
    (hb : ∀ k a (x : letterSpace t S false) v,
      G (repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k v)=
        repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k (B k v))
    (hb' : ∀ k a (x : letterSpace t S false) v,
      G.adjoint (repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k v)=
        repeatedOutput t S f hf hS hfree Bool.false_ne_true a x k ((B k).adjoint v))
    (sl : letterSpace t S true) (sn : ‖sl‖=1) :
    (ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
      (baseEmbedding t S hfree).toContinuousLinearMap).comp
      ((G*L-L*G).comp (baseEmbedding t S hfree).toContinuousLinearMap)=
    B 0*((ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
      (baseEmbedding t S hfree).toContinuousLinearMap).comp
      (L.comp (baseEmbedding t S hfree).toContinuousLinearMap))-
      ((ContinuousLinearMap.adjoint (𝕜:=ℂ) (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
      (baseEmbedding t S hfree).toContinuousLinearMap).comp
      (L.comp (baseEmbedding t S hfree).toContinuousLinearMap))*B 0 := by
  sorry

end Base

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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