Prove2Me

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

Proved
OAI.Kadison.FreeBlocks.innerRepresentation_word

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 InnerProductSpace TensorProduct, open UniformSpace, open FreeWordTensor FreeMoments FreeWordAction):

lemma innerRepresentation_word (u : unitary D) (d : D) (n : ℕ)
    (x : BlockParams t S i j n) :
    innerRepresentation t ht f S i hS hfree hgen u
      (sourceLevelEmbedding t f hf S i j hS hfree hij n
        (levelWord t f hf S i j hS d n x)) =
    sourceLevelEmbedding t f hf S i j hS hfree hij n
      (levelWord t f hf S i j hS ((u:D)*d*star (u:D)) n
        (mapBlockD t S i j
          (letterIso t S (innerFamily f i u) (innerFamily_mem f S i hS u)
            (innerFamily_trace t ht f i u) i) n x))

Definitions. The statement is written with OAI.Kadison.FreeMoments.Free, OAI.Kadison.FreeWordTensor.FaithfulState, OAI.Kadison.FreeWordTensor.letterSpace, OAI.Kadison.FreeWordTensor.preSpace, OAI.Kadison.PreHilbert, OAI.Kadison.PreHilbert.power and OAI.Kadison.FreeWordTensor.Space (definitions bundle OAIQ039C010 of this split); OAI.Kadison.FreeWordAction.innerFamily and OAI.Kadison.FreeWordAction.innerFamily_mem (definitions bundle OAIQ039C011 of this split); OAI.Kadison.FreeBlocks.BlockParams (definitions bundle OAIQ039C025 of this split); OAI.Kadison.FreeBlocks.multiplicityLevel, OAI.Kadison.FreeBlocks.multiplicityTensor, OAI.Kadison.FreeBlocks.pullbackState and OAI.Kadison.FreeWordTensor.generatedOperators (definitions bundle OAIQ039C036 of this split); OAI.Kadison.FreeWordAction.innerFamily_trace (definitions bundle OAIQ039C037 of this split); OAI.Kadison.FreeWordAction.letterIso (definitions bundle OAIQ039C046 of this split); OAI.Kadison.FreeBlocks.factorFull, OAI.Kadison.FreeBlocks.factorZero, OAI.Kadison.FreeBlocks.levelWord, OAI.Kadison.FreeBlocks.mapBlockD, OAI.Kadison.FreeBlocks.sourceLevel, OAI.Kadison.FreeBlocks.sourceLevelEmbedding and OAI.Kadison.FreeBlocks.sourceScalar (definitions bundle OAIQ039C056 of this split); OAI.Kadison.FreeBlocks.innerRepresentation (definitions bundle OAIQ039C059 of this split).

Used by. The proof of OAI.Kadison.FreeBlocks.free_word_structure 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

open scoped ComplexOrder InnerProductSpace TensorProduct

open UniformSpace

namespace Kadison.FreeBlocks

open FreeWordTensor FreeMoments FreeWordAction

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

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

variable (f : D →⋆ₐ[ℂ] A) (hf : Function.Injective f)

variable {ι : Type*} (S : ι → StarSubalgebra ℂ A) (i j : ι) (hS : S i=f.range)

variable (hfree : Free S t.functional.toLinearMap)
  (hgen : ∀ a : A, t.functional.gnsStarAlgHom a ∈
    Set.centralizer (Set.centralizer (generatedOperators t S : Set _)))

variable (hij : i≠j)

lemma innerRepresentation_word (u : unitary D) (d : D) (n : ℕ)
    (x : BlockParams t S i j n) :
    innerRepresentation t ht f S i hS hfree hgen u
      (sourceLevelEmbedding t f hf S i j hS hfree hij n
        (levelWord t f hf S i j hS d n x)) =
    sourceLevelEmbedding t f hf S i j hS hfree hij n
      (levelWord t f hf S i j hS ((u:D)*d*star (u:D)) n
        (mapBlockD t S i j
          (letterIso t S (innerFamily f i u) (innerFamily_mem f S i hS u)
            (innerFamily_trace t ht f i u) i) n x)) := by
  sorry

end Kadison.FreeBlocks

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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