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

Proved

OAI.Kadison.MatrixCorners.normalizedStageAlphabet_orthonormal

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.MatrixCorners, with open scoped BigOperators ComplexOrder InnerProductSpace):

lemma normalizedStageAlphabet_orthonormal :
    Orthonormal ℂ (fun g : G => TracialGNS.algebraVector τ (normalizedStageAlphabet φ p hp e α g))

Definitions. The statement is written with OAI.Kadison.TracialGNS.algebraVector (definitions bundle OAIQ039C003 of this split); OAI.Kadison.MatrixCorners.normalizedStageAlphabet and OAI.Kadison.MatrixCorners.projectionIndices (definitions bundle OAIQ039C013 of this split).

Used by. The proof of OAI.Kadison.TracialGNS.corner_arbitrarily_large_alphabet of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C003
import Definitions.Def_OAIQ039C013

Formal statement

section

attribute [-instance] _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 BigOperators ComplexOrder InnerProductSpace

namespace Kadison.MatrixCorners

attribute [local instance] Classical.propDecidable

variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
  (τ : A →ₚ[ℂ] ℂ) (hτ : ∀ a b, τ (a*b)=τ (b*a)) (h1 : τ 1=1)
  {I : Type*} [Fintype I] [DecidableEq I] [Nonempty I]
  (φ : Matrix I I ℂ →⋆ₐ[ℂ] A) (p : Matrix I I ℂ) (hp : IsStarProjection p)
  {G : Type*} [Fintype G] [DecidableEq G] [AddCommGroup G]
  (e : G ≃ projectionIndices p hp)

variable (α : ℝ) (hα : 0<α) (hατ : τ (φ p)=(α:ℂ))

include hτ h1 hα hατ in
lemma normalizedStageAlphabet_orthonormal :
    Orthonormal ℂ (fun g : G => TracialGNS.algebraVector τ (normalizedStageAlphabet φ p hp e α g)) := by
  sorry

end Kadison.MatrixCorners

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)

View graph