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

Proved

OAI.Kadison.RectangularColumn.column_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.RectangularColumn, with open scoped InnerProductSpace BigOperators):

theorem column_bound [PartialOrder A] [StarOrderedRing A] {ι : Type*} [Fintype ι]
    (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K)) (lam : A →⋆ₐ[ℂ] (F →L[ℂ] F))
    (Δ : A →L[ℂ] (K →L[ℂ] F))
    (hΔ : ∀ a b, Δ (a*b)=(Δ a).comp (σ b)+(lam a).comp (Δ b))
    (a : ι → A) (ξ : K) :
    ‖WithLp.toLp 2 (fun i => Δ (a i) ξ)‖≤
      (4*‖Δ‖)*CyclicColumns.columnNorm a*‖ξ‖

Definitions. The statement is written with OAI.Kadison.CyclicColumns.columnNorm (definitions bundle OAIQ039C004 of this split); OAI.Kadison.RectangularColumn.instNormedSpaceComplexContinuousLinearMapId_solutions (definitions bundle OAIQ039C016 of this split).

Used by. The proofs of OAI.Kadison.CyclicSpatial.exists_embedded_cyclic_intertwiner and OAI.Kadison.commutator_column_bound of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C004
import Definitions.Def_OAIQ039C016

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier
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 InnerProductSpace BigOperators

namespace Kadison.RectangularColumn

variable {A K F : Type*} [CStarAlgebra A]
  [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]
  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]

attribute [local instance] OAI.Kadison.RectangularColumn.instNormedAddCommGroupContinuousLinearMapComplexId_solutions
attribute [local instance] OAI.Kadison.RectangularColumn.instNormedSpaceComplexContinuousLinearMapId_solutions
theorem column_bound [PartialOrder A] [StarOrderedRing A] {ι : Type*} [Fintype ι]
    (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K)) (lam : A →⋆ₐ[ℂ] (F →L[ℂ] F))
    (Δ : A →L[ℂ] (K →L[ℂ] F))
    (hΔ : ∀ a b, Δ (a*b)=(Δ a).comp (σ b)+(lam a).comp (Δ b))
    (a : ι → A) (ξ : K) :
    ‖WithLp.toLp 2 (fun i => Δ (a i) ξ)‖≤
      (4*‖Δ‖)*CyclicColumns.columnNorm a*‖ξ‖ := by
  sorry

end Kadison.RectangularColumn

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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