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

Proved
OAI.Kadison.InfiniteAmplification.separable_space

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.InfiniteAmplification, with open scoped InnerProductSpace Topology):

theorem separable_space : TopologicalSpace.SeparableSpace (Space ι H)

Definitions. The statement is written with OAI.Kadison.InfiniteAmplification.Space (definitions bundle OAIQ039C006 of this split).

Used by. The proofs of OAI.Kadison.CentralSpectral.exists_stable_center_spectral_model, OAI.Kadison.CentralSpectral.separable_uniform_bound and OAI.Kadison.CentralSpectral.stable_model_bound of this split import it.

Preamble
import Mathlib
import Definitions.Def_OAIQ039C006
Formal statement
section

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 Topology

namespace Kadison.InfiniteAmplification

variable {ι H : Type*} [Countable ι] [NormedAddCommGroup H] [InnerProductSpace ℂ H]
  [CompleteSpace H] [TopologicalSpace.SeparableSpace H]

theorem separable_space : TopologicalSpace.SeparableSpace (Space ι H) := by
  sorry

end Kadison.InfiniteAmplification

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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