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Kadison’s similarity conjecture

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Proves Kadison’s similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a \(*\)-homomorphism after conjugation by a bounded invertible operator. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1 (Similarity)

In Kadison’s similarity theorem through uniform derivation estimates, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Corollary 1.3 (Universal hyperreflexivity) in the same paper () and Theorem 1.2 (Uniform commutator estimate) in the same paper ().

Let A be any unital complex C^*-algebra, let H be any complex Hilbert space, and let \pi:A\to\mathcal B(H) be a bounded, complex-linear, unital algebra homomorphism. There exists an operator S\in\mathcal B(H), invertible with bounded inverse, such that

\rho(a)=S\pi(a)S^{-1}\qquad(a\in A)

is a *-homomorphism. In particular, \rho(a^*)=\rho(a)^* for every a\in A.

Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

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Kadison’s similarity theorem through uniform derivation estimates

23 September 2026, current edition.

Read it herePDF

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a \(*\)-homomorphism. This resolves Kadison’s similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

KadisonSimilarity

The paper states

Theorem 1.1 (Similarity)

In Kadison’s similarity theorem through uniform derivation estimates, 23 September 2026

Let A be any unital complex C^*-algebra, let H be any complex Hilbert space, and let \pi:A\to\mathcal B(H) be a bounded, complex-linear, unital algebra homomorphism. There exists an operator S\in\mathcal B(H), invertible with bounded inverse, such that

\rho(a)=S\pi(a)S^{-1}\qquad(a\in A)

is a *-homomorphism. In particular, \rho(a^*)=\rho(a)^* for every a\in A.

The Lean source’s comment reads: /-- Theorem thm:similarity: every bounded unital complex-algebra representation of a unital C-star algebra is similar to a star representation. -/

The Lean statement whose source cites this theorem

The Lean source cites this theorem. No review of whether the statements mean the same thing is recorded.

Kadison’s similarity theorem and uniform hyperreflexivity

KadisonSimilarity.lean, the release lists it as a supporting statement.

theorem universalCommutatorTheorem : UniversalCommutatorTheorem.{u} := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

The file states two more theorems: universalHyperreflexivity, similarityTheorem.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.

Read the complete Lean statement

UniformCommutator

The paper states

Theorem 1.2 (Uniform commutator estimate)

In Kadison’s similarity theorem through uniform derivation estimates, 23 September 2026

There is an absolute constant C<\infty such that, for every complex Hilbert space K, every unital von Neumann algebra P\subset\mathcal B(K), every Y\in\mathcal B(K), every integer h\ge1, and every X\in M_h(P),

\|[Y^{(h)},X]\|\le C\,g_P(Y)\,\|X\|.

Here M_h(P) acts naturally on K^h, and [T,U]=TU-UT.

The words they share: commutator, integer, uniform, von, neumann, complex, algebra, space, constant, estimate, hilbert.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Uniform amplified commutator estimate

UniformCommutator.lean, the release lists it as a supporting statement.

theorem uniform_commutator :    ∃ C : ℝ, 0 ≤ C ∧ ∀ (K : Type u) [NormedAddCommGroup K] [InnerProductSpace ℂ K]      [CompleteSpace K] (P : VonNeumannAlgebra K) (Y : K →L[ℂ] K)      (h : ℕ), 1 ≤ h → ∀ (X : Matrix (Fin h) (Fin h) (K →L[ℂ] K)),      (∀ i j, X i j ∈ P) →      ‖diagonalAmplification h Y * matrixOperator h X -        matrixOperator h X * diagonalAmplification h Y‖ ≤        C * commutatorSeminorm P Y * ‖matrixOperator h X‖ := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: similarityTheorem : SimilarityTheorem.{u,v} := ∀ (A : Type u) [CStarAlgebra A] (K : Type v) [Hilbert space K] (π : BoundedUnitalHom A K), SimilarToStar A K π with BoundedUnitalHom A H := {π : A →ₐ[ℂ] (H →L[ℂ] H) // Continuous π} and SimilarToStar π := ∃ S : (H →L[ℂ] H)ˣ, ∀ a, S π(a⋆) S⁻¹ = (S π(a) S⁻¹)⋆: Kadison’s similarity conjecture for arbitrary Hilbert spaces, plus universalHyperreflexivity (one constant for all von Neumann algebras) and the uniform amplified commutator estimate.

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

Kadison’s similarity problem asks whether every bounded unital homomorphism from a unital complex $C^*$-algebra into operators on a Hilbert space is similar to a $*$-homomorphism. The formalization proves this for arbitrary Hilbert spaces. It also gives one universal hyperreflexivity constant for all unital von Neumann algebras.

The linked commutator estimate is uniform over all finite matrix amplifications: its constant is independent of the algebra, Hilbert space, and amplification size. This is the quantitative derivation estimate supporting the similarity result.

AI-drafted comparisons, with no recorded human review

  • Kadison’s similarity theorem and uniform hyperreflexivity KadisonSimilarity

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main similarity theorem for every unital C*-algebra and Hilbert space, together with the uniform commutator estimate and universal hyperreflexivity.

  • Uniform amplified commutator estimate UniformCommutator

    Unreviewed AI draft: states a supporting result. States the uniform amplified commutator estimate, which the paper states as a theorem in its introduction and the note describes as the estimate supporting the similarity result; similarity itself is KadisonSimilarity.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Kadison’s similarity theorem and uniform hyperreflexivity KadisonSimilarity

One report, at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report

Uniform amplified commutator estimate UniformCommutator

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

References outside the release

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Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

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Identify the theorem the catalogue describes

  • Family 288 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Compare the proposed paper and Lean matches

  • Family 288 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form