Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
In Vanishing of higher bounded Hochschild cohomology, 23 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let M be a complex von Neumann algebra and let k\geq2. Every f\in C_b^k(M,M) satisfying df=0 has the form f=dg for some g\in C_b^{k-1}(M,M). Consequently H_b^k(M,M)=0.
No Lean statement in the release is paired with this theorem.
In Vanishing of higher bounded Hochschild cohomology, 23 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let M be a von Neumann algebra and k\geq2.
If f\in C_b^k(M,M) is a cocycle, there are a separately normal bounded cocycle f_0 and a bounded (k-1)-cochain b such that f=f_0+db.
If M has no type \mathrm{II}_1 central summand, then H_b^k(M,M)=0.
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KadisonRingroseThe paper states
In Vanishing of higher bounded Hochschild cohomology, 23 September 2026
Let M be a von Neumann algebra and k\geq2.
If f\in C_b^k(M,M) is a cocycle, there are a separately normal bounded cocycle f_0 and a bounded (k-1)-cochain b such that f=f_0+db.
If M has no type \mathrm{II}_1 central summand, then H_b^k(M,M)=0.
The words they share: cochain, algebra, cocycle, bounded, von, neumann.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
KadisonRingrose.lean, the release lists it as a supporting statement.
theorem main_result {M : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] [WStarAlgebra M] (n : ℕ) (f : Cochain M (n + 2)) (hf : ∀ x : Fin (n + 3) → M, differentialValue f x = 0) : ∃ g : Cochain M (n + 1), ∀ x : Fin (n + 2) → M, differentialValue g x = f x := by sorryThis 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
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: main_result {M : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] [WStarAlgebra M] (n) (f : Cochain M (n+2)) (hf : ∀ x, differentialValue f x = 0) : ∃ g : Cochain M (n+1), ∀ x, differentialValue g x = f x with Cochain M n := ContinuousMultilinearMap ℂ (fun _ : Fin n => M) M (bounded multilinear) and the standard Hochschild differential v₀ f(v₁..vₙ) + Σ_j (-1)^{j+1} f(.., v_j v_{j+1},..) + (-1)^{n+1} f(v₀..v_{n-1}) vₙ with coefficients in M itself: vanishing of bounded Hochschild cohomology in every degree >= 2 for every von Neumann algebra (Mathlib WStarAlgebra), no separability or type assumption. Config note: KadisonRingrose.json has no enable_nanoda key (index.json shows null).
“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.
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.
OpenAI’s own note on what its Lean covers
The formalization proves vanishing of bounded Hochschild cohomology in every degree at least two for a complex von Neumann algebra with coefficients in itself. Every bounded multilinear cocycle of such a degree is the Hochschild differential of a bounded multilinear cochain one degree lower. No separability or type restriction is imposed. The degree-one inner-derivation theorem is outside this selected statement.
Bounded primitives for all Hochschild cocycles of degree at least two KadisonRingrose
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: bounded Hochschild cohomology of every von Neumann algebra with coefficients in itself vanishes in each degree k ≥ 2. The degree-one theorem is outside the statement.
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KadisonRingroseTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original 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 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
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Why this passage is shown
Theorem 1.1
In Vanishing of higher bounded Hochschild cohomology, 23 September 2026
Why this passage is shown
Proposition 1.2 (Classical reductions)
In Vanishing of higher bounded Hochschild cohomology, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, KadisonRingrose
Proof-verification report
MathVet, KadisonRingrose
Lean statement
KadisonRingrose