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Math-2391-A1 · revision r1 · on Math-2391

OpenAI claim summary for Math-2391

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 289, 2026-09-23): The primary manuscript establishes uniform near-identity unitary conjugacy of sufficiently two-sided close unital von Neumann algebras. The theorem gives spatial isomorphism and near-identity conjugacy for the von Neumann, common-identity subcase. This problem also quantifies over general C*-algebras and asks about their structural invariants at completely bounded distance; that full class is not covered.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Rigidity of Completely Close \$C^*\$-Algebras
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Rigidity of Completely Close \$C^*\$-Algebras

OpenAI's claim

For any prescribed small unitary displacement, the paper claims one uniform Kadison--Kastler tolerance guaranteeing conjugacy of close unital von Neumann algebras on any complex Hilbert space. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $\varepsilon>0$, OpenAI claims a single $\delta(\varepsilon)>0$ valid for every complex Hilbert space $H$ and all unital von Neumann algebras $M,N\subset B(H)$ with common identity: if their norm-unit-ball Hausdorff distance $d(M,N)<\delta$, then \[ \exists u\in B(H):\quad u^*u=uu^*=I_H,\quad uMu^*=N,\quad\|u-I_H\|<\varepsilon. \] No separability, type or injectivity hypothesis is imposed on these von Neumann algebras.

Outline of the argument

Manuscript 1 (family 289, 2026-09-23): For finite factors, free-word estimates control amplification while boundary averaging corrects nearby unitaries into a measurable cocycle. Double ergodicity makes the resulting correspondence multiplicative and yields a spatial implementation. Properly infinite algebras are handled through matrix amplification and modular preparation; alignment of centers and limits over separable restrictions assemble the universal conclusion.

Formal verification

For family 289, the scope page documents the following selections. The selection gives a tolerance depending only on the requested distance of an implementing unitary from identity, for two close common-identity von Neumann algebras. The separate one-way near-inclusion and general norm-closed examples are outside it. The documented statement covers the main claim of manuscript 1. Comparator files: StrongKadisonKastler.lean. Declarations: OAI.KadisonKastler.universal_strong_stability.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources