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The strong Kadison-Kastler conjecture

Theorem 1.1: for every $\varepsilon>0$ there is $\delta(\varepsilon)>0$ , independent of the algebras, their type, representations and the Hilbert space, such that unital von Neumann algebras $M,N\subseteq B(H)$ with $d(M,N)<\delta$ satisfy $uMu^*=N$ with $\|u-1\|<\varepsilon$ . The proof combines amplification estimates, a modular and core analysis for type III, and reductions across types. It concerns von Neumann algebras only; the companions show that one-sided near inclusions and close separable C*-algebras behave differently. No explicit $\delta(\varepsilon)$ is given.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The strong Kadison-Kastler conjecture · Strong Kadison-Kastler conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The strong Kadison-Kastler conjecture

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it with the 1972 reference; it claims a tolerance $\delta(\varepsilon)$ uniform over all unital von Neumann algebras, representations and Hilbert spaces. The proof was not refereed. The challenge ComparatorChallenges/StrongKadisonKastler.lean is not in lean/formalization.yaml; it is found through lean/docs/289.md and its solution module OAI.Analysis.KadisonKastler.StrongStability exists at the pinned commit. Its statement was read here: for every $\varepsilon>0$ there is $\delta>0$ such that for every complex Hilbert space $H$ and von Neumann algebras $M,N$ on $H$ with Hausdorff distance of unit balls below $\delta$ there is a unitary $v$ with $vMv^*=N$ and $\|v-1\|<\varepsilon$ . This states the headline. Not rebuilt here. The paper cites the release's similarity companion for a related commutator estimate.