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Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations

Theorem 1.1: for every irrational $\theta\in(0,1)$ there is a continuous $f:\mathbb T\to[0,1]$ vanishing only at $1$ , with $\int\log f\,dm=-\infty$ , such that every projection $p\in R_\theta$ with $(1-p)Uf(V)p=0$ is $0$ or $1$ ; the operator is nonzero and norm-quasinilpotent. The angle is prescribed in advance with no Diophantine condition. Invariant subspaces of the Hilbert-space representation whose projections lie outside the factor are not excluded. An appendix gives a second realization on an infinite product of circles.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 27, 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: Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations · Zhu-Fang-Shi invariant projection question

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Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations

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No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the question as quoted from Zhu-Fang-Shi. The obstruction and transfer arguments were not refereed. Lean-checked on the release's Comparator challenge IrrationalRotation, found through lean/docs/293.md; this challenge is not in the release's formalization catalogue, but its solution module OAI.Analysis.IrrationalRotation.Main exists at the pinned commit. Its statement was read here: prescribed_irrational_counterexample gives, for every irrational theta in (0,1), a continuous f with values in [0,1], zero exactly at one point and log integral minus infinity, whose weighted rotation lies in the generated von Neumann algebra, has only trivial invariant projections there, is nonzero, norm-quasinilpotent and has vector Fuglede-Kadison determinant zero. That covers the headline. Not rebuilt here.

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Zhu, Fang, Shi 2017, On a class of operators in the hyperfinite II1 factor

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Backward intertwiners and a transitive commutant (companion)

paper · pending

paper link supplied by the VibeMathed registry.

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Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean Comparator challenge IrrationalRotation (OpenAI math release)

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lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge IrrationalRotation (OpenAI math release)

lean artifact · pending

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VibeMathed record: Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations evidence for this event

This event attributed to Unreleased internal OpenAI model

Zhu, Fang, Shi 2017, On a class of operators in the hyperfinite II1 factor evidence for this event

Lean Comparator challenge IrrationalRotation (OpenAI math release) evidence for this event

Backward intertwiners and a transitive commutant (companion) evidence for this event

Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations evidence for this event

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