What was already known
The general hyperinvariant-subspace question remained open; earlier weighted-shift results and Arveson’s density theorem had additional hypotheses.
AI involvement
OpenAI is the credited author and attributes its manuscript and proof artifacts to an internal model. Item-specific autonomy is not documented.
The original manuscript credits OpenAI as author, and the repository explicitly attributes manuscript and proof-artifact production to an internal OpenAI model; result-specific autonomy is not documented.
Importance
Negatively answers the hyperinvariant-subspace question in the stated Hilbert-space setting.
Limitations
OpenAI reports Lean checking of the stated operator and commutant properties. No named human whole-proof check or explanation of the construction is documented.
Technical verification
OpenAI reports a Lean formalization of the unrestricted separable infinite-dimensional complex Hilbert-space existence claim. The comparator challenge states nonzero, quasinilpotent S with a transitive, proper, strongly closed commutant. The public challenge has a sorry placeholder, while the configured solution module is the claimed proof.
Formal checking
Existence, on every separable infinite-dimensional complex Hilbert space, of a nonzero quasinilpotent operator with a proper strongly closed transitive commutant; Source-reported formal proof in OAI.Analysis.Hyperinvariant.Main compared with HyperinvariantSubspaces.lean; permitted axioms propext, Quot.sound and Classical.choice
- Checker
- Lean 4 kernel and OpenAI Comparator configuration
- Scope
- Existence, on every separable infinite-dimensional complex Hilbert space, of a nonzero quasinilpotent operator with a proper strongly closed transitive commutant
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No identified person's explanation of the model-origin weighted-fiber and backward-intertwiner construction was found; Few is the documentation default, not a report of confusion.
Correction and publication history
Negatively answers the hyperinvariant-subspace question in the stated Hilbert-space setting.
Read this published revision