The Hilbert-Smith Conjecture
Hilbert's fifth problem asked how far differentiability can be removed from the theory of continuous transformation groups. Gleason, Montgomery-Zippin and Yamabe settled the case of locally Euclidean groups. The Hilbert-Smith conjecture asks about groups acting on manifolds: if a locally compact group acts faithfully and continuously on a connected topological manifold , must be a Lie group? Equivalently, can the -adic integers act faithfully on a manifold? It was known in dimensions at most 2, in dimension 3 (Pardon 2013), and for Lipschitz, quasiconformal and some Holder actions. Is every such group a Lie group, in every dimension?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric topology; transformation groups
- Posed by
- Named for David Hilbert (fifth problem) and P. A. Smith; the manuscript cites Hilbert (1902) and Pardon (2013) for the statement
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 65 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
For every , every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable topological -manifold without boundary is a Lie group. The core theorem is that every continuous -action on such a manifold has nonzero kernel, by a contradiction between equal integral classes summing to a fixed class and a fixed denominator bound in a Witt-group lattice. Noncompact, nonorientable and nontriangulable manifolds and arbitrary stabilizers are allowed. Manifolds with boundary are not treated.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the main theorem and its p-adic reduction were read against the conjecture as stated by the manuscript's references. The proof, built on Witt groups of sheaf categories and a fixed signature lattice, was not refereed. No Lean formalization is listed. Scope stated by the paper: second-countable Hausdorff manifolds without boundary, and second-countable groups.