The Hilbert–Smith conjecture in every dimension

Claims a proof of Hilbert–Smith: a locally compact group acting faithfully on a connected manifold is a Lie group.

1 paper September 23, 2026 Explained by AI

A Lie group is a group of symmetries that is smooth, like the rotations of a circle. The Hilbert–Smith conjecture says that any reasonable group that moves the points of a manifold faithfully must be a Lie group. It was known in dimensions up to three and under extra smoothness. This paper claims to prove it in every finite dimension, by ruling out actions of the p-adic integers. There is no Lean check.

First step

1902

Hilbert's fifth problem on continuous transformation groups (published as "Mathematical problems", 1902)

Significance

Landmark

It would resolve the Hilbert–Smith conjecture, the open remaining form of Hilbert's fifth problem, in every dimension.

Trust

Claimed, not checked

Field

Topology

1 paper

The explanation

Start with the 2-adic integers. An element is a list of remainders: one mod 2, one mod 4, one mod 8, and so on, each agreeing with the last. The list 1, 3, 7, 15, … describes −1, since 7 + 1 = 8 leaves remainder 0 mod 8. Two elements are close when they agree mod a high power of 2. The elements with remainder 0 mod 8 form a subgroup, and so do those with remainder 0 mod 1024. So this group has subgroups inside every small neighbourhood of zero.

A Lie group is a smooth group of symmetries, like the rotations of a circle. Lie groups never have such small subgroups: near the identity, the only subgroup is the trivial one. A manifold is a space that looks like ordinary n-dimensional space up close. An action is faithful when no element other than the identity fixes every pointSource 1. The Hilbert–Smith conjecture asks whether a locally compact group acting faithfully and continuously on a connected manifold must be a Lie groupSource 2.

The question grew out of Hilbert's fifth problem. Gleason, Montgomery and Zippin settled the version where the group itself is a manifold in 1952Source 3. Dimensions one and two were classical, and Pardon proved dimension threeSource 4. Other work added smoothness, such as Lipschitz actionsSource 5. Known structure theory reduces the conjecture to one case: ruling out faithful actions of the p-adic integers, for a prime pSource 6. The obstacle is that such an action is merely continuous, so there are no derivatives to work with.

The paper claims that every continuous action of the p-adic integers on such a manifold has nonzero kernelSource 7. So it claims the conjecture in every finite dimensionSource 8. The proof builds a group of signature classes, from sheaf duality, that lies inside a fixed lattice. A faithful action would give p^k equal classes adding up to 4 times a fixed class u. Each would be (4/p^k)·u. The lattice only allows multiples of u/L for a fixed whole number LSource 9. Say L were 10 and p = 3. For k = 4, p^k = 81 exceeds 4·10 = 40. Then 4/81 is not a multiple of 1/10, a contradictionSource 10.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof is long and relies on delicate sheaf theory and Witt groups. A claim of this size needs careful expert reading, and no specialist review is published.

What it doesn't show. This does not cover manifolds with boundary or infinite-dimensional spaces, which lie outside the theorem's hypotheses.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark-level claim with no Lean check. It rests on a long argument in sheaf theory and Witt groups, with no published specialist review.

Words used

Lie group: A group that is also a smooth space, with smooth multiplication, like the rotations of a circle.
Manifold: A space that looks like ordinary n-dimensional space near each point.
Faithful action: An action where only the identity element fixes every point.
Locally compact group: A group with a topology in which every point has a small neighbourhood with compact closure.
p-adic integers: Compatible lists of remainders mod p, p², p³, …, for a prime p; two are close when they agree mod a high power.
Kernel: The set of group elements that act as the identity on every point.
Witt group: A group of symmetric forms where forms with a half-sized isotropic part count as zero; it records signature data.
Signature: A whole number from a symmetric form: positive directions minus negative directions.

The original paper

The Hilbert–Smith conjecture in every finite dimension (PDF, opens in a new tab)
September 23, 2026

All quoted lines (10)

An action is faithful when its kernel is trivial: no nonidentity element fixes every point.
The Hilbert–Smith conjecture in every finite dimension Back to text
The Hilbert–Smith conjecture asks whether a locally compact group acting faithfully and continuously on a connected topological manifold must be a Lie group.
The Hilbert–Smith conjecture in every finite dimension Back to text
One central formulation concerns a group that is itself locally Euclidean. Gleason and Montgomery–Zippin resolved that formulation in 1952, and Yamabe extended the structure theory to general locally compact groups.
The Hilbert–Smith conjecture in every finite dimension Back to text
Pardon proved the three-dimensional conjecture.
The Hilbert–Smith conjecture in every finite dimension Back to text
Repov\vs–\vS\vcepin proved the Lipschitz case on Riemannian manifolds, and Maleshich obtained an obstruction for H\"older actions on closed manifolds when a common exponent exceeds $n/(n+2)$ .
The Hilbert–Smith conjecture in every finite dimension Back to text
The mechanism is the exclusion of faithful actions of the additive group of $p$ -adic integers, for a prime $p$ , $\Zp=\varprojlim_k\Z/p^k\Z,$ with its profinite topology.
The Hilbert–Smith conjecture in every finite dimension Back to text
Every jointly continuous action of $\Zp$ on a connected Hausdorff second-countable topological $n$ -manifold without boundary has nonzero kernel.
The Hilbert–Smith conjecture in every finite dimension Back to text
We prove the Hilbert–Smith conjecture in every finite dimension: every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable finite-dimensional topological manifold without boundary is a Lie group.
The Hilbert–Smith conjecture in every finite dimension Back to text
We construct an abelian group $E_d(S^d)$ , an orientation class $u_d$ , and a positive integer $L_d$ .
The Hilbert–Smith conjecture in every finite dimension Back to text
Each image is therefore $(4/p^k)\overline{u}_d$ , which contradicts … as soon as $p^k>4L_d$ .
The Hilbert–Smith conjecture in every finite dimension Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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