# 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. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-304/ Last updated: 2026-10-14 **Claims a proof of Hilbert–Smith: a locally compact group acting faithfully on a connected manifold is a Lie group.** **Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Topology · **Entry:** 304 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. 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. ## At a glance - **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 point[1]. The Hilbert–Smith conjecture asks whether a locally compact group acting faithfully and continuously on a connected manifold must be a Lie group[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 1952[3]. Dimensions one and two were classical, and Pardon proved dimension three[4]. Other work added smoothness, such as Lipschitz actions[5]. Known structure theory reduces the conjecture to one case: ruling out faithful actions of the p-adic integers, for a prime p[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 kernel[7]. So it claims the conjecture in every finite dimension[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 L[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 contradiction[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](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf) - September 23, 2026 ## Quoted lines from the paper (10) [1] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > An action is \emph{faithful} when its kernel is trivial: no nonidentity element fixes every point. [2] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > The Hilbert--Smith conjecture asks whether a locally compact group acting faithfully and continuously on a connected topological manifold must be a Lie group. [3] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > 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. [4] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > Pardon proved the three-dimensional conjecture. [5] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > Repov\v{s}--\v{S}\v{c}epin proved the Lipschitz case on Riemannian manifolds, and Maleshich obtained an obstruction for H\"older actions on closed manifolds when a common exponent exceeds $n/(n+2)$. [6] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > 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. [7] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > Every jointly continuous action of \(\Zp\) on a connected Hausdorff second-countable topological \(n\)-manifold without boundary has nonzero kernel. [8] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > 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. [9] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > We construct an abelian group \(E_d(S^d)\), an orientation class \(u_d\), and a positive integer \(L_d\). [10] From [The Hilbert–Smith conjecture in every finite dimension](https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf): > Each image is therefore \((4/p^k)\overline{u}_d\), which contradicts … as soon as \(p^k>4L_d\). ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Four-dimensional disk embedding and Wall's conjecture](https://closedform.nishachay.tech/e/openai-math-2026-305/) - Topology - [The purely cosmetic surgery conjecture](https://closedform.nishachay.tech/e/openai-math-2026-306/) - Topology - [Quillen's conjecture in rational homology](https://closedform.nishachay.tech/e/openai-math-2026-310/) - Topology - [The Hovey–Strickland and Chai conjectures](https://closedform.nishachay.tech/e/openai-math-2026-311/) - Topology - [The four-dimensional Singer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-315/) - Topology - [Topology](https://closedform.nishachay.tech/science/mathematics/topology/) - Field · 18 results - [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results - [Open since before 1940](https://closedform.nishachay.tech/open-since/before-1940/) - Era · 16 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20304%5D%20)