Always Lie?
An AI preprint claims the Hilbert–Smith conjecture in every finite dimension
Covers result 304 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:38. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Hilbert’s fifth problem, from 1900, asked how far Lie’s theory of continuous symmetry survives without smoothness. Gleason, and Montgomery and Zippin, settled it in 1952 for groups that are themselves manifolds, and Yamabe extended the theory to all locally compact groups. The Hilbert–Smith conjecture, named for P. A. Smith, asks the same for groups that act on a manifold: a locally compact group acting faithfully on a connected manifold must be a Lie group. By classical work, it is enough to show that the p-adic integers cannot act faithfully. They form a group shaped like a Cantor set, with whole copies of themselves in every small ball, which Lie groups never have. The conjecture was known in dimensions up to 2, by Montgomery and Zippin, and in dimension 3, by Pardon in 2013, and for Lipschitz, Hölder and quasiconformal actions. Yang showed that a faithful action would make the orbit space jump from dimension n to n + 2, in cohomological dimension: strange, but not a contradiction.
Result 304 is one OpenAI paper, 46 pages, dated September 23, 2026, and made public October 6. It claims that the p-adic integers have no faithful, jointly continuous action on any connected, Hausdorff, second-countable manifold without boundary, in any finite dimension. With the classical reduction, this gives the Hilbert–Smith conjecture for second-countable groups in every finite dimension. The manifold may be noncompact, nonorientable or not triangulable.
What it doesn’t claim
Anything about infinite-dimensional manifolds, where the p-adic integers do act faithfully, on the Hilbert cube; groups or manifolds that are not second-countable; or homology manifolds. The reduction to the p-adic integers is cited, not reproved.
Where it stands
- Posted as a preprint
- OpenAI, 46 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No formalization for this item
Last checked October 19, 2026. The paper has not been revised, and there is no Lean formalization; a community attempt stalled for lack of library support. OSVerify has not reviewed it. We found no specialist comment and no reported error; two AI-assisted read-throughs list steps that deserve expert checking. The day after the release, a second AI-assisted proof by a different method appeared, with a Lean proof of the second-countable statement that a reviewer reproduced. That supports the statement, but it does not check OpenAI’s argument.
How the proof goes
- A signature with bounded denominators. The proof shrinks to a tiny subgroup acting on one chart, crosses with Euclidean space to reach an odd dimension d, and averages a degree-one pinch over the group to get a test map through the orbit space. It builds a new real signature invariant, from Witt groups of sheaves pushed forward along arbitrary continuous maps, and proves that on the odd sphere its integral classes have denominators bounded by a fixed power of 2.
- Pieces that cannot fit. Splitting the sphere’s signature class by characters of the group gives pᵏ integral pieces of equal size, summing to four times the orientation class, for every k. Each piece would be 4/pᵏ, which, once k is large enough, does not fit. The orbit space never needs a dimension bound, so the proof never goes through Yang’s dimension jump.
The numbers in the video
- Ours: the 2-adic integers drawn as a Cantor set have dimension about 0.63 in that picture; a rotation of the circle by 0.037 leaves the band |θ| < 0.5 after 14 steps; the multiples of 2ᵏ lie within 2⁻ᵏ of zero.
- Ours: the paper’s unoptimized bound in the smallest case, d = 3, is 2⁵⁴⁶, a number with 165 digits, so for p = 2 the contradiction needs k of at least 549; for four-manifolds the bound has over 13,000 digits.
Read the source
- The Hilbert–Smith Conjecture in Every Finite Dimension (PDF), 46 pages.
- The rival AI-assisted proof and its Lean development (GitHub).
- The collection’s catalog, where this is result 304.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Hilbert’s fifth problem, from 1900, asked how far Lie’s theory of continuous symmetry survives without smoothness. Gleason, and Montgomery and Zippin, settled it in 1952 for groups that are themselves manifolds, and Yamabe extended the theory to all locally compact groups. The Hilbert–Smith conjecture, named for P. A. Smith, asks the same for groups that act on a manifold: a locally compact group acting faithfully on a connected manifold must be a Lie group. By classical work, it is enough to show that the p-adic integers cannot act faithfully. The p-adic integers form a group shaped like a Cantor set. Our computer draws the two-adic integers as one, with dimension about 0.63 in that picture. Lie groups have no small subgroups: our computer rotates a circle by a small angle, and its powers leave a small band after 14 steps. But the p-adic integers contain whole copies of themselves in every small ball: in the two-adic integers, the multiples of 2ᵏ lie within 2⁻ᵏ of zero. And they do act faithfully on spaces that are not manifolds, such as a Cantor set, where adding 3ᵏ moves every point, but by at most 3⁻ᵏ. The conjecture was known in dimensions up to 2, by Montgomery and Zippin, and in dimension 3, by Pardon in 2013, and for actions that are Lipschitz, Hölder above a threshold, or quasiconformal. Yang showed that a faithful action would make the orbit space jump from dimension n to n + 2, in cohomological dimension: strange, but not a contradiction. From dimension 4 on, the conjecture was open.
1:44The claim
Result 304 is one OpenAI paper, 46 pages, dated September 23 and made public October 6, 2026. It claims that the p-adic integers have no faithful, jointly continuous action on any connected, Hausdorff, second-countable manifold without boundary, in any finite dimension. With the classical reduction, this gives the Hilbert–Smith conjecture for second-countable groups in every finite dimension. The manifold may be noncompact, nonorientable or not triangulable. Not claimed: infinite-dimensional manifolds, where the p-adic integers do act faithfully, on the Hilbert cube; groups or manifolds that are not second-countable; or homology manifolds. The reduction to the p-adic integers is cited, not reproved.
2:37How the proof goes
The proof shrinks to a tiny subgroup acting on one chart, crosses with Euclidean space to reach an odd dimension d, and averages a degree-one pinch over the group to get a test map through the orbit space. It then builds a new real signature homology, from Witt groups of sheaves pushed forward along arbitrary continuous maps, and proves that on the odd sphere its integral classes have denominators bounded by a fixed power of two. Splitting the sphere’s signature class by characters of the group gives pᵏ integral pieces of equal size, summing to four times the orientation class, for every k. Each piece would be 4/pᵏ, which, once k is large enough, does not fit. Our computer works out the paper’s unoptimized bound in the smallest case, d = 3: 2⁵⁴⁶, a number with 165 digits, so for p = 2 the contradiction needs k of at least 549; for four-manifolds the bound has over 13,000 digits. The orbit space never needs a dimension bound, so the proof never goes through Yang’s dimension jump.
3:46Where it stands
The paper is unrefereed and unchanged since release, and there is no Lean formalization; a community attempt stalled for lack of library support. We found no specialist comment and no reported error; two AI-assisted read-throughs list steps that deserve expert checking. The day after the release, a second AI-assisted proof by a different method appeared, with a Lean proof of the second-countable statement that a reviewer reproduced. That supports the statement, but it does not check OpenAI’s argument. If it holds, every second-countable, locally compact group acting faithfully on a connected finite-dimensional manifold is a Lie group: in every finite dimension, a manifold’s locally compact symmetry groups are never wilder than Lie groups.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.