Short··0:21·Topology
Hilbert-Smith Conjecture: OpenAI Claims a Proof in Every Dimension
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About this video
Hilbert's fifth problem asked which groups can act continuously on a space. Gleason, Montgomery and Zippin showed that the answer is Lie groups, like rotations. One gap remained: could a strange group like the p-adic integers act faithfully on a manifold? That gap is the Hilbert–Smith conjecture, previously known only in low dimensions. OpenAI claims a proof in every finite dimension: every locally compact, second-countable Hausdorff group acting faithfully and continuously on a connected finite-dimensional manifold (without boundary) is a Lie group. This is OpenAI's claim and has not yet been peer reviewed.
The solenoid shown is the 3-adic solenoid, built from loops wrapping 3 times inside each torus, level after level. Adding 1 is the 3-adic odometer. At the end it collapses to a single circle turning, which is a Lie group action.
The animation illustrates the result; it is not a proof.
Paper:
- The Hilbert–Smith conjecture in every finite dimension: github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-…
Original visuals (three.js) and synthesized sound: one drone per solenoid level, each an octave above the last; every chime is a level turning over.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/304
About the problem
Topology·OpenAI Math·Result 304
The Hilbert–Smith conjecture in every dimension
Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.
Topics
Hilbert-Smith conjectureHilbert Smith conjectureHilbert's fifth problemLie groupp-adic integers3-adic integerssolenoidp-adic solenoidodometergroup actiontransformation groupsmanifoldtopologyMontgomery ZippinGleasonOpenAI mathAI proofAI mathematicsmath visualizationmaths shortsGradient Damage
