>>187692, >>187693, >>187694, >>187695, >>187696, >>187697, >>187698, >>187699, >>187700, >>187701, >>187702, >>187703, >>187704, >>187705, >>187706, >>187707, >>187708, >>187709, >>187710, >>187711, >>187712, >>187713, >>187714, >>187715, >>187716, >>187717, >>187718, >>187719, >>187720, >>187721, >>187722
Elon Musk @elonmusk - Grok only gets better from here
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Paata Ivanisvili @PI010101
Grok 4.5 just constructed an explicit counterexample to hypercontractivity for the Poisson semigroup (the square root of the Laplace–Beltrami operator) on the 4-sphere.
Back in 2021, with Rupert Frank
https://arxiv.org/abs/2101.06209
we proved that hypercontractivity holds in dimensions ≤3 and fails in sufficiently large dimensions (for example, in dimension 13). Grok's example shows that it already fails in dimension 4, making our earlier result sharp.
I also tested this problem on several other frontier AI models. One of them also managed to find a counterexample, but I particularly like Grok 4.5's solution: it is explicit, simple, and elegant.
The attached files were generated entirely by Grok 4.5 build (with zero intervention on my side).
https://x.com/elonmusk/status/2075387273797853612

Elon Musk @elonmusk - Grok gaming
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tetsuo @tetsuoai
Video: Grok 4.5 in Grok Build created an FPS game in under an hour.
The prompt was simple. I told it to write a game design document and pull free assets from the web. Then I had it create a TODO.md with implementation phases and run a loop to build out each phase.