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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://t.co/AvXd1zWIpX 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).
PI010101 ↗