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  1. Rohan Paul67

    Grok 4.5 成功构造出 Poisson 半群在 4 维球面上超收缩性的显式反例。2021 年有论文证明该性质在维度 ≤3 时成立、在足够大维度(如 13 维)时失效,Grok 4.5 的反例表明其在 4 维即已失效,使此前结果达到最优。该反例由 Grok 4.5 完全自主生成(零人工干预),简洁优雅。Rohan Paul 指出,未来数学家试图证明优美猜想前,应先用前沿 AI 尝试推翻它。

    引用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).

  2. Elon Musk58

    Elon Musk 发推称 Grok 只会变得更好。引用数学家 @PI010101 的推文:Grok 4.5 为四维球面上的 Poisson 半群构造了一个显式反例,证明超压缩性在维度 4 即已失效。此前 2021 年与 Rupert Frank 的论文仅证明维度 ≤3 成立、足够大维度失效,Grok 4.5 的结果使该结论达到最优。该数学家还测试了其他前沿 AI 模型,其中有一个也找到了反例,但 Grok 4.5 的解更显式、简洁且优雅,附件完全由 Grok 4.5 生成,无人工干预。

    引用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).