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Grok 4.6 (build) solved the “Greedy is least speedy” conjecture of Holmes–Holroyd–Ramírez by proving their stronger Conjecture 5 (shown in the figure), which immediately implies it. chat transcript: https://grok.com/share/c2hhcmQtNA_73c57e1f-2ff7-467b-a325-250dcd6b9e96 polished paper: https://drive.google.com/file/d/1nNj8pwN8U5gnR8AqrBUEWCm-VbR5RBqV/view?usp=sharing I was chatting with my colleague (one of the strongest mathematicians I know) about AI’s capabilities in mathematics, and he seemed a bit skeptical. So I proposed a test: give me a problem that seems within reach for an expert but beyond current AI. He suggested Conjecture 5 from the Holmes–Holroyd–Ramírez paper. After some experimenting I told him, “Look, I think AI solved it.” He was a little surprised and, out of curiosity, gave me two other questions that he believes he can solve, to see whether AI could solve them as well. So far, AI has not solved either of them. This conjecture (now theorem) has an interesting application: imagine a particle on the integer line ℤ. At each position x, it moves one step right with probability pₓ in (0,1) and one step left with probability 1 − pₓ. The environment is m-periodic, so p₀, …, pₘ₋₁ repeat forever. Starting from 0, let Xₙ be its position after n steps. It is known that lim Xₙ/n exists almost surely, call it velocity v. Now fix the probabilities p₀, …, pₘ₋₁, but allow yourself to rearrange them within one period. How should they be arranged to make the particle escape from its starting point as slowly as possible? It turns out that an optimal arrangement is the pendulum (greedy) arrangement z* shown in the figure (except the degenerate case ∏ⱼ (1 − pⱼ) = ∏ⱼ pⱼ when the random walk is recurrent so every arrangement has v=0).