OpenAI announced on May 20 that an internal general-purpose reasoning model had autonomously disproved the planar unit-distance conjecture posed by Paul Erdős in 1946 — a problem that asks how many pairs among n points in a plane can sit exactly distance one apart. [1] The model produced a counterexample improving the lower bound from the square-grid construction to n^(1+δ) for a fixed δ greater than zero; Princeton's Will Sawin subsequently refined the explicit improvement to δ = 0.014. [2] Fields medalist Tim Gowers and seven co-authors published a companion remarks paper, with Gowers writing on X that it solves "one of Erdős's favourite questions and one that many mathematicians had tried."

Earlier in the same week, Anthropic co-founder Jack Clark gave the Oxford Martin School annual lecture in which he predicted a "Nobel-worthy breakthrough" from AI inside twelve months and explicitly acknowledged that there is a "non-zero chance it could kill everyone." [3] The two events landed inside the same news cycle the paper has been tracking as Anthropic's Stainless acquisition and the closing of the $30 billion round.

The structural read is that the public-frontier capability discourse is now Vatican-encyclical-relevant in the literal sense — Magnifica Humanitas publishes Monday. Erdős set the conjecture in 1946; an autonomous system solved it in May 2026; an AI-lab co-founder told an Oxford audience the same week that there is a real probability the path forward kills everyone. The next test is Monday's encyclical text and whether the Vatican explicitly cites a capability event of this size.

-- ANNA WEBER, Berlin