Model research and search ad updates #25
Today's Letter
OpenAI model disproves Erdős unit distance conjecture
- OpenAI said an internal reasoning model found a proof that disproves the long-standing conjecture on the planar unit distance problem, first posed by Paul Erdős in 1946.
- The result gives an infinite family of point configurations with at least n^{1+δ} unit-distance pairs for infinitely many n, exceeding the previously believed n^{1+o(1)} growth pattern.
- OpenAI said the original model proof did not provide an explicit δ, and a forthcoming refinement by Princeton mathematician Will Sawin sets δ = 0.014.
- The company said the proof was produced by a new general-purpose reasoning model rather than a system trained only for mathematics or a workflow targeted at this problem.
- External mathematicians checked the proof, and OpenAI linked both the proof itself and a companion paper with background and remarks on the result.
- The unit distance problem has been studied for about 80 years and is considered a central open question in combinatorial geometry because it is easy to state but difficult to resolve.
- OpenAI framed the result as the first autonomous AI solution to a prominent open problem central to a mathematics subfield, with the argument drawing on algebraic number theory techniques.
- The previously cited upper bound remains O(n^{4/3}), while the new construction challenges the long-held view that square-grid-style constructions were essentially optimal.
Source: openai.com
More: cnbc.com · wsj.com · news.hada.io
Google Search, Gemini ad formats introduced
- Google introduced new Search ad formats built with Gemini, according to a May 20 Google Ads blog post.
- The rollout includes Conversational Discovery ads and Highlighted Answers in AI Mode, both labeled as Sponsored.
- Google said both formats add an independent AI explainer that synthesizes product or service context alongside advertiser creative.
- Conversational Discovery ads are designed to answer specific search questions with tailored product details based on query intent.
- Highlighted Answers can place eligible high-quality ads inside AI Mode recommendation lists during product research.
- Google also plans AI-powered Shopping ads in Search, where Gemini surfaces relevant products and generates a custom explainer for each result.
- Business Agent for Leads adds a chat-based brand agent inside ads, using website content to answer questions instead of static lead forms.
- Google said the Direct Offers pilot, launched in January 2026, is expanding with promotion bundling, native checkout for Universal Commerce Protocol merchants, and travel offers from partners including Booking and Expedia.
Source: blog.google
More: dbushell.com · tante.cc · news.hada.io
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