Just a week after an AI disproved an 80-year-old conjecture and astonished mathematicians, another conjecture that had stood for half a century has fallen, inspired by the same techniques, but this time written entirely by humans. Last week, an unreleased AI model from OpenAI disproved an important conjecture first posed by Hungarian mathematician Paul Erdős, called the unit distance problem. The puzzle, which Erdős considered his "most striking contribution to geometry" and which many mathematicians had failed to unravel, concerns the number of similar-sized connections you can make between dots arranged on a flat surface. Erdős had set an upper ceiling on this number, which many experts had assumed was correct. But the AI model showed that this number could in fact be much larger, using an obscure trick from algebraic number theory to make complex structures with extremely high dimensions, which could then be used to arrange the dots in a very different arrangement than humans had considered.

We show that the maximum expected inner product between a random vector and the standard normal vector over all couplings subject to a mutual information constraint or regularization is equivalent to a truncated integral involving the rate-distortion function, up to universal multiplicative constants. The proof is based on a lifting technique, which constructs a Gaussian process indexed by a random subset of the type class of the probability distribution involved in the information-theoretic inequality, and then applying a form of the majorizing measure theorem.

We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only $O(n^d)$ and respectively $\tilde O(1/\varepsilon^d)$ upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and $k$-Sum problems. Our reductions yield matching lower bounds of $\tildeΩ(n^d)$ and respectively $\tildeΩ(1/\varepsilon^d)$ based on Affine Degeneracy testing, and $\tildeΩ(n^{d/2})$ and respectively $\tildeΩ(1/\varepsilon^{d/2})$ conditioned on $k$-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.

Effective communication requires the ability to refer to specific parts of an observation in relation to others.

Answers to these questions must begin by formalizing the specification for what a generative algorithm for language should be doing.

SelfCodeAlign employs the same base model for inference throughout the data generation process. It first extracts diverse coding concepts from high-quality seed snippets to generate new tasks.

The keywords were extracted using the procedure described in SectionC. The restricted part of the Muharaf dataset has 428 images distributed under a proprietary license.