Showcasing practical examples, projects, and case studies demonstrating how AI models and tools are being applied across various fields.
A greedy pairing can stop after one edge even when two fit. Finding the best pairing is easy to describe, yet odd cycles have made fast, exact solutions surprisingly difficult. OpenAI’s September 24, 2026 manuscript presents a maximum cardinality matching algorithm running in almost-linear time in the graph’s explicit input size, including the cost of … Read more
An algorithm can lose badly even when it may toss coins, adapt its questions, and perform unlimited computation between them. OpenAI’s October 5, 2026 preprint constructs precisely that situation: total Boolean functions where classical randomized algorithms need almost the fourth power of the quantum query count. The result concerns quantum query complexity, the number of … Read more
Two sets of integers, one addition sign, and every sufficiently large prime: that is the deceptively simple target of the Inverse Goldbach Conjecture. OpenAI’s September 24, 2026 manuscript, The Additive Indecomposability of the Primes, presents a proof that this construction is impossible when each set contains at least two nonnegative integers. Even allowing finitely many … Read more
A fractal can contain infinitely many points and still occupy no area. Can its pairwise distances nevertheless occupy a positive length on a ruler? The Falconer Distance Conjecture predicts exactly when that must happen. OpenAI’s September 23, 2026 manuscript presents a proof in every integer dimension d≥2d\ge2: a compact set whose Hausdorff dimension exceeds d2\frac … Read more
Randomness can trap a quantum particle. The harder question is whether weak randomness must trap it. The Anderson localization OpenAI proof addresses that second question: its September 23, 2026 manuscript establishes a nonempty energy interval with purely absolutely continuous spectrum for the nearest-neighbor Anderson model with independent uniform disorder, in every fixed dimension d≥3d\geq3. The … Read more
A regular language can require arbitrarily deep repetition in an ordinary regular expression. Add complement, and OpenAI’s September 25, 2026 manuscript gives every regular language a three-level ceiling. Its answer to the Generalized Star Height Problem is a uniform upper bound over the original alphabet, not a proof that three levels are necessary. Whether one … Read more
A graph can demand many colors while refusing to collapse into an equally large complete graph. OpenAI’s September 23, 2026 manuscript constructs precisely that separation, presenting a disproof of the Hadwiger Conjecture, proposed in 1943. Its counterexamples have no independent set larger than two vertices, yet their largest clique minors fall well below the resulting … Read more