Showcasing practical examples, projects, and case studies demonstrating how AI models and tools are being applied across various fields.
A black hole cannot be arbitrarily large while the universe around it carries arbitrarily little mass. Turning that intuition into a theorem has occupied relativists since 1973. OpenAI’s September 27, 2026 manuscript presents a proof of the spacetime Penrose inequality for a broad, precisely defined class of initial data, including moving configurations and disconnected trapped … Read more
Six squares were enough to leave a gap in a decades-old classification of tic-tac-toe variants. The shape was Snaky: a straight four-cell body with a two-cell kink. OpenAI’s September 25, 2026 paper, Snaky in 21 Maker Moves, supplies a guaranteed winning strategy from an empty board against every legal defensive response. Maker can finish within … Read more
Knowing the lowest-energy state of a quantum system does not tell you how cheaply it can be disturbed. For the Laughlin state, that missing guarantee has challenged mathematical physics for more than four decades. OpenAI’s September 24, 2026 paper presents a Laughlin spectral gap proof for the full fermionic Haldane interaction on expanding spheres: sufficiently … Read more
An infinite network can become connected enough to support endless open paths while remaining fragmented into infinitely many endless components. That is the phenomenon behind the Benjamini–Schramm nonuniqueness conjecture. OpenAI’s September 24, 2026 preprint presents a proof for every infinite, connected, locally finite, nonamenable quasi-transitive graph under Bernoulli bond percolation. The result separates the arrival … Read more
A finite shape can fill three-dimensional space perfectly and still refuse every arrangement with three independent repeating directions. OpenAI’s September 23, 2026 manuscript presents exactly that counterexample to the periodic tiling conjecture: one finite lattice tile that tiles, yet admits no fully periodic tiling. Its unit-cube version has the same property in ordinary space, even … Read more
A computer can quickly factor a polynomial by making random choices. Requiring it to make every choice predictably has resisted a comparable guarantee for arbitrary prime fields. OpenAI’s October 4, 2026 manuscript presents a way through that barrier, using geometry to force unknown roots to behave differently. Deterministic polynomial factorization over prime fields means finding … Read more
Finding one valid assignment can be easy while estimating how many assignments exist remains stubbornly hard. A four-edge bipartite square captures the difference: selecting one perfect matching answers feasibility, but the graph has two. Scale up the constraints, and enumeration quickly becomes the wrong tool. Counting Common Bases of Two Matroids asks for the number … Read more