AI News & Trends

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

At time one, the force stops changing. The computation does not. OpenAI’s September 27, 2026 preprint constructs a viscous fluid whose particle motion can simulate an arbitrarily long Turing-machine run, even after its external force becomes stationary. For readers searching “Navier Stokes Turing complete,” the precise answer is yes for these specially constructed forced flows. … Read more

A finite list of rules can describe an infinite group. The hard question is whether that group can also be simple while containing every operation of another group faithfully. OpenAI’s September 23, 2026 manuscript presents a positive proof of the Boone Higman conjecture: a finitely generated group has a decidable word problem exactly when it … Read more

Choose the integer 2. Can its powers generate every nonzero residue modulo infinitely many primes? Artin’s primitive root conjecture predicts yes. OpenAI’s October 4, 2026 manuscript Primitive roots for every admissible integer base claims an unconditional proof, covering every integer except −1 and perfect squares. Its quantitative guarantee is a lower bound of order x/(logx)2x/(\log … Read more