How many fractions does a fraction really need? For 76 years, a conjecture of Paul Erdős asked whether every rational number below one could be assembled from remarkably few distinct reciprocals. OpenAI’s September 25, 2026 manuscript, Short Egyptian Fractions, presents a proof that the worst-case minimum length grows like the logarithm of the logarithm of … Read more

Imagine stepping from prime to prime across the complex plane, always keeping the next jump below the same fixed limit. OpenAI’s September 26, 2026 manuscript answers that challenge negatively: the Gaussian moat conjecture admits no infinite walk through distinct Gaussian primes with bounded steps. Its stronger conclusion caps every connected component, wherever the walk starts. … Read more

A phase transition can be known to exist while its most distinctive prediction remains unproved. That was the gap facing the BKT transition in the original square-lattice XY model: exactly how fast does the distance over which spins remain correlated grow near criticality? OpenAI’s October 5, 2026 preprint proves that the logarithm of this correlation … Read more

A formula can give the right answer for twenty-two years without anyone knowing how to prove it. The Mezard Parisi formula faced precisely that problem: a hierarchy of cavity-field distributions supplied rigorous upper bounds for sparse spin systems, but the matching lower bound remained missing. OpenAI’s September 23, 2026 paper proves that equality for the … Read more

A magnet should not need a mathematical permission slip to stay magnetized. Yet proving positive-temperature order in the quantum Heisenberg ferromagnet required much more than showing that neighboring spins prefer alignment. OpenAI’s September 24, 2026 paper supplies that proof for the nearest-neighbor isotropic model on the integer lattice in every dimension d≥3d\geq3, for every positive … Read more

Three measurement bases fit. A fourth has resisted every attempt to build it for decades. OpenAI’s new paper tackles that missing basis by covering the possible complex matrices, then eliminating every configuration that could support it. The Zauner MUB conjecture says that dimension six admits exactly three pairwise mutually unbiased orthonormal bases. The supplied manuscript … Read more

A particle can ricochet inside a triangle forever, obeying one elementary reflection rule, while hiding a difficult question: does its motion explore position and direction in the correct statistical proportions? OpenAI’s September 25, 2026 manuscript presents a proof that irrational triangular billiards are ergodic whenever at least one interior angle is an irrational multiple of … Read more