Math enthusiast says remaining cases of 57-year circle packing problem proven with GPT-6 Astra

Math enthusiast nor used GPT-6 Astra to rigorously prove the cases N=15, 16, 17, 18, and 20 in the circle packing problem; previously only N≤14 and N=19 had been verified. The problem requires placing N disks of radius 1 into a larger circle of as small a radius as possible. In 1969, mathematicians had given an arrangement scheme, but for 57 years no one had rigorously proven its optimality. According to nor's blog, GPT-6 Astra no longer directly searches for the optimal solution, but instead uses geometric constraints to reduce the infinite number of arrangements to a finite set of cases, then rules out other possibilities through computer enumeration and geometric rigidity arguments; the case of 15 disks additionally used an odd-cycle argument. Nor said their own contributions included determining the problem's feasibility, guiding the model, manually verifying results, and rechecking everything with Astra and Opus 5.5.