OpenAI's sphere-packing claim lands in a field that took 387 years to confirm Kepler

Kepler wrote down the three-dimensional answer in 1611 ^[6]. The proof came in 1998, when Thomas Callister Hales announced a massive computer-assisted argument for it ^[8]. Three hundred and eighty-seven years passed between the stated arrangement and the confirmed one ^[1].

The two-dimensional case shows how little can separate a plausible arrangement from the proved best. Lay quarters out in aligned rows and they cover about 78.5 percent of an infinite tabletop, the fraction pi/4 ^[4]. Nestle each row into the hollows of the one below and you get the honeycomb, about 90.7 percent, which is known to be optimal ^[5]. The gap is 12.2 percentage points ^[2].

Above three dimensions the pictures go away and mathematicians work in coordinates, with distances computed by adding one more term to a familiar formula ^[15]. Optimal packings are known there in two cases, dimensions 8 and 24 ^[9]. Maryna Viazovska settled dimension 8 in 2016, and a week later she and four co-authors settled dimension 24 ^[10]. Both of those dimensions carry special symmetries the authors used, and about other high-dimensional spaces much less is known ^[12]. Some of those optima may not even repeat ^[13].

Four dimensions in all have proved optima: 2, 3, 8 and 24 ^[4]. Everywhere else, a new packing is compared with the best packing built before it. Scientific American does not say which dimension Astra's claimed advance concerns, or by how much it improved on the prior record ^[16].

That detail decides what verification means here. If the claim is a construction, another mathematician can write the packing down and compute its density. If it is a statement about the optimum, the checking is the kind Hales needed a computer and hundreds of pages for ^[8]. Viazovska's route ran through symmetry ^[12]. Her Fields Medal arrived in 2022, six years after the dimension-8 proof ^[11][3].

Sphere packing underpins coding theory. Efficiency in the abstract space maps onto how much error a code can absorb, and high-dimensional sphere packings model digital communication, the business of turning words into bits and getting them through a noisy channel intact ^[14].

OpenAI put ten claims into one announcement ^[6], for a model it had not released ^[1], in an area Scientific American describes as beloved and full of notoriously difficult open questions ^[2]. Mathematicians will judge each of the ten on its write-up.