Full explainer:
Multiplying two n-digit numbers the school way costs n² digit products. Since Schönhage and Strassen (1971), n log n has been conjectured optimal on a multitape Turing machine, where a fast transform has about log n levels that each read all n bits; Harvey and van der Hoeven reached n log n in 2019. OpenAI’s preprint “Integer multiplication below n log n” (September 23, 2026; not yet peer reviewed) gives one multitape Turing machine that multiplies two n-bit integers in O(n (lg n)^(1−κ)) time with κ = 2^−182, which refutes the conjecture in this model. Since then an open community repository, maintained by Douglas Colkitt with many contributors, people and AI agents, has pushed the reviewed κ to 0.001672660197076918 ≈ 2^−9.22: PR 1094 by Rohan Arun on Jeremy Kothe’s PR 1087 packing, with the bit supplier DreamingOfClouds’ PR 589 on eumemic’s PR 555.
The community repository:
What is what:
• OpenAI’s preprint, not yet peer reviewed: Theorem 1 (κ = 2^−182) and its Turing-machine construction.
• The community repository: every κ after the paper’s in the ladder (5.8·10^−33 by parameter tuning alone, then 2^−75, 2^−59, 2^−30, 3.9·10^−5, 0.00072, 0.00152 and 0.001672660197076918), as its README, current status, result history and selected result report them (read October 17, 2026). Its finite certificates are exact; the all-size steps remain conditional on OpenAI’s framework, and it is not a complete formal proof. We did not replay its certificates. Proposals near κ ≈ 0.00173 (PRs 3042, 3051, 3064) are archived there as sources, not reviewed results; they are not shown.
• Ours, a side road: an anonymous addendum’s design (October 7, 2026) tops out at K* ≈ 2^−73.7 (with Codex; by hand and exact computation; both ends kernel-checked in our checker Satori from named axioms). The community’s new networks left that design far behind.
• Not claimed: any speed-up in practice. A factor of two over n lg n needs (lg n)^κ ≥ 2, i.e. lg lg n ≥ 1/κ ≈ 598, so n ≥ 2^(2^598) bits; for n = 10^80 bits, lg lg n ≈ 8. The gain of OpenAI’s κ = 2^−182 is under one part in 10^54 for any n ≤ 10^80 bits.
• Classical: schoolbook multiplication, Schönhage–Strassen and its conjecture, Harvey–van der Hoeven, the multitape model. The tapes and the butterfly diagram are illustrations.
Code, proofs and an interactive explorer:
The preprint, in OpenAI’s openai/math repository:
Tungsten, the language every computation and frame is written in:
Satori, our proof checker:
Animation in Tungsten · narration ElevenLabs.
Independent channel, not affiliated with @OpenAI.
#Shorts #math #openai #algorithms
source