Integer multiplication below n log n

What OpenAI claims

Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log n)^1-κ), with κ=2^-182, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen n log n optimality conjecture in the ordinary multitape bit model.

Paper

1. Integer multiplication below n log n

   23 September 2026 · Preprint, not peer reviewed · PDF

   We give a deterministic algorithm that multiplies two n-bit integers in O(n(lg n)^1-κ) worst-case time, with κ=2^-182, on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the n log n optimality conjecture of Schönhage and Strassen in this model.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.