OpenAI · 109-01 · 2026-09-23 · Theoretical computer science

Integer multiplication below n log n

preprints/Integer-multiplication-below-n-log-n-September-23-2026

Family 109. Integer multiplication below $n lo g n$ .

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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

We give a deterministic algorithm that multiplies two n-bit integers in $O (n (l g n)^{1 - κ})$ 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 $n lo g n$ optimality conjecture of Schönhage and Strassen in this model.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.