Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 109-01. Integer multiplication below n log n
Manuscript map · lines 4605–4614
OpenAI@2026-10-07#rem:oai-p-109-01
Authors: OpenAI. Family 109.
Title: Integer multiplication below n log n.
Abstract: We give a deterministic algorithm that multiplies two n-bit integers in $O(n({}lg n){}{1-{}kappa})$ worst-case time, with ${}kappa=2{}{-182}$, 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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