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\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={Integer multiplication below n log n},pdfauthor={OpenAI}}
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\title{Integer multiplication below \texorpdfstring{$n\log n$}{n log n}}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{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 Sch\"onhage and Strassen in this model.
\end{abstract}
\tableofcontents
\clearpage
\input{sections/00-introduction.tex}
\input{sections/02-streams.tex}
\input{sections/03-motifs.tex}
\input{sections/04-swap.tex}
\input{sections/05-layers.tex}
\input{sections/06-transforms.tex}
\input{sections/07-resampling.tex}
\input{sections/08-assembly.tex}
\input{sections/09-exact-arithmetic.tex}
\input{sections/10-transposition.tex}
\appendix
\input{sections/11-grouped-rectangles.tex}
\bibliographystyle{plain}
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\end{document}