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\hypersetup{pdftitle={At least one of beta(4), beta(6), ..., beta(32) is irrational},
 pdfauthor={twinheterodox},pdfsubject={An isolated-pole denominator refinement for even Dirichlet beta values}}
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\title{At least one of $\beta(4),\beta(6),\ldots,\beta(32)$ is irrational}
\author{twinheterodox\\[2pt]\normalsize Prepared with Codex}
\date{11 November 2026}
\begin{document}
\maketitle
\begin{abstract}
For the Dirichlet beta function, we prove that at least one of
$\beta(4),\beta(6),\ldots,\beta(32)$ is irrational. This reduces
Krisman's seventeen-value set $\beta(4),\beta(6),\ldots,\beta(36)$
\cite{krisman} to fifteen values. The improvement comes from a sharper
denominator estimate for his construction.
\end{abstract}
\input{paper/sections/01-introduction}
\input{paper/sections/02-forms}
\input{paper/sections/03-denominators}
\input{paper/sections/04-decay}
\input{paper/sections/05-nonvanishing}
\input{paper/sections/06-computations}
\appendix
\input{paper/sections/appendix-small-primes}
\input{paper/sections/references}
\end{document}