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\hypersetup{pdftitle={Explicit lower bounds for the number of irrational even beta values},
 pdfauthor={twinheterodox},pdfcreator={LaTeX},
 pdfsubject={An explicit irrationality count for every even endpoint at least 160}}
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\title{\Large Explicit lower bounds for the number of irrational even beta values}
\author{twinheterodox\\[2pt]\normalsize Prepared with Codex}
\date{4 November 2026}
\begin{document}
\maketitle
\begin{abstract}
For every even integer $s\ge160$, we prove that at least
\(\left\lfloor\pi\sqrt{2/135}\,\sqrt{s/\log s}\right\rfloor\)
of the numbers $\beta(2),\beta(4),\ldots,\beta(s)$ are irrational,
where $\beta$ is the Dirichlet beta function. We also give sharper
finite endpoints for specified counts of irrational values.
\end{abstract}
\input{sections/introduction}
\input{sections/construction}
\input{sections/gram}
\input{sections/arithmetic}
\input{sections/decay}
\input{sections/dilation}
\input{sections/contradiction}
\input{sections/endpoints}
\input{sections/count}
\appendix
\input{sections/checks}
\input{sections/earlier-count}
\input{references}
\end{document}