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\hypersetup{pdftitle={Irrational even beta values in shifted intervals},
 pdfauthor={twinheterodox},pdfcreator={LaTeX},
 pdfsubject={Explicit irrationality counts in intervals with a variable starting index}}
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\title{\Large Irrational even beta values in shifted intervals}
\author{twinheterodox\\[2pt]\normalsize Prepared with Codex}
\date{6 November 2026 \quad Version 2}
\begin{document}
\maketitle
\begin{abstract}
For every positive odd integer $m$, we prove that at least one of
\[
 \beta(m+1),\beta(m+3),\ldots,\beta(19m-1)
\]
is irrational, where $\beta$ is the Dirichlet beta function.
More generally, for each positive integer $B$ we give an explicit
constant $C_B$ such that at least $B$ even beta values with indices
between $m+1$ and $C_Bm-1$ are irrational, for every positive odd $m$.
Specific parameter choices give at least two irrational values among
$\beta(2),\beta(4),\ldots,\beta(70)$, at least three through
$\beta(186)$, and at least four through $\beta(360)$.
\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}
\appendix
\input{sections/checks}
\input{sections/optimization}
\input{references}
\end{document}