\section{Computer checks}
\label{rw:app:computer}

Some of the computations in this paper were also carried out by
computer. No proof depends on such a computation: in each case the
argument is given in the text, and the program is a check on it.
The programs are written in Python, with the library SymPy, and are
in the directory \texttt{anc} of the source of this paper. They
compute with matrices, for subspaces of Lie algebras.

\begingroup
\small
\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{.25\textwidth}
                    >{\raggedright\arraybackslash}p{.31\textwidth}
                    >{\raggedright\arraybackslash}p{.38\textwidth}@{}}
\toprule
Statement & What is checked & Program and range\\
\midrule
\endfirsthead
\toprule
Statement & What is checked & Program and range\\
\midrule
\endhead
\bottomrule
\endfoot
Lemma~\ref{fl:lem:fixed-maximal-detectors}(2) &
the primes dividing the discriminant of the invariant form on the
span of the simple coroots of each standard Levi divide $|W_G|$ &
\texttt{\footnotesize check\_kappa.py}; $A_n$, $B_n$, $C_n$, $D_n$ for $n\le6$,
and all exceptional types\\
Lemma~\ref{rw:lem:hecke-cotangent-projections}, step~3;
Lemma~\ref{fl:lem:quadric-detection};
Lemma~\ref{fl:lem:graded-morse-chart}, step~1;
Lemma~\ref{rw:lem:poincare-kernel} &
the rank of the pairing \eqref{fl:eq:root-hessian}, with
degenerate cases as controls; the identities of the chart of the
universal conic; the graded Lagrangian complement; the recurrence
and the rank for a truncation in Laurent polynomials &
\texttt{\footnotesize check\_transverse\_chart.py}; five blocks
and orders of pole in
$\mathfrak{gl}_2,\mathfrak{gl}_3,\mathfrak{gl}_4$, $66$ graded cases
with initial dimension at most $10$, and the chart identities
symbolically\\
\end{longtable}
\endgroup