\section*{Acknowledgements}
The paper builds on the theorems of Eyal Markman on secant sheaves and Weil classes, on the work of Bert van Geemen, Matthias Sch\"utt and Mauro Varesco on real multiplication of K3 surfaces, on Nikolay Buskin's theorem on Hodge isometries, on the semiregularity theory of Spencer Bloch, Ragnar-Olaf Buchweitz and Hubert Flenner, and on the descriptions of Hodge classes on abelian varieties by Ben Moonen, Yuri Zarhin, Yves Andr\'e and James Milne. We thank Priyabrata Mandal for the collaboration on the earlier paper \cite{BMB26} whose results are used here.

\section*{Data and code availability}
The \LaTeX{} source of the paper, the Python, Julia, C, C\texttt{++} and shell programs of Appendix~\ref{app:checks} and the Lean~4 file are published as GitHub release v2.7.0, \url{\codeRelease} \cite{BB26code}; the compiled paper is the only file of the folder \texttt{preprints} of that release. The candidate configurations of Section~\ref{ssec:e06count} and the first enumerations of their chains, among them that of Remark~\ref{rem:sevenbundles}, were produced with Anaxia, the construction engine of the BLLM project \cite{Bha26bllm}, release v3.0.5, \url{https://github.com/creelie/eBLLM/releases/tag/v3.0.5}, and the support results of Section~\ref{ssec:supports} with release v3.1.0 \cite{Bha26bllmb}, \url{https://github.com/creelie/eBLLM/releases/tag/v3.1.0}; the point counts of Remark~\ref{rem:frickepicard} follow the Frobenius test of its module \texttt{k3frob} (commit \texttt{8782bd3}), which first showed that a Fricke self-map is a complex and not a real multiplication, at level $13$; every statement of the paper is decided by the proofs in the text and by the programs of the release, not by the engine. Deep Bhattacharjee carried out the constructions on \Fii{} and \Fiii{} and the Lean, Julia, C, C\texttt{++}, Python and shell verifications with the help of AI assistance (Claude, Anthropic). Both authors approved the final version.

\section*{Ethical declarations}
Funding: none. Competing interests: none.