\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 archived at Zenodo, release v1.0.0, under \href{https://doi.org/\zenodoCodeID}{\nolinkurl{doi:\zenodoCodeID}} \cite{BB26code}; the compiled paper is the only file of the folder \texttt{preprintHodgeConjecture} of that archive. 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.