\begin{thebibliography}{99}
\bibitem{HistoricalClaude}
E. F. W. Heffern, \emph{Geometric Descent on the Markov Surface},
working draft v32, archived October 2026, pp.~38--40, historical
computational reports. The earlier search was assisted by Claude.

\bibitem{HistoricalAnnouncement}
E. F. W. Heffern, public LinkedIn announcement of a 24-minute Markov-number
search, 23 October 2026, 14:54:57.771 UTC, activity
7519411048115003392. Platform metadata and the source URL are retained in
the verification supplement's evidence packet; see
\href{https://www.linkedin.com/posts/elleard-felix-webster-heffern-joseph-by-confirmation-2953a3104_markovnumbers-activity-7519411048115003392-4NaJ}{the public announcement}.

\bibitem{Verification}
E. F. W. Heffern, \emph{Markov 50 Billion Verification Package}, version 1,
completed and sealed 28 October 2026; public derivative version 1.1 in
\emph{Computational supplements for Meridians on the Markov Surface},
Zenodo, 2 November 2026.
\href{https://doi.org/10.5281/zenodo.23874075}{doi:10.5281/zenodo.23874075}.
\bibitem{TrillionVerification}
E. F. W. Heffern, \emph{Certified magnitude-band verification of the first
one trillion Markov values}, completed 9 November 2026 (CST).
Final independent and portable replay reports, retained receipt chains,
resource records and \texttt{v16\_Trillion\_Verifier\_Recovery.zip},
included with this release. \href{https://doi.org/10.5281/zenodo.24096109}{doi:10.5281/zenodo.24096109}.
\bibitem{SelectedValues}
E. F. W. Heffern, \emph{Six selected Markov values: full decimal expansions
and certified numerical ranks}. Digits first posted 9 November 2026,
\href{https://doi.org/10.5281/zenodo.24093257}{doi:10.5281/zenodo.24093257};
rank certification and unchanged digit files accompany the present release.
\bibitem{BandVerification}
E. F. W. Heffern, \emph{Certified magnitude-band, residue-first verification
of the first 500 billion Markov maxima}, completed 9 November 2026.
Final evidence: \texttt{v1\_Band\_500\_Billion\_Final\_Evidence.json};
frozen recovery archive: \texttt{v20\_Band\_500\_Billion\_Recovery.zip}.
The comparison includes 550,693,634,649 configurations.
The final evidence and reconciliation accompany this manuscript's source;
the recovery archive is identified by SHA-256 in
Section~\ref{sec:verification-evidence}.

\bibitem{FiftySupplement}
E. F. W. Heffern, \emph{Computational chronology and proofs for the
fifty-billion verification}, supplement to the present manuscript,
version 4, 7 November 2026.
\bibitem{Baragar1996}
A.~Baragar, \emph{On the unicity conjecture for Markoff numbers},
Canadian Mathematical Bulletin \textbf{39} (1996), no.~1, 3--9.
\href{https://doi.org/10.4153/CMB-1996-001-x}{doi:10.4153/CMB-1996-001-x}.
The computational benchmark $M<10^{140}$ is reported in Section~4, p.~9.

\bibitem{Aigner}
M. Aigner, \emph{Markov's Theorem and 100 Years of the Uniqueness Conjecture:
A Mathematical Journey from Irrational Numbers to Perfect Matchings},
Springer, Cham, 2013.
\href{https://doi.org/10.1007/978-3-319-00888-2}{doi:10.1007/978-3-319-00888-2}.
\bibitem{CS}
\.{I}. \c{C}anak\c{c}{\i} and R. Schiffler,
\emph{Snake graphs and continued fractions}, European J. Combin.
\textbf{86} (2020), 103081.
\href{https://doi.org/10.1016/j.ejc.2020.103081}{doi:10.1016/j.ejc.2020.103081}.
\bibitem{DegreeVerification}
E. F. W. Heffern, \emph{Meridian Degree 10,000 Reproducibility Package},
computational supplement, 31 October 2026; address-convention revision,
7 November 2026. Original computational release in
\emph{Computational supplements for Meridians on the Markov Surface},
version 1.0, Zenodo, 2 November 2026.
\href{https://doi.org/10.5281/zenodo.23874075}{doi:10.5281/zenodo.23874075}.
\bibitem{CorridorRecord}
E. F. W. Heffern, \emph{Markov Meridian Residue Supplement}, version 1.18,
and \emph{Corridor-to-Corridor Induction}, version 1.3,
unpublished supplementary research records, 7 November 2026.
The corridor results used here are proved in Section~\ref{sec:corridors}.
\bibitem{LPTV}
C. Lagisquet, E. Pelantov\'a, S. Tavenas and L. Vuillon,
\emph{On the Markov numbers: fixed numerator, denominator, and sum conjectures},
Adv. Appl. Math. \textbf{130} (2021), 102227.
\href{https://doi.org/10.1016/j.aam.2021.102227}{doi:10.1016/j.aam.2021.102227}.
Preprint: \href{https://arxiv.org/abs/2010.10335}{arXiv:2010.10335}.
\bibitem{LLRS}
K. Lee, L. Li, M. Rabideau and R. Schiffler,
\emph{On the ordering of the Markov numbers}, Adv. Appl. Math.
\textbf{143} (2023), 102453.
\href{https://doi.org/10.1016/j.aam.2022.102453}{doi:10.1016/j.aam.2022.102453}.
\bibitem{McShane}
G. McShane, \emph{Convexity and Aigner's conjectures}, preprint (2021),
\href{https://arxiv.org/abs/2101.03316}{arXiv:2101.03316}.
\bibitem{RS}
M. Rabideau and R. Schiffler, \emph{Continued fractions and orderings on the
Markov numbers}, Adv. Math. \textbf{370} (2020), 107231.
\href{https://doi.org/10.1016/j.aim.2020.107231}{doi:10.1016/j.aim.2020.107231}.
\bibitem{Musiker2002}
G.~Musiker, \emph{A conjectured combinatorial interpretation for Markov
numbers}, REACH memorandum, 27 June 2002.
\href{https://faculty.uml.edu/jpropp/reach/musiker/NewResults.ps}{Original memorandum};
\href{https://faculty.uml.edu/jpropp/reach/index.html}{REACH archive}.
\bibitem{IMPV2003}
A.~Itsara, G.~Musiker, J.~Propp, and R.~Viana,
\emph{Combinatorial interpretations for the Markov numbers},
REACH manuscript, 10 January 2003, Theorem~3.1.
\href{https://faculty.uml.edu/jpropp/reach/Itsara/markovversion1.pdf}{Original manuscript}.
\bibitem{Propp}
J.~Propp, \emph{The combinatorics of frieze patterns and Markoff numbers},
\href{https://arxiv.org/abs/math/0511633}{arXiv:math/0511633}, first posted
25 November 2005; published in \emph{Integers} \textbf{20} (2020), A12.

\bibitem{TanZhang2026}
H.~Tan and Y.~Zhang,
\emph{A computational verification of the Markov uniqueness conjecture
for $z\le10^{500000}$},
\href{https://arxiv.org/abs/2611.01007v1}{arXiv:2611.01007v1}
(31 October 2026), Sections~2.3--2.4 and~3.2.
Computational evidence archived 26 October 2026:
\href{https://doi.org/10.5281/zenodo.23670834}{doi:10.5281/zenodo.23670834}.

\end{thebibliography}