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\hypersetup{pdftitle={Verification of the Markov Uniqueness Conjecture for the First One Trillion Markov Values: Meridians on the Markov Surface and Farey Addresses},pdfauthor={Elleard Felix Webster Heffern},pdfsubject={Version 22: Completed one-trillion verification, Meridian growth, and certified Farey grouping},pdfkeywords={Markov numbers, Meridians, Farey addresses, magnitude bands, computational verification}}
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\title[Verification of the First One Trillion Markov Values]{Verification of the Markov Uniqueness Conjecture\\for the First One Trillion Markov Values:\\Meridians on the Markov Surface and Farey Addresses}
\author{Elleard Felix Webster Heffern}
\address{Unaffiliated, St. Louis, Missouri, USA}
\email{soldier@alum.mit.edu}
\urladdr{https://orcid.org/0009-0000-8831-0289}
\date{Version 22 --- 12 November 2026}
\subjclass[2020]{Primary 11D25; Secondary 11B39, 05C70, 11Y50}
\keywords{Markov numbers, unicity, geometric descent, Meridians, corridors, snake graphs}
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  \scriptsize\itshape\centering
  for Tian, my life, and for Mom and for Dad, my water and my soil;\\
  for Juan, for Charles, for Marie, for the girls, my starry nights;\\
  and for all the puppies, for all the kittens, for all the fish:\\
  my sunny sunny days.\par
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    {\scriptsize\itshape
    and for my fellow Sunflower: \textquotesingle Chat\textquotesingle.\\[0.5em]
    together, we might stand tall as trees.}
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\begin{document}
\raggedbottom
\begin{abstract}
The Markov unicity problem asks whether the largest entry of a positive
integer solution determines the other two. We organize the triples into
families called Meridians, with compact addresses and explicit evaluation
rules. The construction starts with catenary profiles: a fixed host sets
the curve, and two heights one unit apart solve the Markov equation with
that host. Selected positions give integer triples. Reversing geometric
descent gives an exact indexing of these triples. Remainder patterns,
growth bounds, corridor paths, and Fibonacci-matrix and snake-graph
arguments then restrict possible collisions.

The Meridian description of starting scales and logarithmic spacing
guided the organization of a large verification into magnitude bands.
The implementation uses Farey addresses and certified matrix bounds
to select each band; its correctness does not require the Meridian
collision theorems. Production finished on 10 November 2026 at
03:54:13 UTC (9 November, 21:54:13 CST). The combined computation
compared 1,101,388,071,945 configurations and certified uniqueness of the
first one trillion Markov values in numerical order. The trillionth value
has 1,021,609 decimal digits. All 12 repeated
64-bit residue groups were resolved by exact arithmetic, with no equal
maxima. On eight workers, retained active band processing took
23 h 57 min 23.146 s, including the inherited 500-billion certificate;
the maximum recorded temporary key-file footprint was 1,065,664,392 bytes
(0.992 GiB). This elapsed processing time is distinct from summed CPU
time and the calendar span. Both final replays passed. A separate computer-checked
certificate covers all 5,067,077 Meridians of degree at most 10,000,
at every integer phase. The general unicity problem remains open.
\end{abstract}
\maketitle
\input{sections/01_introduction}
\input{sections/02_geometry}
\input{sections/03_indexing}
\input{sections/04_residues}
\input{sections/05_snakes}
\input{sections/06_windows}
\input{sections/07_corridors}
\input{sections/08_certificate}
\input{sections/09_verification}
\input{sections/09_discussion}
\input{sections/10_disclosure}
\input{references}
\input{sections/11_sneak_peak}
\end{document}