\section{All-phase verification through degree 10,000}\label{sec:certificate}
\begin{theorem}[Degree-10,000 separation]\label{thm:degreebound}
Among canonical Meridians of Laurent degree at most $10{,}000$, distinct
canonical integer-phase marks have distinct maxima. There are exactly
$5{,}067{,}077$ Meridians in this family.
\end{theorem}
The computational part separates different Meridians at all integer phases.
The fixed-degree theorem also separates marks on a single Meridian, so the
two conclusions combine as stated. The bound is on \emph{degree}, not on the
size of a Markov number, and it does not exclude a competitor of higher degree.

\subsection{Why a finite residue test covers infinitely many phases}
For a period $T$, define the complete image
\[
\mathcal R_\alpha(T)=\{M_\alpha(x)\bmod q(T):0\le x<T\}.
\]
Theorem~\ref{thm:period} makes this the image at \emph{every} integer phase.
Disjoint images therefore separate two entire Meridians. If images overlap
for several periods, a true equality must use compatible phases in all of
them. Define the witness relation
\[
W_T(\alpha,\beta)=\{(u,v)\in(\Z/T\Z)^2:
M_\alpha(u)\equiv M_\beta(v)\pmod{q(T)}\}.
\]
\begin{proposition}[Compatible witnesses]\label{prop:witnesses}
Equality of two integer values requires one witness in each selected
$W_T$, with first coordinates agreeing modulo every pairwise gcd of the
periods, and second coordinates agreeing in the same way. If no compatible
choice exists, the two Meridians are disjoint at all integer phases.
\end{proposition}
\begin{proof}
A true equality uses the same integers $x,y$ in every modulus and hence
supplies the required residues. Conversely, the generalized Chinese
remainder theorem lifts compatible coordinates to a phase pair modulo the
least common multiple. This converse concerns modular witnesses only;
it does not imply equality of the full values.
\end{proof}
The moduli need not be independent. In particular, the periods $36,38,40$
are all even, so contradictory parity requirements alone can exclude a
pair that survives each separate image test.

\subsection{Complete inventory and the final exclusion}
\begin{proof}[Proof of Theorem~\ref{thm:degreebound}]
Proposition~\ref{prop:helm} identifies the inventory with coprime positive
odd pairs $r\le s$, $r+s\le10{,}000$. It contains $5{,}067{,}077$
Meridians and $12{,}837{,}632{,}128{,}426$ unordered pairs.
The producer enumerates the count-vector address tree; an independent
checker enumerates the coprime pairs directly and compares the inventories.
The child degree is $2|k|D\pm(|v|-|u|)>D$, where $|u|,|v|$ are the
endpoint word lengths. This gives finite exhaustive index ranges.

Every one of the 36 phase residues of every Meridian is evaluated modulo
$q(36)$. Equal-residue grouping lists all pairs whose images intersect;
its complement is excluded at once. Periods 38 and 40 are then evaluated
on every endpoint of the preceding survivor sets. The complete image
tests leave 87 pairs. For each of them, project every full witness relation
onto the four parity pairs $(u\bmod2,v\bmod2)$. The three projected sets
have empty intersection in all 87 cases. Proposition~\ref{prop:witnesses}
therefore excludes every pair of different Meridians at every integer
phase. Theorem~\ref{thm:equaldegree} also excludes different marks within
each individual Meridian, completing the statement.
\end{proof}

\begin{table}[htbp]\centering
\caption{Complete cumulative survivor counts from the archived certificates.
``Images'' forgets meeting phases; ``phases'' also imposes compatibility.
Survivors are modular coincidences, not integer collisions.}
\label{tab:degreecert}
\begin{tabular}{lrr}\toprule
Tests retained&$D\le2{,}000$&$D\le10{,}000$\\\midrule
36: images&7,121&320,755\\
36, 38: images&612&17,427\\
36, 38: phases&2&2,052\\
36, 38, 40: images&0&87\\
36, 38, 40: phases&\textbf{0}&\textbf{0}\\
36, 38, 40, 42: images&---&1\\
36, 38, 40, 42, 44: images&---&0\\\bottomrule
\end{tabular}\end{table}

\subsection{What was checked independently}
The degree-10,000 producer uses signed endpoint-orbit updates.
The separate evaluator uses Fibonacci staircase words with Euclidean
matrix reduction. They agree on all $188{,}369{,}412$ retained phase
evaluations. The checker verifies the $182{,}414{,}772$ baseline phase tags
bijectively, so an omitted or repeated phase record cannot pass unnoticed.
Independent sorting of the original unsorted buckets reconstructs all
70,009 cross-Meridian residue groups and the subsequent complete pair sets.
Coverage is checked by exact records, not inferred merely from their counts
or from a matching digest.

The phase result is checked three ways: full generalized-CRT joins,
an independent 16-bit mask computation on $(x\bmod4,y\bmod4)$, and the
weaker four-state parity projection, already sufficient for the final 87.
The degree-2,000 certificate, containing 202,862 Meridians, was completed
first; its common survivor sets agree exactly with the larger inventory
when identified by helm labels.

Five deliberate corruptions---an altered value, an incorrect modulus, a
duplicate tag, an omitted record and an invented pair---were all rejected.
The degree-100 control compares 18,684 residues with literal Fibonacci
words and the polynomial record of \eqref{eq:record}. The optional
image-only extension through periods 42 and 44 also ends with zero pairs;
all 88 period-44 values were separately checked using arbitrary-size Python
integers because $q(44)>2^{63}$.

\paperfigure{degreeprofile}{1}{figures/degree_survivors.pdf}{Exact
unresolved-pair profiles in the degree-10,000 inventory. The phase-compatible
sieve through period 40 has no survivors in the entire displayed range.
Lines join discrete even-degree bounds. The figure is a record of the
finite certificate, with no extrapolation to larger degrees.}{htbp}

\subsection{Reproducibility and scope}
The degree reproducibility package \cite{DegreeVerification} contains the
inventory, compact residue groups, phase witnesses, producer and checker
sources, audit receipts, and a regeneration driver. Large baseline tables
are regenerated; their recorded sizes and hashes identify the original
run. The reindexed edition retains the original computational kernels with
an explicit address-convention adapter. Reversing the written address tail
does not change a helm label, degree, phase, or certified numerical value.

This is a deterministic computer-assisted certificate, not an external
referee report or proof-assistant formalization. Its scope is a bounded
set of Meridians with unbounded phase. The numerical-order verification
in Section~\ref{sec:verification} is a separate computation with its own
coverage proof and execution record.