\section{Corridor walls and numerator--denominator windows}\label{sec:corridors}
\subsection{A second way to travel through the same tree}
The corridor observation groups the walls between neighboring phasors.
After an initial entrance, a wall follows a fixed-numerator family
\[
m_{p/(q_0+np)},\qquad n\ge0.
\]
Both Meridian and phasor edges occur in such a route. Their common law is
the advance $q\mapsto q+p$, not a rule that distinguishes the two edge colors.
For the $5\to13$ phasor-capped corridor (equivalently, the
$1/2\to1/3$ phasor-capped corridor), the fixed-numerator portions are
$3/(4+3n)$ and $3/(5+3n)$, tending to the historical coordinate $c=1/3$.
The first wall begins
\[
169\xrightarrow{\rm phasor}2897\xrightarrow{\rm Meridian}51641
\xrightarrow{\rm phasor}925765\longrightarrow\cdots,
\]
and the other begins $433,7561,135137,2423525,\ldots$ with the opposite
edge alternation.

Corridor role and edge color must be kept separate. With corridor-wall
travel designated \emph{standard} and a new entrance \emph{exotic}, the
two entry routes here are
\[
5\xrightarrow{\rm exotic}29\xrightarrow{\rm standard}433
\xrightarrow{\rm standard}7561\longrightarrow\cdots,
\qquad
13\xrightarrow{\rm exotic}169\xrightarrow{\rm standard}2897
\longrightarrow\cdots.
\]
Both exotic entrances are Meridian edges. In the first route the initial
standard step $29\to433$ changes the numerator from two to three; the
fixed-numerator formula starts at $433$. Thus excluding the concave ceiling
arc from a wall's recurrence does not move the exotic entrance to that arc.
The graph and its actual depth \eqref{eq:depth} are unchanged.

\subsection{The exact recurrence behind the observed regularity}
The same Fibonacci run blocks give the corridor recurrence directly.
For a reduced label $0<p/q<1$, the successive horizontal run lengths are
$\lceil jq/p\rceil-\lceil(j-1)q/p\rceil$. Equation~\eqref{eq:fibword}
therefore reads
\begin{equation}\label{eq:corridorfib}
m_{p/q}=\left[
\mathcal A_{\lceil q/p\rceil-1}
\prod_{j=2}^{p}
\mathcal A_{\lceil jq/p\rceil-\lceil(j-1)q/p\rceil}
\right]_{11}.
\end{equation}
The shortened first run is the boundary convention already used for the
six-piece matching count. For example,
$m_{3/4}=[\mathcal A_1^3]_{11}=169$ and
$m_{3/7}=[\mathcal A_2^3]_{11}=2897$.

Replacing $q$ by $q_0+np$ adds exactly $n$ to each of the $p$ block indices.
In the positive expansion \eqref{eq:fibpositive}, each plus choice
contributes $\rho^n$ and each minus choice contributes $\rho^{-n}$.
Grouping the terms by the number of minus choices gives
\begin{equation}\label{eq:corridorexact}
m_{p/(q_0+np)}=\sum_{j=0}^{p}c_{p,q_0,j}\rho^{(p-2j)n},
\qquad c_{p,q_0,j}>0.
\end{equation}
Every mode occurs. The least order of a constant-coefficient recurrence
is therefore $p+1$, with characteristic polynomial
\[
\prod_{j=0}^{p}(X-\rho^{p-2j}).
\]
Pairing conjugate roots shows that its coefficients are integers.
For $p=3$ the polynomial is
$(X^2-18X+1)(X^2-3X+1)$, so either wall of this corridor satisfies
\begin{equation}\label{eq:phasorcappedcorridor}
m_{3/(q+12)}=21m_{3/(q+9)}-56m_{3/(q+6)}
              +21m_{3/(q+3)}-m_{3/q}.
\end{equation}
This equality holds at every step, for both colors of edge.

The exact regularity is a shared recurrence, not a constant output ratio.
The positive spectral sum gives strict log-convexity, hence
\[
\log_{10}\frac{m_{p/(q_0+(n+1)p)}}{m_{p/(q_0+np)}}
\nearrow p\log_{10}\rho.
\]
The small deviation is retained in the collision argument, rather than
discarded as an asymptotic error. Figure~\ref{fig:corridor} shows both
the geometric walls and that deviation.
\paperfigure{corridor}{.96}{figures/corridor_growth.pdf}{The two fixed-numerator
walls of the $5\to13$ phasor-capped corridor
($1/2\to1/3$ in fraction labels). The concave ceiling arc is omitted from this
recurrence illustration. Both walls obey \eqref{eq:phasorcappedcorridor}. In the
lower panel, the positive difference between $3\log_{10}\rho$ and the
actual logarithmic increment decreases at every step. Squares denote
Meridian increments and circles phasor increments. The values are exact
integers; only the plotted curves and logarithms are numerical.}{tbp}

\subsection{A positive correction with a uniform bound}
The sum of the block indices in \eqref{eq:corridorfib} is $q-1$.
Selecting every plus term in \eqref{eq:fibpositive} therefore gives
$3^{p-1}\rho^{q+p/2}/(\sqrt5)^p$ and hence
\begin{equation}\label{eq:corridorleading}
3m_{p/q}=
\left(\frac3{\sqrt5}\right)^p
\left(\frac{1+\sqrt5}{2}\right)^{2q+p}
(1+\varepsilon_{p,q}),
\end{equation}
where
\begin{equation}\label{eq:corridorerror}
0<\varepsilon_{p,q}\le
\left(1+\rho^{-2\lfloor q/p\rfloor}\right)^p-1.
\end{equation}
For $p>1$, coprimality implies that $q/p$ is not an integer.
Every block index in \eqref{eq:corridorfib}, including the shortened first
one, is at least $\lfloor q/p\rfloor$. The column and boundary comparison
in Lemma~\ref{lem:tail} bounds a specified set of $k$ minus choices by
$\rho^{-k}(\rho^{-2\lfloor q/p\rfloor})^k$, which is no larger than
$(\rho^{-2\lfloor q/p\rfloor})^k$. Summing over all nonempty subsets
proves \eqref{eq:corridorerror}. For $p=1$, the single entry
$[\mathcal A_{q-1}]_{11}=F_{2q+1}$ has relative correction
$\rho^{-2q-1}<\rho^{-2q}$, giving the same bound.
This estimate is independent of the remainder class of $q$ modulo $p$.

Along a fixed wall the correction decreases strictly to zero. Consequently
its logarithmic deviations telescope exactly:
\begin{equation}\label{eq:telescoping}
\begin{split}
\sum_{k=0}^{n-1}\left[
p\log\rho-\log\frac{m_{p/(q+(k+1)p)}}{m_{p/(q+kp)}}\right]
=\log\frac{1+\varepsilon_{p,q}}{1+\varepsilon_{p,q+np}}.
\end{split}
\end{equation}
Thus there is no accumulated error proportional to the path length.
For a route changing corridors, subtraction of the endpoint formula still
telescopes, but monotone decay is claimed only within a fixed wall.

\begin{theorem}[Necessary correction gap]\label{thm:corridorgap}
If $p\ne p'$ and $m_{p/q}=m_{p'/q'}$ for reduced labels, then
\begin{equation}\label{eq:correctiongap}
\frac{|\varepsilon_{p,q}-\varepsilon_{p',q'}|}
{1+\min(\varepsilon_{p,q},\varepsilon_{p',q'})}
>9^{-|p-p'|}.
\end{equation}
\end{theorem}
\begin{proof}
Assume $p>p'$. The ratio of the leading terms is
\[
R=\left(\frac{3(1+\sqrt5)}{2\sqrt5}\right)^{p-p'}\rho^{q-q'}.
\]
Its quadratic-field norm is $(9/5)^{p-p'}$, so $R\ne1$.
The number $(\sqrt5)^{p-p'}(R-1)$ is an algebraic integer, because the
powers of $(1+\sqrt5)/2$ and $\rho$ are units. Its nonzero norm gives
\[
|(R-1)(\overline R-1)|\ge5^{-(p-p')},\qquad
\overline R=(9/5)^{p-p'}/R>0.
\]
Put $\eta=\max(R,R^{-1})-1$. If $R\ge1$, then
$|R-1|=\eta$ and $|\overline R-1|<(9/5)^{p-p'}$.
If $R<1$, their product is
$\eta((9/5)^{p-p'}-1/(1+\eta))<\eta(9/5)^{p-p'}$.
Either case forces $\eta>9^{-(p-p')}$.
At equality of the Markov values,
$R=(1+\varepsilon_{p',q'})/(1+\varepsilon_{p,q})$,
so $\eta$ is precisely the left side of \eqref{eq:correctiongap}.
\end{proof}

\Needspace{13\baselineskip}
\subsection{The corridor collision window}
\begin{theorem}[Numerator--denominator window]\label{thm:corridorwindow}
Let $p\ne p'$, $P=\max(p,p')$, and let $K$ be the least nonnegative integer
satisfying
\begin{equation}\label{eq:K}
\rho^{2K}>P(9^{|p-p'|}+1).
\end{equation}
Every equality $m_{p/q}=m_{p'/q'}$ requires
\begin{equation}\label{eq:corridorwindow}
\max(q,q')\le P(K+1)-1.
\end{equation}
For bounded numerators a simpler, weaker consequence is
$\max(q,q')\le2P^2+P-1$.
\end{theorem}
\begin{proof}
If both quotient floors are at least $K$, then
$P\rho^{-2K}<1/(9^{|p-p'|}+1)$.
The elementary estimate $(1+u)^p\le(1-pu)^{-1}$ for $pu<1$
makes both corrections strictly smaller than $9^{-|p-p'|}$, contradicting
Theorem~\ref{thm:corridorgap}. Thus one denominator, say $q$, is at most
$Kp-1$.

The Fibonacci formula \eqref{eq:fibpositive} gives the entrywise bound
$\mathcal A_t\le\rho^t\mathcal A_0$ for every integer $t\ge0$.
Products preserve that bound, and $\mathcal A_0=\mat2111$ has operator
norm $\rho$. Since the indices sum to $q-1$, \eqref{eq:corridorfib} gives
$m_{p/q}\le\rho^{q-1}[\mathcal A_0^p]_{11}\le\rho^{q+p-1}$.
Together with the positive leading term, this yields
\[
\frac13\rho^{q'}<m_{p'/q'}=m_{p/q}\le\rho^{q+p-1}.
\]
Now $3<\rho^2$ implies $q'<q+p+1$. The denominators are integers,
so $q'\le q+p$.
It follows that both denominators are at most $P(K+1)-1$.
For the simpler bound it suffices to use $K=2P$:
$\rho^{4P}>36^P>P(9^{P-1}+1)$ for $P\ge2$, and $|p-p'|\le P-1$.
\end{proof}

For example, numerators at most 64 permit $K=75$, so every remaining
comparison lies at denominators at most 4863. With $P=64$ and numerator
difference five, $K=8$ gives 575. These are bounds on candidates, not new
numerical verification claims.

Theorem~\ref{thm:equaldegree} first discards equal Meridian degrees.
For the rest, the phase rectangle \eqref{eq:window} and denominator cap
\eqref{eq:corridorwindow} can be imposed together through the dictionary
\eqref{eq:degree-from-label}. Fixed-numerator ordering already separates
all labels with $p=p'$. What remains is a comparison of different
numerators and degrees inside their joint windows. Neither numerator nor
degree is globally bounded by these theorems, so this is not yet a finite
reduction of the entire unicity conjecture.