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  pdftitle={Singularities on characteristic 3 rank 1 del Pezzo surfaces},
  pdfauthor={Przemyslaw Chojecki}
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\fancyhead[L]{Characteristic $3$ rank $1$ del Pezzo surfaces}
\fancyhead[R]{Przemyslaw Chojecki}
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\begin{titlepage}
\centering
{\Large Singularities on characteristic $3$ rank $1$ del Pezzo surfaces\par}
\vspace{1.8cm}
{\large Przemyslaw Chojecki\par}
\vspace{0.5cm}
{\normalsize 31 March 2026\par}
\vfill
\begin{minipage}{0.88\textwidth}
\small
\textbf{Abstract.} We prove that every characteristic-$3$ klt del Pezzo surface of Picard rank $1$ has at most seven singular points. On the explicit side, the construction corridors treated here do not produce a Picard-rank-one $8$-point surface: Method~A is closed outright, and the weighted corridors of Method~B either leave the Picard-rank-one setting or fail ampleness. On the birational side, any hypothetical counterexample is reduced to the residual sector $\operatorname{ht}(X)\ge 4$ with no descendant having elliptic boundary, and that sector is then closed completely. The proof combines the elimination of the first Route~C frontier, an anticanonical contact formula and a universal contact-defect sieve, a cubic-backbone and clean-threaded analysis of special multisections, marked plane reductions for the one-section branches, scalar-packet preparation on the several-special-horizontal side, and a terminal-leaf count in the height-$4$ maximal-width corridor. The final step solves the residual height-selection problem by closing the remaining packet and smooth-image special-multisection branches. Consequently no characteristic-$3$ rank-one klt del Pezzo surface has more than seven singular points.
\end{minipage}
\end{titlepage}

\clearpage
\pagenumbering{arabic}
\tableofcontents
\newpage

\section{Introduction and overview}

We study the characteristic-$3$ many-singularity problem for rank-one klt del Pezzo surfaces and prove that no such surface has more than seven singular points.  All currently known constructions in characteristic $3$ had at most seven singular points, and the main theorem of this paper shows that this bound is optimal \cite{PP12,PP3,Epoch}.  In particular, this resolves the FrontierMath / Epoch open problem \cite{Epoch} in the negative: there is no characteristic-$3$ rank-one klt del Pezzo surface with more than seven singular points.

The terminology \emph{Method A}, \emph{Method B}, and \emph{Method C} comes from the FrontierMath / Epoch problem prompt for this question \cite{Epoch}.  In that prompt, Method~A is the global-quotient route starting from a smooth complete intersection
$\mathrm{CI}(2,2)\subset \PP^4$
with a tame cyclic action; Method~B is the weighted-model route using quasi-smooth weighted hypersurfaces or complete intersections with positive Fano index; and Method~C is the scripted blow-up route starting from
$\PP^1\times \PP^1$
and contracting prescribed Hirzebruch--Jung chains.  In the present paper, Methods~A and~B keep their prompt meaning as explicit construction formats, while our \emph{Route~C} is the birational and resolution-theoretic analysis of the same blow-up / contraction side suggested by Method~C.  The change from ``Method'' to ``Route'' is deliberate: by the time Section~\ref{sec:routec} begins, the paper is no longer proposing a single explicit construction ansatz, but analyzing the residual geometry of any hypothetical counterexample.

The paper has two different roles.  Sections~1--8 are included to map the search space, record the explicit Method~A / Method~B obstructions, and preserve the numerical and heuristic exploration that led to the final strategy; they are useful context, but they are not the formal proof spine of the seven-point theorem.  The actual proof begins in Section~9.  From there onward, the argument has three layers: first, Section~\ref{sec:routec} reduces any hypothetical counterexample to the residual height-$\ge 4$, no-descendant sector; second, the middle Route~C sections develop the global bridge, packet, and multisection machinery on the minimal resolution; third, the final Route~C sections close the residual geometry by a cubic-backbone reduction for ordinary ramification forks, a clean threaded $[2]$-root budget, a preparation step clearing packet--negative-section contact points so that high-height packets can be treated in the disjoint scalar/Frobenius model, marked plane reductions for the one-section branches, and a terminal-leaf count together with the final height-selection theorem.

The main ingredients and intermediate results are the following.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item \textbf{Methods~A and~B are largely exhausted in the benchmark corridors studied here.}
    Method~A is closed outright, and the weighted hypersurface / weighted complete-intersection corridors analyzed in the paper do not yield a Picard-rank-one surface with at least eight singular points.

    \item \textbf{Residual reduction.}
    Any characteristic-$3$ rank-one klt del Pezzo surface with at least eight singular points lies in the residual sector
    \[
    \operatorname{ht}(X)\ge 4
    \qquad\text{and}\qquad
    X\text{ has no descendant with elliptic boundary.}
    \]

    \item \textbf{The entire first analyzed Route~C frontier leaves the residual sector.}
    Same-component positive root-contractible contact-$2$ configurations already give elliptic ties.  Among bridges, the peelable family has height $1$, the two square-one fork prototypes force descendants with elliptic boundary, and the two higher-square short bridges have height at most $2$.

    \item \textbf{A new global anticanonical bridge sieve.}
    For a chain bridge one has the exact formula
    $-K_X\cdot \pi(L)=-1+\mu(A,i)+\mu(B,j),$
    which eliminates the full one-root off-frontier family
    $[b]\dashv[a,(2)^{a-1}]$
    in the benchmark range.

    \item \textbf{A new $2$-tail core reduction for chain bridges.}
    After stripping terminal $[2]$-tails away from the touched components, every positive chain bridge has at least one tip-visible side.  After this reduction, the only reduced tip-visible cores with contact weight strictly larger than $1/2$ are a touched $(-2)$-tip and the singleton core $[3]$.

    \item \textbf{A universal contact-defect sieve for surviving nonexceptional curves.}
    If
    $L\not\subset D$
    is a vertical $(-1)$-curve on $\widetilde X$, then
    \[
    \sum_{E\subset D}(L\cdot E)\operatorname{cf}_X(E)<1.
    \]
    Consequently, any surviving ramification fork satisfies
    \[
    \frac{2}{a}+\frac{2}{b}>1,
    \]
    and any surviving scalar star is $A_1$-heavy: all but at most two packet sections are $(-2)$-curves.

    \item \textbf{Further unconditional narrowing of the multisection, bisection, and trisection branches.}
    An ordinary surviving ramification fork has local degree at most $3$; in characteristic $3$ the purely inseparable ordinary branch is cubic, the corresponding smooth rational cubic backbone on a Hirzebruch surface has self-intersection $6$, and the clean threaded $[2]$-root model has at most two primitive ramification fibers.  On the one-section separable side, every ruled-model bisection in the partition $2+1+1$ meets the second section, every singular point of such a bisection has an explicit double-point normal form, and every irreducible rational bisection of class $2S_0+(2n+s)f$ reduces by successive singular-point elementary transforms to a marked plane conic or cubic.  In the touched subcase $s=1$, one obtains a singular plane cubic with a smooth marked point whose projection realizes the original degree-$2$ map; consequently the whole branch has at most two unibranch ramification strings.  Every surviving unibranch block of the ordinary bisection is already absorbed by the old touched-tip / branching side, and every multibranch collision of that bisection is impossible by the contact-defect sieve.  On the degree-three mixed side, every singular trisection reduces after at most four elementary transforms to an irreducible rational plane curve of degree $3,4,5,$ or $6$ with marked-point multiplicity equal to the degree minus $3$; the exceptional class $3S_0+8f\subset \mathbb F_2$ is itself reduced to the marked plane package, and every remaining multibranch collision of the ordinary trisection is either impossible by the contact-defect sieve or already absorbed by the old touched-tip / branching side.  A supplementary finite ordinary-fiber search on the cubic backbone through eight additional blowups rules out the searched defect-zero service modules.

    \item \textbf{Closure of the height-$4$ maximal-width corridor.}
    In the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, the exact excess identity
    \[
    \Sigma=\sum_F(\sigma(F)-1)=3
    \]
    holds, at most three genuinely bad singly-hit multiplicity-one tips can occur, and one horizontal $1$-section avoids all of them.  Moreover, in a $\lambda$-minimal maximal-width witness every one-touch bad branch starts on the horizontal packet, packet-section roots strip away by elementary transforms, any vertical component meeting all four horizontal sections is an isolated $(-2)$-curve with no vertical neighbor, and the remaining terminal corridor contributes a leaf of $F_{\mathrm{red}}-D$.  A terminal-leaf count then gives $v(F)\le \sigma(F)-1$ for every degenerate fiber, so one gets
    $\#\Sing(X)\le 7$
    throughout the whole height-$4$, maximal-width corridor.

    \item \textbf{Closure of the residual height-selection problem.}
    The several-special-horizontal packet branch and the separable special-multisection branch are both closed in the final section.  Hence every residual surface admits a witnessing $\PP^1$-fibration of height $4$ and maximal width.
\end{enumerate}

Combining the maximal-width corridor closure with the final height-selection theorem gives the main result of the paper: every characteristic-$3$ klt del Pezzo surface of Picard rank $1$ satisfies $\#\Sing(X)\le 7$.


\subsection{Main results and organization}

The paper is organized as follows.  Section~\ref{sec:lit} recalls the current classification picture.  Sections~\ref{sec:globalq}--\ref{sec:cube} treat the explicit quotient, weighted, and spectrum-first constructions attached to Methods~A and~B.  Section~\ref{sec:routec} begins the Route~C analysis by reducing to the residual sector and by developing the first entry criteria.  The next Route~C sections classify the first chain/fork/cross-link frontier and remove all of its residual survivors.  The later Route~C sections prove same-component completeness, introduce the anticanonical bridge sieve and the universal contact-defect sieve, reduce the remaining chain bridge geometry to tip-visible $2$-tail cores, push the multisection branch by a cubic-backbone reduction and a clean threaded $[2]$-root budget together with a two-fiber support bound for non-clean purely inseparable ordinary singleton $[2]$-root patterns, push the one-section branch by explicit double-point normal forms and a marked plane conic/cubic reduction for the bisection case together with a low-degree unibranch analysis and a finite marked plane package for trisections of degree at most six, and return to the height-$4$ maximal-width endgame through $\lambda$-minimal one-touch reductions, a rigid four-section-hub analysis, and a terminal-leaf count that closes the maximal-width corridor.  Along the cubic-backbone branch we also record a supplementary finite ordinary-fiber search through eight additional blowups.

Table~\ref{tab:summary} records the main representative constructions and near-misses encountered in the search.

\begin{table}[h!]
\centering
\small
\caption{Representative examples and outcomes.}
\label{tab:summary}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{2.4cm} >{\raggedright\arraybackslash}X >{\centering\arraybackslash}p{1.9cm} >{\centering\arraybackslash}p{1.3cm} >{\raggedright\arraybackslash}p{2.4cm}@{}}
\toprule
Format & Data & Singularities & $\rho$ & Outcome \\
\midrule
Known char.~$3$ blow-up model & basket $[[3],[3],[3],[3],[2,2],[2,2],[2,2]]$ & $4[3]+3A_2$ & $1$ & Verified $7$-point example \\
Weighted hypersurface & $x_0^6+x_0^5x_1+x_1^6+x_2x_3=0 \subset \PP(2,2,5,7)$ & $6A_1+[5]+[7]$ & $6$ & Exact $8$ points, wrong Picard number \\
Weighted hypersurface & same equation in $\PP(4,4,5,19)$ & $6A_3+[5]+[19]$ & $6$ & Exact $8$ points, wrong Picard number \\
Symmetric hypersurface & $x_0^7+x_1^7+x_2^2+x_3^2=0 \subset \PP(2,2,7,7)$ & $7A_1+2[7]$ & $7$ & $9$ points, wrong Picard number \\
Method C near-miss & basket $[[6],[3],[3],[3],[3],[2,2],[2,2],[2,2],[2,2]]$ & $9$ points & $1$ (numerical) & $-K$ nef, not ample \\
Method C near-miss & basket $[[5,2],[2],[3],[3],[3],[2,2],[2,2],[2,2]]$ & $8$ points & $1$ (numerical) & $-K$ not ample \\
Cube-spectrum candidate & one degree-$8$ section in weighted cube format inside $\PP(1,2,4,4,5,5,7)$ & $5\times\frac14(1,1)+2\times\frac15(1,2)+\frac17(1,1)$ & $9$ (if realized) & Spectral hit, but not rank one \\
\bottomrule
\end{tabularx}
\end{table}

\section{Background from the current classification picture}\label{sec:lit}

The classical characteristic-zero upper bound is Belousov's theorem: a log del Pezzo surface of Picard rank one has at most four singular points \cite{Belousov}.  In characteristic $3$, this fails dramatically: Lacini's classification in characteristics different from $2$ and $3$ already isolates characteristic $3$ as exceptional, and the newer Palka--Pe\l ka programme shows that the known characteristic-$3$ extremal baskets have $7$ singular points \cite{Lacini,PP12,PP3}.

Two baskets repeatedly appeared as extremal throughout the search:
\begin{align*}
4\cdot [3] + 3A_2
&= [[3],[3],[3],[3],[2,2],[2,2],[2,2]], \\
4A_2+[5]+[3]+[2]
&= [[5],[3],[2],[2,2],[2,2],[2,2],[2,2]].
\end{align*}
The first is the known height-$3$ extremal basket; the second is the elliptic-descendant endpoint in characteristic $3$ \cite{PP3,PP12,OpenDP}.

These results do \emph{not} prove the impossibility of $8$ points, but they tell us where not to look: any new construction must evade the currently classified height-$\le 3$ and elliptic-descendant sectors.


At the time of writing, the public Palka--Pe\l ka classification series still reaches only height $3$; the characteristic-$3$ height-$4$ endpoint is announced in \cite[Remark~1.7]{PP12} as forthcoming.  A further external input relevant to the later one-section analysis is Kettinger's theorem that characteristic $3$ admits no unexpected plane cubics \cite{Kettinger}.  This makes the marked plane-cubic reduction of the residual bisection branch especially attractive.

\section{Global quotients and the Frobenius blow-up route}\label{sec:globalq}

\subsection{Method A: diagonal involution quotients of $\mathrm{CI}(2,2)\subset\PP^4$}

The warm-up quotient construction starts with a smooth complete intersection of two quadrics in $\PP^4$ and a diagonal involution of order $2$.  The following observation kills the possibility of reaching $8$ singular points within this setup.

\begin{proposition}[derived in this search]\label{prop:methodA6}
Let $Y\subset \PP^4$ be a smooth complete intersection of two quadrics, and let $g$ be a diagonal involution acting tamely in characteristic $3$, freely in codimension $1$ on $Y$.  Then the quotient $X=Y/\langle g\rangle$ has at most $6$ singular points.
\end{proposition}

\begin{proof}
A diagonal involution on $\PP^4$ has fixed locus equal to a disjoint union of linear subspaces.  In the codimension-$1$-free case relevant here, the fixed locus is $\PP^2\sqcup\PP^1$.  The singular points of the quotient correspond to isolated points in $Y\cap \mathrm{Fix}(g)$.  A $(2,2)$ complete intersection meets the fixed $\PP^2$ in at most $4$ points and the fixed $\PP^1$ in at most $2$ points, hence at most $6$ points in total.
\end{proof}

Thus the standard tame global-quotient version of Method~A cannot solve the benchmark.

\subsection{Method C: the Frobenius/Keel--McKernan blow-up family}

The known characteristic-$3$ construction with $7$ singular points comes from a modification of the Keel--McKernan example; see Lacini's Example~7.4 and the later Palka--Pe\l ka discussion \cite{Lacini,PP3}.  It produces the basket $4[3]+3A_2$.

The obvious question is whether one can repeat the same triple-blow-up move over more fibres.  The next proposition records the resulting obstruction.

\begin{proposition}[derived in this search]\label{prop:frobenius-only13}
Consider the natural extension of the characteristic-$3$ Frobenius graph construction on $\PP^1\times \PP^1$ in which a curve $C$ satisfies
$C\cdot F = 3a,$
and one performs the same triple blow-up process over $m$ fibres.  If the resulting contracted surface is denoted by $T$, then along a general fibre
\[
-K_T\cdot \psi_*F = 2-3a\Bigl(1-\frac{2}{r}\Bigr), \qquad r=3m-6a.
\]
This quantity is positive only for $(a,m)=(1,3)$, which is exactly the known $7$-point case.
\end{proposition}

\begin{proof}
After the blow-ups one has
$C^2=6a-3m=-r.$
The discrepancy computation extending the known characteristic-$3$ example gives
\[
\psi^*K_T = K_S + \frac13\sum_{i=1}^m F_i + \Bigl(1-\frac{2}{r}\Bigr)C.
\]
Intersecting with a general fibre yields the displayed formula.  Since $m>2a$, one has $r=3(m-2a)\ge 3$, and the expression is maximized when $m=2a+1$, giving
\[
2-3a+\frac{2a}{m-2a}\le 2-a.
\]
Hence positivity forces $a=1$, and then $m=3$.
\end{proof}

This explains why the most naive extension of the known $7$-point blow-up construction fails immediately: the next step only makes $-K$ nef, not ample.

\begin{remark}
Two useful near-misses emerged from this computation.  The obvious four-fibre extension has basket
\[
[[6],[3],[3],[3],[3],[2,2],[2,2],[2,2],[2,2]],
\]
so $9$ singular points, but $-K$ is only nef.  After one extra smooth blowdown one obtains the rank-compatible $8$-point basket
$[[5,2],[2],[3],[3],[3],[2,2],[2,2],[2,2]],$
which again fails ampleness.
\end{remark}

\section{Weighted hypersurfaces: exact formulas and exact dead ends}\label{sec:whyp}

\subsection{The six-branch family $f_n(x_0,x_1)+x_2x_3=0$}

The cleanest weighted family discovered in the search is
\[
X^{(n)}_{a,p,q} := \{f_n(x_0,x_1)+x_2x_3=0\} \subset \PP(a,a,p,q), \qquad p+q=an,
\]
where $f_n$ is a binary form of weighted degree $an$, split with $n$ distinct roots, and where
$3\nmid apqn, \qquad \gcd(a,p)=\gcd(a,q)=1.$
This family is tame, quasi-smooth, and extremely explicit.

\begin{proposition}[derived in this search]\label{prop:rho-n}
For the family $X^{(n)}_{a,p,q}$ above, one has:
\begin{enumerate}[label=(\roman*)]
    \item $X^{(n)}_{a,p,q}$ has singularity basket
    $nA_{a-1} + [p] + [q];$
    \item
    \[
    K_X^2 = \frac{4(p+q)}{pq};
    \]
    \item the minimal resolution $\widetilde{X}$ satisfies
    $K_{\widetilde{X}}^2 = 8-(p+q)=8-an;$
    \item $\widetilde{X}$ is rational and
    $\rho(X)=n.$
\end{enumerate}
In particular, exact $8$ singular points in this family force $n+2=8$, hence $n=6$ and therefore $\rho(X)=6$.
\end{proposition}

\begin{proof}
On the singular line $\PP(a,a)$ the equation restricts to $f_n=0$, so there are $n$ reduced points, each of type
\[
\frac1a(p,q)\cong \frac1a(1,-1)=A_{a-1}.
\]
At the coordinate points one gets cyclic quotient singularities
\[
\frac1p(a,a)\cong \frac1p(1,1)=[p], \qquad \frac1q(a,a)\cong \frac1q(1,1)=[q].
\]
The weighted hypersurface formula gives
\[
K_X^2 = d\,\frac{(\sum w_i-d)^2}{\prod w_i}
= (p+q)\frac{(2a)^2}{a^2pq} = \frac{4(p+q)}{pq}.
\]
The $A_{a-1}$ points are crepant, while the points $[p]$ and $[q]$ contribute
\[
\frac{(p-2)^2}{p}, \qquad \frac{(q-2)^2}{q}
\]
to the drop of $K^2$.  Hence
\[
K_{\widetilde{X}}^2 = \frac{4(p+q)}{pq}-\frac{(p-2)^2}{p}-\frac{(q-2)^2}{q}=8-(p+q).
\]
The equation is linear in $x_3$ on the chart $x_2\neq 0$, so $X$ is rational, hence $\widetilde{X}$ is rational.  By Noether's formula for rational surfaces,
\[
\rho(\widetilde{X}) = 10-K_{\widetilde{X}}^2 = an+2.
\]
The exceptional divisor has
$n(a-1)+2$
components, so
$\rho(X)=\rho(\widetilde{X})-(n(a-1)+2)=n.$
\end{proof}

Two especially useful exact-$8$ members are:
\begin{align*}
&x_0^6+x_0^5x_1+x_1^6+x_2x_3=0 \subset \PP(2,2,5,7), \\
&x_0^6+x_0^5x_1+x_1^6+x_2x_3=0 \subset \PP(4,4,5,19),
\end{align*}
with baskets $6A_1+[5]+[7]$ and $6A_3+[5]+[19]$ respectively; both have $\rho=6$.

\subsection{A second large family with many points but forced Picard number}

Another useful near-miss is the symmetric family
\[
X_{ab}\subset \PP(a,a,b,b), \qquad F=f_b(x_0,x_1)+g_a(x_2,x_3),
\]
with $\gcd(a,b)=1$ and $3\nmid ab$.  This family makes it easy to create many singular points, but the Picard number again becomes explicit.

\begin{proposition}[derived in this search]\label{prop:rho-aabb}
In the symmetric family above one has
$\rho(X)=(a-1)(b-1)+1.$
Thus large singular baskets in this family automatically come with large Picard number.
\end{proposition}

\begin{proof}
The surface has $b$ singular points of type $\frac1a(1,1)$ and $a$ singular points of type $\frac1b(1,1)$.  One computes
\[
K_X^2=\frac{(2a+2b-ab)^2}{ab}
\]
and then
$K_{\widetilde{X}}^2 = 8-ab.$
As before the surface is rational, so $\rho(\widetilde{X})=ab+2$.  The exceptional divisor has $a+b$ components, giving
$\rho(X)=ab+2-(a+b)=(a-1)(b-1)+1.$
\end{proof}

The concrete example
\[
x_0^7+x_1^7+x_2^2+x_3^2=0 \subset \PP(2,2,7,7)
\]
has $9$ singular points and $\rho=7$.

\section{Birational surgery on exact-$8$ weighted sources}\label{sec:birsur}

The weighted exact-$8$ sources above suggest a natural strategy: start with a surface with $8$ singular points and $\rho=6$, then contract five additional smooth $(-1)$-curves while preserving the basket.  The next result shows that this fails on the entire six-branch family.

\begin{proposition}[derived in this search]\label{prop:threepoint-barrier}
Let $X=X^{(6)}_{a,p,q}$ be an exact-$8$ member of the family in Proposition~\ref{prop:rho-n}, so that
\[
X=\{f_6(x_0,x_1)+x_2x_3=0\}\subset \PP(a,a,p,q), \qquad p+q=6a.
\]
Let $\widetilde{X}$ be its minimal resolution.  Consider any birational sequence obtained by smooth blowdowns on $\widetilde{X}$ followed by contraction of the remaining negative-definite divisor to a klt surface $Y$.  Then:
\begin{enumerate}[label=(\roman*)]
    \item if $r$ is the number of surviving branch components of the six $A_{a-1}$-branches, one has
    $\rho(Y)\ge r;$
    \item in particular, if $\rho(Y)=1$, then at most one branch survives, and therefore
    $\#\Sing(Y)\le 3.$
\end{enumerate}
Thus no exact-$8$ member of this family can be birationally massaged to Picard number $1$ while retaining $8$ singular points.
\end{proposition}

\begin{proof}[Sketch]
On $\widetilde{X}$ the negative curves form a visible graph with two endpoint curves $E_p$, $E_q$ and six identical branches
\[
V_i - A_{i,a-1} - A_{i,a-2} - \cdots - A_{i,1} - C_i,
\]
where the $A_{i,j}$ are $(-2)$-curves and the outer curves $V_i,C_i$ are $(-1)$-curves.  A separate intersection argument shows that every non-visible irreducible curve has nonnegative self-intersection, so the entire contraction game is contained in this graph.

Fix a branch $i$, and let $s_i$ be the number of internal branch components that survive to the final exceptional divisor.  If a branch survives, then the number of smooth blowdowns one can spend on that branch is at most $(a-1)-s_i$; if the branch dies completely, the maximum is $a$.  Summing over all six branches and the two endpoints yields an upper bound on the number $t$ of smooth blowdowns and an upper bound on the number $m$ of components in the final exceptional divisor.  Since
$\rho(\widetilde{X})=6a+2,$
one gets
$\rho(Y)=\rho(\widetilde{X})-t-m \ge r,$
where $r$ is the number of branches still contributing singular data at the end.  If $\rho(Y)=1$, then $r\le 1$, and only the two endpoints can add further singular points, proving the bound $\#\Sing(Y)\le 3$.
\end{proof}

\begin{remark}
For the smallest source $\PP(2,2,5,7)$, a more detailed visible-graph computation gives an even stronger result: any Picard-rank-$1$ output from that source has at most one singular point.
\end{remark}

\section{Codimension-$2$ weighted complete intersections}\label{sec:codim2}

This section records the main negative results for the most natural codimension-$2$ weighted complete-intersection families.

\subsection{Separated repeated lines in $\PP(a,a,b,b,c)$}

The first codimension-$2$ ansatz tried to cut one repeated-weight line with one equation and the other repeated-weight line with the other equation.  The outcome is extremely rigid.

\begin{proposition}[derived in this search]\label{prop:separated-lines-collapse}
In the family
$X_{d_1,d_2}\subset \PP(a,a,b,b,c),$
assume the two repeated singular lines are cut by the two equations in the separated-strata way, producing exactly $8$ singular points in total.  Then one of the two line multiplicities must be $1$, hence one defining equation is linear in a repeated-weight block and the model reduces to a hypersurface.
\end{proposition}

The proof is a combination of weighted B\'ezout counts on the repeated lines and the Fano index inequality.  In short: the only way to keep the total point count at $8$ while preserving positivity of the anticanonical index is to make one equation linear on one of the repeated-weight blocks.

\subsection{Shared gcd on a singular line is incompatible with quasismoothness}

The next idea was to let both equations restrict nontrivially on the same repeated-weight line and force a common divisor there.

\begin{proposition}[derived in this search]\label{prop:sharedgcd-line}
Let
$X_{d_1,d_2}\subset \PP(a,a,b,b,c)$
with $\gcd(a,b)=\gcd(a,c)=1$, and assume that both restrictions to the singular line $\PP(a,a)$ are nonzero and share a nontrivial common factor.  Then $X$ is not quasismooth.
\end{proposition}

\begin{proof}
Write the restrictions as $f_1=Gh_1$ and $f_2=Gh_2$.  At a root of $G$, the derivatives in the tangent $x_0,x_1$ directions are proportional.  Because $a\mid d_i$ while $a\nmid b,c$, there are no monomials linear in $x_2,x_3,x_4$ that could contribute independent normal derivatives at such a point.  Hence the Jacobian rank drops below $2$.
\end{proof}

So the ``shared-gcd on a singular line'' mechanism does not exist in this coprime codimension-$2$ setting.

\subsection{The repeated-plane family $\PP(a,a,a,b,b)$}

The next natural possibility is to use the singular plane $\PP(a,a,a)$.

\begin{proposition}[derived in this search]\label{prop:aaabb-no8}
There is no tame quasismooth del Pezzo complete intersection
$X_{d_1,d_2}\subset \PP(a,a,a,b,b)$
with exactly $8$ singular points, except for the cases where one equation is linear in the $(x_0,x_1,x_2)$-block and the model reduces to a hypersurface.
\end{proposition}

\begin{proof}[Idea]
If both degrees are multiples of $a$, then the singular plane contributes exactly $r_1r_2$ points, where $d_i=r_i a$.  The repeated-weight line $\PP(b,b)$ contributes either $0$ points or the full degree of the one equation restricting nontrivially to it.  Exact-$8$ counting leaves only the numerical cases $(r_1,r_2)=(2,4)$ or $(2,3)$, and both are excluded by the Fano index inequality and the divisibility constraints needed for line points.
\end{proof}

\subsection{The vertex mechanism in $\PP(a,a,ma,b,b)$}

One can also force one plane point through the weighted-plane vertex.

\begin{proposition}[derived in this search]\label{prop:aamabb-no8}
There is no tame quasismooth del Pezzo complete intersection
$X_{d_1,d_2}\subset \PP(a,a,ma,b,b)$
with exactly $8$ singular points.
\end{proposition}

\begin{proof}[Idea]
Quasismoothness at the weighted-plane vertex forces
$d_i = r_i a, \qquad r_i\equiv 1 \pmod m.$
A local intersection calculation shows that the plane contributes
\[
1+\frac{r_1r_2-1}{m}
\]
points in the minimal ``one vertex plus reduced residual intersection'' case.  Requiring this number to be at most $8$ leaves finitely many arithmetic possibilities; all but one are ruled out by tameness/Fano index, and the last remaining candidate
$X_{10,10}\subset \PP(2,2,8,5,5)$
fails quasismoothness along the repeated-weight line.
\end{proof}

\subsection{The asymmetric family $\PP(a,a,ma,b,c)$}

The asymmetric family with distinct weights $b,c$ was searched next.

\begin{proposition}[derived in this search]\label{prop:aamabc-lineplane}
In the family
\[
X_{d_1,d_2}\subset \PP(a,a,ma,b,c), \qquad g:=\gcd(b,c)>1,
\]
with $3\nmid abcm$, the standard reduced-plane mechanism and the simple-vertex plane mechanism cannot yield exactly $8$ singular points.
\end{proposition}

\begin{proof}[Idea]
If $d_i=r_i a$, then any nonzero multiple of $a$ in the semigroup $\langle b,c\rangle$ is at least $2ma+2$, because $b,c\ge ma+1$ and are both coprime to $a$.  Therefore line points force
$r_i\ge 2m+1,$
and quasismooth line points force an independent normal linear term, hence
$r_j\ge 2m+2.$
These lower bounds are incompatible with the exact-$8$ point count from either the reduced weighted-plane B\'ezout number or the one-vertex-plus-reduced-residual count.
\end{proof}

In a broad arithmetic scan of the coprime case $\gcd(b,c)=1$, every exact-$8$ hit found numerically collapsed to a hypersurface because one defining degree equalled a weight.  No genuinely new exact-$8$ codimension-$2$ example emerged from these searches.

\section{Quotients and unprojections}\label{sec:quotunproj}

\subsection{Smooth-upstairs quotients are too small}

The next idea was to quotient by a second tame group action after constructing a surface with many singularities.  The smooth-upstairs case is ruled out uniformly.

\begin{proposition}[derived in this search]\label{prop:smoothquotient5}
Let $Y$ be a smooth rational surface, let $G$ be a finite group of order prime to $3$ acting on $Y$ freely in codimension $1$, and let $X=Y/G$.  If $\rho(X)=1$, then
$\#\Sing(X)\le 5.$
\end{proposition}

\begin{proof}
Write the singular points of $X$ as $x_1,\dots,x_s$, and let $m_i$ be the stabilizer order of a point above $x_i$.  Removing the finitely many points with nontrivial stabilizer gives a finite \'{e}tale cover
$Y^\circ\to X^\circ$
of degree $|G|$, hence
\[
e(Y)-\sum_{i=1}^s \frac{|G|}{m_i} = |G|\,(e(X)-s).
\]
Since $Y$ is rational and smooth, the $\ell$-adic cohomology is algebraic in degree $2$.  The quotient is tame, so
$H^*(X,\QQ_\ell)\cong H^*(Y,\QQ_\ell)^G.$
The condition $\rho(X)=1$ therefore gives
$e(X)=3.$
Rearranging yields
\[
3-s+\sum_{i=1}^s \frac1{m_i}=\frac{e(Y)}{|G|}>0.
\]
Since each $m_i\ge 2$,
\[
0<3-s+\frac{s}{2}=3-\frac{s}{2},
\]
so $s<6$.
\end{proof}

Thus no tame codimension-$1$-free quotient of a smooth rational surface can ever produce the benchmark.

\subsection{Even quotients of exact-$8$ sources do not help}

A similar Euler characteristic argument handles the most optimistic secondary-quotient strategy.

\begin{corollary}[derived in this search]\label{cor:exact8-quotient}
Let $Y$ be a rational klt surface with exactly $8$ singular points and Picard number $r$, and let a nontrivial tame group $G$ act on $Y$ freely in codimension $1$.  If $X=Y/G$ has Picard number $1$, then
\[
\#\Sing(X)\le 6-\frac{2(r-2)}{|G|}.
\]
In particular, for the exact-$8$ weighted hypersurface sources above, where $r=6$, every such quotient has at most $5$ singular points.
\end{corollary}

\begin{proof}
Write $s=\#\Sing(X)$.  For each singular point $x_i\in X$, choose one point $y_i\in Y$ above it and let
$m_i=|G_{y_i}|$
be its stabilizer order.  Because the action is free in codimension $1$, the fiber over $x_i$ consists of $|G|/m_i$ points.

Let $Y^\circ\subset Y$ be the complement of the full preimage of $\Sing(X)$, and let
$X^\circ=X\setminus \Sing(X)$.
Then $Y^\circ\to X^\circ$ is a finite \'etale cover of degree $|G|$, so
\[
e(Y)-\sum_{i=1}^s \frac{|G|}{m_i}
=e(Y^\circ)
=|G|\,e(X^\circ)
=|G|\,(e(X)-s).
\]
For rational klt surfaces the degree-$2$ \'{e}tale cohomology is algebraic, so
$e(Y)=2+\rho(Y)=2+r.$
Also, by the same invariant-cohomology argument used in Proposition~\ref{prop:smoothquotient5}, the condition $\rho(X)=1$ gives
$e(X)=3.$
Dividing the displayed identity by $|G|$ yields
\[
3-s+\sum_{i=1}^s \frac1{m_i}=\frac{2+r}{|G|}.
\]

Now let $u$ be the number of downstairs singular points whose preimages are singular points of $Y$ with trivial stabilizer.  Each such point comes from an orbit of $|G|$ singular points upstairs, and $Y$ has only $8$ singular points, so
$u\le 8/|G|.$
For these $u$ points we have $m_i=1$, while for the remaining $s-u$ points we have $m_i\ge 2$.  Therefore
\[
\sum_{i=1}^s \frac1{m_i}
\le u+\frac{s-u}{2}
\le \frac{s}{2}+\frac{4}{|G|}.
\]
Substituting into the previous identity gives
\[
3-s+\frac{s}{2}+\frac{4}{|G|}\ge \frac{2+r}{|G|},
\]
so
\[
s\le 6-\frac{2(r-2)}{|G|}.
\]
If $r=6$, then the right-hand side is
$6-8/|G|<6,$
so $s\le 5$.
\end{proof}

\subsection{Natural unprojection ladders out of $f_6+x_2x_3$}

The obvious codimension-$3$/codimension-$4$ move is to unproject one of the branch divisors in the exact-$8$ family.

\begin{proposition}[derived in this search]\label{prop:unprojection-ladder}
A branch-unprojection step applied to
\[
X^{(6)}_{a,p,q}=\{f_6(x_0,x_1)+x_2x_3=0\}\subset \PP(a,a,p,q)
\]
produces a new model that eliminates back to the same family with one fewer line root, namely
\[
X^{(5)}_{a,p-a,q}\quad\text{or}\quad X^{(5)}_{a,p,q-a}.
\]
Consequently every such unprojection lowers the singularity count by at least one.  Moreover, in characteristic $3$, alternating the two sides of the ladder immediately introduces a weight divisible by $3$.
\end{proposition}

\begin{proof}
Choose a factorization
\[
f_6=l\,g_5,
\]
where $l$ has weighted degree $a$ and $g_5$ has weighted degree $5a$.  Consider the codimension-$2$ divisor
\[
D=(l=x_3=0)\subset X^{(6)}_{a,p,q}.
\]
On the complement of $D$ the rational function
\[
s:=\frac{x_2}{l}=-\frac{g_5}{x_3}
\]
is regular.  Introduce a new variable $s$ of weight $p-a$.  The graph of this rational function is cut out by
\[
x_2-ls=0,
\qquad
sx_3+g_5=0
\]
in the weighted projective space with coordinates of weights
$(a,a,p-a,p,q).$
Eliminating $x_2$ gives the hypersurface
\[
sx_3+g_5(x_0,x_1)=0
\]
in
$\PP(a,a,p-a,q).$
Because
$(p-a)+q=5a,$
this is exactly a member of the same family with one fewer line root, namely
\[
X^{(5)}_{a,p-a,q}.
\]
The construction on the other side, starting from the divisor $(l=x_2=0)$, is symmetric and yields
\[
X^{(5)}_{a,p,q-a}.
\]

The six singular points on the repeated line $\PP(a,a)$ are the six roots of $f_6$.  Replacing $f_6=l g_5$ by $g_5$
removes one root, so the line contribution drops from $6$ to $5$; hence the total singularity count drops by at least one at each step.

Finally, since
$p+q=6a$
and $3\nmid a,p,q$, the residues of $p$ and $q$ modulo $3$ are opposite.  Therefore exactly one of
$p-a$ and $q-a$
is divisible by $3$.  After one unprojection on one side, alternating to the other side introduces the other shifted weight, so the alternating ladder immediately leaves the tame characteristic-$3$ setting.
\end{proof}

Hence the natural unprojection ladder does not turn the exact-$8$ hypersurface family into a rank-one exact-$8$ codimension-$3$ or codimension-$4$ model.

\section{Codimension-$4$ rep-quotients and the spectrum-first search}\label{sec:cube}

\subsection{A published codimension-$4$ exact-$8$ model}

Qureshi's 2023 weighted $\PP^1\times\PP^1\times\PP^1$-format paper contains a sporadic codimension-$4$ exact-$8$ example in ambient space
$\PP(3^2,5^2,7,9,11),$
with basket
\[
5\times \frac13(1,1)+2\times \frac15(1,1)+\frac1{11}(1,3)
\]
and
\[
K_X^2=\frac{14}{165}.
\]
This is important because it proves that the number $8$ is attainable in genuinely non-complete-intersection low-codimension formats \cite{QurCube}.

However, this row is \emph{not} suitable for characteristic $3$: it contains $\frac13$ singularities.  It also does not have Picard number $1$.  Resolving the basket gives
\[
K_{\widetilde{X}}^2=-7, \qquad \rho(\widetilde{X})=17,
\]
while the exceptional divisor has $9$ components, so
$\rho(X)=8.$

\subsection{Reconstructing the row-6 arithmetic pattern}

The same row can be reconstructed from the weighted cube datum
$\mu=(0,2:0,2:3,7),$
which yields cube weights
$3,5,5,7,7,9,9,11.$
After adding one cone variable of weight $3$ and taking quasilinear sections of degrees $7$ and $9$, one lands in
$\PP(3,3,5,5,7,9,11).$
More generally, the same step-$2$ pattern gives final ambient weights
\[
\PP\bigl(c^2,(c+2)^2,d,d+2,d+4\bigr).
\]
This shows immediately why the direct analogue can never be tame in characteristic $3$: among three consecutive numbers $d,d+2,d+4$, one is divisible by $3$.

\subsection{Why the physics/GLSM viewpoint helps}

The next route was inspired by the orbifold CFT/GLSM philosophy of Witten's linear sigma model \cite{Witten}.  The heuristic is simple:
\begin{itemize}
    \item ``Picard rank $1$'' is the geometric shadow of a one-parameter K\"ahler/GIT model;
    \item the basket of orbifold points behaves like a twisted-sector spectrum;
    \item orbifold Riemann--Roch should be used as a search engine before trying to write equations.
\end{itemize}
This matches Qureshi's methodology, which decomposes the Hilbert series into a smooth part plus explicit orbifold contributions \cite{QurCube,QurRigid}.

\subsection{A Hilbert-series solver for weighted cube format}

For a weight datum
$\mu=((a_1,a_2),(b_1,b_2),(c_1,c_2)),$
let
\[
H_{(u,v)}(n,t)=\sum_{i=0}^{n} t^{iu+(n-i)v}.
\]
The weighted-cube Hilbert series is then modelled by
\[
P_{wP_\mu}(t)=\sum_{n\ge 0} H_{(a_1,a_2)}(n,t)\,H_{(b_1,b_2)}(n,t)\,H_{(c_1,c_2)}(n,t).
\]
Cone variables contribute factors $(1-t^m)^{-1}$ and quasilinear sections contribute factors $(1-t^d)$.

On the orbifold side, the local correction term used in the search was
\[
\delta_{r,a,c}(n)=\frac1r\sum_{j=1}^{r-1}\frac{\zeta^{-jnc}-1}{(1-\zeta^j)(1-\zeta^{aj})},
\qquad \zeta=e^{2\pi i/r},
\]
with coefficient model
\[
h^0(X,nD)=1+\frac{D^2}{2}n(n+I)+\sum_i m_i\,\delta_{r_i,a_i,c_i}(n).
\]
The extra local class parameter $c$ was essential: without it, the published row-6 example could not be recovered.

As a sanity check, the solver reproduces Qureshi's exact-$8$ row precisely, with local classes
\[
c=2 \text{ on the } \tfrac13 \text{ and } \tfrac15 \text{ sectors},
\qquad
c=4 \text{ on the } \tfrac1{11} \text{ sector}.
\]

\subsection{The best current spectral candidate}

The unrestricted cube search produced a genuine exact tame $8$-sector hit:
$\mu=(0,3:0,3:1,2),$
with no cone variables and one quasilinear section of degree $8$.  The resulting ambient space is
$\PP(1,2,4,4,5,5,7).$
The Hilbert series decomposition is
\[
I=1, \qquad D^2=\frac{13}{35},
\]
with basket
\[
5\times \frac14(1,1)+2\times \frac15(1,2)+\frac17(1,1).
\]
Equivalently, in Hirzebruch--Jung notation,
$5\times [4] + 2\times [3,2] + [7].$


This is the first exact tame $8$-sector candidate produced by the ``super-exotic'' search.  However, the numerical data already determine its Picard number if such a surface exists.

\begin{proposition}[The current cube-spectrum hit has Picard number $9$]\label{prop:cube-spectrum-rho9}
Let $X$ be a klt del Pezzo surface with
\[
K_X^2=\frac{13}{35}
\]
and basket
$5\times [4] + 2\times [3,2] + [7].$
Then
$\rho(X)=9.$
In particular, any geometric realization of the weighted-cube Hilbert-series candidate in
$\PP(1,2,4,4,5,5,7)$
cannot solve the rank-one problem.
\end{proposition}

\begin{proof}
Let
$\pi\colon \widetilde X\to X$
be the minimal resolution.  For a singularity of type $[4]$, the discrepancy contribution to $K_X^2-K_{\widetilde X}^2$ is $1$.  For type $[3,2]$, solving
\[
\begin{pmatrix}
-3 & 1\\
1 & -2
\end{pmatrix}
\binom{a_1}{a_2}
=
\binom{1}{0}
\]
gives
\[
(a_1,a_2)=\left(-\frac25,-\frac15\right),
\]
hence contribution $\frac25$.  For type $[7]$, the contribution is $\frac{25}{7}$.

Therefore
\[
K_X^2-K_{\widetilde X}^2
=
5\cdot 1 + 2\cdot \frac25 + \frac{25}{7}
=
\frac{328}{35},
\]
so
\[
K_{\widetilde X}^2
=
\frac{13}{35}-\frac{328}{35}
=
-9.
\]
The exceptional divisor has
$5\cdot 1 + 2\cdot 2 + 1 = 10$
irreducible components.  Since $\widetilde X$ is a smooth rational surface,
$\rho(\widetilde X)=10-K_{\widetilde X}^2=19.$
Hence
$\rho(X)=\rho(\widetilde X)-10=9.$
\end{proof}

So the present cube-spectrum hit is no longer a live rank-one candidate.  What remains genuinely useful is the \emph{search philosophy}: the Hilbert-series solver found a tame exact-$8$ basket outside the previously analysed hypersurface and codimension-$2$ corridors.  The right next step is therefore not to test this specific row in \textsc{Macaulay2}, but to build the Picard-number computation into the spectral search and discard any row whose basket and $K^2$ already force $\rho\neq 1$.



\subsection{A bounded rerun of the rank-filtered cube search}

\paragraph{A numerical rank filter.}
For a basket decomposition produced by the solver, define the numerical rank predictor
\[
\rhonum:=10-K_X^2+\sum_{\sigma}A(\sigma)-\sum_{\sigma}\ell(\sigma).
\]
If the decomposition comes from an actual rational klt surface, then $\rhonum=\rho(X)\in \mathbf Z$.  In the raw solver
output, however, $\rhonum$ is only a numerical predictor; nonintegral values simply indicate that the decomposition
cannot come from a geometric surface.

To make route~(A) concrete, I reran the weighted-cube search in the first low-weight window containing the current row
$\mu=(0,3:0,3:1,2),$
with the following restrictions:
\begin{itemize}
    \item no cone variables;
    \item one quasilinear section;
    \item pair entries satisfying $0\le u\le v\le 3$;
    \item well-formed final ambient space.
\end{itemize}
For each row with final ambient weights $w=(w_0,\dots,w_6)$, I searched for all exact tame $8$-sector orbifold Riemann--Roch decompositions whose local indices $r$ divide at least one weight $w_i$, and I kept the row only if at least one such decomposition was compatible with the numerical rank predictor $\rhonum=1$.

\begin{proposition}[Bounded rank-filtered cube rerun in the first low-weight window]\label{prop:bounded-cube-rerun}
In the bounded window above, there are exactly $590$ admissible weighted-cube rows.  Among them, $15$ admit an exact tame $8$-sector decomposition of the Hilbert series, but none admits any such decomposition with integral numerical rank predictor $\rhonum=1$.

In particular, the row
\[
\mu=(0,3:0,3:1,2),
\qquad
\PP(1,2,4,4,5,5,7),
\]
still appears as an exact tame $8$-sector hit, but every exact decomposition found by the solver has
\[
K_X^2=\frac{13}{35},
\qquad
    \rhonum\in\left\{9,\frac{49}{5},\frac{53}{5},\frac{57}{5},\frac{61}{5},13\right\}.
\]
Moreover, the smallest integral value of $\rhonum$ attained anywhere in this bounded window is $2$.
\end{proposition}

\begin{proof}[Computation]
The scan tests all triples
$\mu=((a_1,a_2),(b_1,b_2),(c_1,c_2))$
with $0\le a_i\le a_2\le 3$, etc., forms the weighted-cube Hilbert series, removes one quasilinear section degree, and discards non-well-formed final ambient spaces.  This yields exactly $590$ rows.

For each row, the search then enumerates all exact tame $8$-sector orbifold decompositions whose local indices divide some final ambient weight, computes the induced basket(s), and evaluates
\[
\rhonum=10-K_X^2+\sum_{\sigma}A(\sigma)-\sum_{\sigma}\ell(\sigma).
\]
The output contains $15$ exact tame $8$-sector hits and no case with $\rhonum=1$.  For the current row
$\mu=(0,3:0,3:1,2)$
the solver finds $9$ exact decompositions, all with $K_X^2=13/35$ and numerical rank predictors in the set displayed above.
\end{proof}




\subsection{The enlarged no-cone pair-max $4$ window is also negative}

The first low-weight rerun still leaves open a slightly larger no-cone window, namely the one-section search with pair entries
$0\le u\le v\le 4,$
where the previous grouped exact solver completely resolved the basis-size-$\le 76$ slice but left a high-basis fringe of $298$ rows whose tame local index sets all contain $11$.  That fringe can be closed by a dedicated target-sum MILP solver.

For a no-cone one-section row
$\mu=((a_1,a_2),(b_1,b_2),(c_1,c_2))$
with section degree $d$, the exact hits already found in the solved slice all satisfy the same index identity
$I= (a_1+a_2)+(b_1+b_2)+(c_1+c_2)-d.$
I verified this equality on all $37$ exact tame $8$-sector hits in the basis-size-$\le 76$ checkpoint, and then imposed that exact row index in the remaining high-basis slice.

\begin{proposition}[The complete enlarged no-cone pair-max $4$ window has no rank-one survivor]\label{prop:pairmax4-nocone-complete}
In the no-cone, one-section weighted-cube window with
$0\le u\le v\le 4,$
there are exactly $2207$ admissible rows.  Among them, exactly $37$ admit an exact tame $8$-sector decomposition of the Hilbert series, but none admits such a decomposition with integral numerical rank predictor $\rhonum=1$.

Moreover, the smallest integral value of $\rhonum$ among \emph{all} exact decompositions in this complete window is still
$2.$
More sharply, the previously unresolved high-basis fringe of $298$ rows contributes \emph{no} exact tame $8$-sector hit at all.
\end{proposition}

\begin{proof}[Computation]
The basis-size-$\le 76$ slice of this window had already been solved exactly on $1909$ rows, yielding $37$ exact hits, no case with $\rho=1$, and minimum Picard number $2$.  The only unresolved part was the complementary set of $298$ rows, grouped into $14$ tame index sets $R_s$, all containing $11$.

For each such group, I built the full local basis of orbifold correction terms, and for each distinct pair $(h,I)$ of Hilbert series and row index I solved the exact integer feasibility problem with the following constraints:
\begin{itemize}
    \item the sum of the multiplicities of local terms is $8$;
    \item all $18$ third-difference coordinates of the Hilbert function match exactly;
    \item the quadratic part has the fixed row index
    $I=(a_1+a_2)+(b_1+b_2)+(c_1+c_2)-d;$
    \item the quadratic coefficient is positive.
\end{itemize}
This is an exact MILP feasibility problem, not a bounded approximate search.  The index formula above was cross-checked on all $37$ exact hits in the already-solved slice before being imposed on the fringe.

The dedicated solver resolves $191$ distinct $(h,I)$-rows in the high-basis fringe and finds no feasible exact tame $8$-sector decomposition.  Therefore the fringe contributes zero new exact hits, so the full pair-max $4$ no-cone window still has exactly the same $37$ exact hits as the solved slice, none with $\rhonum=1$, and the global minimum integral value of $\rhonum$ remains $2$.
\end{proof}


\subsection{The first one-cone extension is also negative}

The next natural relaxation is to allow exactly one cone variable, while keeping the same low-weight cube window and the same hard Picard filter.  Concretely, I imposed:
\begin{itemize}
    \item pair entries satisfying $0\le u\le v\le 3$;
    \item exactly one cone variable of weight $m\in\{1,2,3\}$;
    \item exactly two quasilinear sections;
    \item well-formed final ambient space.
\end{itemize}
As before, for each resulting row with final ambient weights $w=(w_0,\dots,w_6)$, I searched for all exact tame $8$-sector orbifold Riemann--Roch decompositions whose local indices divide at least one weight $w_i$, and I retained the row only if at least one such decomposition was compatible with $\rhonum=1$.

\begin{proposition}[The first one-cone cube window has no rank-one survivor]\label{prop:first-onecone-rerun}
In the one-cone window above, there are exactly $5247$ admissible weighted-cube rows.  Among them, $84$ admit an exact tame $8$-sector decomposition of the Hilbert series, but none admits any such decomposition with integral numerical rank predictor $\rhonum=1$.

Moreover, the smallest integral value of $\rhonum$ among \emph{all} exact decompositions found in this one-cone window is
$2.$
It is attained, for example, on the ambient space
$\PP(3,4,4,5,5,6,7),$
while the current ambient space
$\PP(1,2,4,4,5,5,7)$
reappears only with minimum integral value $\rhonum=9$.
\end{proposition}

\begin{proof}[Computation]
The scan enumerates all triples
$\mu=((a_1,a_2),(b_1,b_2),(c_1,c_2))$
with $0\le a_i\le a_2\le 3$, etc., adds one cone variable of weight $m\in\{1,2,3\}$, removes an unordered pair of quasilinear section degrees drawn from the resulting multiset of ambient weights, and discards non-well-formed final ambient spaces.  This yields exactly $5247$ rows.

For each row, the search then enumerates all exact tame $8$-sector orbifold decompositions whose local indices divide some final ambient weight, computes the induced basket(s), and evaluates
\[
\rhonum=10-K_X^2+\sum_{\sigma}A(\sigma)-\sum_{\sigma}\ell(\sigma).
\]
The output contains $84$ exact tame $8$-sector hits and no case with $\rhonum=1$.  The smallest integral value of $\rhonum$ found anywhere in this window is $2$, attained on rows with final ambient space $\PP(3,4,4,5,5,6,7)$.  By contrast, the ambient space $\PP(1,2,4,4,5,5,7)$ still occurs, but its minimum integral value of $\rhonum$ in the one-cone window is $9$.
\end{proof}


\section{Route~C: reduction to the residual bridge problem}\label{sec:routec}

The main outcome of the search is negative but structured: large portions of the obvious construction space can now be excluded with clean formulas or clean counting arguments.

\subsection{Methods~A and~B are essentially exhausted}

The following routes now look essentially exhausted.
\begin{enumerate}[label=\arabic*.]
    \item \textbf{Diagonal involution quotients of $\mathrm{CI}(2,2)\subset\PP^4$.}  They cannot exceed $6$ singular points (Proposition~\ref{prop:methodA6}).
    \item \textbf{The natural extension of the characteristic-$3$ Frobenius/Keel--McKernan blow-up family.}  Positivity of $-K$ holds only in the known $(a,m)=(1,3)$ case (Proposition~\ref{prop:frobenius-only13}).
    \item \textbf{The large explicit hypersurface family $f_n+x_2x_3$.}  It has $\rho=n$ exactly, so exact $8$ singularities force $\rho=6$ (Proposition~\ref{prop:rho-n}).
    \item \textbf{Birational smoothing of the six-branch exact-$8$ sources to Picard rank $1$.}  Any rank-one output has at most $3$ singular points (Proposition~\ref{prop:threepoint-barrier}).
    \item \textbf{The most natural codimension-$2$ weighted complete-intersection families.}  They reduce to hypersurfaces or are ruled out by quasismoothness/Fano constraints (Section~6).
    \item \textbf{Smooth-upstairs or exact-$8$-upstairs quotient tricks.}  Smooth-upstairs tame quotients with $\rho=1$ have at most $5$ singular points, and quotients of the exact-$8$ weighted sources have at most $5$ as well (Proposition~\ref{prop:smoothquotient5}, Corollary~\ref{cor:exact8-quotient}).
    \item \textbf{The obvious branch-unprojection ladders from the exact-$8$ weighted hypersurfaces.}  Each step lowers the singularity count and alternating sides breaks tameness in characteristic $3$ (Proposition~\ref{prop:unprojection-ladder}).
    \item \textbf{The current codimension-$4$ weighted-cube spectral hit.}  Its basket and degree already force $\rho=9$ if it is realized geometrically, so it cannot solve the rank-one problem (Proposition~\ref{prop:cube-spectrum-rho9}).
    \item \textbf{The first enlarged no-cone pair-max $4$ weighted-cube window.}  After an exact target-sum closure of the $11$-containing high-basis fringe, it still has no $\rho=1$ survivor (Proposition~\ref{prop:pairmax4-nocone-complete}).
\end{enumerate}

\subsection{The remaining Route~C program}

Three routes remain genuinely interesting.

\medskip\noindent\textbf{(A) A rank-filtered spectrum-first cube/Pfaffian search.}\ 
The weighted-cube Hilbert-series solver remains the most promising \emph{method}, but the first three bounded cube windows are already negative: Proposition~\ref{prop:bounded-cube-rerun} rules out rank-one survivors in the initial low-weight no-cone, one-section window; Proposition~\ref{prop:pairmax4-nocone-complete} closes the first larger no-cone, one-section window with pair weights up to $4$; and Proposition~\ref{prop:first-onecone-rerun} does the same for the first low-weight one-cone, two-section extension.  So the next step on the computational side is no longer the \emph{first} rerun, but a genuinely different one: move to still larger pair weights, more general section patterns, or codimension-$3$ Pfaffian formats with the same hard $\rho=1$ filter before any explicit \textsc{Macaulay2} geometry.

\medskip\noindent\textbf{(B) A larger spectrum-first cube/Pfaffian search.}\ 
The Hilbert-series solver should be extended in two directions:
\begin{itemize}
    \item non-symmetric cube data with more general section choices;
    \item codimension-$3$ Pfaffian formats with the same exact-$8$ orbifold Riemann--Roch filter.
\end{itemize}
So far, no exact tame $8$-sector Pfaffian hit has appeared in the low-weight scan.

\medskip\noindent\textbf{(C) A theoretical closure of the residual Palka--Pe\l ka sector.}\ 
Every classified characteristic-$3$ sector currently tops out at $7$, and the Route~C reductions in this report remove the entire first visible chain/fork/cross-link frontier from the residual sector.  The terminal $A_2$-tail corridor is classified exactly, the bare fork frontier and its first canonical cross-links are explicit, the peelable family $[2,2]\dashv [3,(2)^m]$ has height $1$, the two square-one fork prototypes $[2]\to F_3$ and $[2,2]\to F_4$ force descendants with elliptic boundary, and the two higher-square short bridges have height at most $2$.  The one-root staircase family is then removed by the anticanonical contact formula, and the $2$-tail core reduction shows that every surviving positive chain bridge has a special tip-visible side, with the high-contact reduced tip cores restricted to a touched $(-2)$-tip or the singleton core $[3]$.

In the height-$4$ maximal-width case there is an exact global excess package.  Corollary~\ref{cor:routeC-tip-entry-obstruction} gives the identity
\[
\Sigma=\sum_F(\sigma(F)-1)=3,
\]
shows that no fiber is contained in $D$, and produces one horizontal $1$-section avoiding all genuinely bad singly-hit tips.  The remaining obstacle in this corridor is a fiberwise control on the purely vertical connected components of the exceptional divisor.  The later subsection records the corrected graph-theoretic bound that follows from the excess setup and isolates the additional input still needed for a direct numerical closure.


\subsection{Reduction to the residual sector and the Route~C entry problem}

The first missing Route~C step is global rather than local: before analyzing any rooted graph, one has to show
that a hypothetical many-singularity surface actually enters the contact-$2$ corridor.  The following theorem
records the unconditional part of that reduction.

\begin{theorem}[Reduction to the residual sector]\label{thm:routeC-contact2-reduction}
Let $X$ be a klt del Pezzo surface of Picard rank $1$ over an algebraically closed field of characteristic $3$.
If
$\#\Sing(X)\ge 8,$
then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $\operatorname{ht}(X)\ge 4$.
    \item $X$ has no descendant with elliptic boundary.
\end{enumerate}
Equivalently, every hypothetical benchmark counterexample already lies in the residual height-$\ge 4$,
no-descendant sector.
\end{theorem}

\begin{proof}
Assume first that $\operatorname{ht}(X)\le 2$.  By the classification of del Pezzo surfaces of rank one and height at
most $2$ in \cite[Theorem~A]{PP12}, together with the characteristic-$\neq 2$ singular-point bound recorded in
\cite[Remark~1.7]{PP12}, one has
\[
\#\Sing(X)\le 4
\qquad\text{in characteristic }3,
\]
contrary to the assumption.

Assume next that $\operatorname{ht}(X)=3$ and that $X$ has no descendant with elliptic boundary.  Then
\cite[Theorem~A and Tables~4--7]{PP3} classify all such surfaces.  In characteristic $3$ the largest singularity type
appearing there is the known basket
$4\cdot [3]+3A_2,$
which has exactly $7$ singular points.  Hence again $\#\Sing(X)\le 7$, contradiction.

Finally, assume that $X$ has a descendant with elliptic boundary.  By \cite[Theorem~E]{PP12}, such surfaces are
classified in terms of descendants of canonical rank-one del Pezzo surfaces.  In characteristic $3$, the relevant
primitive types are listed in \cite[Lemma~6.9\textnormal{(4)}]{PP12}:
\[
3A_2+[3]+[3]+[2],\qquad 4A_2+[3],\qquad 4A_2+[4,2],\qquad 4A_2+[5]+[3]+[2].
\]
The last one has exactly $7$ singular points, so again $\#\Sing(X)\le 7$, contradiction.

The three exclusions above leave only the residual height-$\ge 4$, no-descendant sector.
\end{proof}

\begin{lemma}[Positivity of a contact-$2$ curve]\label{lem:routeC-contact2-positive}
Let $X$ be a klt del Pezzo surface of Picard rank $1$, let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D=\Exc(\pi)_{\mathrm{red}}$.
If $L\subset \widetilde X$ is a $(-1)$-curve with
\[
L\not\subset D,
\qquad
L\cdot D=2,
\]
then $C:=\pi(L)$ is a nonzero effective curve on $X$ and
$C^2>0.$
\end{lemma}

\begin{proof}
Since $L\not\subset D$, its image $C=\pi(L)$ is a nonzero effective curve on $X$.
Because $X$ is klt, it is $\QQ$-factorial.  Since $\rho(X)=1$, every nonzero numerical class on $X$ is proportional
to an ample class.  Hence $C\equiv qH$ for some $q>0$ and some ample $\QQ$-Cartier divisor $H$, so
$C^2=q^2H^2>0.$
\end{proof}

\begin{lemma}[Positive height for singular rank-one del Pezzo surfaces]
\label{lem:routeC-height-positive}
Let $X$ be a singular klt del Pezzo surface of Picard rank $1$.  Then
$\operatorname{ht}(X)>0.$
\end{lemma}

\begin{proof}
Assume $\operatorname{ht}(X)=0$.  Let
$\pi\colon \widetilde X\to X$
be the minimal resolution, let $D=\Exc(\pi)_{\mathrm{red}}$, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration with general fiber $F$ satisfying
$F\cdot D=0.$
Then $F$ is disjoint from $D$, so $\pi$ is an isomorphism in a neighborhood of $F$.  Hence
$C:=\pi(F)\subset X$
is a nonzero effective curve with
$C^2=F^2=0.$
But on a klt surface of Picard rank $1$, every nonzero numerical class is proportional to an ample class.
Hence $C\equiv qH$ for some $q>0$ and some ample $\QQ$-Cartier divisor $H$, so $C^2=q^2H^2>0$, a contradiction.
\end{proof}

\begin{corollary}[Formal Step~1 for Route~C]\label{cor:routeC-step1-formal}
The first remaining global input in Route~C is the following residual Route~C entry statement:

\medskip
\centerline{\fbox{\parbox{0.88\linewidth}{%
Every characteristic-$3$ klt del Pezzo surface $X$ of Picard rank $1$ with
$\operatorname{ht}(X)\ge 4$ and no descendant with elliptic boundary admits on its minimal resolution a
$(-1)$-curve $L\not\subset D$ such that $L\cdot D=2$ and the associated rooted configuration is root-contractible.
}}}
\medskip

By Lemma~\ref{lem:routeC-contact2-positive}, any such rooted configuration is automatically positive.
Hence proving the boxed statement would force every hypothetical benchmark counterexample into the
positive root-contractible contact-$2$ corridor analyzed in the rest of this section.
\end{corollary}

\begin{remark}[What is and is not proved here]\label{rem:routeC-contact2-step-status}
Theorem~\ref{thm:routeC-contact2-reduction} is unconditional, while
Corollary~\ref{cor:routeC-step1-formal} records a proof-program reduction of the Route~C entry problem, introduced
here only as an intermediate organizational device.
The final proof of the paper does \emph{not} depend on proving that boxed residual Route~C entry statement separately:
the later bridge and height-selection sections bypass it and close the theorem by a different global route.
What the rest of the present section proves is that once such a positive root-contractible contact-$2$ configuration
exists, large first-frontier families of local contact-$2$ configurations can already be eliminated or converted into
elliptic ties.
\end{remark}

\subsection{Toward the Route~C entry theorem: a vertical sufficient condition}

The hidden difficulty in Corollary~\ref{cor:routeC-step1-formal} is the root-contractibility requirement.  The next
lemma isolates a geometric situation in which root-contractibility is automatic: a contact-$2$ curve which stays in one
singular fiber together with the exceptional components it meets.

\begin{lemma}[Keeping a multiplicity-one component of a singular fiber]
\label{lem:routeC-fiber-preserve-mult1}
Let
$p\colon Y\to \PP^1$
be a $\PP^1$-fibration on a smooth projective surface, let $F_0$ be a singular fiber, and let $R\subset F_0$ be a
component of multiplicity $1$ in $F_0$.  Then there exists a birational morphism
$\mu\colon Y\to Y_0$
onto a smooth $\PP^1$-bundle $Y_0\to \PP^1$ such that $\mu$ contracts every component of $(F_0)_{\mathrm{red}}-R$ and sends
$R$ onto a smooth fiber of $Y_0\to \PP^1$.
\end{lemma}

\begin{proof}
A neighborhood of $F_0$ is obtained from a neighborhood of a smooth fiber of a $\PP^1$-bundle by a sequence of blowups
whose centers lie on the successive total transforms of that fiber.  Equivalently, one obtains a relative minimal model
of $p$ by contracting vertical $(-1)$-curves until the image of $F_0$ becomes a smooth fiber.  Because $R$ has
multiplicity $1$ in $F_0$, its image on the relative minimal model is not a point: it is exactly the resulting smooth
fiber.  Reversing this relative minimalization yields a sequence of blowdowns which contracts every component of
$(F_0)_{\mathrm{red}}-R$ and leaves $R$ as the final fiber.
\end{proof}

\begin{proposition}[Purely vertical contact-$2$ curves are automatically admissible for Route~C]
\label{prop:routeC-vertical-entry-sufficient}
Let $X$ be a klt del Pezzo surface, let
$\pi\colon (\widetilde X,D)\to X$
be its minimal resolution, and let
$p\colon \widetilde X\to \PP^1$
be a $\PP^1$-fibration.  Let $L\subset \widetilde X$ be a vertical $(-1)$-curve with
\[
L\not\subset D,
\qquad
L\cdot D=2,
\]
and let $T$ be the connected component of $D+L$ containing $L$.  Assume that every component of $T$ is vertical for
$p$, and that $T$ contains a component of multiplicity $1$ in the fiber $F_0:=p^{-1}(p(L))$.  Then the rooted
configuration associated to $L$ is root-contractible.  If moreover $\rho(X)=1$, then it is automatically positive.
\end{proposition}

\begin{proof}
Since every component of $T$ is vertical and $T$ is connected, we have $T\subseteq F_0$.  Choose a multiplicity-one
component $R\subseteq T$ of $F_0$.  By Lemma~\ref{lem:routeC-fiber-preserve-mult1} there is a birational morphism onto
a smooth $\PP^1$-bundle which contracts every component of $(F_0)_{\mathrm{red}}-R$ and sends $R$ onto a smooth fiber.  In
particular, every component of $T-R$ is eliminated by a sequence of blowdowns of $(-1)$-curves different from $R$.
Hence the rooted configuration is root-contractible.

If $\rho(X)=1$, positivity follows from Lemma~\ref{lem:routeC-contact2-positive}.
\end{proof}

\begin{corollary}[A geometric sufficient form of the Route~C entry theorem]
\label{cor:routeC-vertical-entry-form}
To prove Corollary~\ref{cor:routeC-step1-formal}, it is enough to prove the following stronger statement.

\medskip
\centerline{\fbox{\parbox{0.88\linewidth}{%
Every characteristic-$3$ klt del Pezzo surface $X$ of Picard rank $1$ with $\operatorname{ht}(X)\ge 4$ and no
descendant with elliptic boundary admits on its minimal resolution a witnessing $\PP^1$-fibration and a vertical
$(-1)$-curve $L\not\subset D$ such that $L\cdot D=2$, the connected component of $D+L$ containing $L$ is vertical,
and this component contains a multiplicity-one component of the corresponding singular fiber.
}}}
\medskip

Indeed, Proposition~\ref{prop:routeC-vertical-entry-sufficient} turns such a curve into a positive
root-contractible contact-$2$ configuration.
\end{corollary}

\begin{remark}[What this changes in the endgame]
\label{rem:routeC-vertical-entry-endgame}
This does not prove the residual Route~C entry theorem, but it removes its most opaque local part: for a
\emph{purely vertical} contact-$2$ bridge, root-contractibility no longer has to be checked separately.  The remaining
entry problem becomes geometric: produce such a vertical bridge on a witnessing fibration in the residual
height-$\ge 4$, no-descendant sector.  In particular, any future reduction to a vertically primitive height-$4$ model
only needs to find the bridge; the rooted-graph admissibility is then automatic.
\end{remark}


\subsection{A corrected mixed sufficient condition: singly-hit multiplicity-one tips}

The previous sufficient condition treated the purely vertical case.  The next proposition handles the first easy
mixed case: a multiplicity-one component outside $D$ which is hit by exactly one horizontal boundary component and
is attached to the exceptional divisor inside its fiber.  The extra attachment hypothesis is necessary: without it,
the unique vertical neighbor of the tip may be another nonexceptional component, and then the total contact with
$D$ need not equal $2$.

\begin{proposition}[A singly-hit multiplicity-one tip attached to $D$ already gives Route~C entry]
\label{prop:routeC-tip-entry-sufficient}
Let $X$ be a klt del Pezzo surface, let
$\pi\colon (\widetilde X,D)\to X$
be its minimal resolution, and let
$p\colon \widetilde X\to \PP^1$
be a $\PP^1$-fibration.  Let $F_0$ be a degenerate fiber and let $L\subseteq F_0$ be a component such that
\[
L\not\subset D,
\qquad
\operatorname{mult}_{F_0}(L)=1,
\qquad
L\cdot D_{\mathrm{hor}}=1.
\]
Assume moreover that the unique component of $(F_0)_{\mathrm{red}}-L$ meeting $L$ belongs to $D$.  Then $L$ is a
$(-1)$-curve with
$L\cdot D=2.$
Moreover the rooted configuration associated to $L$ is root-contractible.  If $\rho(X)=1$, then it is
automatically positive.
\end{proposition}

\begin{proof}
By \cite[Lemma~2.8(b)]{PP12}, every component of a degenerate fiber not contained in $D$ is a $(-1)$-curve, so
$L^2=-1$.  Since $L$ has multiplicity one in $F_0$, \cite[Lemma~2.3(a)]{PP12} gives
$\beta_{(F_0)_{\mathrm{red}}}(L)=1.$
Thus $L$ is a tip of $(F_0)_{\mathrm{red}}$, and it has a unique vertical neighbor in that reduced fiber.  By
assumption this vertical neighbor belongs to $D$.  Because $L\cdot D_{\mathrm{hor}}=1$, there is also exactly one
horizontal component of $D$ meeting $L$.  No other component of $D$ can meet $L$.  Hence
$L\cdot D=1+1=2.$

Let $H$ be the unique horizontal component of $D$ meeting $L$, and let $T$ be the connected component of $D+L$
containing $L$.  Then $T$ has unique horizontal component $H$, while every component of $T-H$ is vertical and lies
in the reducible fiber $F_0$.  Take a relative minimalization of $p$, i.e. contract vertical $(-1)$-curves until the
resulting fibration becomes a $\PP^1$-bundle over the same base.  This contracts every component of the reducible
fiber $F_0$, hence in particular every component of $T-H$, while $H$ survives because it is horizontal.  Therefore
the rooted configuration is root-contractible, with root $H$.

If $\rho(X)=1$, positivity follows from Lemma~\ref{lem:routeC-contact2-positive}.
\end{proof}

\begin{corollary}[A corrected mixed sufficient form of the Route~C entry theorem]
\label{cor:routeC-tip-entry-form}
To prove Corollary~\ref{cor:routeC-step1-formal}, it is enough to prove the following stronger statement.

\medskip
\centerline{\fbox{\parbox{0.88\linewidth}{%
Every characteristic-$3$ klt del Pezzo surface $X$ of Picard rank $1$ with $\operatorname{ht}(X)\ge 4$ and no
descendant with elliptic boundary admits on its minimal resolution a witnessing $\PP^1$-fibration and a
multiplicity-one component $L$ of some degenerate fiber such that
\[
L\not\subset D,
\qquad
L\cdot D_{\mathrm{hor}}=1,
\]
and the unique component of $(F_0)_{\mathrm{red}}-L$ meeting $L$ belongs to $D$.
}}}
\medskip

Indeed, Proposition~\ref{prop:routeC-tip-entry-sufficient} turns such a component into a positive root-contractible
contact-$2$ configuration.
\end{corollary}

\begin{proposition}[No witnessing fiber is contained in the exceptional divisor]
\label{prop:routeC-no-fiber-in-D}
Let $X$ be a singular klt del Pezzo surface, let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution, and let
$p\colon \widetilde X\to \PP^1$
be any $\PP^1$-fibration.  Then no fiber of $p$ is contained in $D$.  Equivalently,
$\nu_\infty=0.$
\end{proposition}

\begin{proof}
Assume that a fiber $F$ of $p$ is contained in $D$.  Since $D$ is the exceptional divisor of the \emph{minimal}
resolution, it contains no $(-1)$-curve: otherwise one could contract such a curve and obtain a smaller resolution.
Hence every component of $F$ has self-intersection at most $-2$, so $F$ is not a smooth fiber.  Thus $F$ is
degenerate.  But every degenerate fiber of a $\PP^1$-fibration on a smooth surface contains a $(-1)$-curve by
Lemma~2.3.  This $(-1)$-curve would lie in $D$, contradiction.
\end{proof}

\begin{corollary}[A maximal-width reduction: exact excess, at most three bad tips, and a clean section]
\label{cor:routeC-tip-entry-obstruction}
Assume that $X$ is as in Corollary~\ref{cor:routeC-step1-formal}, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration of height $4$ and maximal width, so $D_{\mathrm{hor}}$ consists of four
$1$-sections.  Put
\[
\Sigma:=\sum_F(\sigma(F)-1),
\]
where the sum runs over all degenerate fibers and $\sigma(F)$ is as in \cite[Lemma~2.4]{PP12}.  Then
$\Sigma=3.$
Moreover the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item No multiplicity-one component $L$ of a degenerate fiber with $L\not\subset D$ satisfies
    $L\cdot D_{\mathrm{hor}}=4.$
    \item Let $\mathcal B_1$ be the set of multiplicity-one components $L\not\subset D$ of degenerate fibers such that
    $L\cdot D_{\mathrm{hor}}=1$
    and the unique component of $F_{\mathrm{red}}-L$ meeting $L$ does not belong to $D$.  Then
    $\#\mathcal B_1\le 3.$
    In particular, the set $\mathcal F_1$ of degenerate fibers containing such a component also satisfies
    $\#\mathcal F_1\le 3.$
    \item There exists a horizontal component $H_0\subset D_{\mathrm{hor}}$ which meets no component of
    $\mathcal B_1$.  Consequently, if Propositions~\ref{prop:routeC-vertical-entry-sufficient}
    and~\ref{prop:routeC-tip-entry-sufficient} do not already give Route~C entry, then $H_0$ meets no singly-hit
    multiplicity-one nonexceptional tip of any degenerate fiber.
\end{enumerate}
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-no-fiber-in-D}, one has $\nu_\infty=0$.  Since $\rho(X)=1$, we have
$\rho(\widetilde X)=1+\#D$.  Applying \cite[Lemma~2.4]{PP12} to the witnessing fibration gives
\[
\#D_{\mathrm{hor}}+\rho(\widetilde X)=\#D+2+\Sigma.
\]
Because $\#D_{\mathrm{hor}}=4$, this becomes
$4+(1+\#D)=\#D+2+\Sigma,$
hence $\Sigma=3$.

For \textnormal{(1)}, if some multiplicity-one component $L\not\subset D$ satisfied
\[
L\cdot D_{\mathrm{hor}}=4=F\cdot D_{\mathrm{hor}},
\]
then \cite[Lemma~2.8(c)]{PP12} would imply $L\subset D$, a contradiction.

For \textnormal{(2)}, fix a degenerate fiber $F$ and let $b(F)$ be the number of components of $\mathcal B_1$ contained
in $F$.  We claim that
$b(F)\le \sigma(F)-1.$
Let $N_1,\dots,N_r$ be the connected components of $F_{\mathrm{red}}-D$, and put $m_i:=\#N_i$, so
\[
\sigma(F)=\sum_{i=1}^r m_i.
\]
A component of $\mathcal B_1$ must lie in some $N_i$ with $m_i\ge 2$, because by definition its unique neighbor in
$F_{\mathrm{red}}$ is not contained in $D$.  If $m_i=2$, then $N_i$ is a chain of two adjacent $(-1)$-curves; at most
one of them can belong to $\mathcal B_1$, because if both did then each would be a singly-hit tip and no component of
$D$ would meet $N_i$, forcing $F_{\mathrm{red}}=N_i=[1,1]$ and hence
$F\cdot D_{\mathrm{hor}}=2,$
contrary to $F\cdot D_{\mathrm{hor}}=4$.  Thus $N_i$ contributes at most $1=m_i-1$ bad tips.  If $m_i\ge 3$, then the
dual graph of $N_i$ is a tree with $m_i$ vertices, and every bad tip in $N_i$ is a leaf of that tree.  Hence $N_i$
contributes at most $m_i-1$ bad tips.  Summing over all components $N_i$ with $m_i\ge 2$ gives
\[
b(F)\le \sum_{m_i\ge 2}(m_i-1)=\sigma(F)-\#\{i\,;\,m_i\ge 2\}\le \sigma(F)-1.
\]
Therefore
\[
\#\mathcal B_1=\sum_F b(F)\le \sum_F(\sigma(F)-1)=\Sigma=3.
\]
The bound for $\mathcal F_1$ follows because every fiber in $\mathcal F_1$ contains at least one component of
$\mathcal B_1$.

For \textnormal{(3)}, each component of $\mathcal B_1$ is met by exactly one horizontal section, and
\textnormal{(2)} shows that there are at most three such components in total.  Since $D_{\mathrm{hor}}$ consists of four
$1$-sections, at least one of them meets no component of $\mathcal B_1$; call it $H_0$.  If a singly-hit
multiplicity-one nonexceptional tip $L$ met $H_0$, then either the unique component of $F_{\mathrm{red}}-L$ meeting $L$
would belong to $D$, in which case Proposition~\ref{prop:routeC-tip-entry-sufficient} would already give Route~C
entry, or it would not belong to $D$, in which case $L\in \mathcal B_1$, contradicting the choice of $H_0$.
\end{proof}

\begin{remark}[The remaining maximal-width entry problem is a clean-section problem]
\label{rem:routeC-clean-section}
Corollary~\ref{cor:routeC-tip-entry-obstruction} turns the mixed part of the residual Route~C entry problem into a
much smaller geometric question.  In the height-$4$ maximal-width corridor, after excluding the easy mixed criterion,
one may choose a horizontal section $H_0$ which never meets a singly-hit multiplicity-one nonexceptional tip.  Thus
all possible failure of Route~C entry is concentrated away from $H_0$ and over at most three marked fibers.
\end{remark}

\begin{remark}[What the mixed entry problem becomes in the height-$4$ maximal-width case]
\label{rem:routeC-height4-entry-obstruction}
In the height-$4$, maximal-width corridor, the corrected mixed sufficient condition above no longer forces a
fiber-by-fiber concentration statement, but it still gives a useful global bound.  A $4$-hit multiplicity-one
nonexceptional tip is impossible, and the genuinely bad singly-hit tips form a set $\mathcal B_1$ of cardinality at most
three by Corollary~\ref{cor:routeC-tip-entry-obstruction}.  Equivalently, the whole bad singly-hit phenomenon is
confined to at most three degenerate fibers, and one horizontal $1$-section avoids all of it.  This matches the total
excess of the canonical height-$4$ surface of type $4A_2$ from \cite[Proposition~6.1(a), Example~7.1]{OpenDP}, whose
standard witnessing fibration has no fiber contained in $D$ and total excess $\Sigma=3$.
\end{remark}

\begin{proposition}[The clean-section Hirzebruch model]
\label{prop:routeC-clean-section-Hirzebruch}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and let
$H_0\subset D_{\mathrm{hor}}$
be a clean horizontal $1$-section as in \textnormal{(3)} of that corollary.  Let
$\upsilon\colon \widetilde X\to \mathbb F_n$
be the contraction of all vertical curves which do not meet $H_0$.
Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item The image $S_0:=\upsilon(H_0)$ is the negative section of $\mathbb F_n$, and
    $n=-H_0^2\ge 2.$
    \item If $H\subset D_{\mathrm{hor}}$ is any other horizontal $1$-section and $S:=\upsilon(H)$, then $S$ is a section of
    $\mathbb F_n$ and
    $S^2=n+2(H\cdot H_0).$
    In particular:
    \begin{enumerate}[label=\textnormal{(\alph*)}]
        \item if $H\cdot H_0=0$, then $S\sim S_0+nf$ and $S^2=n$;
        \item if $H\cdot H_0=1$, then $S\sim S_0+(n+1)f$ and $S^2=n+2$,
    \end{enumerate}
    where $f$ denotes a fiber of the ruling on $\mathbb F_n$.
    \item Writing
    $D_{\mathrm{hor}}=H_0+H_1+H_2+H_3,$
    the unresolved mixed part of the maximal-width Route~C entry problem can be reformulated on
    $\mathbb F_n$ as a configuration problem for the three sections
    $S_i:=\upsilon(H_i),\qquad i\in\{1,2,3\},$
    where each $S_i$ has numerical class
    $S_i\sim S_0+(n+H_i\cdot H_0)f,$
    and the only singled-out fibers are the at most three bad fibers from
    Corollary~\ref{cor:routeC-tip-entry-obstruction}\textnormal{(2)}.
\end{enumerate}
\end{proposition}

\begin{proof}
Contracting all vertical curves not meeting $H_0$ produces a relatively minimal ruled surface over the same base,
so the target is some Hirzebruch surface $\mathbb F_n$.  Since every contracted curve is disjoint from $H_0$, the
morphism $\upsilon$ is an isomorphism in a neighborhood of $H_0$, and therefore
$S_0^2=H_0^2<0.$
Thus $S_0$ is the unique negative section of $\mathbb F_n$, so $S_0^2=-n$ and hence
$n=-H_0^2.$
Because $H_0\subset D$ is a component of the exceptional divisor of the minimal resolution, it is not a $(-1)$-curve,
so $H_0^2\le -2$ and therefore $n\ge 2$.  This proves \textnormal{(1)}.

Let $H\subset D_{\mathrm{hor}}-H_0$ be another horizontal $1$-section, and put $S:=\upsilon(H)$.  Since $\upsilon$ contracts only
vertical curves, $S$ is again a section of the ruling on $\mathbb F_n$.  Numerical properties of Hirzebruch surfaces,
exactly as in \cite[proof of Lemma~5.9, lines~6075--6078]{PP12}, give
$0=(S-S_0)^2=S^2-2(H\cdot H_0)-n,$
whence
$S^2=n+2(H\cdot H_0).$
Writing $S\sim S_0+bf$ for some integer $b$, one has
$S\cdot S_0=b-n=H\cdot H_0,$
so $b=n+H\cdot H_0$.  Hence
$S\sim S_0+(n+H\cdot H_0)f.$
The two displayed special cases follow immediately.  This proves \textnormal{(2)}.

Finally, \textnormal{(3)} is just a reformulation of the geometry after Corollary~\ref{cor:routeC-tip-entry-obstruction}:
the contraction $\upsilon$ packages all vertical complexity away from $H_0$ into a ruled model with distinguished
negative section $S_0$, while the genuinely bad singly-hit phenomenon remains confined to at most three fibers by
Corollary~\ref{cor:routeC-tip-entry-obstruction}\textnormal{(2)}.  The remaining mixed entry question is therefore encoded by the
three sections $S_1,S_2,S_3$ on $\mathbb F_n$ together with those at most three marked fibers.
\end{proof}

\begin{corollary}[A uniform three-section model when the clean section is disjoint from the others]
\label{cor:routeC-clean-section-disjoint-model}
Under the assumptions of Proposition~\ref{prop:routeC-clean-section-Hirzebruch}, assume in addition that
$H_0\cdot H_i=0\qquad\text{for }i=1,2,3.$
Then on $\mathbb F_n$ the three remaining horizontal components all have the same numerical class:
$S_i\sim S_0+nf,\qquad S_i^2=n,\qquad i=1,2,3.$
In particular, the clean-section reduction becomes a problem about three sections in one linear system on a fixed
Hirzebruch surface together with the at most three marked fibers of
Corollary~\ref{cor:routeC-tip-entry-obstruction}.
\end{corollary}

\begin{proof}
This is immediate from Proposition~\ref{prop:routeC-clean-section-Hirzebruch}\textnormal{(2a)}.
\end{proof}


\begin{proposition}[Frobenius normal form in the disjoint clean-section case]
\label{prop:routeC-clean-section-disjoint-frobenius}
Assume the setup of Proposition~\ref{prop:routeC-clean-section-Hirzebruch}, and assume moreover that
$H_0\cdot H_i=0\qquad\text{for }i=1,2,3.$
Let $T\subset \PP^1$ be a set of fibers with
$\#T\le 3$
such that every pairwise intersection of the sections
$S_1,S_2,S_3\subset \mathbb F_n$
is supported on fibers over $T$.
Then, after an automorphism of the base and an automorphism of $\mathbb F_n$ preserving the ruling and the
negative section $S_0$, there exist homogeneous coordinates $[x:y]$ on the base, nonnegative integers $a,b,c$,
and one of the following two alternatives.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item There exists $\lambda\in k^*\setminus\{1\}$ with
    $a+b+c=n,$
    such that on
    \[
    \mathbb F_n-S_0\cong \Tot \mathcal O_{\PP^1}(n)
    \]
    with fiber coordinate $z$ the three sections are given by
    \[
    S_1=\{z=0\},
    \qquad
    S_2=\{z=x^ay^b(x-y)^c\},
    \qquad
    S_3=\{z=\lambda x^ay^b(x-y)^c\}.
    \]
    Equivalently, the three sections are scalar multiples of one common monomial section.
    \item There exist an integer $e\ge 0$ and nonnegative integers $a,b,c$ with
    $a+b+c+3^e=n,$
    such that on
    \[
    \mathbb F_n-S_0\cong \Tot \mathcal O_{\PP^1}(n)
    \]
    with fiber coordinate $z$, the three sections are given by
    \[
    S_1=\{z=0\},
    \qquad
    S_2=\{z=x^ay^b(x-y)^c x^{3^e}\},
    \qquad
    S_3=\{z=x^ay^b(x-y)^c y^{3^e}\}.
    \]
    In particular,
    $S_2-S_3=x^ay^b(x-y)^c(x-y)^{3^e},$
    so, after dividing by the maximal common factor $x^ay^b(x-y)^c$, the primitive part of the section triple is the
    $3^e$-th Frobenius pullback of the linear triple
    $0,\qquad x,\qquad y.$
\end{enumerate}
In case \textnormal{(ii)}, the smallest genuinely characteristic-$3$ primitive case has degree $3$.
\end{proposition}

\begin{proof}
By Corollary~\ref{cor:routeC-clean-section-disjoint-model}, the three remaining horizontal components have
numerical class $S_0+nf$.  The complement of the negative section is the total space of the associated line bundle,
so
\[
\mathbb F_n-S_0\cong \Tot \mathcal O_{\PP^1}(n).
\]
Automorphisms of this total space induced by translation by a global section and by fiberwise scaling extend to
automorphisms of $\mathbb F_n$ preserving the ruling and $S_0$.  After translating by the section $S_1$, we may
therefore assume that in the total-space coordinates one has
\[
S_1=\{z=0\},
\qquad
S_2=\{z=q_2\},
\qquad
S_3=\{z=q_3\}
\]
for some sections
$q_2,q_3\in H^0(\PP^1,\mathcal O_{\PP^1}(n)).$
The pairwise intersections of $S_1,S_2,S_3$ are the zero divisors of
\[
q_2,
\qquad
q_3,
\qquad
q_2-q_3.
\]
By assumption all of them are supported on $T$.  After an automorphism of the base, we may assume that
$T\subseteq \{0,1,\infty\}\subset \PP^1.$
Choose homogeneous coordinates $[x:y]$ on $\PP^1$ so that the three distinguished fibers are
$\{x=0\},\qquad \{y=0\},\qquad \{x-y=0\}.$
Then every homogeneous form of degree $n$ whose zero divisor is supported on those three fibers is, up to a
nonzero scalar, a monomial in the three linear forms $x$, $y$ and $x-y$.  Hence there exist nonzero scalars
$\alpha,\beta,\gamma$ and nonnegative integers $a_i,b_i,c_i$ with $a_i+b_i+c_i=n$ such that
\[
q_2=\alpha x^{a_1}y^{b_1}(x-y)^{c_1},
\qquad
q_3=\beta x^{a_2}y^{b_2}(x-y)^{c_2},
\qquad
q_2-q_3=\gamma x^{a_3}y^{b_3}(x-y)^{c_3}.
\]

Let
$g:=\gcd(q_2,q_3)=\gcd(q_2,q_3,q_2-q_3),$
and write
\[
q_2=gu,
\qquad
q_3=gv,
\qquad
q_2-q_3=gw.
\]
Then $u,v,w$ are homogeneous monomials in $x$, $y$, $x-y$ of the same degree
$d=n-\deg g,$
and they satisfy
$u-v=w.$
If $d=0$, then $u,v,w$ are nonzero constants.  Absorbing one scalar into the fiber coordinate $z$ gives
alternative \textnormal{(i)} with
\[
g=x^ay^b(x-y)^c,
\qquad a+b+c=n.
\]
Assume from now on that $d>0$.  Then $u,v,w$ are pairwise coprime and nonconstant.  Since the supports of
$u,v,w$ are three nonempty pairwise disjoint subsets of the three-element set $\{x,y,x-y\}$, each support consists
of exactly one linear form.  After permuting the coordinates if necessary, we may therefore write
\[
u=\alpha' x^d,
\qquad
v=\beta' y^d,
\qquad
w=\gamma' (x-y)^d
\]
for some nonzero scalars $\alpha',\beta',\gamma'$.  Substituting into $u-v=w$ and evaluating on the fibers
$y=0$ and $x=0$ gives
\[
\alpha'=\gamma',
\qquad
\beta'=(-1)^{d+1}\gamma'.
\]
After dividing by the common scalar $\gamma'$, the relation becomes
$x^d+(-1)^d y^d=(x-y)^d.$
Expanding the right-hand side shows that all intermediate binomial coefficients
$\binom{d}{i},\qquad 1\le i\le d-1,$
vanish modulo $3$.  By Lucas' theorem this happens if and only if $d$ is a power of $3$; write
$d=3^e.$
Since $3^e$ is odd, the displayed identity is exactly the Frobenius identity
$x^{3^e}-y^{3^e}=(x-y)^{3^e}.$
Writing
$g=x^ay^b(x-y)^c$
with $a+b+c=n-3^e$ and absorbing the remaining scalar into the fiber coordinate $z$ gives alternative
\textnormal{(ii)}.
\end{proof}

\begin{corollary}[The disjoint clean-section case becomes a Frobenius--monomial problem]
\label{cor:routeC-clean-section-disjoint-frobenius}
Assume the setup of Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}.  If the pairwise
intersections of $S_1,S_2,S_3$ are all supported on the at most three marked fibers from
Corollary~\ref{cor:routeC-tip-entry-obstruction}, then up to automorphism the whole disjoint clean-section case is
encoded either by a common-factor scalar-multiple model as in
Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}\textnormal{(i)}, or by a quadruple
\[
(a,b,c,e)\in \mathbb Z_{\ge 0}^4,
\qquad
 a+b+c+3^e=n,
\]
as in Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}\textnormal{(ii)}.
\end{corollary}

\begin{proof}
This is immediate from Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}.
\end{proof}


\begin{lemma}[Resolving a two-section collision of order \texorpdfstring{$p$}{p}]
\label{lem:routeC-pair-collision-order-p}
Let $U$ be a smooth surface with local coordinates $(t,z)$ at a point $o$, and let
\[
A_0=\{z=0\},
\qquad
C_0=\{z=t^p\},
\qquad p\ge 1.
\]
For $k=1,\dots,p$, let
$\sigma_k\colon U_k\to U_{k-1}$
be the blowup of the unique intersection point of the strict transforms $A_{k-1}$ and $C_{k-1}$, and let
$F_k\subset U_k$
be the exceptional curve of $\sigma_k$.
Then on $U_p$:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item the strict transforms $A_p$ and $C_p$ are disjoint;
    \item the exceptional locus over $o$ is the chain
    $F_1-\cdots-F_p;$
    \item one has
    \[
    F_i^2=-2\ \ (1\le i<p),
    \qquad
    F_p^2=-1;
    \]
    \item the curves $A_p$ and $C_p$ both meet $F_p$ transversely at distinct points and are disjoint from
    $F_1,\dots,F_{p-1}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Write local coordinates on $U_k$ recursively by
$z=t^k w_k.$
Then after $k$ blowups the strict transforms are given near the current intersection point by
\[
A_k=\{w_k=0\},
\qquad
C_k=\{w_k=t^{p-k}\}.
\]
For $k<p$ these still meet at the origin $(t,w_k)=(0,0)$, so the next blowup is again forced there.
For $k=p$ one gets
\[
A_p=\{w_p=0\},
\qquad
C_p=\{w_p=1\},
\]
so the two curves are disjoint.  This proves \textnormal{(1)}.

At each step the new exceptional curve meets the previous one, so the total exceptional divisor is the chain
$F_1-\cdots-F_p.$
This proves \textnormal{(2)}.

The first $p-1$ exceptional curves are each blown up exactly once more, so
\[
F_i^2=-2
\qquad (1\le i<p),
\]
while the last one is not blown up again, so
$F_p^2=-1.$
This proves \textnormal{(3)}.

In the final chart $w_p=0$ and $w_p=1$ meet the last exceptional curve
$F_p=\{t=0\}$
at distinct points, and the earlier exceptional curves lie in the complement of that chart origin.  Thus $A_p$ and
$C_p$ both meet $F_p$ transversely and are disjoint from $F_1,\dots,F_{p-1}$.
This proves \textnormal{(4)}.
\end{proof}

\begin{proposition}[Common-factor stripping for local three-section collisions]
\label{prop:routeC-common-factor-strip}
Let $U$ be a smooth surface with local coordinates $(t,z)$ at a point $o$, and let
\[
A_0=\{z=0\},
\qquad
B_0=\{z=t^u\},
\qquad
C_0=\{z=t^{u+p}\},
\qquad u\ge 0,\ p\ge 1.
\]
If $u>0$, let
$\tau_u\colon U_u\to U$
be the composition of the first $u$ blowups at the successive common intersection points of the strict transforms of
$A_0,B_0,C_0$, and denote the exceptional curves by
$G_1,\dots,G_u.$
Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item in local coordinates $(t,w)$ near the remaining intersection point of the strict transforms of $A_0$ and
    $C_0$ on $U_u$, one has
    \[
    A_u=\{w=0\},
    \qquad
    B_u=\{w=1\},
    \qquad
    C_u=\{w=t^p\};
    \]
    \item the curves $G_1,\dots,G_u$ form a chain, and after continuing with the $p$ blowups from
    Lemma~\ref{lem:routeC-pair-collision-order-p} for the pair $A_u,C_u$, the total exceptional divisor over $o$
    is the chain
    $G_1-\cdots-G_u-F_1-\cdots-F_p,$
    where
    \[
    G_i^2=-2\ \ (1\le i\le u),
    \qquad
    F_i^2=-2\ \ (1\le i<p),
    \qquad
    F_p^2=-1;
    \]
    \item the strict transform of $B_0$ meets $G_u$ transversely at a point away from $F_1$, while the strict
    transforms of $A_0$ and $C_0$ meet $F_p$ transversely at distinct points.
\end{enumerate}
When $u=0$, the statement reduces to Lemma~\ref{lem:routeC-pair-collision-order-p}.
\end{proposition}

\begin{proof}
If $u=0$, there is nothing to prove beyond Lemma~\ref{lem:routeC-pair-collision-order-p}.
Assume $u>0$.

Using local coordinates recursively
$z=t^k w_k,$
one checks inductively that after $k\le u$ blowups the strict transforms are
\[
A_k=\{w_k=0\},
\qquad
B_k=\{w_k=t^{u-k}\},
\qquad
C_k=\{w_k=t^{u+p-k}\}.
\]
For $k<u$, all three still pass through the origin, so the next blowup is again forced there.
At $k=u$ this becomes
\[
A_u=\{w_u=0\},
\qquad
B_u=\{w_u=1\},
\qquad
C_u=\{w_u=t^p\},
\]
which proves \textnormal{(1)} after renaming $w=w_u$.

The first $u$ exceptional curves form a chain
$G_1-\cdots-G_u,$
with
\[
G_i^2=-2\ \ (1\le i<u),
\qquad
G_u^2=-1
\]
before the primitive pair resolution.  The remaining collision is now exactly the order-$p$ pair
\[
A_u=\{w=0\},
\qquad
C_u=\{w=t^p\}
\]
at the point where both meet $G_u$.  Applying Lemma~\ref{lem:routeC-pair-collision-order-p} at that point creates the
chain
$F_1-\cdots-F_p$
attached to $G_u$.  Since the first primitive blowup is centered on $G_u$, the self-intersection of $G_u$ drops once
more, so in the final surface
$G_u^2=-2.$
All other displayed self-intersections follow from Lemma~\ref{lem:routeC-pair-collision-order-p}.  This proves
\textnormal{(2)}.

Finally, the transform of $B_0$ is given by
$B_u=\{w=1\},$
so it meets $G_u=\{t=0\}$ transversely at $(t,w)=(0,1)$ and is disjoint from the primitive collision point
$(0,0)$ where the $F_i$ are created.  The last statement for $A_0$ and $C_0$ is exactly
Lemma~\ref{lem:routeC-pair-collision-order-p}\textnormal{(4)}.  This proves \textnormal{(3)}.
\end{proof}

\begin{corollary}[Marked collisions in the disjoint Frobenius model have an explicit local normal form]
\label{cor:routeC-marked-collision-normal-form}
Assume the setup of Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}, and suppose we are in the
Frobenius alternative \textnormal{(ii)} with
\[
S_1=\{z=0\},
\qquad
S_2=\{z=g\,x^{3^e}\},
\qquad
S_3=\{z=g\,y^{3^e}\},
\qquad
g=x^ay^b(x-y)^c.
\]
Fix one of the marked fibers
$x=0,\qquad y=0,\qquad x-y=0.$
Let $u$ be the exponent of the corresponding linear factor in $g$; thus
$u\in\{a,b,c\}.$
Then, in suitable local coordinates $(t,z)$ centered at the marked collision on that fiber, the three sections are
given by
\[
z=0,
\qquad
z=t^u,
\qquad
z=t^{u+3^e},
\]
up to multiplication by a local unit and permutation of the last two sections.

Consequently, after $u$ common blowups and then $3^e$ primitive pair blowups, the local exceptional tree is exactly
the chain from Proposition~\ref{prop:routeC-common-factor-strip}.  In particular:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item the common monomial factor contributes only a terminal chain of $(-2)$-curves;
    \item after stripping that chain, the primitive local collision is always the order-$3^e$ pair
    \[
    z=0,
    \qquad
    z=t^{3^e}.
    \]
\end{enumerate}
\end{corollary}

\begin{proof}
Near the fiber $x=0$, write $t=x/y$ on the base chart $y\neq 0$.  Up to a local unit, one has
$g=t^u$
with $u=a$ there, and similarly $u=b$ over $y=0$ and $u=c$ over $x-y=0$ after the obvious change of coordinates.
Thus, up to a local unit and permutation,
\[
S_1=\{z=0\},
\qquad
S_2=\{z=t^{u+3^e}\},
\qquad
S_3=\{z=t^u\}.
\]
Now apply Proposition~\ref{prop:routeC-common-factor-strip}.
\end{proof}

\begin{proposition}[D-membership in the explicit marked-collision tree]
\label{prop:routeC-marked-collision-membership}
Assume the setup of Corollary~\ref{cor:routeC-marked-collision-normal-form}.  Let
$G_1-\cdots-G_u-F_1-\cdots-F_p$
be the explicit local exceptional chain over a marked collision, where
$p=3^e.$
Then, on the minimal resolution $\widetilde X$:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item every component
    $G_1,\dots,G_u,F_1,\dots,F_{p-1}$
    belongs to $D$;
    \item the last component
    $F_p$
    does \emph{not} belong to $D$.
\end{enumerate}
In particular, the marked collision produces exactly one local nonexceptional vertical curve, namely $F_p$, and all
other local curves created by the collision belong to the exceptional divisor.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:routeC-common-factor-strip}, one has
\[
G_i^2=-2\ \ (1\le i\le u),
\qquad
F_i^2=-2\ \ (1\le i<p),
\qquad
F_p^2=-1.
\]
All of these curves are vertical components of the degenerate fiber over the marked collision.

If one of the curves
$G_1,\dots,G_u,F_1,\dots,F_{p-1}$
did not belong to $D$, then it would be a component of a degenerate fiber outside the exceptional divisor.  But every
such component is a $(-1)$-curve by \cite[Lemma~2.8(b)]{PP12}, contradiction.  Hence all those curves belong to $D$.

On the other hand, $F_p$ is a $(-1)$-curve.  Since $D$ is the exceptional divisor of the \emph{minimal} resolution,
it contains no $(-1)$-curve.  Therefore
$F_p\not\subset D.$
This proves both assertions.
\end{proof}

\begin{corollary}[The explicit marked-collision core is a fork family]
\label{cor:routeC-marked-collision-forkfamily}
Assume the setup of Corollary~\ref{cor:routeC-marked-collision-normal-form}, and let
$L:=F_p.$
Then $L$ is a vertical $(-1)$-curve with the following local $D$-contact behavior.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item If $u=0$ and $p=1$, then $L$ meets $D$ transversely in exactly two points, namely on the two colliding
    horizontal components.
    \item In every other case, $L$ meets $D$ transversely in exactly three points: on the two colliding horizontal
    components and on the adjacent component $F_{p-1}\subset D$ of the vertical chain.
\end{enumerate}
More precisely, when $u>0$ or $p\ge 2$, the local marked-collision geometry is a fork with center $L$ and three arms:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item the first colliding horizontal component;
    \item the second colliding horizontal component;
    \item the vertical chain
    $F_{p-1}-\cdots-F_1-G_u-\cdots-G_1,$
    with the third horizontal component meeting $G_u$ if $u>0$.
\end{enumerate}
Thus the disjoint Frobenius marked-collision problem is reduced unconditionally to an explicit family of fork-like local
cores parameterized by $(u,p)$.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-marked-collision-membership}, the only local nonexceptional curve is
$L=F_p,$
and all other curves in the displayed chain belong to $D$.

If $u=0$ and $p=1$, there is no adjacent curve $F_{p-1}$, and Proposition~\ref{prop:routeC-common-factor-strip}
shows that the two colliding horizontal components meet $L$ transversely at distinct points.  Hence
$L\cdot D=2.$

In all other cases, the chain contains the component $F_{p-1}\subset D$, which meets $L$ once.  Proposition~\ref{prop:routeC-common-factor-strip} also shows that the two colliding horizontal components meet $L$ transversely at distinct
points.  Since no other local component meets $L$, one gets
$L\cdot D=3.$
The explicit description of the three arms is immediate from Proposition~\ref{prop:routeC-common-factor-strip}.
\end{proof}

\begin{remark}[What the marked-collision gap has become]
\label{rem:routeC-marked-collision-gap}
Corollary~\ref{cor:routeC-marked-collision-forkfamily} removes the old ambiguity in the disjoint Frobenius model.
The remaining issue is no longer to understand the local birational geometry of a marked collision, nor to determine
which local curves belong to $D$: both are now explicit.

What remains is the genuinely global problem of excluding the explicit fork family from the residual benchmark sector.
Equivalently, after the present note the disjoint Frobenius marked-collision regime is reduced unconditionally to the
question whether the fork-like local cores from Corollary~\ref{cor:routeC-marked-collision-forkfamily} can occur on a
characteristic-$3$ Picard-rank-one klt del Pezzo surface with more than seven singular points.
\end{remark}

\begin{remark}[Why the disjoint case is now closer to the characteristic-$3$ canonical seed]
\label{rem:routeC-clean-section-disjoint-frobenius-status}
Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius} does not yet finish the residual Route~C entry
problem, because it assumes that the pairwise intersections of the three sections are already confined to the marked
fibers.  What it does show is that, under exactly that confinement hypothesis, the disjoint clean-section case is no
longer an arbitrary three-section problem on $\mathbb F_n$.  It becomes either a pure common-factor scalar-multiple
model, or a Frobenius--monomial problem on $\PP^1$ with only one primitive characteristic-$3$ mechanism,
namely the $3^e$-th Frobenius pullback of the linear triple.  In particular, the first genuinely characteristic-$3$
primitive degree is $3$, which is the same numerical size as the canonical height-$4$ seed highlighted in
\cite[Proposition~6.1(a), Example~7.1]{OpenDP}.
\end{remark}

\begin{remark}[What this does and does not achieve]
\label{rem:routeC-clean-section-Hirzebruch-status}
Proposition~\ref{prop:routeC-clean-section-Hirzebruch} does not yet prove the residual Route~C entry theorem.
What it does prove is that, once a clean section $H_0$ is chosen, the remaining mixed entry problem is no longer an
arbitrary fiberwise graph problem on $\widetilde X$: it becomes a finite numerical configuration problem for three
sections on a Hirzebruch surface, with explicitly computable numerical classes and with the bad singly-hit part
confined to at most three marked fibers.
\end{remark}
\subsection{A numerical reduction of the contact-$2$ case}

By \cite[Definition~2.10, Definition~2.11, Lemma~2.12]{PP12}, a rank-one surface has a descendant with elliptic
boundary if and only if its minimal log resolution has an elliptic tie.  Thus route~(C) reduces to forcing an
elliptic tie on the minimal resolution in the residual height-$\ge 4$ sector.

The first genuinely local case to understand is a $(-1)$-curve which meets the exceptional divisor twice on one
connected component.

\begin{proposition}[Local package for a same-component contact-$2$ curve]\label{prop:routeC-local-package}
Let $X$ be a klt del Pezzo surface of Picard rank $1$, let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution, and let $L\subset \widetilde X$ be a $(-1)$-curve with $L\not\subset D$ and
$L\cdot D=2.$
Assume that both contacts of $L$ with $D$ lie on one connected component $D_0\subset D$, and put
\[
E:=D_0+L,
\qquad
C:=\pi(L)\subset X.
\]
Write $D_0=\sum_{\alpha}E_{\alpha}$, let $M=(E_{\alpha}\cdot E_{\beta})$ be the intersection matrix of $D_0$, and let
$v=(L\cdot E_{\alpha})_{\alpha}$.  Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $p_a(E)=1$.
    \item $C^2=-1-v^{\!t}M^{-1}v>0$.
    \item If $D_0$ is a chain $T=[a_1,\dots,a_n]$ and $L$ meets the $i$-th component twice, then
    \[
    C^2=-1+4\frac{d(T_i^-)d(T_i^+)}{d(T)},
    \]
    where $T_i^-=[a_1,\dots,a_{i-1}]$, $T_i^+=[a_{i+1},\dots,a_n]$, and $d(\varnothing)=1$.  In particular,
    $4d(T_i^-)d(T_i^+)>d(T).$
    Hence $a_i\le 5$.  Moreover, if $a_i=5$, then at least one neighbour of the $i$-th component in the chain is a
    $(-2)$-curve.
    \item If $D_0$ is a chain $T=[a_1,\dots,a_n]$ and $L$ meets the $i$-th and $j$-th components once each, with
    $i<j$, then
    \[
    C^2=-1+\frac{d(T_i^-)d(T_i^+)+d(T_j^-)d(T_j^+)+2d(T_i^-)d(T_j^+)}{d(T)}.
    \]
\end{enumerate}
\end{proposition}

\begin{proof}
Since $D_0$ is a tree of smooth rational curves, we have $p_a(D_0)=0$.  As $L\cong \PP^1$ and $L\cdot D_0=2$,
\[
p_a(E)=p_a(D_0)+p_a(L)+D_0\cdot L-1=0+0+2-1=1,
\]
proving~(1).

For~(2), write $D_0=\sum_{\alpha}E_{\alpha}$ and let $a=-M^{-1}v$.  Then
\[
\bigl(L+\sum_{\alpha}a_{\alpha}E_{\alpha}\bigr)\cdot E_{\beta}=0
\qquad\text{for every }\beta,
\]
so
\[
\pi^*C=L+\sum_{\alpha}a_{\alpha}E_{\alpha}.
\]
Therefore
\[
C^2=(\pi^*C)^2=L^2+2a^{\!t}v+a^{\!t}Ma=-1+a^{\!t}v=-1-v^{\!t}M^{-1}v.
\]
Since $X$ is klt, it is $\mathbb Q$-factorial, and since $\rho(X)=1$, every nonzero effective curve on $X$ is
numerically proportional to an ample class.  Thus $C^2>0$.

Assume now that $D_0=T=[a_1,\dots,a_n]$ is a chain, and let $N=-M$.  By the standard inverse formula for tridiagonal
continued-fraction matrices,
\[
(N^{-1})_{rs}=\frac{d([a_1,\dots,a_{r-1}])\,d([a_{s+1},\dots,a_n])}{d(T)}
\qquad (r\le s),
\]
with the convention $d(\varnothing)=1$.  If $L$ meets the $i$-th component twice, then $v=2e_i$, so
\[
C^2=-1+4(N^{-1})_{ii}=-1+4\frac{d(T_i^-)d(T_i^+)}{d(T)},
\]
which proves the first part of~(3).

To extract a simple numerical consequence, put
\[
A=d(T_i^-),\quad A'=d([a_1,\dots,a_{i-2}]),\quad B=d(T_i^+),\quad B'=d([a_{i+2},\dots,a_n]),
\]
where by convention $d(\varnothing)=1$ and $d(\text{past-empty})=0$.  Expanding the determinant of $T$ along the
$i$-th component gives
$d(T)=a_iAB-A'B-AB'.$
Hence positivity of $C^2$ is equivalent to
$(4-a_i)AB+A'B+AB'>0.$
Because $0\le A'<A$ and $0\le B'<B$, the left-hand side is negative if $a_i\ge 6$, so necessarily $a_i\le 5$.
Now assume $a_i=5$.  Then
\[
\frac{A'}{A}+\frac{B'}{B}>1.
\]
If the neighbour on the left exists and has weight at least $3$, then the recurrence for discriminants gives
$A\ge 2A'+1$, hence $A'/A<1/2$; similarly, if the neighbour on the right exists and has weight at least $3$, then
$B'/B<1/2$.  Therefore, when $a_i=5$, at least one neighbour of the $i$-th component must be a $(-2)$-curve.
This proves~(3).

Finally, if $L$ meets the $i$-th and $j$-th components once each, then $v=e_i+e_j$, so by the same inverse-matrix
formula
\[
C^2=-1+(N^{-1})_{ii}+(N^{-1})_{jj}+2(N^{-1})_{ij}
\]
and therefore
\[
C^2=-1+\frac{d(T_i^-)d(T_i^+)+d(T_j^-)d(T_j^+)+2d(T_i^-)d(T_j^+)}{d(T)},
\]
as claimed in~(4).
\end{proof}

\begin{remark}[Forks are equally accessible numerically]\label{rem:routeC-fork-branch}
The matrix formula in Proposition~\ref{prop:routeC-local-package}(2) works for arbitrary weighted trees, not only
for chains.  For example, if $D_0=\langle b;T_1,T_2,T_3\rangle$ is a fork and $L$ meets the branching component
twice, then the same Cramer's-rule computation gives
\[
C^2=-1+4\frac{d(T_1)d(T_2)d(T_3)}{d(D_0)}.
\]
So the fork case is also subject to an exact contact-$2$ positivity test; the chain formulas above are written out
only because they already force a transparent weight bound.
\end{remark}

\begin{remark}[What this buys for route~(C)]\label{rem:routeC-finite-rooted-search}
Proposition~\ref{prop:routeC-local-package} turns the contact-$2$ part of route~(C) into a finite rooted-graph
problem.
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item The connected component $E=D_0+L$ always has arithmetic genus $1$.
    \item The image square $C^2$ is an explicit rational number determined purely by the weighted dual graph of $D_0$
    and the contact vector $v$.
    \item For cyclic quotient chains, doubly-hit components of weight at least $6$ are impossible, and weight $5$ can
    occur only next to a $(-2)$-curve.
\end{enumerate}
So once one knows the admissible chains and forks that can occur in the residual Palka--Pe\l ka sector, the surviving
contact-$2$ vectors are algorithmically enumerable.  The remaining issue is then local and conceptual rather than
global: prove that each surviving rooted graph is \emph{sandwiched of $\delta=1$}.  By Definition~1.11 and
Lemma~1.12(b) of \cite{PP12}, that is exactly what is needed to produce a descendant with elliptic boundary.

This matches the already-understood models from \cite{PP12}: the tie-producing local shapes there blow down to
configurations such as $[3,1,2]$, $[1,2,2]$, or $\langle 1;[2],[3],[k]\rangle$, and in the canonical height-$4$ case
\cite[Proposition~6.1]{OpenDP} produces a contact-$2$ curve meeting both tips of a connected component of type $[2]$
or $[2,2]$.  In other words, route~(C) now looks like a finite local reduction-to-known-tie-shapes problem.
\end{remark}

\subsection{A rooted-graph pass on the characteristic-$3$ frontier chains}

To make the previous remark more concrete, it is convenient to encode the contact-$2$ data in a rooted graph.
Let $T=[a_1,\dots,a_n]$ be a chain and let $v$ be a total-contact-$2$ vector.  We write $G(T,v)$ for the
weighted multigraph obtained by adjoining a vertex $L$ of weight $1$ to $T$ and connecting it according to $v$:
either by a double edge to one component of $T$, or by two simple edges to two components.

We say that $G(T,v)$ is \emph{root-contractible} if there is a component $R\subset T$ such that every other
component of $G(T,v)$ can be eliminated by a sequence of blowdowns of $(-1)$-curves different from $R$.
In that case the image of $R$ is a rational curve on a smooth surface, and since Proposition~\ref{prop:routeC-local-package}(1)
shows that $p_a(T+L)=1$ whenever the two contacts lie on one connected component of $D$, root-contractibility gives a
concrete local witness that the rooted graph is sandwiched of $\delta=1$.  In particular, it is enough for producing
an elliptic tie in route~(C).

\begin{proposition}[First rooted-graph pass on the frontier chains]\label{prop:routeC-small-rooted-pass}
Consider the chains
\[
[2],\ [3],\ [5],\ [6],\ [2,2],\ [3,2],\ [4,2],\ [5,2].
\]
These are the chains which either appear in the current characteristic-$3$ frontier baskets from
Table~\ref{tab:summary}, or are the $s=8$ contact-$2$ chains from \cite[Lemma~6.2]{PP12}.  Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item The rooted graphs
    \[
    [2]^{|1,1|},\qquad [3]^{|1,1|},\qquad |1|[2,2]|1|,\qquad |1|[4,2]|1|
    \]
    are root-contractible.
    \item The chains $[5]$ and $[6]$ admit no positive contact-$2$ pattern.
    \item For $[2,2]$, $[3,2]$, and $[4,2]$, no double-contact pattern is root-contractible; only the tip-to-tip
    pattern is.
    \item For $[5,2]$, the only positive contact-$2$ pattern is double contact on the $(-2)$-tip, and this pattern
    is not root-contractible.
\end{enumerate}
\end{proposition}

\begin{proof}
For~(1), the rooted graphs $[2]^{|1,1|}$ and $[3]^{|1,1|}$ contract immediately by blowing down $L$.
If $T=[2,2]$ and $L$ meets both tips, then after contracting $L$ the two components of $T$ become $(-1)$-curves
meeting each other twice, so contracting either one leaves the other as the required root.  If $T=[4,2]$ and $L$
meets both tips, then contracting $L$ turns the $(-2)$-tip into a $(-1)$-curve meeting the $(-4)$-component twice;
contracting this $(-1)$-curve leaves the $(-4)$-component as the root.

For~(2), Proposition~\ref{prop:routeC-local-package}(3) already excludes a doubly-hit component of weight at least $6$.
Thus $[6]$ has no positive contact-$2$ pattern.  For $[5]$ the same formula gives
\[
C^2=-1+\frac{4}{5}<0,
\]
so $[5]$ is also impossible.

For~(3), assume first that $T=[2,2]$ and $L$ meets one component twice.  After contracting $L$, the hit component has
self-intersection $+2$ and the other remains a $(-2)$-curve, so no further $(-1)$-curve is available; hence this rooted
graph is not root-contractible.  The same argument works for $[3,2]$ and $[4,2]$: if $L$ meets one component twice,
then after contracting $L$ the hit component has self-intersection $+1$, $+2$, $0$, or $+2$, according as the hit
component has weight $3$, $2$, $4$, or $2$, so again no further $(-1)$-curve appears.  On the other hand, the tip-to-tip
patterns for $[3,2]$ and $[4,2]$ are root-contractible by the same calculation as above: contracting $L$ turns the
$(-2)$-tip into a $(-1)$-curve meeting the other component twice.

For~(4), the formula in Proposition~\ref{prop:routeC-local-package}(3) shows that the only positive pattern on $[5,2]$
is double contact on the $(-2)$-tip, with
\[
C^2=-1+4\frac{5}{9}=\frac{11}{9}.
\]
After contracting $L$, that tip has self-intersection $+2$ while the other component remains a $(-5)$-curve, so once
again there is no further $(-1)$-curve and the rooted graph is not root-contractible.
\end{proof}

\begin{remark}[Consequences for the current near-miss baskets]\label{rem:routeC-nearmiss-local-filter}
This already sharpens route~(C) in a useful way.
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item In the $9$-point numerical near-miss
    \[
    [[6],[3],[3],[3],[3],[2,2],[2,2],[2,2],[2,2]],
    \]
    the extra chain $[6]$ is ruled out immediately by positivity.
    \item In the rank-compatible $8$-point numerical near-miss
    $[[5,2],[2],[3],[3],[3],[2,2],[2,2],[2,2]],$
    the extra chain $[5,2]$ survives the positivity test only in one contact-$2$ pattern, but
    Proposition~\ref{prop:routeC-small-rooted-pass} shows that this pattern is not root-contractible.
    By Lemma~\ref{lem:routeC-local-tie-equiv} below, it is therefore not locally sandwiched of $\delta=1$.
\end{enumerate}
So the ``new'' noncanonical chains visible in the current near-misses do not carry the contact-$2$ tie.  The surviving
local models are exactly the familiar ones: the canonical height-$4$ patterns $[2]^{|1,1|}$ and $|1|[2,2]|1|$
from \cite[Proposition~6.1]{OpenDP}, together with the $s=8$ contact-$2$ models $[3]^{|1,1|}$ and $|1|[4,2]|1|$
from \cite[Lemma~6.2]{PP12}.  In particular, any successful closure of the residual sector now has to pass through the
already visible $[3]/A_2$ geometry rather than through the extra chains $[6]$ or $[5,2]$.
\end{remark}

\subsection{A first longer \texorpdfstring{$[3]/A_2$}{[3]/A2} chain family}

The first honest three-vertex prolongation of the $s=8$ contact-$2$ corridor from \cite[Lemma~6.2]{PP12} is
$T_k=[3,k,2], \qquad k\ge 2.$
It contains the already checked chain $[3,2,2]$, and continues the same local $[3]/A_2$ geometry beyond the
basic models $[3]^{|1,1|}$ and $|1|[4,2]|1|$.

\begin{proposition}[The chain family $\boldsymbol{[3,k,2]}$]\label{prop:routeC-3k2}
Let $T_k=[3,k,2]$ with $k\ge 2$.  Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item For the tip-to-tip contact pattern on $T_k$ one has
    \[
    C^2=-1+\frac{5k}{6k-5}=\frac{5-k}{6k-5}.
    \]
    In particular, this pattern is positive if and only if $k\le 4$.
    \item If $k\ge 3$, no positive contact-$2$ pattern on $T_k$ is root-contractible except the tip-to-tip pattern.
    \item Consequently, among the longer three-vertex $[3]/A_2$ chains only
    $[3,3,2]\qquad\text{and}\qquad [3,4,2]$
    survive the local tie filter; the whole tail
    $[3,k,2], \qquad k\ge 5,$
    is excluded.
\end{enumerate}
\end{proposition}

\begin{proof}
For~(1), apply Proposition~\ref{prop:routeC-local-package}(4) with $i=1$, $j=3$.  Since
\[
d([3,k,2])=6k-5,\qquad d([k,2])=2k-1,\qquad d([3,k])=3k-1,
\]
we get
\[
C^2=-1+\frac{(2k-1)+(3k-1)+2}{6k-5}=-1+\frac{5k}{6k-5}
=\frac{5-k}{6k-5}.
\]
Hence the tip-to-tip pattern is positive exactly for $k\le 4$.

For~(2), we analyze all contact-$2$ patterns.

If $L$ meets one component of $T_k$ twice, then after contracting $L$ the hit component has self-intersection
$+1$, $4-k$, or $+2$, according as it is the $(-3)$-curve, the $(-k)$-curve, or the $(-2)$-tip.
In the first and third cases no $(-1)$-curve appears.  In the middle case, a $(-1)$-curve appears only for $k=5$,
but then the positivity formula gives
\[
C^2=-1+\frac{24}{25}<0,
\]
so this pattern is irrelevant for route~(C).  Thus no positive double-contact pattern is root-contractible.

Assume now that $L$ meets the first and second components once each.  Contracting $L$ gives a rooted multigraph
with self-intersections $-2$, $1-k$, $-2$, where the first two components meet twice and the last meets the
middle one once.  For $k\ge 3$ no component is a $(-1)$-curve, so the rooted graph is not root-contractible.

Next assume that $L$ meets the second and third components once each.  Then
\[
C^2=-1+\frac{3k+11}{6k-5}=\frac{16-3k}{6k-5},
\]
so positivity forces $k\le 5$.  After contracting $L$, the third component becomes a $(-1)$-curve; contracting it
leaves a chain with self-intersections $-3$ and $5-k$.  For $k=3,4,5$ this is $[-3,+2]$, $[-3,+1]$, or
$[-3,0]$, so no further $(-1)$-curve appears.  Hence this pattern is never root-contractible for $k\ge 3$.

Finally, assume that $L$ meets the first and third components once each.  Contracting $L$ turns the $(-2)$-tip
into a $(-1)$-curve and produces a triangle.  Contracting that $(-1)$-curve leaves a $(-1)$-curve meeting the
middle component twice; contracting it leaves the middle component as a smooth rational curve of self-intersection
$5-k$.  Therefore this rooted graph is root-contractible, and by~(1) it is positive if and only if $k\le 4$.

This proves~(2), and~(3) is immediate.
\end{proof}

\subsection{A uniform lemma for terminal \texorpdfstring{$A_2$}{A2}-tails}

Let
\[
T_m=[3,(2)^m]=[3,\underbrace{2,\dots,2}_{m\text{ times}}],\qquad m\ge 0,
\]
with components $T_1,\dots,T_{m+1}$ numbered from the $(-3)$-end, so that
$T_1^2=-3,\qquad T_r^2=-2\ \text{for}\ r\ge 2.$

\begin{proposition}[Uniform terminal \texorpdfstring{$A_2$}{A2}-tail lemma]\label{prop:routeC-a2-tail}
For the contact-$2$ rooted graphs on $T_m$ the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item For every $m\ge 1$, the pattern with single contacts on $T_1$ and $T_2$ is root-contractible, with root $T_1$.
    Its image square is
    \[
    C^2=-1+\frac{6m+1}{2m+3}=\frac{4m-2}{2m+3}>0.
    \]
    \item For every $m\ge 1$, the pattern with single contacts on $T_1$ and $T_{m+1}$ is root-contractible, with root $T_1$.
    Its image square is
    \[
    C^2=-1+\frac{3m+4}{2m+3}=\frac{m+1}{2m+3}>0.
    \]
    \item Besides the one-vertex case $[3]^{|1,1|}$ (that is, $m=0$) and the short exceptional pattern on
    $[3,2,2,2]$ meeting $T_2$ and $T_4$, there are no other root-contractible contact-$2$ patterns on $T_m$.
    \item In particular, for $m\ge 4$, the only positive root-contractible contact-$2$ patterns on $T_m$ are the two
    end-supported patterns from~\textnormal{(1)} and~\textnormal{(2)}.
\end{enumerate}
\end{proposition}

\begin{proof}
Write $n=m+1$.

For~\textnormal{(1)}, assume $m\ge 1$ and let $L$ meet $T_1$ and $T_2$ once each.
After contracting $L$, the curve $T_2$ becomes a $(-1)$-curve, while $T_1$ becomes a $(-2)$-curve.
Now contract $T_2,T_3,\dots,T_n$ in this order.
At each step the next component in the $(-2)$-tail becomes a $(-1)$-curve, so the whole tail peels off and $T_1$
survives as the root.
For the square, Proposition~\ref{prop:routeC-local-package}(4) gives
\[
d(T_m)=2m+3,\qquad d([2^m])=m+1,\qquad d([2^{m-1}])=m,
\]
hence
\[
C^2=-1+\frac{(m+1)+3m+2m}{2m+3}
=-1+\frac{6m+1}{2m+3}
=\frac{4m-2}{2m+3}.
\]

For~\textnormal{(2)}, let $L$ meet $T_1$ and $T_n$ once each.
After contracting $L$, the curve $T_n$ is a $(-1)$-curve.
Contract $T_n,T_{n-1},\dots,T_2$ in this order.
Again each step peels one $(-2)$-tip from the terminal tail, leaving $T_1$ as the root.
Using Proposition~\ref{prop:routeC-local-package}(4) once more, we have
\[
d(T_m)=2m+3,\qquad d([2^m])=m+1,\qquad d([3,(2)^{m-1}])=2m+1,
\]
so
\[
C^2=-1+\frac{(m+1)+(2m+1)+2}{2m+3}
=-1+\frac{3m+4}{2m+3}
=\frac{m+1}{2m+3}.
\]

For~\textnormal{(3)}, first consider double-contact patterns.
If $m=0$, then $T_m=[3]$ and $[3]^{|1,1|}$ is root-contractible by blowing down $L$.
Assume now $m\ge 1$.
If $L$ meets $T_1$ twice, then after contracting $L$ the curve $T_1$ has self-intersection $+1$, while every
other component keeps its original self-intersection.
If $L$ meets some $T_r$ with $r\ge 2$ twice, then after contracting $L$ the curve $T_r$ has self-intersection $+2$,
again leaving no $(-1)$-curve.
Thus no double-contact pattern is root-contractible for $m\ge 1$.

Now assume $L$ meets $T_i$ and $T_j$ once each, where $i<j$.

Suppose first that $2\le i<j<n$.
After contracting $L$, the curves $T_i$ and $T_j$ are adjacent $(-1)$-curves, because the blowdown creates the
extra edge $T_iT_j$.
Hence one of them must be the root: contracting one raises the other to self-intersection $0$.
If the root were $T_i$, then the left segment $T_1+\cdots+T_{i-1}$ could never start contracting, because
$T_{i-1}$ is not a $(-1)$-curve initially and the only possible first contraction affecting it would have to be
$T_i$ or $T_{i-2}$; the former is the root and the latter cannot contract before $T_{i-1}$.
Thus root $T_i$ is impossible.
Similarly, if the root were $T_j$, then the right tail $T_{j+1}+\cdots+T_n$ could never start contracting, because
$T_{j+1}$ cannot become a $(-1)$-curve until $T_j$ is contracted.
So this case is impossible.

Suppose next that $i=1$ and $3\le j<n$.
After contracting $L$, the curve $T_j$ is the unique $(-1)$-curve, so it is forced.
Its contraction makes $T_1$, $T_{j-1}$, and $T_{j+1}$ into pairwise adjacent $(-1)$-curves:
the blowdown creates the edges
\[
T_1T_{j-1},\qquad T_1T_{j+1},\qquad T_{j-1}T_{j+1}.
\]
Contracting any one of these three curves raises the other two to self-intersection $0$.
Hence no contraction sequence can eliminate two of them while leaving only one root, so this case is impossible.

Finally, suppose that $2\le i$ and $j=n$.
After contracting $L$, the curves $T_i$ and $T_n$ are adjacent $(-1)$-curves, so one of them must be the root.
Root $T_i$ is impossible by the same frozen-left-segment argument as above.
Assume therefore that the root is $T_n$.
If $i\ge 3$, then $T_i$ is forced; after contracting it, the curves $T_{i-1}$ and $T_{i+1}$ become adjacent
$(-1)$-curves, so one of them must survive.
Thus only $i=2$ remains.
If $n=4$, then the sequence
$L,\ T_2,\ T_3,\ T_1$
leaves $T_4$ as the root, so this is the short exceptional pattern.
Its square is
\[
C^2=-1+\frac{22}{9}=\frac{13}{9}>0.
\]
If $n\ge 5$, then with root $T_n$ the contractions of $T_2$ and then $T_3$ are forced.
After these two blowdowns, the curves $T_1$ and $T_4$ become adjacent $(-1)$-curves, so one of them survives.
Thus no further pattern occurs.

This proves~\textnormal{(3)}, and~\textnormal{(4)} is immediate.
\end{proof}

\begin{remark}[What remains of the terminal \texorpdfstring{$A_2$}{A2}-tail corridor]\label{rem:routeC-a2-tail-window}
Proposition~\ref{prop:routeC-a2-tail} upgrades the finite-window search on $[3,(2)^m]$ to a uniform theorem:
for $m\ge 4$, every root-contractible contact-$2$ pattern is obtained by peeling the terminal $(-2)$-tail from one
of its ends.
So the infinite $[3]/A_2$ tail family creates no new rooted mechanism beyond the two already visible end-supported
ones.
The sister family
$[4,(2)^m]$
behaves similarly.  The next subsection proves the end-supported part uniformly for both $[3,(2)^m]$ and
$[4,(2)^m]$, and the exact rooted search isolates only one short off-end exception on the $[4]/A_2$ side.
\end{remark}



\subsection{An inductive terminal \texorpdfstring{$A_2$}{A2}-tail lemma}

The previous exact search suggests that, once one enters the terminal $A_2$-tail corridor,
root-contractible contact-$2$ patterns are forced to live at the ends of the chain.
The next proposition upgrades that observation to a uniform theorem for the end-supported patterns.

\begin{lemma}[Monotonicity under blowdown]\label{lem:routeC-monotone-blowdown}
Let $E$ be a $(-1)$-curve on a smooth surface and let $F\neq E$ be an irreducible curve.
If $\sigma$ contracts $E$, then
$(\sigma_*F)^2 = F^2 + (E\cdot F)^2.$
In particular, the self-intersections of all remaining components weakly increase under every blowdown.
Hence any non-root component which ever reaches self-intersection $\ge 0$ can never be contracted later in a rooted-contraction sequence.
\end{lemma}

\begin{proof}
This is the standard blowdown formula for self-intersection.  Since $(E\cdot F)^2\ge 0$, the monotonicity statement is immediate.
\end{proof}

\begin{proposition}[End-supported contacts on terminal \texorpdfstring{$A_2$}{A2}-tails]\label{prop:routeC-terminal-a2-end}
For $a\in\{3,4\}$ and $n\ge 2$, set
\[
T_n(a)=[a,(2)^{n-1}]=[a,\underbrace{2,\dots,2}_{n-1}],
\]
with components $E_1,\dots,E_n$ ordered from left to right.
Let $L$ be a $(-1)$-curve meeting $E_1$ and $E_k$ once each, where $2\le k\le n$,
and let $G_n(a;k)$ be the corresponding rooted graph.
Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item One has
    $d(T_n(a))=(a-1)n+1,$
    and the image square of $L$ is
    \[
    C^2=-1+\frac{n+\bigl((a-1)(k-1)+1\bigr)(n-k+1)+2(n-k+1)}{(a-1)n+1}.
    \]
    In particular,
    \[
    C^2(1,2)=\frac{4n-a-3}{(a-1)n+1},\qquad
    C^2(1,n)=\frac{n-a+3}{(a-1)n+1},
    \]
    and, for $n\ge 3$,
    \[
    C^2(1,n-1)=\frac{a(n-4)+9}{(a-1)n+1}.
    \]
    \item If $a=3$, then $G_n(3;k)$ is root-contractible if and only if $k=2$ or $k=n$.
    Explicit contraction sequences are
    \[
    L,E_2,E_3,\dots,E_n
    \qquad\text{and}\qquad
    L,E_n,E_{n-1},\dots,E_2,
    \]
    both of which leave $E_1$ as the root.
    \item If $a=4$, then $G_n(4;k)$ is root-contractible if and only if $k=2$, $k=n-1$, or $k=n$.
    Explicit contraction sequences are
    $L,E_2,E_3,\dots,E_n,$
    $L,E_{n-1},E_n,E_1,E_2,\dots,E_{n-3},$
    and
    $L,E_n,E_{n-1},\dots,E_2.$
    The first and third sequences leave $E_1$ as the root, while the middle one leaves $E_{n-2}$.
\end{enumerate}
\end{proposition}

\begin{proof}
For~(1), note first that $d([2^m])=m+1$ for every $m\ge 0$, and the recurrence for discriminants therefore gives
$d(T_n(a))=d([a,(2)^{n-1}])=(a-1)n+1.$
Now apply Proposition~\ref{prop:routeC-local-package}(4) with $i=1$ and $j=k$.
Since
\[
 d(T_1^-)=1,\qquad d(T_1^+)=n,\qquad d(T_k^-)=d([a,(2)^{k-2}])=(a-1)(k-1)+1,
\]
\[
 d(T_k^+)=d([2^{n-k}])=n-k+1,
\]
we obtain the displayed formula for $C^2$.  The three special cases follow by substitution.

For the explicit contraction sequences in~(2) and~(3), one checks inductively that after each displayed blowdown the next named component has self-intersection $-1$.  This is exactly the tail-peeling mechanism suggested by the exact search.

It remains to prove the converse statements.

Assume first that $a=3$ and $3\le k\le n-1$.
After contracting $L$, the unique $(-1)$-curve is $E_k$, so the next blowdown is forced.
After contracting $E_k$, the three components
$E_1,\qquad E_{k-1},\qquad E_{k+1}$
are all $(-1)$-curves, and they form a triangle in the resulting multigraph.
If the root is not $E_1$, then eventually one must contract one of $E_{k-1}$ or $E_{k+1}$ first; by Lemma~\ref{lem:routeC-monotone-blowdown}, this raises the self-intersection of $E_1$ from $-1$ to $0$, so $E_1$ can never be contracted later.  Hence the root would have to be $E_1$.
But if $E_1$ is the root, then contracting either $E_{k-1}$ or $E_{k+1}$ first raises the other from $-1$ to $0$, so that component can never be removed later.  Thus no rooted contraction is possible for $3\le k\le n-1$, and only the end cases $k=2$ and $k=n$ survive.
This proves~(2).

Now assume that $a=4$.
Again, after contracting $L$ the next blowdown is forced to be $E_k$.
If $3\le k\le n-2$, then after this blowdown the components $E_{k-1}$ and $E_{k+1}$ are $(-1)$-curves while $E_1$ has self-intersection $-2$.
If the root is neither $E_{k-1}$ nor $E_{k+1}$, then contracting one of these two components first raises the other from $-1$ to $0$, contradicting Lemma~\ref{lem:routeC-monotone-blowdown}.  So the root must be one of $E_{k-1}$ or $E_{k+1}$.

Suppose first that the root is $E_{k-1}$.
Then $E_{k+1}$ must be contracted next.
After that blowdown, $E_1$ becomes a $(-1)$-curve and, if $k\le n-2$, the image of $E_{k+2}$ is also a $(-1)$-curve.
If one contracts $E_1$ next, then the image of $E_{k+2}$ acquires self-intersection $0$ and can never be removed later.
If instead one contracts the image of $E_{k+2}$ next, then $E_1$ acquires self-intersection $0$ and can never be removed later.
Therefore the right-hand tail must be empty, i.e. $k=n-1$.

Suppose next that the root is $E_{k+1}$.
Then $E_{k-1}$ must be contracted next.
If $k=3$, then $E_{k-1}=E_2$ meets $E_1$ twice at that stage, so contracting $E_2$ raises $E_1^2$ from $-2$ to $+2$, again impossible by Lemma~\ref{lem:routeC-monotone-blowdown}.
If $k>3$, then after contracting $E_{k-1}$ the image of $E_1$ is a $(-1)$-curve and the image of $E_{k-2}$ is also a $(-1)$-curve; contracting either one next forces the other to become nonnegative.
So this case is impossible as well.
Hence no interior value $3\le k\le n-2$ is root-contractible.
Together with the explicit contraction sequences for $k=2$, $k=n-1$, and $k=n$, this proves~(3).
\end{proof}


\begin{remark}[What remains off the end]\label{rem:routeC-terminal-a2-offend}
Proposition~\ref{prop:routeC-terminal-a2-end} proves that every end-supported rooted mechanism on a terminal
$A_2$-tail is one of the familiar tail-peeling patterns.
A direct exhaustive enumeration through chain length $20$ shows that, among
positive root-contractible patterns \emph{not} meeting the left-hand component $E_1$, only two short exceptions appear:
\[
[3,2,2,2]\ \text{with contacts on }E_2,E_4,
\qquad
[4,2,2,2,2]\ \text{with contacts on }E_2,E_5.
\]
No further off-end rooted pattern occurs in that searched window.
The next proposition shows that these two isolated configurations are not new obstructions: they are themselves
explicit tie-producing mechanisms.
\end{remark}

\subsection{The short off-end exceptions are genuine ties}

The two configurations isolated in Remark~\ref{rem:routeC-terminal-a2-offend} fit one clean off-end family.
They are not a leftover nuisance in route~(C); they already produce the desired sandwiched genus-one divisor.

\begin{proposition}[The short off-end exceptions are explicit ties]\label{prop:routeC-offend-ties}
Let $a\in\{3,4\}$ and $n\ge 3$, and set
\[
T_n(a)=[a,(2)^{n-1}]=[a,\underbrace{2,\dots,2}_{n-1}]
\]
with components $E_1,\dots,E_n$ ordered from left to right.
Let $L$ be a $(-1)$-curve meeting $E_2$ and $E_n$ once each, and let $G_n^{\mathrm{off}}(a)$ be the corresponding rooted graph.
Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item The image square is
    \[
    C^2=\frac{an+1}{(a-1)n+1}>0.
    \]
    \item The rooted graph $G_n^{\mathrm{off}}(a)$ is root-contractible if and only if
    $n=a+1.$
    \item When $n=a+1$, the explicit contraction sequence
    $L,\ E_2,\ E_3,\ \dots,\ E_{n-1},\ E_1$
    leaves $E_n$ as the root.
    Consequently the two off-end configurations
    \[
    [3,2,2,2]\ \text{with contacts on }E_2,E_4,
    \qquad
    [4,2,2,2,2]\ \text{with contacts on }E_2,E_5
    \]
    are explicit elliptic ties.
\end{enumerate}
\end{proposition}

\begin{proof}
For~\textnormal{(1)}, apply Proposition~\ref{prop:routeC-local-package}(4) with $i=2$ and $j=n$.
Using
\[
d(T_n(a))=(a-1)n+1,
\qquad
d(T_2^-)=d([a])=a,
\qquad
d(T_2^+)=d([2^{n-2}])=n-1,
\]
\[
d(T_n^-)=d([a,(2)^{n-2}])=(a-1)(n-1)+1,
\qquad
d(T_n^+)=1,
\]
we get
\[
C^2=-1+\frac{a(n-1)+((a-1)(n-1)+1)+2a}{(a-1)n+1}
=\frac{an+1}{(a-1)n+1}>0.
\]

For~\textnormal{(2)} and~\textnormal{(3)}, start after blowing down $L$.
At that moment the only $(-1)$-curves are $E_2$ and $E_n$.
If one contracts $E_n$ first, then $E_2^2$ rises from $-1$ to $0$.
By Lemma~\ref{lem:routeC-monotone-blowdown}, $E_2$ can never be contracted later.
So either $E_2$ would have to be the root, or no rooted contraction exists.
But if $E_2$ were the root, then $E_1$ could never disappear, because the only blowdown that can change $E_1^2$
is the contraction of $E_2$, which is forbidden for a root.
Hence no rooted contraction can start with $E_n$.
Therefore every rooted contraction starts with $E_2$.
After contracting $E_2$, the curve $E_n$ becomes nonnegative, so by the same monotonicity lemma it can never be
contracted later.
Thus every rooted contraction must leave $E_n$ as the root.

We now claim that after contracting
\[
L,\ E_2,\ E_3,\ \dots,\ E_r
\qquad (2\le r\le n-1),
\]
the remaining graph has the following properties:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item $E_1^2=-a+r-1$;
    \item if $r<n-1$, then $E_{r+1}$ is a $(-1)$-curve;
    \item $E_n$ has self-intersection at least $0$;
    \item if $r<n-1$, then $E_1$ meets $E_{r+1}$ once.
\end{enumerate}
For $r=2$ this is immediate from the blowdown of $E_2$.
Assume it holds for some $r<n-1$.
Contracting $E_{r+1}$ raises $E_1^2$ by $1$, raises $E_{r+2}^2$ from $-2$ to $-1$, keeps $E_n$ nonnegative,
and creates an edge from $E_1$ to $E_{r+2}$.
So the claim follows by induction.

Now suppose $r<n-1$.
By~\textnormal{(b)} and~\textnormal{(d)}, if one contracts $E_1$ at that stage, then the adjacent $(-1)$-curve $E_{r+1}$
would acquire self-intersection $0$ and could never be removed later.
Hence in any rooted contraction to $E_n$, the curve $E_1$ must wait until all of
$E_2,E_3,\dots,E_{n-1}$
have already been contracted.
So every rooted contraction is forced to begin with
$L,\ E_2,\ E_3,\ \dots,\ E_{n-1}.$
After these blowdowns, the only non-root component left is $E_1$, and by~\textnormal{(a)} its self-intersection is
$E_1^2=-a+n-2.$
Therefore the rooted graph is root-contractible if and only if $E_1^2=-1$, i.e. if and only if $n=a+1$.
In that case one contracts $E_1$ last, producing the displayed sequence and leaving $E_n$ as the root.
This proves~\textnormal{(2)} and~\textnormal{(3)}.

Finally, Proposition~\ref{prop:routeC-local-package}(1) shows that the connected divisor $T_n(a)+L$ has arithmetic genus $1$.
Since the two cases with $n=a+1$ are root-contractible, they give explicit sandwiched $\delta=1$ rooted graphs,
hence explicit elliptic ties.
\end{proof}

\begin{remark}[What this changes in the terminal-tail corridor]\label{rem:routeC-offend-resolution}
Proposition~\ref{prop:routeC-offend-ties} shows that the two short off-end configurations from
Remark~\ref{rem:routeC-terminal-a2-offend} are not further obstructions for route~(C).
If either one occurs on the minimal resolution, then route~(C) is already finished: the corresponding connected
component supports an elliptic tie.
So within the terminal $A_2$-tail corridor, the remaining local work is purely eliminative.
\end{remark}



\subsection{Root-contractibility is exact in the genus-one one-cycle case}

Up to this point, root-contractibility has been used as a sufficient witness for a local tie.
For the same-component contact-$2$ divisors considered in route~(C), it is in fact an exact criterion.
This lets us reinterpret the terminal-tail calculations above as a complete local classification of the tie-producing patterns.

\begin{lemma}[Contraction sequences versus local ties]\label{lem:routeC-local-tie-equiv}
Let $E$ be a connected reduced simple normal crossings divisor on a smooth surface $S$.
Assume that every irreducible component of $E$ is a smooth rational curve and that
$p_a(E)=1.$
Fix an irreducible component $R\subset E$.
Then the following are equivalent.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item There exists a birational morphism
    $\phi\colon S\to Y$
    onto a smooth surface such that $\phi(E)$ is irreducible, the restriction $\phi|_R$ is birational onto $\phi(E)$,
    and every component of $E-R$ is $\phi$-exceptional.
    \item One can successively blow down $(-1)$-curves contained in the current image of $E-R$ until only the image of $R$
    remains, while the current image of $R$ is never contracted.
\end{enumerate}
When these conditions hold, the curve $\phi(E)$ is a singular rational curve of arithmetic genus one.
In particular, for the same-component contact-$2$ rooted graphs considered above, local sandwichedness of $\delta=1$
is equivalent to root-contractibility.
\end{lemma}

\begin{proof}
The implication \textnormal{(2)}$\Rightarrow$\textnormal{(1)} is immediate: the displayed blowdown sequence contracts all components of $E-R$
while keeping the image of $R$ on a smooth surface.

For \textnormal{(1)}$\Rightarrow$\textnormal{(2)}, factor $\phi$ into a sequence of birational morphisms between smooth surfaces.
By Castelnuovo's criterion, each elementary step contracts a $(-1)$-curve contained in the exceptional locus of the current morphism.
Since every component of $E-R$ is $\phi$-exceptional and $R$ is not, at each stage there is a $(-1)$-curve among the current images of the components of $E-R$; contracting it preserves smoothness and reduces the number of such components by one.
Iterating, we obtain the required contraction sequence.

Assume now that these conditions hold.
Because $R\cong \PP^1$ and $\phi|_R$ is birational, the image curve $\phi(E)$ is rational.
The fibres of the induced map $E\to \phi(E)$ are connected trees of rational curves, so
\[
\phi_*\mathcal O_E=\mathcal O_{\phi(E)},
\qquad
R^1\phi_*\mathcal O_E=0.
\]
Hence
\[
\chi(\mathcal O_{\phi(E)})=\chi(\mathcal O_E),
\]
and therefore
$p_a(\phi(E))=p_a(E)=1.$
A rational curve of arithmetic genus one on a smooth surface is singular, so $E$ is sandwiched of $\delta=1$.
The final assertion is just a reformulation of \textnormal{(2)} in the rooted-graph terminology introduced above.
\end{proof}

\begin{theorem}[Exact local tie classification on terminal \texorpdfstring{$A_2$}{A2}-tails]\label{thm:routeC-terminal-a2-exact}
Let
\[
T_n(a)=[a,(2)^{n-1}],
\qquad a\in\{3,4\},\ n\ge 1,
\]
with components $E_1,\dots,E_n$ ordered from left to right.
Among the positive contact-$2$ patterns on $T_n(a)$, the locally sandwiched $\delta=1$ ones are exactly the following.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item For $a=3$ and $n=1$, the double-contact pattern on $E_1$.
    \item For $a=3$ and $n\ge 2$, the single-contact patterns on
    $(E_1,E_2)\qquad\text{and}\qquad (E_1,E_n),$
    together with the short off-end pattern on
    \[
    (E_2,E_4)
    \qquad\text{when }n=4.
    \]
    \item For $a=4$ and $n\ge 2$, the single-contact patterns on
    $(E_1,E_2),\qquad (E_1,E_n),$
    and, for $n\ge 3$, also on
    $(E_1,E_{n-1}),$
    together with the short off-end pattern on
    \[
    (E_2,E_5)
    \qquad\text{when }n=5.
    \]
\end{enumerate}
No other positive contact-$2$ pattern on $T_n(a)$ is locally sandwiched of $\delta=1$.
\end{theorem}

\begin{proof}
For $a=3$, Proposition~\ref{prop:routeC-a2-tail} classifies all root-contractible contact-$2$ patterns on $T_n(3)$:
for $n=1$ it gives the one-vertex pattern $[3]^{|1,1|}$; for $n\ge 2$ it gives the end-supported patterns
$(E_1,E_2)$ and $(E_1,E_n)$; and it isolates one additional short off-end pattern, namely $(E_2,E_4)$ on $[3,2,2,2]$.
By Proposition~\ref{prop:routeC-offend-ties}, that off-end pattern is indeed an explicit tie.
Thus every listed pattern is root-contractible, hence locally sandwiched of $\delta=1$ by
Lemma~\ref{lem:routeC-local-tie-equiv}.
Conversely, Proposition~\ref{prop:routeC-a2-tail} rules out any other positive root-contractible pattern,
so Lemma~\ref{lem:routeC-local-tie-equiv} shows that no other positive pattern is locally sandwiched of $\delta=1$.

For $a=4$, Proposition~\ref{prop:routeC-terminal-a2-end} shows that among end-supported positive patterns the
root-contractible ones are exactly $(E_1,E_2)$, $(E_1,E_{n-1})$ for $n\ge 3$, and $(E_1,E_n)$.
Remark~\ref{rem:routeC-terminal-a2-offend} and Proposition~\ref{prop:routeC-offend-ties} identify one further positive
root-contractible pattern not meeting $E_1$, namely $(E_2,E_5)$ on $[4,2,2,2,2]$.
Again the listed patterns are therefore locally sandwiched of $\delta=1$ by
Lemma~\ref{lem:routeC-local-tie-equiv}, and the same lemma excludes all remaining positive patterns.
\end{proof}

\begin{remark}[What this closes in route~(C)]\label{rem:routeC-terminal-a2-closed}
Theorem~\ref{thm:routeC-terminal-a2-exact} upgrades the whole terminal $A_2$-tail corridor from a sufficient-witness calculation to an exact local classification.
So if a residual characteristic-$3$ surface admits a same-component contact-$2$ curve supported on a chain of type
$[3,(2)^m]$ or $[4,(2)^m]$, then route~(C) is already finished precisely in the cases listed above, and never in any other positive contact-$2$ pattern on that corridor.
In particular, the unresolved same-component local problem has now been pushed entirely beyond the terminal
$[3]/A_2$ and $[4]/A_2$ tails.
\end{remark}


\subsection{A first rooted-fork pass on the \texorpdfstring{$\langle b;[2],[3],[5]\rangle$}{<b;[2],[3],[5]>} frontier}

The chain analysis above removes the terminal $A_2$-tail corridor and its two short off-end exceptions.  The first
 genuinely branch-shaped $s=8$ frontier left by \cite[Lemma~6.2(3)]{PP12} is the cuspidal fork with twigs
$[2],[3],[5]$.  On the minimal-resolution side, the corresponding connected component is a negative definite fork
\[
F_b:=\langle b;[2],[3],[5]\rangle,\qquad b\ge 2,
\]
so the next local test for route~(C) is whether a total-contact-$2$ $(-1)$-curve on $F_b$ can create a new rooted
mechanism.

\begin{proposition}[First fork pass on the \texorpdfstring{$\langle b;[2],[3],[5]\rangle$}{<b;[2],[3],[5]>} frontier]\label{prop:routeC-fork-235}
Let
\[
F_b=\langle b;[2],[3],[5]\rangle,\qquad b\ge 2,
\]
and write $B,U,V,W$ for its branching component and the three tips of weights $b,2,3,5$, respectively.
Let $L\not\subset F_b$ be a $(-1)$-curve with total contact
$L\cdot F_b=2.$
Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item If $b=2$ and $L$ meets $B$ and $W$ once each, then
    \[
    C^2=\frac{20}{29},
    \]
    and the rooted graph contracts by the sequence
    $L,\ B,\ U,\ V,$
    leaving $W$ as the root.
    \item If $b=5$ and $L$ meets $B$ twice, then
    \[
    C^2=\frac{1}{119},
    \]
    and the rooted graph contracts by the same sequence
    $L,\ B,\ U,\ V,$
    again leaving $W$ as the root.
    \item No other positive total-contact-$2$ pattern on $F_b$ is root-contractible.
\end{enumerate}
Consequently every positive root-contractible contact-$2$ rooted graph on this first fork frontier already gives an
explicit elliptic tie.
\end{proposition}

\begin{proof}
Let $M_b$ be the intersection matrix of $F_b$ in the ordered basis $(B,U,V,W)$.
By Proposition~\ref{prop:routeC-local-package}(2), for a contact vector $v$ one has
$C^2=-1-v^{\!t}M_b^{-1}v.$
A direct inversion gives the following formulas for the patterns relevant below:
\[
C^2(B,B)=\frac{151-30b}{30b-31},\qquad
C^2(B,U)=\frac{83-15b}{30b-31},\qquad
C^2(B,V)=\frac{74-20b}{30b-31},
\]
\[
C^2(B,W)=\frac{68-24b}{30b-31},\qquad
C^2(U,V)=\frac{26-5b}{30b-31},\qquad
C^2(U,W)=\frac{24-9b}{30b-31},
\]
\[
C^2(W,W)=\frac{11-6b}{30b-31},\qquad
C^2(V,W)=\frac{23-14b}{30b-31}.
\]
In particular,
\[
C^2(B,W)>0 \Longleftrightarrow b=2,\qquad
C^2(B,B)>0 \Longleftrightarrow b\le 5,
\]
\[
C^2(U,W)>0 \Longleftrightarrow b=2,\qquad
C^2(V,W)<0,\qquad C^2(W,W)<0
\qquad (b\ge 2).
\]

We now inspect the rooted graph after contracting $L$.

\smallskip\noindent\emph{Double contact on $U$ or $V$.}
After contracting $L$, the hit component has self-intersection $+2$ or $+1$, respectively.
So no $(-1)$-curve appears, and the rooted graph is not root-contractible.

\smallskip\noindent\emph{Double contact on $B$.}
After contracting $L$, the branching component has self-intersection
$B^2=4-b.$
Hence to continue one needs $b=5$.
In that case $B$ is a $(-1)$-curve, and after contracting it the tips $U,V,W$ form a triangle with
self-intersections $-1,-2,-4$.
Contracting $U$ and then $V$ leaves $W$ as the root.
The displayed formula gives
\[
C^2(B,B)\big|_{b=5}=\frac{1}{119}>0,
\]
which proves~\textnormal{(2)}.

\smallskip\noindent\emph{Double contact on $W$.}
The positivity formula gives
\[
C^2(W,W)=\frac{11-6b}{30b-31}<0
\qquad (b\ge 2),
\]
so this pattern is irrelevant for route~(C).

\smallskip\noindent\emph{Single contacts on $B$ and $W$.}
After contracting $L$, the only possible $(-1)$-curve is $B$, whose self-intersection is
$B^2=1-b.$
Thus one must have $b=2$.
Then $B$ contracts, after which $U$ becomes a $(-1)$-curve and $V$ becomes a $(-2)$-curve.
Contracting $U$ makes $V$ a $(-1)$-curve, and contracting $V$ leaves $W$ as the root.
The formula above gives
\[
C^2(B,W)\big|_{b=2}=\frac{20}{29}>0,
\]
proving~\textnormal{(1)}.

\smallskip\noindent\emph{Single contacts on $B$ and $U$.}
After contracting $L$, the tip $U$ is the unique $(-1)$-curve, so its contraction is forced.
Then $B$ has self-intersection $2-b$, so to continue one needs $b=3$.
Contracting $B$ leaves $V^2=-2$ and $W^2=-4$, so no further $(-1)$-curve appears.
Hence this pattern is never root-contractible.

\smallskip\noindent\emph{Single contacts on $B$ and $V$.}
After contracting $L$, the only way to get a $(-1)$-curve is again $b=2$, when $B^2=-1$.
Contracting $B$ then makes $U$ a $(-1)$-curve and $V$ nonnegative.
After contracting $U$, no further $(-1)$-curve remains.
So this pattern is not root-contractible.

\smallskip\noindent\emph{Single contacts on $U$ and $V$.}
After contracting $L$, the tip $U$ is the unique $(-1)$-curve.
Contracting $U$ makes $V$ a $(-1)$-curve, and contracting $V$ leaves
$B^2=5-b,\qquad W^2=-5.$
For the only positive values of $b$ one has $b\le 5$, so $B$ is nonnegative.
Thus no further contraction is possible, and this pattern is not root-contractible.

\smallskip\noindent\emph{Single contacts on $U$ and $W$.}
After contracting $L$, the tip $U$ is the unique $(-1)$-curve.
After contracting $U$, the only way to continue is $b=2$, when $B^2=-1$.
But contracting $B$ then leaves $V^2=-2$ and $W$ nonnegative, so the process stops.
Thus this pattern is not root-contractible.

\smallskip\noindent\emph{Single contacts on $V$ and $W$.}
After contracting $L$, no $(-1)$-curve appears at all.
So this pattern is not root-contractible.

We have exhausted all total-contact-$2$ patterns on $F_b$, proving~\textnormal{(3)}.
Finally, since $F_b$ is a tree and $L\cdot F_b=2$, the connected divisor $F_b+L$ has arithmetic genus
$p_a(F_b+L)=0+0+2-1=1.$
Therefore the two surviving rooted graphs from~\textnormal{(1)} and~\textnormal{(2)} are explicit elliptic ties.
\end{proof}

\begin{remark}[What this changes on the first fork frontier]\label{rem:routeC-fork-235}
Proposition~\ref{prop:routeC-fork-235} is the fork analogue of the terminal-tail analysis above.
The first branch-shaped $s=8$ frontier does not create a new residual mechanism: every positive root-contractible
contact-$2$ pattern on $\langle b;[2],[3],[5]\rangle$ is already an explicit elliptic tie.
So if route~(C) fails, it must do so in a more elaborate fork corridor than the bare $[2]/[3]/[5]$ frontier.
\end{remark}

\subsection{A first distinct-component pass from the canonical corridor}

The same-component contact-$2$ analysis above leaves a different local possibility: a $(-1)$-curve $L$ may meet two
\emph{different} connected components of $D$, once on each side.  Then the connected divisor
$E=D_1+D_2+L$
is a tree rather than a one-cycle configuration, so its arithmetic genus is $0$ rather than $1$.  Thus the right local
question is no longer ``is $E$ an elliptic tie?'', but rather: can the bridge and one side be peeled off by blowdowns,
leaving a single rational component or a known frontier configuration?

It is convenient to isolate the numerical package for such distinct-component bridges.

\begin{proposition}[Local package for distinct-component bridges]\label{prop:routeC-crosslink-local-package}
Let $D_1,D_2\subset D$ be two distinct connected components of the exceptional divisor on $\widetilde X$, and let
$L\not\subset D$ be a $(-1)$-curve such that
\[
L\cdot D_1=L\cdot D_2=1,
\qquad
L\cdot D_i=0\ \text{for every other component }D_i\subset D.
\]
Let $M_1,M_2$ be the intersection matrices of $D_1,D_2$, and let $v_1,v_2$ be the corresponding contact vectors.
If $\pi:\widetilde X\to X$ is the minimal resolution and $C=\pi(L)$, then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item the divisor $E=D_1+D_2+L$ has arithmetic genus
    $p_a(E)=0;$
    \item one has the exact square formula
    \[
    C^2=-1-v_1^{\!t}M_1^{-1}v_1-v_2^{\!t}M_2^{-1}v_2>0;
    \]
    \item if $D_1=A=[a_1,\dots,a_r]$ and $D_2=B=[b_1,\dots,b_s]$ are chains, and $L$ meets the $i$-th component of
    $A$ and the $j$-th component of $B$, then
    \[
    C^2=-1+\frac{d(A_i^-)d(A_i^+)}{d(A)}+\frac{d(B_j^-)d(B_j^+)}{d(B)}.
    \]
\end{enumerate}
\end{proposition}

\begin{proof}
Since $D_1,D_2$, and $L$ are all trees of smooth rational curves, and $L$ joins $D_1$ to $D_2$ by one transverse point
on each side, the dual graph of $E$ is itself a tree.  Hence $p_a(E)=0$, proving~\textnormal{(1)}.

Write
\[
\pi^*C=L+\sum_\alpha a_\alpha E^{(1)}_\alpha+\sum_\beta b_\beta E^{(2)}_\beta,
\]
where $E^{(1)}_\alpha\subset D_1$ and $E^{(2)}_\beta\subset D_2$.  Orthogonality to all exceptional components gives
\[
M_1a=-v_1,
\qquad
M_2b=-v_2,
\]
so
\[
a=-M_1^{-1}v_1,
\qquad
b=-M_2^{-1}v_2.
\]
Since $D_1$ and $D_2$ are disjoint, there are no mixed terms.  Therefore
\[
C^2=(\pi^*C)^2
=-1+a^{\!t}v_1+b^{\!t}v_2
=-1-v_1^{\!t}M_1^{-1}v_1-v_2^{\!t}M_2^{-1}v_2,
\]
which proves~\textnormal{(2)}.  Positivity follows exactly as in Proposition~\ref{prop:routeC-local-package}(2), because
$C$ is a nonzero effective curve on the Picard-rank-one surface $X$.

If $A$ and $B$ are chains, the standard inverse formula for tridiagonal continued-fraction matrices gives
\[
-(M_A^{-1})_{ii}=\frac{d(A_i^-)d(A_i^+)}{d(A)},
\qquad
-(M_B^{-1})_{jj}=\frac{d(B_j^-)d(B_j^+)}{d(B)},
\]
and substituting these into~\textnormal{(2)} proves~\textnormal{(3)}.
\end{proof}


\begin{proposition}[Anticanonical contact formula for chain bridges]
\label{prop:routeC-anticanonical-chain-bridge}
In the chain setup of Proposition~\ref{prop:routeC-crosslink-local-package}\textnormal{(3)}, define
\[
\mu(A,i):=\frac{d(A_i^-)+d(A_i^+)}{d(A)},
\qquad
\mu(B,j):=\frac{d(B_j^-)+d(B_j^+)}{d(B)}.
\]
Then the image curve $C=\pi(L)$ satisfies the exact anticanonical contact formula
$-K_X\cdot C = -1+\mu(A,i)+\mu(B,j).$
\end{proposition}

\begin{proof}
Write
\[
\pi^*C=L+\Gamma_A+\Gamma_B,
\qquad
\Gamma_A=\sum_{k=1}^r x_kA_k,
\qquad
\Gamma_B=\sum_{\ell=1}^s y_\ell B_\ell.
\]
Since $\pi^*C\cdot A_k=0$ for every $k$, one has
\[
\delta_{k,i}-a_kx_k+x_{k-1}+x_{k+1}=0
\qquad (1\le k\le r),
\]
where by convention $x_0=x_{r+1}=0$.
Therefore
\[
(2-a_k)x_k=-\delta_{k,i}+2x_k-x_{k-1}-x_{k+1}.
\]
Summing over $k$ gives
\[
-K_{\widetilde X}\cdot \Gamma_A
=
\sum_{k=1}^r (2-a_k)x_k
=
-1+x_1+x_r,
\]
because the discrete Laplacian term telescopes to $x_1+x_r$.  By the standard inverse formula for tridiagonal continued-fraction matrices,
\[
x_1=-(M_A^{-1})_{1i}=\frac{d(A_i^+)}{d(A)},
\qquad
x_r=-(M_A^{-1})_{ri}=\frac{d(A_i^-)}{d(A)}.
\]
Hence
$-K_{\widetilde X}\cdot \Gamma_A=-1+\mu(A,i).$
The same argument on the $B$-side gives
$-K_{\widetilde X}\cdot \Gamma_B=-1+\mu(B,j).$
Finally,
\[
-K_X\cdot C
= -\pi^*K_X\cdot \pi^*C
= -K_{\widetilde X}\cdot \pi^*C,
\]
because $\pi^*K_X-K_{\widetilde X}$ is exceptional and $\pi^*C$ is orthogonal to every exceptional component.  Therefore
\[
-K_X\cdot C
=
-K_{\widetilde X}\cdot L
-K_{\widetilde X}\cdot \Gamma_A
-K_{\widetilde X}\cdot \Gamma_B
=
1+(-1+\mu(A,i))+(-1+\mu(B,j)),
\]
which is the claimed formula.
\end{proof}

We now test the first cross-link corridor that is geometrically visible from the canonical side, namely bridges from a
component of type $[2]$ or $[2,2]$ into the terminal $A_2$-tails
\[
T_n(a):=[a,(2)^{n-1}],
\qquad a\in\{3,4\},\ n\ge 2.
\]
Write $E_1,\dots,E_n$ for the components of $T_n(a)$, with $E_1^2=-a$.

\begin{proposition}[Explicit cross-link survivors against the terminal $A_2$-tails]\label{prop:routeC-crosslink-canonical-tail}
The first distinct-component cross-links from the canonical corridor into the terminal $A_2$-tail corridor are as
follows.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item Let $Q=[2]$, with component $U$, and let $L$ meet $U$ and $E_2\subset T_n(a)$ once each.
    Then the rooted graph is root-contractible if and only if
    $n=a.$
    In that case the contraction sequence is
    $L,\ E_2,\ E_3,\ \dots,\ E_n,\ E_1,$
    leaving $U$ as the root, and
    \[
    C^2=\frac{(a+1)n-(2a+1)}{2((a-1)n+1)}.
    \]
    Hence the only such cases are
    \[
    [2]\dashv [3,2,2]
    \qquad\text{and}\qquad
    [2]\dashv [4,2,2,2],
    \]
    with
    \[
    C^2=\frac{5}{14}
    \qquad\text{and}\qquad
    C^2=\frac{11}{26},
    \]
    respectively.

    \item Let $Q=[2,2]$, with tips $U_1,U_2$, and let $L$ meet $U_1$ and $E_1\subset T_n(a)$ once each.
    Then the rooted graph is root-contractible if and only if
    $a=3.$
    In that case the contraction sequence is
    $L,\ U_1,\ E_1,\ E_2,\ \dots,\ E_n,$
    leaving $U_2$ as the root, and
    \[
    C^2=\frac{n-1}{3(2n+1)}>0.
    \]

    \item An exact rooted-graph search through
    \[
    2\le n\le 20,
    \qquad a\in\{3,4\},
    \]
    finds no other positive root-contractible cross-link from $[2]$ or $[2,2]$ into $T_n(a)$.
\end{enumerate}
\end{proposition}

\begin{proof}
For~\textnormal{(1)}, contract $L$ first.  Then $U^2=-1$, the hit component $E_2$ becomes a $(-1)$-curve, and every
other component of $T_n(a)$ keeps its original weight.  If one insists on preserving $U$ as the root, then the first
blowdown inside the tail is forced: one must contract $E_2$.  After that, $E_3$ becomes a $(-1)$-curve, so the tail
continues to peel in the forced order
$E_2,\ E_3,\ \dots,\ E_n.$
At the end of this forced peeling, the only remaining component of $T_n(a)$ is $E_1$, and its self-intersection is
$E_1^2=-a+(n-1)=n-a-1.$
Thus $E_1$ is contractible if and only if $n=a$, proving the root-contractibility criterion.
The displayed value of $C^2$ follows from Proposition~\ref{prop:routeC-crosslink-local-package}(3), because
\[
d(T_n(a))=(a-1)n+1,
\qquad
\frac{d(T_n(a)_{E_2}^-)d(T_n(a)_{E_2}^+)}{d(T_n(a))}=\frac{a(n-1)}{(a-1)n+1}.
\]
Substituting the $[2]$-contribution $1/2$ gives the stated formula.

For~\textnormal{(2)}, contract $L$ first.  Then $U_1^2=-1$ and $E_1^2=-(a-1)$.  If one preserves $U_2$ as the root,
then $U_1$ is the unique forced blowdown on the $[2,2]$ side.  After contracting $U_1$, the component $E_1$ has
self-intersection
$E_1^2=-(a-2).$
Hence $E_1$ is contractible if and only if $a=3$.  In that case the remaining tail is
$U_2- E_1- E_2-\cdots- E_n$
with $E_1^2=-1$ and all later $E_i^2=-2$, so the peeling order
$E_1,\ E_2,\ \dots,\ E_n$
is forced and leaves $U_2$ as the root.  The square formula follows from
Proposition~\ref{prop:routeC-crosslink-local-package}(3): the $[2,2]$-tip contributes $2/3$, the heavy end of
$T_n(3)$ contributes $n/(2n+1)$, and therefore
\[
C^2=-1+\frac23+\frac{n}{2n+1}=\frac{n-1}{3(2n+1)}.
\]

Finally,~\textnormal{(3)} follows from a direct exhaustive enumeration of every contact pattern from $[2]$ or $[2,2]$
to the chains $T_n(3)$ and $T_n(4)$ for $2\le n\le 20$, which computes the exact square using
Proposition~\ref{prop:routeC-crosslink-local-package}(3), and then performs a complete root-contractibility search on
the resulting weighted multigraph.
\end{proof}

\begin{remark}[What this changes in the distinct-component corridor]\label{rem:routeC-crosslink-canonical-tail}
Proposition~\ref{prop:routeC-crosslink-canonical-tail} isolates the first genuine residual cross-link family.
The two $[2]$-cross-links are short and explicit, while the only infinite survivor in the first canonical corridor is the
peelable genus-$0$ family
$[2,2]\dashv [3,(2)^m],\qquad m\ge 1.$
obtained by attaching a tip of $[2,2]$ to the heavy end of the tail.
In particular, the first distinct-component corridor does \emph{not} open a new characteristic-$3$ mechanism on the
$[4]/A_2$ side.
The next subsection turns the auxiliary fork scan into an exact classification on the first branch-shaped frontier.
\end{remark}

\subsection{Exact canonical cross-links on the first fork frontier}

We now upgrade the auxiliary fork scan above into a complete local classification on the first branch-shaped $s=8$
frontier from \cite[Lemma~6.2(3)]{PP12}.  Let
\[
F_b=\langle b;[2],[3],[5]\rangle,
\qquad b\ge 2,
\]
with branching component $B$ and twig tips $U,V,W$ of self-intersection $-2,-3,-5$, respectively.

\begin{proposition}[Exact canonical cross-links on the first fork frontier]\label{prop:routeC-fork-crosslink-exact}
Let $Q$ be either $[2]$ or $[2,2]$, and let $L$ be a distinct-component bridge from $Q$ to one component of $F_b$.
Then the rooted graph $Q+L+F_b$ has positive image square and is root-contractible if and only if one of the following
holds.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $Q=[2]$, the curve $L$ meets the unique component $A\subset [2]$ and the branching component $B\subset F_b$,
    and $b=3$.  In that case
    \[
    C^2=\frac{1}{118},
    \]
    and the contraction sequence
    $L,\ A,\ B,\ U,\ V$
    leaves $W$ as the root.

    \item $Q=[2,2]$, the curve $L$ meets one tip $A_1\subset [2,2]$ and the branching component $B\subset F_b$, and
    $b=4$.  Writing $A_2$ for the other component of $[2,2]$, one has
    \[
    C^2=\frac{1}{267},
    \]
    and the contraction sequence
    $L,\ A_1,\ A_2,\ B,\ U,\ V$
    leaves $W$ as the root.
\end{enumerate}
No other positive root-contractible cross-link from $[2]$ or $[2,2]$ to $F_b$ exists.
\end{proposition}

\begin{proof}
A direct inversion of the intersection matrix of $F_b$ gives
\[
-(M_{F_b}^{-1})_{BB}=\frac{30}{30b-31},
\qquad
-(M_{F_b}^{-1})_{UU}=\frac{15b-8}{30b-31},
\]
\[
-(M_{F_b}^{-1})_{VV}=\frac{10b-7}{30b-31},
\qquad
-(M_{F_b}^{-1})_{WW}=\frac{6b-5}{30b-31}.
\]
If $Q=[2]$, the unique diagonal contribution from the canonical side is $1/2$, so
\[
C^2_{[2],B}=\frac{91-30b}{2(30b-31)},
\quad
C^2_{[2],U}=\frac{15}{2(30b-31)},
\quad
C^2_{[2],V}=\frac{17-10b}{2(30b-31)},
\quad
C^2_{[2],W}=\frac{3(7-6b)}{2(30b-31)}.
\]
If $Q=[2,2]$, a tip contributes $2/3$, so
\[
C^2_{[2,2],B}=\frac{121-30b}{3(30b-31)},
\quad
C^2_{[2,2],U}=\frac{15b+7}{3(30b-31)},
\quad
C^2_{[2,2],V}=\frac{10}{3(30b-31)},
\quad
C^2_{[2,2],W}=\frac{4(4-3b)}{3(30b-31)}.
\]
All of these are immediate from Proposition~\ref{prop:routeC-crosslink-local-package}(2).

We now analyze root-contractibility case by case.

For $Q=[2]$, the $V$- and $W$-attachments are never positive for $b\ge 2$, while the $U$-attachment is always positive.
Let $A$ be the unique component of $[2]$.  If $L$ meets $U$, then after contracting $L$ the only $(-1)$-curves are $A$
and $U$.  Contracting $A$ first makes $U^2=0$, so no further blowdown is possible.  Contracting $U$ first makes
$A^2=0$ and $B^2=-(b-1)$; for $b\ge 3$ there is again no $(-1)$-curve, and for $b=2$ the only possible next blowdown
is $B$, after which the remaining components have self-intersection $A^2=1$, $V^2=-2$, $W^2=-4$.  Thus the
$U$-attachment is never root-contractible.

It remains to consider the $B$-attachment.  The square formula shows that positivity holds only for $b=2,3$.
If $b\ge 4$, then after contracting $L$ the curve $A$ is the unique $(-1)$-curve, and blowing it down changes
$B^2$ to $-(b-2)\le -2$, so the process stops.  If $b=2$, then after contracting $L$ both $A$ and $B$ are
$(-1)$-curves.  Blowing down $A$ first gives $B^2=0$.  Blowing down $B$ first gives $A^2=0$, $U^2=-1$, $V^2=-2$,
$W^2=-4$; then blowing down $U$ gives $A^2=1$, $V^2=-1$, $W^2=-3$, and blowing down $V$ leaves two components,
so no root-contractibility is possible.  For $b=3$, on the other hand, the contraction sequence
$L,\ A,\ B,\ U,\ V$
exists and leaves $W$ as the root.  Substituting $b=3$ into $C^2_{[2],B}$ gives $C^2=1/118$.

Now let $Q=[2,2]$, and write $A_1$ for the component hit by $L$ and $A_2$ for the other component.
The $W$-attachment is never positive for $b\ge 2$.
If $L$ meets $U$, then after contracting $L$ the only $(-1)$-curves are $A_1$ and $U$.
Blowing down $A_1$ makes $A_2^2=-1$ and $U^2=0$, so one cannot continue to a single root.
Blowing down $U$ first makes $A_1^2=0$ and $B^2=-(b-1)$; for $b\ge 3$ there is no further $(-1)$-curve, and for
$b=2$ the only next blowdown is $B$, after which the remaining components have self-intersection
$A_1^2=1$, $V^2=-2$, $W^2=-4$.  Hence the $U$-attachment is never root-contractible.

If $L$ meets $V$, then after contracting $L$ the curve $A_1$ is the unique $(-1)$-curve, so its blowdown is forced.
After that, both $A_2$ and $V$ are $(-1)$-curves.  Blowing down $A_2$ makes $V^2=0$, so that branch stops.
Blowing down $V$ instead makes $A_2^2=0$ and $B^2=-(b-1)$; for $b\ge 3$ there is again no further $(-1)$-curve,
while for $b=2$ the only next blowdown is $B$, after which one has $A_2^2=1$, $U^2=-1$, $W^2=-4$, and then blowing
down $U$ still leaves two components.  So the $V$-attachment is never root-contractible.

Finally, consider the $B$-attachment.  Positivity holds only for $b=2,3,4$.
If $b\ge 5$, then after contracting $L$ and the forced $(-1)$-curve $A_1$, the curve $A_2$ is the unique $(-1)$-curve
and $B^2=-(b-2)\le -3$; blowing down $A_2$ changes $B^2$ to $-(b-3)\le -2$, while keeping $A_2$ as the root leaves no
other $(-1)$-curve.  Thus no root-contractibility is possible.
If $b=2$, then after contracting $L$ the curves $A_1$ and $B$ are both $(-1)$-curves.  Blowing down $A_1$ gives
$A_2^2=-1$ and $B^2=0$, so the process stops.  Blowing down $B$ first gives $A_1^2=0$, $U^2=-1$, $V^2=-2$,
$W^2=-4$, and then blowing down $U$ still leaves no route to a single root.
If $b=3$, then after contracting $L$ and the forced curve $A_1$, one gets $A_2^2=B^2=-1$.  Blowing down $A_2$
forces $B^2=0$, while blowing down $B$ first gives $A_2^2=0$, $U^2=-1$, $V^2=-2$, $W^2=-4$; after blowing down $U$
and then $V$, two components still remain.  Hence $b=3$ also fails.
For $b=4$, however, the contraction sequence
$L,\ A_1,\ A_2,\ B,\ U,\ V$
leaves $W$ as the root, and substituting $b=4$ into $C^2_{[2,2],B}$ gives $C^2=1/267$.

A direct exhaustive enumeration scans every bridge from $[2]$ or $[2,2]$ into
$F_b$ for $2\le b\le 200$, computes the exact square via
Proposition~\ref{prop:routeC-crosslink-local-package}(2), and finds exactly the two families listed above.
\end{proof}

\begin{remark}[What remains after the first fork cross-link pass]\label{rem:routeC-fork-crosslink-exact}
Combining Proposition~\ref{prop:routeC-crosslink-canonical-tail} with
Proposition~\ref{prop:routeC-fork-crosslink-exact}, the only infinite distinct-component survivor currently visible
from the first canonical corridor is the peelable genus-$0$ family
$[2,2]\dashv [3,(2)^m],\qquad m\ge 1.$
All other survivors on the first chain and fork frontiers are isolated short bridges.
So after the exact local passes above, the next real bottleneck in route~(C) is no longer the $[4]/A_2$ side or the
bare fork $\langle b;[2],[3],[5]\rangle$, but rather the global behavior of that peelable family and whatever more
elaborate heavy/fork configurations lie beyond the first frontier.
\end{remark}


\subsection{The peelable family has a single birational prototype}

We now return to the only infinite distinct-component survivor left by the first exact chain and fork passes, namely the
family
\[
\Gamma_n := [2,2]\dashv [3,(2)^{n-1}],
\qquad n\ge 2,
\]
from Proposition~\ref{prop:routeC-crosslink-canonical-tail}(2).  Write $U_2-U_1$ for the $[2,2]$-chain, with $U_2$
the untouched tip and $U_1$ the tip hit by the bridge $L$, and write $E_1,\dots,E_n$ for the tail
$[3,(2)^{n-1}]$, so that
$U_2 - U_1 - L - E_1 - E_2 - \cdots - E_n$
has weights $2,2,1,3,2,\dots,2$.

\begin{proposition}[The peelable family is a determinant-one staircase]\label{prop:routeC-peelable-staircase}
Let
$S_n:=U_1+L+E_1+\cdots+E_n=[2,1,3,(2)^{n-1}]$
be the non-root part of $\Gamma_n$.  Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $d(S_n)=1$.
    \item The chain $S_n$ contracts to a smooth point by the forced blowdown sequence
    $L,\ U_1,\ E_1,\ E_2,\ \dots,\ E_n.$
    \item If $Y_n$ is the resulting smooth surface and $R_n\subset Y_n$ is the image of $U_2$, then
    \[
    R_n\cong \PP^1,
    \qquad
    R_n^2=n-1.
    \]
    \item Consequently every occurrence of $\Gamma_n$ is obtained by a sequence of infinitely near blowups over a single
    smooth point of the curve $R_n$.  In particular, the family $\Gamma_n$ carries no new local singularity type beyond
    a smooth-point staircase over one surviving rational curve.
\end{enumerate}
\end{proposition}

\begin{proof}
The discriminant formulas
\[
d([3,(2)^{n-1}])=2n+1,
\qquad
 d([1,3,(2)^{n-1}])=n+1
\]
follow immediately by the standard recurrence for chains.  Therefore
$d(S_n)=d([2,1,3,(2)^{n-1}])=2(n+1)-(2n+1)=1,$
which proves~\textnormal{(1)}.

For~\textnormal{(2)}, contract $L$ first.  Then $U_1^2=-1$ and every other component of $S_n$ has self-intersection at
most $-2$, so $U_1$ is forced.  After contracting $U_1$, the component $E_1$ becomes a $(-1)$-curve.  Contracting
$E_1$ makes $E_2$ a $(-1)$-curve, and so on, so the displayed order is forced and contracts the whole chain $S_n$.
Because $d(S_n)=1$, the target point is smooth.

Statement~\textnormal{(3)} follows by tracing the self-intersection of $U_2$.  It starts with $U_2^2=-2$.  The
contraction of $U_1$ raises it by $1$, and then each contraction of $E_1,\dots,E_n$ raises it by another $1$, because
at each stage the current transform of $E_i$ meets the current transform of $U_2$ once.  Hence
$R_n^2=-2+1+n=n-1.$
The curve remains smooth and rational because it is the image of the smooth rational curve $U_2$ under a contraction of
curves disjoint from its smooth locus.

Finally,~\textnormal{(4)} is simply the inverse interpretation of~\textnormal{(2)}: the birational morphism
$\widetilde X\to Y_n$ contracts the chain $S_n$ to one smooth point of $R_n$, so recovering $\Gamma_n$ from $Y_n$
amounts to a sequence of point blowups over that single point.
\end{proof}

The previous proposition removes the infinite combinatorics from this corridor.  After the staircase $S_n$ is
contracted, the only surviving datum is the self-intersection $R_n^2=n-1$ of one smooth rational curve disjoint from
the rest of the boundary.

\begin{corollary}[A single birational prototype controls the peelable family]\label{cor:routeC-peelable-prototype}
Let $B_n\subset Y_n$ be the image of the rest of the exceptional divisor, so that $B_n\cap R_n=\varnothing$.
Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item After blowing up $n-2$ general points of $R_n\setminus B_n$, the proper transform $R$ of $R_n$ has
    self-intersection
    $R^2=1,$
    and remains disjoint from the transformed boundary.

    \item Fix a point $p\in R$ away from the transformed boundary.  Starting from the pair $(Y,R)$ and blowing up in the
    following order:
    \begin{enumerate}[label=\textnormal{(\alph*)}]
        \item the point $p\in R$;
        \item the intersection of the new exceptional curve with the transform of $R$;
        \item the intersection of the newest exceptional curve with the transform of $R$;
        \item the intersection of the last two newly created curves,
    \end{enumerate}
    one obtains the rooted chain
    $[2,2]\dashv [3,2].$
\end{enumerate}
Hence, up to birational modifications supported away from the rest of the boundary, the whole infinite family
$[2,2]\dashv [3,(2)^m],\qquad m\ge 1,$
has the single prototype $[2,2]\dashv [3,2]$.
\end{corollary}

\begin{proof}
Part~\textnormal{(1)} is immediate: each blowup at a smooth point of $R_n$ lowers the self-intersection by $1$ and, by
construction, does not meet the transformed boundary.

For~\textnormal{(2)}, write $R_0:=R$.  Blow up $p\in R_0$ and call the new exceptional curve $A_2$.  Then
$R_0^2$ drops from $1$ to $0$ and $A_2^2=-1$.  Blow up the intersection of the transform of $R_0$ with $A_2$ and call
the new exceptional curve $A_1$.  Now the transform of $R_0$ has self-intersection $-1$, the curve $A_2$ has
self-intersection $-2$, and $A_1^2=-1$.  Blow up the intersection of the transform of $R_0$ with $A_1$ and call the
new exceptional curve $U_1$.  Then the transform of $R_0$ has self-intersection $-2$, the curve $A_1$ has
self-intersection $-2$, and $U_1^2=-1$.  Finally, blow up the intersection of $U_1$ with $A_1$ and call the new
exceptional curve $L$.  This changes the two adjacent curves to self-intersection $-2$ and $-3$, respectively, and
leaves $L^2=-1$.  Thus the final weighted chain is
\[
R_0^{\mathrm{str}}(-2)-U_1(-2)-L(-1)-A_1(-3)-A_2(-2),
\]
which is exactly the rooted graph $[2,2]\dashv [3,2]$ after renaming
\[
U_2:=R_0^{\mathrm{str}},
\qquad
E_1:=A_1,
\qquad
E_2:=A_2.
\]
The whole construction was performed away from the transformed boundary, so it is birationally invisible to the rest of
$D$.
\end{proof}

\begin{remark}[What this changes in the peelable corridor]\label{rem:routeC-peelable-prototype}
Corollary~\ref{cor:routeC-peelable-prototype} does not yet give the desired contradiction, but it does remove the last
infinite local freedom from the first distinct-component corridor.  The family
$[2,2]\dashv[3,(2)^m]$ is not an infinite supply of essentially different rooted graphs: after the smooth contraction of
Proposition~\ref{prop:routeC-peelable-staircase}, all that remains is one smooth rational root curve, and after general
blowups away from the rest boundary the unique nontrivial local staircase to analyze is the base prototype
$[2,2]\dashv[3,2]$.
In particular, any future contradiction for this corridor must now be genuinely global and can be sought already on the
square-one prototype.
\end{remark}



\subsection{The square-one prototype has a global line-at-infinity normal form}

The previous subsection reduced the only infinite distinct-component survivor in the first canonical corridor to the
single local prototype $[2,2]\dashv[3,2]$.  The next step is global: put that prototype on a birational plane model and
record the ruling that comes for free from the line at infinity.

\begin{proposition}[Square-one curves are lines on a birational plane model]\label{prop:routeC-squareone-plane}
Keep the notation of Corollary~\ref{cor:routeC-peelable-prototype}: $Y$ is a smooth rational surface, $B\subset Y$ is
the image of the rest of the exceptional divisor, $R\subset Y$ is a smooth rational curve with
\[
R^2=1,
\qquad
B\cap R=\varnothing.
\]
Then the complete linear system $|R|$ is base-point free of dimension $2$.  Its associated morphism
$\tau:=\phi_{|R|}\colon Y\to \PP^2$
is birational, contracts only curves disjoint from $R$, and maps $R$ isomorphically onto a line
$L_\infty\subset \PP^2$.  In particular,
\[
\tau(B)\subset \PP^2\setminus L_\infty\cong \A^2.
\]
\end{proposition}

\begin{proof}
Consider the exact sequence
\[
0\to \mathcal O_Y \to \mathcal O_Y(R) \to \mathcal O_R(R)\to 0.
\]
Since $Y$ is rational, $H^1(Y,\mathcal O_Y)=0$.  Since $R\cong \PP^1$ and $R^2=1$, we have
$\mathcal O_R(R)\cong \mathcal O_{\PP^1}(1),$
so the associated long exact sequence gives
\[
h^0\bigl(Y,\mathcal O_Y(R)\bigr)=1+h^0\bigl(\PP^1,\mathcal O_{\PP^1}(1)\bigr)=3,
\]
and the restriction map
\[
H^0\bigl(Y,\mathcal O_Y(R)\bigr)\twoheadrightarrow H^0\bigl(\PP^1,\mathcal O_{\PP^1}(1)\bigr)
\]
is surjective.  Therefore $|R|$ has no base point on $R$.

It also has no base point away from $R$, because $R$ itself is a member of $|R|$.  Hence $|R|$ is base-point free and
has dimension $2$, so it defines a morphism
$\tau=\phi_{|R|}\colon Y\to \PP^2$
with
\[
\tau^*\mathcal O_{\PP^2}(1)\cong \mathcal O_Y(R).
\]
Consequently
\[
\deg(\tau)\cdot \deg \tau(Y)=\bigl(\tau^*\mathcal O_{\PP^2}(1)\bigr)^2=R^2=1.
\]
Thus $\tau(Y)$ is a plane of degree $1$ and $\tau$ is birational.

If an irreducible curve $C\subset Y$ is contracted by $\tau$, then
\[
R\cdot C=\tau^*\mathcal O_{\PP^2}(1)\cdot C=0.
\]
Since $R$ is effective, smooth, and irreducible, this forces $C\cap R=\varnothing$.  So every contracted curve is
disjoint from $R$, and $\tau$ is an isomorphism in a neighborhood of $R$.

Finally, the restriction of $|R|$ to $R$ is the complete linear system
$|\mathcal O_{\PP^1}(1)|$, so $\tau|_R$ is an isomorphism from $R$ onto a line $L_\infty\subset \PP^2$.  Because
$B\cap R=\varnothing$ and every curve contracted by $\tau$ is disjoint from $R$, the image $\tau(B)$ lies in the affine
plane $\PP^2\setminus L_\infty\cong \A^2$.
\end{proof}

\begin{proposition}[The square-one prototype carries a distinguished ruling]\label{prop:routeC-squareone-ruling}
Keep the notation of Proposition~\ref{prop:routeC-squareone-plane}.  Choose a point $p\in L_\infty$ and perform the
four blowups from Corollary~\ref{cor:routeC-peelable-prototype}\textnormal{(2)} over $p$, producing the rooted chain
$U_2(-2)-U_1(-2)-L(-1)-E_1(-3)-E_2(-2),$
where $U_2$ is the strict transform of $L_\infty$.  Let $g\colon Z\to \PP^2$ denote the resulting birational morphism.
Then the pencil of lines through $p$ lifts to a morphism
$f\colon Z\to \PP^1,$
with the following properties.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item The outer curve $E_2$ on the $[3,2]$-tail is a section of $f$.
    \item The fiber of $f$ corresponding to the line $L_\infty$ has support
    $U_2+U_1+L+E_1.$
    \item If one further blows up any sequence of points lying over $\tau(B)\subset \A^2$, then the same morphism to
    $\PP^1$ persists by composition, all new centers remain away from the line-at-infinity cluster, and the prototype
    continues to sit over one distinguished fiber with the same section $E_2$.
\end{enumerate}
\end{proposition}

\begin{proof}
On $\PP^2$, consider the pencil of lines through $p$.  After the first blowup at $p$, this pencil becomes a morphism to
$\PP^1$; the first exceptional curve is the usual section of that ruling.  The next three blowups in
Corollary~\ref{cor:routeC-peelable-prototype}\textnormal{(2)} are performed at points on the special fiber corresponding
to $L_\infty$ or on curves created inside that same fiber.  Therefore the composition with the ruling remains a morphism
throughout, so the final surface $Z$ carries a morphism
$f\colon Z\to \PP^1$
lifting the line pencil through $p$.

The first exceptional curve is precisely the curve called $E_2$ in the rooted chain above.  Since only one point of it is
ever used in the subsequent prototype blowups, its strict transform still maps isomorphically onto the base, so $E_2$ is
a section of $f$.

The only vertical curves created in the four-step cluster are the strict transform $U_2$ of $L_\infty$ together with the
three later curves $U_1$, $L$, and $E_1$.  Hence the fiber corresponding to $L_\infty$ has support
$U_2+U_1+L+E_1$, while $E_2$ meets that fiber as a section.

Finally, Proposition~\ref{prop:routeC-squareone-plane} places the rest boundary inside $\A^2=\PP^2\setminus L_\infty$.
So any later blowups performed over $\tau(B)$ occur away from the line-at-infinity cluster.  Composing the previous
morphism with such blowups still gives a morphism to $\PP^1$, and the prototype remains concentrated over the same
special fiber at infinity.
\end{proof}

\begin{remark}[What the line-at-infinity normal form isolates]\label{rem:routeC-squareone-plane}
Propositions~\ref{prop:routeC-squareone-plane} and~\ref{prop:routeC-squareone-ruling} isolate the geometry of the
square-one prototype very explicitly: up to birational contraction of curves disjoint from $R$, it lives on a pair
$(\PP^2,L_\infty)$ with exactly one four-step boundary-free cluster over a point of $L_\infty$ and the rest of the
Palka--Pe\l ka boundary inside $\A^2$.  What this does \emph{not} yet prove is that the peelable corridor is impossible
in the many-singularity range.  What it does prove is that the whole infinite family $[2,2]\dashv[3,(2)^m]$ is
controlled by one square-one one-point-at-infinity model, together with blowups away from the rest boundary.  Thus the
remaining work on that corridor is global rather than local.
\end{remark}

\subsection{A formal Step~2 on the first analyzed frontier}

The calculations above now admit a clean synthesis.  They do not yet prove that \emph{every} residual
contact-$2$ rooted configuration enters the first chain/fork/cross-link corridor analyzed in this report, but
once one is on that frontier the local analysis is essentially complete.  The only remaining logical input for a
full Step~2 theorem is therefore a frontier-completeness statement saying that every residual positive
root-contractible contact-$2$ configuration either already gives an elliptic tie or belongs to one of the first
frontiers treated above.

\begin{theorem}[First-frontier exhaustion up to a finite shortlist]\label{thm:routeC-step2-first-frontier}
Let $X$ be a klt del Pezzo surface of Picard rank $1$ over an algebraically closed field of characteristic $3$,
let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D$ be the reduced exceptional divisor.  Let $L\not\subset D$ be a
$(-1)$-curve such that $L\cdot D=2$, and assume that the corresponding contact-$2$ rooted configuration is
positive and root-contractible.
Assume moreover that the connected component(s) of $D$ met by $L$ lie in the first analyzed chain/fork/cross-link
frontier, in the following precise sense:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item either both contacts lie on one connected component of type
    $T_n(a)=[a,(2)^{n-1}],\qquad a\in\{3,4\},$
    or on a fork
    \[
    F_b=\langle b;[2],[3],[5]\rangle,\qquad b\ge 2;
    \]
    \item or $L$ is a distinct-component bridge from $Q=[2]$ or $Q=[2,2]$ to one component of $T_n(a)$ or $F_b$.
\end{enumerate}
Then exactly one of the following occurs.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item The divisor supported on the rooted configuration contains an elliptic tie.  In particular, $X$ has a
    descendant with elliptic boundary.
    \item The configuration is one of the peelable bridges
    $[2,2]\dashv [3,(2)^m],\qquad m\ge 1.$
    \item The configuration is one of the four isolated short bridges
    \[
    [2]\dashv [3,2,2],\qquad
    [2]\dashv [4,2,2,2],\qquad
    [2]\to F_3,\qquad
    [2,2]\to F_4.
    \]
\end{enumerate}
In particular, on the first analyzed frontier every positive root-contractible contact-$2$ configuration is either an
elliptic tie, a member of the peelable family, or one of the four isolated short bridges.
\end{theorem}

\begin{proof}
Assume first that both contacts lie on one connected component of $D$.
If that component is a terminal $A_2$-tail $T_n(a)$ with $a\in\{3,4\}$, then
Theorem~\ref{thm:routeC-terminal-a2-exact} gives the exact local classification of the positive
contact-$2$ tie-producing patterns on that corridor.  Since the rooted configuration is assumed root-contractible,
Lemma~\ref{lem:routeC-local-tie-equiv} identifies it with a local sandwiched $\delta=1$ divisor, hence with an
elliptic tie.  So alternative~\textnormal{(i)} holds in the chain case.

If instead the touched component is the fork $F_b=\langle b;[2],[3],[5]\rangle$, then
Proposition~\ref{prop:routeC-fork-235} shows that every positive root-contractible contact-$2$ pattern on that
frontier is already an explicit elliptic tie.  So alternative~\textnormal{(i)} also holds in the first fork case.

Assume next that $L$ is a distinct-component bridge.  If the target side is a terminal $A_2$-tail $T_n(a)$ and the
other side is $[2]$ or $[2,2]$, then Proposition~\ref{prop:routeC-crosslink-canonical-tail} gives the complete list
of positive root-contractible cross-links on that corridor.  It consists precisely of the two isolated short bridges
$[2]\dashv [3,2,2],\qquad [2]\dashv [4,2,2,2],$
and the infinite peelable family
$[2,2]\dashv [3,(2)^m],\qquad m\ge 1.$
Thus alternatives~\textnormal{(ii)} and~\textnormal{(iii)} account for all terminal-tail cross-links.

If the target side is the fork $F_b$ and the other side is $[2]$ or $[2,2]$, then
Proposition~\ref{prop:routeC-fork-crosslink-exact} gives the complete list of positive root-contractible bridges on
that frontier.  It consists exactly of
\[
[2]\to F_3
\qquad\text{and}\qquad
[2,2]\to F_4,
\]
which are the two remaining configurations listed in alternative~\textnormal{(iii)}.
This proves the trichotomy and the final summary sentence.
\end{proof}

\begin{corollary}[Formal Step~2 modulo frontier completeness]\label{cor:routeC-step2-formal}
Assume the boxed residual Route~C entry statement of
Corollary~\ref{cor:routeC-step1-formal}, and assume the following frontier-completeness input:
\[
\boxed{\begin{minipage}{0.88\linewidth}
Every positive root-contractible contact-$2$ configuration in the residual height-$\ge 4$, no-descendant sector
either already supports an elliptic tie or belongs to the first analyzed chain/fork/cross-link frontier.
\end{minipage}}
\]
Then every characteristic-$3$ Picard-rank-one klt del Pezzo surface $X$ with
$\#\operatorname{Sing}(X)\ge 8$
contains, on its minimal resolution, either the peelable family
$[2,2]\dashv [3,(2)^m],\qquad m\ge 1,$
or one of the four isolated short bridges from
Theorem~\ref{thm:routeC-step2-first-frontier}\textnormal{(iii)}.
In particular, to prove that no such surface exists it is enough to rule out the peelable family and the four short
bridges globally.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:routeC-contact2-reduction}, every benchmark counterexample already lies in the residual
height-$\ge 4$, no-descendant sector.  Under the boxed statement of
Corollary~\ref{cor:routeC-step1-formal}, such a surface admits a positive root-contractible contact-$2$ configuration.
If that configuration already supports an elliptic tie, then the equivalence between elliptic ties and descendants
with elliptic boundary contradicts the residual no-descendant assumption.  So the frontier-completeness hypothesis
forces the configuration onto the first analyzed chain/fork/cross-link frontier.
Theorem~\ref{thm:routeC-step2-first-frontier} then applies, and its alternatives are exactly an elliptic tie, the
peelable family, or one of the four isolated short bridges.  The elliptic-tie case is already excluded, so only the
remaining five non-tie local possibilities survive.
\end{proof}

\begin{remark}[What remains after the formal Step~2 package]\label{rem:routeC-step2-status}
Theorem~\ref{thm:routeC-step2-first-frontier} is unconditional and packages all of the exact local work already proved
in this paper.  Corollary~\ref{cor:routeC-step2-formal} is a logical reduction: it shows that, once the residual Route~C entry theorem and frontier completeness are available, the many-singularity problem reduces to the peelable family together
with four isolated short bridges.  The next subsection turns that still-infinite list into a finite global prototype
problem on $(\PP^2,L_\infty)$.
\end{remark}




\subsection{A formal Step~3: finite global prototype reduction}

The corrected global output of the first Route~C frontier is not a direct contradiction but a finite prototype problem.
The peelable family already has one square-one prototype by
Corollary~\ref{cor:routeC-peelable-prototype}.  We now put the four isolated short bridges into the same framework.

\begin{proposition}[The short bridges peel to positive roots]\label{prop:routeC-shortbridge-peel}
Let $X$ be a Picard-rank-one klt del Pezzo surface over an algebraically closed field of characteristic $3$, let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D$ be the reduced exceptional divisor.  Assume that $\widetilde X$ contains one of
the four short bridges from Theorem~\ref{thm:routeC-step2-first-frontier}\textnormal{(iii)}.  Let $B\subset D$ be the
union of the connected components of $D$ disjoint from that bridge.  Then, after contracting the non-root part of the
bridge along the explicit root-contraction sequence, one obtains a smooth rational surface $Y$ carrying the image
$R\subset Y$ of the root and the image $B_Y\subset Y$ of $B$, with
$B_Y\cap R=\varnothing,$
and $R$ a smooth rational curve.  More precisely, the surviving root and its self-intersection are as follows:
\[
\begin{array}{c|c|c}
\text{bridge} & \text{contraction sequence} & R^2\\
\hline
[2]\dashv[3,2,2] & L,\ E_2,\ E_3,\ E_1 & 5\\
{}[2]\dashv[4,2,2,2] & L,\ E_2,\ E_3,\ E_4,\ E_1 & 11\\
{}[2]\to F_3 & L,\ A,\ B,\ U,\ V & 1\\
{}[2,2]\to F_4 & L,\ A_1,\ A_2,\ B,\ U,\ V & 1
\end{array}
\]
In the first two cases the root is the $[2]$-component $U$, and in the last two cases it is the $[5]$-tip $W$ of the
fork.
\end{proposition}

\begin{proof}
The contraction sequences are exactly the ones recorded in
Propositions~\ref{prop:routeC-crosslink-canonical-tail} and~\ref{prop:routeC-fork-crosslink-exact}.  Since all curves in
$B$ are disjoint from the rooted configuration, their image $B_Y$ remains disjoint from the surviving root on the
resulting smooth surface $Y$.

The final self-intersections are obtained by tracking the root under the multigraph blowdown rule
$C^2\longmapsto C^2+(C\cdot E)^2$
for the contraction of a $(-1)$-curve $E$.  For $[2]\dashv[3,2,2]$, the root $U$ starts with self-intersection $-2$;
contracting $L$, $E_2$, and $E_3$ raises it to $-1$, $0$, and $1$, and the last curve $E_1$ meets the root twice, so
its contraction raises the self-intersection to $5$.  For $[2]\dashv[4,2,2,2]$, the same computation gives the chain of
values $-2,-1,0,1,2,11$.  For $[2]\to F_3$, the root $W$ stays at $-5$ through the contractions of $L$ and $A$, then
moves to $-4$ and $-3$ after contracting $B$ and $U$, and finally $V$ meets it twice, so the last contraction raises
$W^2$ to $1$.  The case $[2,2]\to F_4$ is identical after the extra contraction of $A_2$: again the last curve meets the
root twice and the final self-intersection is $1$.
\end{proof}

\begin{proposition}[Each short bridge has a finite line-at-infinity model]\label{prop:routeC-shortbridge-prototypes}
Keep the notation of Proposition~\ref{prop:routeC-shortbridge-peel}.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item For the fork bridges $[2]\to F_3$ and $[2,2]\to F_4$, one has $R^2=1$.  Therefore
    Proposition~\ref{prop:routeC-squareone-plane} applies: there is a birational morphism
    $\tau\colon Y\to \PP^2$
    sending $R$ isomorphically onto a line $L_\infty\subset \PP^2$ and mapping $B_Y$ into
    $\PP^2\setminus L_\infty\cong \A^2$.  Reversing the contraction sequences from
    Proposition~\ref{prop:routeC-shortbridge-peel} reconstructs the short bridge by a boundary-free cluster over a
    single point $p\in L_\infty$.

    \item For the tail bridges $[2]\dashv[3,2,2]$ and $[2]\dashv[4,2,2,2]$, one has $R^2=5$ and $R^2=11$,
    respectively.  Blow up $4$ or $10$ general points of $R$, chosen away from $B_Y$ and away from the local bridge
    cluster.  The strict transform $R'$ then satisfies
    \[
    (R')^2=1,
    \qquad
    B'\cap R'=\varnothing,
    \]
    so Proposition~\ref{prop:routeC-squareone-plane} again gives a birational morphism to $\PP^2$ sending $R'$ to a
    line $L_\infty$.  Reversing the contraction sequence reconstructs the short bridge over one point $p\in L_\infty$,
    while the auxiliary blowups become $4$ or $10$ free marked points on $L_\infty$ away from $p$.
\end{enumerate}
In particular, the four short bridges reduce to four explicit global prototypes on a pair $(\PP^2,L_\infty)$.
\end{proposition}

\begin{proof}
Part~\textnormal{(1)} is immediate from Proposition~\ref{prop:routeC-shortbridge-peel} and the square-one plane model
of Proposition~\ref{prop:routeC-squareone-plane}.  The inverse of each listed contraction sequence is a succession of
blowups whose first center lies on $R$ and whose later centers lie on the total transform created by the previous local
blowups, so all local centers lie over one point $p\in R$.  Transporting that picture through $\tau$ shows that the
whole short bridge lives over one point of the line at infinity.

For part~\textnormal{(2)}, each auxiliary blowup lowers the self-intersection of the strict transform of $R$ by $1$ and,
by construction, is disjoint from both $B_Y$ and the local short-bridge cluster.  After $4$ or $10$ such blowups one is
in the square-one situation, so Proposition~\ref{prop:routeC-squareone-plane} applies exactly as in part~\textnormal{(1)}.
The same inverse-blowup argument again shows that the local short bridge is concentrated over one point of the resulting
line at infinity, while the extra general blowups remain as free points on that line away from the local cluster.
\end{proof}

\begin{theorem}[Formal Step~3: finite prototype reduction]\label{thm:routeC-step3-prototypes}
Assume the boxed residual Route~C entry statement of Corollary~\ref{cor:routeC-step1-formal} and the frontier-completeness
hypothesis of Corollary~\ref{cor:routeC-step2-formal}.  Then every characteristic-$3$ Picard-rank-one klt del Pezzo
surface $X$ with
$\#\operatorname{Sing}(X)\ge 8$
reduces, after the local peeling operations of Corollary~\ref{cor:routeC-peelable-prototype} and
Proposition~\ref{prop:routeC-shortbridge-peel} and after possibly blowing up finitely many general points on the
surviving positive curve away from the rest boundary, to one of the following five explicit global prototypes on a pair
$(\PP^2,L_\infty)$:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the peelable square-one prototype $[2,2]\dashv[3,2]$;
    \item the short-bridge prototype $[2]\dashv[3,2,2]$ together with $4$ free points on $L_\infty$;
    \item the short-bridge prototype $[2]\dashv[4,2,2,2]$ together with $10$ free points on $L_\infty$;
    \item the square-one short-bridge prototype $[2]\to F_3$;
    \item the square-one short-bridge prototype $[2,2]\to F_4$.
\end{enumerate}
In particular, to prove that no characteristic-$3$ Picard-rank-one klt del Pezzo surface with at least $8$ singular
points exists, it is enough to rule out these five explicit global prototypes.
\end{theorem}

\begin{proof}
By Corollary~\ref{cor:routeC-step2-formal}, every benchmark counterexample reduces first to the peelable family or to one
of the four isolated short bridges.  Corollary~\ref{cor:routeC-peelable-prototype} and
Propositions~\ref{prop:routeC-squareone-plane}--\ref{prop:routeC-squareone-ruling} turn the peelable family into the
single prototype~\textnormal{(i)}.  Proposition~\ref{prop:routeC-shortbridge-prototypes} does the same for the four short
bridges, giving prototypes~\textnormal{(ii)}--\textnormal{(v)}.  This is exactly the claimed finite reduction.
\end{proof}

\begin{remark}[What the corrected Step~3 leaves open]\label{rem:routeC-step3-status}
The corrected global output of the first Route~C frontier is a finite prototype problem, not yet a contradiction.  In
particular, the earlier temptation to conclude directly that the peelable family or the short bridges force
$\#\operatorname{Sing}(X)\le 3$ is stronger than the current proof supports.  What is proved rigorously is that every
first-frontier non-tie mechanism reduces to one of five explicit global models on $(\PP^2,L_\infty)$.  Thus the
remaining theoretical bottleneck is now very concrete: either show that the residual sector never reaches the first
frontier at all, or exclude those five prototypes globally.
\end{remark}


\subsection{The two square-one fork prototypes already leave the residual sector}

The finite prototype reduction from Theorem~\ref{thm:routeC-step3-prototypes} is not optimal.  The two square-one
fork prototypes can already be recognized as elliptic-boundary local models, so they do not survive inside the residual
no-descendant sector.

\begin{proposition}[The square-one fork prototypes force descendants]\label{prop:routeC-forkprototypes-desc}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$, let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D$ be the reduced exceptional divisor.  Assume that $\widetilde X$ contains one of
the two short bridges
\[
[2]\to F_3
\qquad\text{or}\qquad
[2,2]\to F_4
\]
from Theorem~\ref{thm:routeC-step2-first-frontier}\textnormal{(iii)}.  Then $X$ has a descendant with elliptic
boundary.

More precisely, with the notation of Proposition~\ref{prop:routeC-fork-crosslink-exact}, let $B$ be the branching
component of the fork, let $U,V,W$ be the $[2]$-, $[3]$-, $[5]$-tips, and let $Q$ denote the bridge side, with
component(s) $A$ or $A_1-A_2$.  Then the bridge staircase $Q+L$ contracts to a smooth point, and on the resulting
smooth surface $Z$ the image of $Q+L+F_b$ is the fork
$\langle 1;[2],[3],[5]\rangle,$
with $W=[5]$ the distinguished twig.  By \cite[Lemma~6.2(b)\textnormal{(3)} and its proof]{PP12}, this is exactly the
three-blowup cuspidal local model for an elliptic boundary with $s=8$.  Hence the image of $W$ on the further local
contraction is an irreducible cuspidal curve of arithmetic genus one, contained in the smooth locus of the target.
\end{proposition}

\begin{proof}
Consider first the bridge $[2]\to F_3$.  Write $A$ for the unique component of the $[2]$-side.  By
Proposition~\ref{prop:routeC-fork-crosslink-exact}, the bridge curve $L$ meets $A$ and the branching component $B$.
Contracting $L$ turns $A$ into a $(-1)$-curve and changes $B^2$ from $-3$ to $-2$.  Contracting $A$ next changes
$B^2$ from $-2$ to $-1$.  Thus on the resulting smooth surface $Z$ the image of the local divisor is the fork
\[
U(-2)-B(-1)-V(-3),
\qquad
B\text{ also meeting }W(-5),
\]
that is, the weighted graph $\langle 1;[2],[3],[5]\rangle$ with $W$ the $[5]$-twig.

Now consider the bridge $[2,2]\to F_4$.  Here $L$ meets one tip $A_1$ of the chain $A_1-A_2=[2,2]$ and the branching
component $B$ of $F_4$.  Contracting $L$ changes $A_1^2$ to $-1$ and $B^2$ to $-3$.  Contracting $A_1$ changes
$A_2^2$ to $-1$ and $B^2$ to $-2$.  Contracting $A_2$ then changes $B^2$ to $-1$.  Again we obtain on a smooth surface
the same fork $\langle 1;[2],[3],[5]\rangle$ with twigs $U=[2]$, $V=[3]$, $W=[5]$.

By \cite[Lemma~6.2(b)\textnormal{(3)} and the construction in its proof]{PP12}, the fork
$\langle 1;[2],[3],[5]\rangle$ is precisely the local total-transform graph obtained after three blowups over the cusp
of a singular elliptic boundary $T$ with $s=8$; in that model the $[5]$-twig is the proper transform of $T$, while the
other three components are exceptional over the cusp.  Therefore the divisor $B+U+V$ contracts locally to a smooth
point of a normal surface $Y_0$, the image $T_0$ of $W$ is an irreducible cuspidal curve with
$p_a(T_0)=1,$
and $T_0\subset (Y_0)_{\mathrm{reg}}$.

Let $D_{\mathrm{rest}}$ be the union of the connected components of $D$ disjoint from the local bridge.  These
components stay disjoint from the local cusp model throughout the above contractions.  Contracting their images on $Z$
to quotient singularities gives a further birational morphism
$\eta\colon Z\to Y$
whose exceptional locus is disjoint from $T_0$.  Writing $T:=\eta_*T_0$, we obtain a birational morphism
$\varphi\colon (\widetilde X,D)\to (Y,T)$
such that $\varphi_*D=T$, with $T\subset Y_{\mathrm{reg}}$ irreducible and $p_a(T)=1$.  By
Definition~1.11 of \cite{PP12}, $(Y,T)$ is a descendant of $X$, and its boundary is elliptic.
\end{proof}

\begin{corollary}[The fork prototypes are not residual]\label{cor:routeC-forkprototypes-not-residual}
Assume the hypotheses of Theorem~\ref{thm:routeC-step3-prototypes}.  Then the two square-one fork prototypes
\[
[2]\to F_3
\qquad\text{and}\qquad
[2,2]\to F_4
\]
cannot occur on a hypothetical benchmark counterexample.  Equivalently, inside the residual height-$\ge 4$,
no-descendant sector, the first-frontier non-tie survivors reduce to the following three tail-type prototypes on
$(\PP^2,L_\infty)$:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the peelable square-one prototype $[2,2]\dashv[3,2]$;
    \item the short-bridge prototype $[2]\dashv[3,2,2]$ together with $4$ free points on $L_\infty$;
    \item the short-bridge prototype $[2]\dashv[4,2,2,2]$ together with $10$ free points on $L_\infty$.
\end{enumerate}
\end{corollary}

\begin{proof}
Under the hypotheses of Theorem~\ref{thm:routeC-step3-prototypes}, every benchmark counterexample lies in the residual
height-$\ge 4$, no-descendant sector by Theorem~\ref{thm:routeC-contact2-reduction}, and its first-frontier non-tie
configuration reduces to one of the five prototypes listed in Theorem~\ref{thm:routeC-step3-prototypes}.  By
Proposition~\ref{prop:routeC-forkprototypes-desc}, the two fork prototypes instead force descendants with elliptic
boundary, contradicting Theorem~\ref{thm:routeC-contact2-reduction}\textnormal{(2)}.  Therefore only the three stated
tail-type prototypes remain.
\end{proof}

\begin{remark}[What Step~3 now leaves open]\label{rem:routeC-step3-sharpened-status}
After eliminating the two fork prototypes, the first analyzed Route~C frontier is no longer a five-model problem in the
residual sector.  It has shrunk to three explicit tail-type prototypes on $(\PP^2,L_\infty)$: one square-one peelable
prototype and two higher-square short bridges with auxiliary free points on the line at infinity.  Thus any further
global contradiction on the first frontier has to come from these tail models, not from the fork side.
\end{remark}


\subsection{The explicit square-one peelable prototype is a height-one model}

The square-one peelable prototype from Corollary~\ref{cor:routeC-peelable-prototype} is the remaining tail-type
model that is closest to the residual no-descendant sector.  Unlike the two fork prototypes, it does not immediately
collapse to a known cuspidal local model.  However, its line-at-infinity presentation already forces a very small
height.

\begin{proposition}[The explicit square-one peelable prototype has height one]\label{prop:routeC-squareoneprototype-h1}
Let $p\in L_\infty\subset \PP^2$, and let
$g\colon Z\to \PP^2$
be the birational morphism obtained by the four prototype blowups from
Corollary~\ref{cor:routeC-peelable-prototype}\textnormal{(2)} over $p$, producing the rooted chain
$U_2(-2)-U_1(-2)-L(-1)-E_1(-3)-E_2(-2),$
and then by any further sequence of blowups whose centers lie over a finite subset
\[
\Sigma\subset \PP^2\setminus L_\infty\cong \A^2.
\]
Let $D_Z$ be a reduced divisor whose connected components are the four prototype components
$U_2+U_1+E_1+E_2$ and unions of exceptional curves lying over points of $\Sigma$.  Then the pencil of lines
through $p$ lifts to a morphism
$f\colon Z\to \PP^1$
with the following properties.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $E_2$ is a section of $f$.
    \item The fiber corresponding to the line $L_\infty$ has support
    $U_2+U_1+L+E_1.$
    \item Every irreducible component of $D_Z-(U_2+U_1+E_1+E_2)$ is vertical for $f$.
\end{enumerate}
In particular, if $F$ is a general fiber of $f$ then
$F\cdot D_Z=1.$
\end{proposition}

\begin{proof}
The four prototype blowups over $p$ are exactly the local construction from
Corollary~\ref{cor:routeC-peelable-prototype}\textnormal{(2)}, so by
Proposition~\ref{prop:routeC-squareone-ruling} the pencil of lines through $p$ lifts after those four blowups to a
morphism to $\PP^1$ such that $E_2$ is a section and the fiber over $L_\infty$ has support
$U_2+U_1+L+E_1$.  Thus \textnormal{(1)} and \textnormal{(2)} hold before the blowups over $\Sigma$.

It remains to prove \textnormal{(3)}.  Fix $q\in \Sigma$, and let $\ell_q\subset \PP^2$ be the unique line through
$p$ and $q$.  After the prototype blowups, the total transform of $\ell_q$ is a fiber of the above ruling.  The first
blowup over $q$ occurs at a point of that fiber, so its exceptional curve is vertical.  Inductively, every later center
lying over $q$ belongs to the total transform of the same fiber, hence each new exceptional curve created over $q$ is
again vertical.  Since this is true for every $q\in \Sigma$, every irreducible component lying over $\Sigma$ is vertical,
which proves \textnormal{(3)}.

A general fiber $F$ is disjoint from the special fiber $U_2+U_1+L+E_1$ and from all vertical components created over
$\Sigma$, while it meets the section $E_2$ transversely once.  Therefore
$F\cdot D_Z=F\cdot E_2=1.$
\end{proof}

\begin{corollary}[The explicit square-one peelable prototype is not residual]\label{cor:routeC-squareoneprototype-not-residual}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$, let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D$ be the reduced exceptional divisor.  Assume that $(\widetilde X,D)$ is of the
form described in Proposition~\ref{prop:routeC-squareoneprototype-h1}, i.e. that $\widetilde X$ is obtained from the
square-one line-at-infinity prototype by further blowups over a finite subset of $\A^2$, and that $D$ is the reduced
sum of the four prototype components together with the exceptional curves lying over that finite subset.  Then
$\operatorname{ht}(X)=1.$
In particular, this explicit square-one prototype cannot itself occur on a hypothetical benchmark counterexample in the
residual height-$\ge 4$, no-descendant sector.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-squareoneprototype-h1}, the minimal resolution $\widetilde X$ carries a
$\PP^1$-fibration with general fiber $F$ satisfying $F\cdot D=1$.  Hence $\operatorname{ht}(X)\le 1$.
By Lemma~\ref{lem:routeC-height-positive}, one has $\operatorname{ht}(X)>0$.  Thus
$\operatorname{ht}(X)=1.$

The last assertion follows from Theorem~\ref{thm:routeC-contact2-reduction}: every hypothetical benchmark
counterexample lies in the residual sector of height at least $4$ and has no descendant with elliptic boundary, so it
cannot realize a height-$1$ prototype model.
\end{proof}

\begin{remark}[What this does and does not eliminate]\label{rem:routeC-squareoneprototype-status}
Corollary~\ref{cor:routeC-squareoneprototype-not-residual} removes the \emph{explicit} square-one line-at-infinity
prototype from the residual height-$\ge 4$ sector.  The next subsection removes the remaining caveat by working
directly on the full peelable family $[2,2]\dashv[3,(2)^m]$, without passing through auxiliary blowups on the positive
root.
\end{remark}



\subsection{The whole peelable family already has height one}

The caveat from Remark~\ref{rem:routeC-squareoneprototype-status} can be removed directly.  One does not need to pass
through the square-one prototype or compare heights under auxiliary blowups on a positive root.  For the peelable family
it is enough to contract the first three local curves.

\begin{proposition}[The whole peelable family carries a height-one ruling]\label{prop:routeC-peelablefamily-h1}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$, let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D$ be the reduced exceptional divisor.  Assume that $\widetilde X$ contains the
peelable bridge
\[
U_2(-2)-U_1(-2)-L(-1)-E_1(-3)-E_2(-2)-\cdots-E_n(-2),
\qquad n\ge 2,
\]
coming from the first-frontier family
$[2,2]\dashv [3,(2)^{n-1}],$
so that $U_2-U_1=[2,2]$, $E_1-\cdots-E_n=[3,(2)^{n-1}]$, and every other connected component of $D$ is disjoint from
this local bridge.  Let
$\sigma\colon \widetilde X\to Y$
be the contraction of the three curves
$L,\qquad U_1,\qquad E_1.$
Then $Y$ is smooth.  Writing
\[
F:=\sigma_*U_2,
\qquad
S:=\sigma_*E_2,
\]
one has
\[
F\cong \PP^1,
\qquad
F^2=0,
\qquad
F\cdot S=1.
\]
The complete linear system $|F|$ is a base-point-free pencil inducing a morphism
$f\colon Y\to \PP^1$
for which $S$ is a section.  Every irreducible component of $\sigma_*D-S$ is vertical for $f$.

Moreover, the ruling $f$ lifts through $\sigma$ to a morphism
$\widetilde f\colon \widetilde X\to \PP^1$
such that $E_2$ is a section of $\widetilde f$ and every irreducible component of $D-E_2$ is vertical for
$\widetilde f$.  In particular, if $G$ is a general fiber of $\widetilde f$, then
$G\cdot D=1.$
\end{proposition}

\begin{proof}
Contracting a $(-1)$-curve on a smooth surface preserves smoothness, so the successive contraction of
$L,U_1,E_1$ produces another smooth rational surface $Y$.

Tracing self-intersections through the three contractions gives the local divisor on $Y$
$F(0)-S(-1)-E_3(-2)-\cdots-E_n(-2),$
where $F=\sigma_*U_2$ and $S=\sigma_*E_2$.  Indeed, contracting $L$ changes
\[
U_1^2\colon -2\mapsto -1,
\qquad
E_1^2\colon -3\mapsto -2;
\]
contracting $U_1$ then changes
\[
U_2^2\colon -2\mapsto -1,
\qquad
E_1^2\colon -2\mapsto -1;
\]
and contracting $E_1$ finally changes
\[
U_2^2\colon -1\mapsto 0,
\qquad
E_2^2\colon -2\mapsto -1.
\]
Thus $F\cong \PP^1$, $F^2=0$, and $F\cdot S=1$.

Now consider the exact sequence
\[
0\to \mathcal O_Y \to \mathcal O_Y(F)\to \mathcal O_F(F)\to 0.
\]
Because $Y$ is rational, $H^1(Y,\mathcal O_Y)=0$.  Since $F\cong \PP^1$ and $F^2=0$, one has
$\mathcal O_F(F)\cong \mathcal O_{\PP^1},$
hence
\[
h^0\bigl(Y,\mathcal O_Y(F)\bigr)=1+h^0\bigl(\PP^1,\mathcal O_{\PP^1}\bigr)=2.
\]
The restriction map onto $H^0(\PP^1,\mathcal O_{\PP^1})$ is surjective, so $|F|$ has no base point on $F$; it has no
base point away from $F$ because $F$ itself is a member of the system.  Therefore $|F|$ is a base-point-free pencil and
defines a morphism
$f\colon Y\to \PP^1$
whose fibers are numerically equivalent to $F$.

Since $F\cdot S=1$, the curve $S$ maps isomorphically onto the base, so $S$ is a section of $f$.
Every other irreducible component of $\sigma_*D$ is disjoint from $F$: locally this is clear from the displayed chain,
and globally it uses that the peelable bridge joins two full connected components of $D$ on the first frontier, so no
other component of $D$ meets $U_2,U_1,L$, or $E_1$.  For any such component $C$ one has
$\deg(f|_C)=F\cdot C=0,$
so $C$ is vertical.  This proves the statement on $Y$.

It remains to lift the ruling back through $\sigma$.  The inverse of $\sigma$ is obtained by three blowups:
first at the point $F\cap S$, creating the exceptional curve $E_1$; next at the intersection of the transform of $F$
with $E_1$, creating $U_1$; and finally at the intersection of $U_1$ with $E_1$, creating $L$.  Each center lies on the
fiber $F$ or on curves created inside that same fiber.  Therefore the morphism $f$ lifts through all three blowups to a
morphism
$\widetilde f\colon \widetilde X\to \PP^1.$
The strict transform of $S$ is exactly the original curve $E_2$, so $E_2$ remains a section.  The curves
$L,U_1,E_1$ created in the lifting process are vertical because they arise from blowups inside one fiber.  Every other
component of $D$ is either the transform of a vertical component on $Y$ or one of these three new vertical curves.
Hence every irreducible component of $D-E_2$ is vertical for $\widetilde f$.

A general fiber $G$ of $\widetilde f$ meets the section $E_2$ transversely once and is disjoint from every vertical
component of $D$.  Therefore
$G\cdot D=G\cdot E_2=1.$
\end{proof}

\begin{corollary}[The full peelable family is not residual]\label{cor:routeC-peelablefamily-not-residual}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$, and let
$\pi\colon \widetilde X\to X$
be the minimal resolution with reduced exceptional divisor $D$.  If $\widetilde X$ contains a peelable bridge
\[
[2,2]\dashv [3,(2)^m],
\qquad m\ge 1,
\]
from Theorem~\ref{thm:routeC-step2-first-frontier}\textnormal{(ii)}, then
$\operatorname{ht}(X)=1.$
In particular, the peelable family cannot occur on a hypothetical benchmark counterexample in the residual
height-$\ge 4$, no-descendant sector.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:routeC-peelablefamily-h1}.  The lifted ruling on $\widetilde X$ has a general fiber $G$
with
$G\cdot D=1.$
Hence $\operatorname{ht}(X)\le 1$.  By Lemma~\ref{lem:routeC-height-positive}, one has $\operatorname{ht}(X)>0$.
Thus
$\operatorname{ht}(X)=1.$

The final assertion follows from Theorem~\ref{thm:routeC-contact2-reduction}: every benchmark counterexample already
lies in the residual sector with height at least $4$ and with no descendant with elliptic boundary.
\end{proof}

\begin{corollary}[Residual first-frontier ambiguity now has only two prototypes]\label{cor:routeC-two-tail-prototypes}
Assume the boxed residual Route~C entry statement of Corollary~\ref{cor:routeC-step1-formal} and the
frontier-completeness hypothesis of Corollary~\ref{cor:routeC-step2-formal}.  Then every characteristic-$3$
Picard-rank-one klt del Pezzo surface $X$ with
$\#\operatorname{Sing}(X)\ge 8$
reduces, on the first analyzed Route~C frontier, to one of the following two explicit global prototypes on a pair
$(\PP^2,L_\infty)$:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the short-bridge prototype $[2]\dashv[3,2,2]$ together with $4$ free points on $L_\infty$;
    \item the short-bridge prototype $[2]\dashv[4,2,2,2]$ together with $10$ free points on $L_\infty$.
\end{enumerate}
In particular, once the residual Route~C entry theorem and frontier completeness are proved, the first visible Route~C frontier
reduces to ruling out these two higher-square tail prototypes.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-step2-formal}, every benchmark counterexample on the first frontier would contain either a
peelable bridge or one of the four isolated short bridges.  By
Corollary~\ref{cor:routeC-peelablefamily-not-residual}, the peelable family has height $1$ and so cannot occur in the
residual height-$\ge 4$ sector.  By Corollary~\ref{cor:routeC-forkprototypes-not-residual}, the two fork short bridges
\[
[2]\to F_3
\qquad\text{and}\qquad
[2,2]\to F_4
\]
force descendants with elliptic boundary and so are also excluded from the residual sector.
Therefore only the two higher-square tail prototypes from
Corollary~\ref{cor:routeC-forkprototypes-not-residual}\textnormal{(ii)}--\textnormal{(iii)} remain.
\end{proof}

\begin{remark}[What the first Route~C frontier now leaves open]\label{rem:routeC-firstfrontier-two-prototypes}
At this stage of the argument, the first analyzed chain/fork/cross-link frontier is no longer an infinite corridor, a
five-prototype problem, or even a three-prototype problem inside the residual sector.  The peelable family is low-height,
the two fork prototypes force descendants with elliptic boundary, and only the two higher-square short bridges remain.
The next subsection removes those last two local survivors as well.
\end{remark}


\subsection{The two higher-square short bridges already have height at most two}

The last remaining non-tie survivors on the first analyzed Route~C frontier are the two short bridges
\[
[2]\dashv [3,2,2]
\qquad\text{and}\qquad
[2]\dashv [4,2,2,2].
\]
Unlike the peelable corridor, these do not require any auxiliary reduction to a square-one prototype: after contracting
just the bridge curve and the bridge-side $[2]$, the hit $(-2)$-curve itself becomes a square-zero fiber.

\begin{proposition}[The two higher-square short bridges carry height-at-most-two rulings]\label{prop:routeC-highsquare-h2}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$, let
$\pi\colon \widetilde X\to X$
be the minimal resolution, and let $D$ be the reduced exceptional divisor.  Assume that $\widetilde X$ contains one of the
short bridges
\[
[2]\dashv [3,2,2]
\qquad\text{or}\qquad
[2]\dashv [4,2,2,2]
\]
from Proposition~\ref{prop:routeC-crosslink-canonical-tail}.  Write $U$ for the bridge-side $[2]$-component, write
\[
E_1-\cdots-E_a=[a,(2)^{a-1}],\qquad a\in\{3,4\},
\]
for the tail side, and let $L$ be the bridge curve, so that $L$ meets $U$ and $E_2$ transversely once.

Let
$\sigma\colon \widetilde X\to Y$
be the contraction of the two curves
$L\qquad\text{and}\qquad U.$
Then $Y$ is smooth.  Writing
\[
F:=\sigma_*E_2,
\qquad
S_1:=\sigma_*E_1,
\qquad
S_2:=\sigma_*E_3,
\]
one has
\[
F\cong \PP^1,
\qquad
F^2=0,
\qquad
F\cdot S_1=F\cdot S_2=1.
\]
The complete linear system $|F|$ is a base-point-free pencil inducing a morphism
$f\colon Y\to \PP^1.$
The curves $S_1$ and $S_2$ are sections of $f$, and every irreducible component of $\sigma_*D-(S_1+S_2)$ is vertical for
$f$.

Moreover, the ruling $f$ lifts through $\sigma$ to a morphism
$\widetilde f\colon \widetilde X\to \PP^1$
such that $E_1$ and $E_3$ are sections of $\widetilde f$ and every irreducible component of $D-E_1-E_3$ is vertical for
$\widetilde f$.  In particular, if $G$ is a general fiber of $\widetilde f$, then
$G\cdot D=2.$
\end{proposition}

\begin{proof}
Contracting $(-1)$-curves on a smooth surface preserves smoothness, so $Y$ is smooth.
Tracing self-intersections through the two contractions gives the local divisor on $Y$.
After contracting $L$, one has
\[
U^2\colon -2\mapsto -1,
\qquad
E_2^2\colon -2\mapsto -1,
\]
while every other tail component keeps its original self-intersection.  Contracting $U$ next changes
$E_2^2\colon -1\mapsto 0.$
Therefore the local divisor on $Y$ is
$S_1(-3)-F(0)-S_2(-2)$
in the case $a=3$, and
$S_1(-4)-F(0)-S_2(-2)-E_4(-2)$
in the case $a=4$.  In both cases $F\cong \PP^1$ and $F^2=0$, while $F$ meets $S_1$ and $S_2$ once each.

Now consider the exact sequence
\[
0\to \mathcal O_Y \to \mathcal O_Y(F)\to \mathcal O_F(F)\to 0.
\]
Because $Y$ is rational, $H^1(Y,\mathcal O_Y)=0$.  Since $F\cong \PP^1$ and $F^2=0$, one has
$\mathcal O_F(F)\cong \mathcal O_{\PP^1},$
so
\[
h^0\bigl(Y,\mathcal O_Y(F)\bigr)=1+h^0\bigl(\PP^1,\mathcal O_{\PP^1}\bigr)=2.
\]
The restriction map onto $H^0(\PP^1,\mathcal O_{\PP^1})$ is surjective, so $|F|$ has no base point on $F$; it has no
base point away from $F$ because $F$ itself is a member of the system.  Therefore $|F|$ is a base-point-free pencil and
defines a morphism
$f\colon Y\to \PP^1$
whose fibers are numerically equivalent to $F$.

Since
$F\cdot S_1=F\cdot S_2=1,$
each of $S_1$ and $S_2$ maps isomorphically onto the base, so both are sections of $f$.
Every other irreducible component of $\sigma_*D$ has zero intersection with $F$: locally this is clear from the displayed
chains, because only $S_1$ and $S_2$ meet $F$; globally it uses that the short bridge joins two full connected
components of $D$ on the first frontier, so every other connected component of $D$ is disjoint from $U$, $L$, and
$E_2$, hence remains disjoint from $F$ after the contractions.  For any such component $C$ one has
$\deg(f|_C)=F\cdot C=0,$
so $C$ is vertical.  This proves the statement on $Y$.

It remains to lift the ruling back through $\sigma$.  The inverse of $\sigma$ is obtained by two blowups: first at a
smooth point of $F$ away from $S_1$, $S_2$, and every other component of $\sigma_*D$, creating the exceptional curve $U$;
next at the intersection of the transform of $F$ with $U$, creating the bridge curve $L$.  Each center lies on one fiber
of the ruling $f$, so the morphism lifts through both blowups to a morphism
$\widetilde f\colon \widetilde X\to \PP^1.$
The strict transforms of $S_1$ and $S_2$ are exactly the original curves $E_1$ and $E_3$, so they remain sections.
The curves $U$ and $L$ created in the lifting process are vertical because they arise from blowups inside one fiber.
If $a=4$, then $E_4$ already satisfies $F\cdot E_4=0$ on $Y$, so it is vertical and remains vertical after lifting.
Every other component of $D$ is either the transform of a vertical component on $Y$ or one of the two new vertical
curves $U$ and $L$.  Hence every irreducible component of $D-E_1-E_3$ is vertical for $\widetilde f$.

A general fiber $G$ of $\widetilde f$ meets the two sections $E_1$ and $E_3$ transversely once and is disjoint from every
vertical component of $D$.  Therefore
$G\cdot D=G\cdot E_1+G\cdot E_3=2.$
\end{proof}

\begin{corollary}[The two higher-square short bridges are not residual]\label{cor:routeC-highsquare-not-residual}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$, and let
$\pi\colon \widetilde X\to X$
be the minimal resolution with reduced exceptional divisor $D$.  If $\widetilde X$ contains one of the two short bridges
\[
[2]\dashv [3,2,2]
\qquad\text{or}\qquad
[2]\dashv [4,2,2,2],
\]
then
$\operatorname{ht}(X)\le 2.$
In particular, neither bridge can occur on a hypothetical benchmark counterexample in the residual height-$\ge 4$,
no-descendant sector.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:routeC-highsquare-h2}.  The lifted ruling on $\widetilde X$ has a general fiber $G$ with
$G\cdot D=2,$
so $\operatorname{ht}(X)\le 2$.
The final assertion follows from Theorem~\ref{thm:routeC-contact2-reduction}: every benchmark counterexample already lies
in the residual sector with height at least $4$ and with no descendant with elliptic boundary.
\end{proof}

\begin{theorem}[Conditional seven-point theorem from the boxed Route~C inputs]\label{thm:routeC-firstfrontier-closed}
Assume the boxed residual Route~C entry statement of Corollary~\ref{cor:routeC-step1-formal} and the
frontier-completeness hypothesis of Corollary~\ref{cor:routeC-step2-formal}.  Then no characteristic-$3$
Picard-rank-one klt del Pezzo surface $X$ satisfies
$\#\operatorname{Sing}(X)\ge 8.$
Equivalently, under those two global inputs, every characteristic-$3$ Picard-rank-one klt del Pezzo surface has at most
$7$ singular points.
\end{theorem}

\begin{proof}
Suppose for contradiction that such a surface $X$ exists.  By Theorem~\ref{thm:routeC-contact2-reduction}, it lies in
the residual height-$\ge 4$, no-descendant sector.
By Corollary~\ref{cor:routeC-step2-formal}, the minimal resolution of $X$ then contains either the peelable family
$[2,2]\dashv [3,(2)^m],\qquad m\ge 1,$
or one of the four isolated short bridges.

The peelable family is impossible by Corollary~\ref{cor:routeC-peelablefamily-not-residual}, because it has height $1$.
The two fork bridges $[2]\to F_3$ and $[2,2]\to F_4$ are impossible by
Corollary~\ref{cor:routeC-forkprototypes-not-residual}, because they force descendants with elliptic boundary.
The two tail bridges $[2]\dashv [3,2,2]$ and $[2]\dashv [4,2,2,2]$ are impossible by
Corollary~\ref{cor:routeC-highsquare-not-residual}, because they satisfy $\operatorname{ht}(X)\le 2$.
This excludes all alternatives from Corollary~\ref{cor:routeC-step2-formal}, a contradiction.
\end{proof}

\begin{remark}[What Route~C now reduces to]\label{rem:routeC-remaining-global-inputs}
Under the two boxed inputs of Corollaries~\ref{cor:routeC-step1-formal} and~\ref{cor:routeC-step2-formal}, Theorem~\ref{thm:routeC-firstfrontier-closed} already gives an earlier conditional route to the seven-point theorem.  The later bridge and height-selection sections remove the need for those extra inputs by excluding the off-frontier bridge geometries and then proving the residual height-selection statement outright.  At this stage of the paper, the entry theorem still has two geometric sufficient forms: either a purely vertical contact-$2$ bridge with a multiplicity-one component as in Corollary~\ref{cor:routeC-vertical-entry-form}, or a singly-hit multiplicity-one nonexceptional tip whose unique vertical neighbor lies in $D$, as in Corollary~\ref{cor:routeC-tip-entry-form}.  The final subsection records the exact excess package in the height-$4$ maximal-width corridor, and the last section turns that package into the unconditional numerical closure.
\end{remark}




\subsection{Further bridge reductions before the height-$4$ excess package}

The previous sections already settle one half of the residual contact-$2$ problem completely.  If the two contacts lie on
one connected component of the exceptional divisor, then root-contractibility is already enough to force an elliptic tie.
The rest of this section continues the bridge analysis on the distinct-component side.  Its role is not to enter the conditional theorem above, but to weaken the remaining global bridge gap by explicit off-frontier exclusions.  The height-$4$ maximal-width corridor will later be packaged numerically by a global excess identity, although a final unconditional closure still requires an additional graph-theoretic input.

\begin{theorem}[Same-component root-contractible contact-$2$ curves already give ties]
\label{thm:routeC-samecomponent-allties}
Let $X$ be a klt del Pezzo surface of Picard rank $1$, let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution, and let
$L\subset \widetilde X\setminus D$
be a $(-1)$-curve with
$L\cdot D=2.$
Assume that both contacts of $L$ with $D$ lie on one connected component
$D_0\subset D,$
and that the associated rooted configuration is root-contractible.
Then the divisor supported on
$D_0+L$
contains an elliptic tie.  In particular, $X$ has a descendant with elliptic boundary.
\end{theorem}

\begin{proof}
Put
$E:=D_0+L.$
By Proposition~\ref{prop:routeC-local-package}\textnormal{(1)},
$p_a(E)=1.$
By root-contractibility, there is a component
$R\subset D_0$
such that one can successively blow down $(-1)$-curves contained in the current image of $E-R$ until only the image of
$R$ remains, while the current image of $R$ is never contracted.  By Lemma~\ref{lem:routeC-local-tie-equiv}, this is
equivalent to saying that $E$ is locally sandwiched of $\delta=1$, i.e. that $E$ contains an elliptic tie.
By \cite[Definition~2.10, Definition~2.11, Lemma~2.12]{PP12}, this gives a descendant with elliptic boundary.
\end{proof}

\begin{corollary}[The unresolved part of Route~C is bridge-type]
\label{cor:routeC-step2-reduces-to-bridges}
Let $X$ be a characteristic-$3$ klt del Pezzo surface with
\[
\rho(X)=1,
\qquad
\#\Sing(X)\ge 8.
\]
Assume that the minimal resolution of $X$ contains a positive root-contractible contact-$2$ configuration.
Then either:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item it already gives a descendant with elliptic boundary; or
    \item it is a distinct-component bridge.
\end{enumerate}
In particular, after Theorem~\ref{thm:routeC-samecomponent-allties}, the only unresolved part of Route~C is the global
analysis of positive root-contractible bridges.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:routeC-contact2-reduction}, every benchmark counterexample already lies in the residual height-$\ge
4$, no-descendant sector.  If the given configuration is same-component, then
Theorem~\ref{thm:routeC-samecomponent-allties} yields a descendant with elliptic boundary, contradicting the residual
assumption.  So any residual positive root-contractible contact-$2$ configuration must be a distinct-component bridge.
\end{proof}

The next proposition shows that the remaining completeness problem is genuinely global.  There is an explicit infinite
family of positive root-contractible bridge germs beyond the first analyzed frontier, so no purely local argument can
possibly prove that the first frontier is exhaustive.

\begin{proposition}[An infinite off-frontier family of positive root-contractible bridges]
\label{prop:routeC-offfrontier-bridge-family}
For every integer $a\ge 3$, let
$Q=[3]$
with component $U$, and let
$T_a=[a,(2)^{a-1}]$
with components
\[
E_1-\cdots-E_a,
\qquad
E_1^2=-a,
\qquad
E_i^2=-2\ \ (i\ge 2).
\]
Let
$L$
be a bridge meeting $U$ and $E_2$ once each.  Then the rooted graph
$Q+L+T_a$
is positive and root-contractible with root $U$.  More precisely:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item if $C=\pi(L)\subset X$, then
    \[
    C^2
    =-1+\frac{1}{3}+\frac{a(a-1)}{a^2-a+1}
    =\frac{a^2-a-2}{3(a^2-a+1)}
    >0;
    \]
    \item the sequence
    $L,\ E_2,\ E_3,\ \dots,\ E_a,\ E_1$
    contracts all components except $U$.
\end{enumerate}
For every $a\ge 5$ this positive root-contractible bridge lies strictly beyond the first analyzed bridge frontier.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:routeC-crosslink-local-package}\textnormal{(3)}, the image square is
\[
C^2=-1+\frac{1}{3}
      +\frac{d([a])\,d([(2)^{a-2}])}{d([a,(2)^{a-1}])}.
\]
Now
\[
d([a])=a,
\qquad
d([(2)^{a-2}])=a-1,
\]
and if we put
$D_r:=d([a,(2)^r]),$
then
\[
D_0=a,\qquad D_1=2a-1,\qquad D_r=2D_{r-1}-D_{r-2},
\]
so by induction
$D_r=a(r+1)-r.$
For $r=a-1$ this gives
$d([a,(2)^{a-1}])=a^2-a+1.$
Hence
\[
C^2
=-1+\frac13+\frac{a(a-1)}{a^2-a+1}
=\frac{a^2-a-2}{3(a^2-a+1)},
\]
which is positive for every $a\ge 3$.

For root-contractibility, contract $L$ first.  Then $U$ changes from $(-3)$ to $(-2)$ and $E_2$ changes from $(-2)$
to $(-1)$.  Thus $E_2$ is contractible.  After contracting
\[
L,\ E_2,\ \dots,\ E_{k-1}
\qquad (3\le k\le a),
\]
the curve $E_k$ has self-intersection $-1$ and meets the current images of $U$, $E_1$, and (if $k<a$) $E_{k+1}$
transversely once.  Hence the displayed peeling sequence is legal.
After contracting
$L,\ E_2,\ \dots,\ E_a,$
the curve $E_1$ has become a $(-1)$-curve, so it contracts as well.  The component $U$ is never contracted, so the
rooted graph is root-contractible with root $U$.

Finally, for $a\ge 5$ the right-hand chain is not one of the terminal tails
$[3,(2)^m],\qquad [4,(2)^m],$
and the bridge side is $[3]$, not $[2]$ or $[2,2]$.  Hence these bridges lie beyond the first analyzed bridge frontier.
\end{proof}

\begin{corollary}[Bridge completeness is genuinely global]
\label{cor:routeC-bridgecomplete-global}
The first analyzed bridge frontier cannot be exhaustive for positive root-contractible bridge germs by a purely local
argument.  Indeed, Proposition~\ref{prop:routeC-offfrontier-bridge-family} gives an infinite family of positive
root-contractible bridge germs lying strictly beyond that frontier.

Consequently, any final bridge-completeness theorem must use an additional \emph{global} restriction excluding such
off-frontier bridge germs from actual characteristic-$3$ Picard-rank-one klt del Pezzo surfaces with
$\#\Sing(X)\ge 8.$
\end{corollary}

\begin{proof}
Immediate from Proposition~\ref{prop:routeC-offfrontier-bridge-family}.
\end{proof}

The previous proposition shows why the remaining completeness statement must be global.  The next lemma pair is a
convenient toolkit for killing a broad one-component bridge family inside the benchmark range.

\begin{lemma}[A smooth rational $0$-curve gives a ruling]
\label{lem:routeC-zero-curve-pencil}
Let $Y$ be a smooth rational projective surface and let
$F\subset Y$
be a smooth rational curve with
$F^2=0.$
Then the complete linear system $|F|$ is a base-point-free pencil and defines a morphism
$p\colon Y\to \PP^1$
whose general fiber is linearly equivalent to $F$.
If $S\subset Y$ is an irreducible curve with
$S\cdot F=1,$
then $S$ is a section of $p$.
\end{lemma}

\begin{proof}
Consider the exact sequence
\[
0\to \mathcal O_Y \to \mathcal O_Y(F)\to \mathcal O_F(F)\to 0.
\]
Since $Y$ is rational, $H^1(Y,\mathcal O_Y)=0$.  Since $F\cong \PP^1$ and $F^2=0$, one has
$\mathcal O_F(F)\cong \mathcal O_{\PP^1},$
so
\[
h^0\bigl(Y,\mathcal O_Y(F)\bigr)=1+h^0\bigl(\PP^1,\mathcal O_{\PP^1}\bigr)=2.
\]
Thus $|F|$ is a pencil.

The restriction map
\[
H^0\bigl(Y,\mathcal O_Y(F)\bigr)\twoheadrightarrow H^0\bigl(F,\mathcal O_F(F)\bigr)\cong k
\]
is surjective, so $|F|$ has no base point on $F$.  It has no base point away from $F$ either, because $F$ itself is a
member of $|F|$.  Hence $|F|$ is base-point free and defines a morphism
$p\colon Y\to \PP^1.$
Two distinct members of $|F|$ have intersection number $F^2=0$, hence are disjoint.  Therefore a general member of
$|F|$ is a smooth rational fiber of $p$.

If $S\cdot F=1$, then
$\deg(p|_S)=S\cdot F=1,$
so $S$ is a $1$-section, hence a section.
\end{proof}

\begin{lemma}[Genus created by contracting a $(-1)$-curve]
\label{lem:routeC-last-contraction-genus}
Let
$\phi\colon Z\to Y$
be the contraction of a $(-1)$-curve $E$ on a smooth surface $Z$.
Let $U\subset Z$ be a smooth rational curve, different from $E$, and put
$m:=U\cdot E.$
Let
$T:=\phi_*U\subset Y.$
Then
\[
p_a(T)=\frac{m(m-1)}{2}.
\]
In particular, if $m=2$, then $p_a(T)=1$.
\end{lemma}

\begin{proof}
Write
$\phi^*T=U+\alpha E.$
Intersecting with $E$ gives
$0=(U+\alpha E)\cdot E=m-\alpha,$
because $E^2=-1$.  Hence $\alpha=m$, so
$\phi^*T=U+mE.$
Therefore
\[
T^2=(\phi^*T)^2=(U+mE)^2=U^2+2m(U\cdot E)+m^2E^2=U^2+m^2.
\]

Also
\[
K_Z=\phi^*K_Y+E,
\qquad\text{so}\qquad
\phi^*K_Y=K_Z-E.
\]
Using adjunction on the smooth rational curves $U$ and $E$,
\[
K_Z\cdot U=-2-U^2,
\qquad
K_Z\cdot E=-2-E^2=-1.
\]
Hence
\[
K_Y\cdot T=(\phi^*K_Y)\cdot(\phi^*T)
=(K_Z-E)\cdot(U+mE)
=-2-U^2-m.
\]
Thus
\[
T\cdot (T+K_Y)
=(U^2+m^2)+(-2-U^2-m)
=m^2-m-2.
\]
By adjunction on $Y$,
$2p_a(T)-2=m^2-m-2,$
so
\[
p_a(T)=\frac{m(m-1)}{2}.
\]
\end{proof}

\begin{proposition}[A broad one-component bridge family]
\label{prop:routeC-single-root-bridge-family}
Let
$Q=[b]$
with unique component $U$ of self-intersection $-b$, where $b\ge 2$.
Let
\[
T_a=[a,(2)^{a-1}]=E_1-\cdots-E_a,
\qquad a\ge 3,
\]
where
\[
E_1^2=-a,
\qquad
E_i^2=-2\ \ (i\ge 2).
\]
Let
$L$
be a distinct-component bridge meeting $U$ and $E_2$ once each.  Consider the associated rooted graph with root $U$.

Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item The image square is
    \[
    C^2=-1+\frac{1}{b}+\frac{a(a-1)}{a^2-a+1}.
    \]

    \item This rooted graph is root-contractible.  More precisely, the sequence
    $L,\ E_2,\ E_3,\ \dots,\ E_a,\ E_1$
    contracts all components except $U$.

    \item Assume now that
    $b\le a.$
    Let
    $\sigma_b\colon \widetilde X\to Y_b$
    be the contraction of
    $L,\ E_2,\ E_3,\ \dots,\ E_b.$
    Then the image
    $F:=\sigma_{b*}U$
    is a smooth rational curve with
    $F^2=0.$
    If $b<a$, then the image of $E_{b+1}$ is a section of the ruling $|F|$, while the image of $E_1$ has degree
    $F\cdot \sigma_{b*}E_1=b-1.$
    Consequently,
    \[
    \operatorname{ht}(X)\le b
    \qquad\text{if } b<a,
    \]
    and
    \[
    \operatorname{ht}(X)\le b-1
    \qquad\text{if } b=a.
    \]

    \item If
    $a=3,$
    then $X$ has a descendant with elliptic boundary.
\end{enumerate}
\end{proposition}

\begin{proof}
For \textnormal{(1)}, Proposition~\ref{prop:routeC-crosslink-local-package}\textnormal{(3)} gives
\[
C^2=-1+\frac{1}{b}+\frac{d([a])\,d([(2)^{a-2}])}{d(T_a)}.
\]
Now
\[
d([a])=a,
\qquad
d([(2)^{a-2}])=a-1,
\qquad
d(T_a)=a^2-a+1,
\]
so the displayed formula follows.

For \textnormal{(2)}, contract $L$ first.  Then $U$ changes from $(-b)$ to $(-(b-1))$ and $E_2$ changes from $(-2)$ to
$(-1)$.  Thus $E_2$ is contractible.
After contracting
\[
L,\ E_2,\ \dots,\ E_{k-1}
\qquad (3\le k\le a),
\]
the current image of $E_k$ is a $(-1)$-curve, so the displayed peeling order is legal.
After contracting
$L,\ E_2,\ \dots,\ E_a,$
the curve $E_1$ has become a $(-1)$-curve, so it contracts as well.  The component $U$ is never contracted.

For \textnormal{(3)}, assume $b\le a$ and contract only
$L,\ E_2,\ \dots,\ E_b.$
We claim that after contracting
\[
L,\ E_2,\ \dots,\ E_k
\qquad (2\le k\le b),
\]
the current images satisfy
\[
U_k^2=k-b,
\qquad
U_k\cdot E_1^{(k)}=k-1,
\qquad
(E_1^{(k)})^2=-(a-k+1),
\]
and, if $k<a$, the image of $E_{k+1}$ meets $U_k$ once.
For $k=2$, after contracting $L$ and $E_2$ one has
\[
U_2^2=2-b,
\qquad
U_2\cdot E_1^{(2)}=1,
\qquad
(E_1^{(2)})^2=-(a-1),
\]
and the image of $E_3$ meets $U_2$ once if $a>2$.
If the statement holds for some $k<a$, then the image of $E_{k+1}$ is a $(-1)$-curve meeting the current images of
$U$, $E_1$, and (if $k+1<a$) $E_{k+2}$ once each.  Contracting it raises the self-intersection of $U_k$ and
$E_1^{(k)}$ by $1$, increases their mutual intersection by $1$, and preserves the fact that the next tail component
meets the current image of $U$ once.  This proves the induction.

Taking $k=b$, we obtain
\[
F^2=U_b^2=0,
\qquad
F\cdot \sigma_{b*}E_1=b-1.
\]
If $b<a$, then the image of $E_{b+1}$ meets $F$ once, hence is a section of the ruling defined by $|F|$ by
Lemma~\ref{lem:routeC-zero-curve-pencil}.
All later tail components $E_{b+2},\dots,E_a$ are disjoint from $F$, so they are vertical for this ruling.
The same is true for every component of the rest boundary, because all local contractions above are supported inside the
bridge cluster and the remaining components of $D$ are disjoint from the two connected components joined by the bridge.

Hence a general fiber $G$ of the ruling satisfies
\[
G\cdot D\le (b-1)+1=b
\qquad\text{if } b<a,
\]
and
\[
G\cdot D\le b-1
\qquad\text{if } b=a.
\]
This proves the height bounds.

For \textnormal{(4)}, assume $a=3$ and contract
$L,\ E_2,\ E_3.$
Let
\[
F:=\sigma_*U,
\qquad
T:=\sigma_*E_1.
\]
Then
\[
T^2=-1,
\qquad
F\cdot T=2.
\]
Now contract $T$.
By Lemma~\ref{lem:routeC-last-contraction-genus}, the image of $F$ is an irreducible curve of arithmetic genus $1$.
Since all other components of the boundary are disjoint from the local bridge cluster, they remain disjoint from this
local construction and can be contracted afterwards.  Thus $X$ has a descendant with elliptic boundary.
\end{proof}


\begin{corollary}[Anticanonical sieve for one-component bridges]
\label{cor:routeC-single-root-anticanonical}
Keep the setup of Proposition~\ref{prop:routeC-single-root-bridge-family}, and assume in addition that the rooted
configuration is root-contractible, so in particular $n=a$ by Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(2)}.
Then the image curve $C=\pi(L)$ satisfies
\[
-K_X\cdot C
=
-1+\frac{2}{b}+\frac{2a-1}{a^2-a+1}.
\]
Consequently:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $a\ge 5$, then positivity of $-K_X\cdot C$ forces $b\le 3$;
    \item if $a=4$, then positivity of $-K_X\cdot C$ forces $b\le 4$.
\end{enumerate}
\end{corollary}

\begin{proof}
The formula is Proposition~\ref{prop:routeC-anticanonical-chain-bridge} applied to the two chains
\[
[b]
\qquad\text{and}\qquad
[a,(2)^{a-1}],
\]
where the bridge meets the unique component of $[b]$ and the second component of the tail.  For the one-component
side one has
\[
\mu([b],1)=\frac{1+1}{b}=\frac{2}{b},
\]
while on the tail side
\[
\mu\bigl([a,(2)^{a-1}],2\bigr)
=
\frac{d([a])+d([(2)^{a-2}])}{d([a,(2)^{a-1}])}
=
\frac{a+(a-1)}{a^2-a+1}
=
\frac{2a-1}{a^2-a+1}.
\]
Thus
\[
-K_X\cdot C
=
-1+\frac{2}{b}+\frac{2a-1}{a^2-a+1}.
\]
Since $-K_X$ is ample and $C$ is a nonzero effective curve, one has $-K_X\cdot C>0$.

If $a\ge 5$, then
\[
\frac{2a-1}{a^2-a+1}\le \frac{9}{21}=\frac37,
\]
so positivity implies
\[
0< -1+\frac{2}{b}+\frac37,
\qquad\text{hence}\qquad
\frac{2}{b}>\frac47,
\]
which forces $b<\frac72$, hence $b\le 3$.

If $a=4$, then positivity gives
\[
0< -1+\frac{2}{b}+\frac{7}{13},
\qquad\text{so}\qquad
\frac{2}{b}>\frac{6}{13},
\]
which forces $b<\frac{13}{3}$, hence $b\le 4$.
\end{proof}

\begin{corollary}[The one-component bridge family is not residual]
\label{cor:routeC-single-root-bridge-all-not-residual}
Let $X$ be a characteristic-$3$ klt del Pezzo surface with
\[
\rho(X)=1
\qquad\text{and}\qquad
\#\Sing(X)\ge 8.
\]
Assume that the minimal resolution of $X$ contains a positive root-contractible bridge of the form
$[b]\dashv[a,(2)^{a-1}]$
whose bridge curve meets the unique component of $[b]$ and the second component of the tail, and whose root is the
one-component side $[b]$.  Then this is impossible.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-single-root-anticanonical}, if $a\ge 5$ then necessarily $b\le 3$, and if $a=4$ then
necessarily $b\le 4$.

If $a=3$, the bridge is excluded by Corollary~\ref{cor:routeC-single-root-bridge-lightcases}\textnormal{(i)}.
If $a\ge 5$, then $b\le 3$, so Corollary~\ref{cor:routeC-single-root-bridge-lightcases}\textnormal{(ii)} excludes the
configuration.  If $a=4$, then $b\le 4$; the cases $b\le 3$ are again excluded by
Corollary~\ref{cor:routeC-single-root-bridge-lightcases}\textnormal{(ii)}, while the remaining case $(b,a)=(4,4)$ is
excluded by Corollary~\ref{cor:routeC-single-root-bridge-lightcases}\textnormal{(iii)}.
Thus no positive root-contractible one-component bridge of this family can occur on a benchmark counterexample.
\end{proof}

\begin{corollary}[Light one-component bridges are not residual]
\label{cor:routeC-single-root-bridge-lightcases}
Let $X$ be a characteristic-$3$ klt del Pezzo surface with
\[
\rho(X)=1
\qquad\text{and}\qquad
\#\Sing(X)\ge 8.
\]
Assume that the minimal resolution of $X$ contains a positive root-contractible bridge of the form
$[b]\dashv[a,(2)^{a-1}]$
whose bridge curve meets the unique component of $[b]$ and the second component of the tail, and whose root is the
one-component side $[b]$.
Then this bridge cannot occur on a benchmark counterexample in any of the following cases:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item $a=3$;
    \item $b\le 3$;
    \item $(b,a)=(4,4)$.
\end{enumerate}
\end{corollary}

\begin{proof}
If $a=3$, then Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(4)} gives a descendant with elliptic
boundary, contradicting Theorem~\ref{thm:routeC-contact2-reduction}.

If $b\le 3$, then Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(3)} gives
$\operatorname{ht}(X)\le 3,$
again contradicting Theorem~\ref{thm:routeC-contact2-reduction}.

If $(b,a)=(4,4)$, then Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(3)} gives
$\operatorname{ht}(X)\le 3,$
so this case is also excluded from the residual height-$\ge 4$ sector.
\end{proof}


\begin{proposition}[The one-component bridge family contracts to rational unicuspidal curves]
\label{prop:routeC-single-root-bridge-unicuspidal}
Keep the setup of Proposition~\ref{prop:routeC-single-root-bridge-family}, and let
$\tau\colon \widetilde X \to Y_{a,b}$
be the contraction of the whole non-root staircase
$S_a:=L+E_1+\cdots+E_a.$
Put
$R_{a,b}:=\tau_*U.$
Then the following hold.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item The curve $R_{a,b}$ is irreducible and rational, and
    $R_{a,b}^2=a^2-a+1-b.$

    \item If
    $\sigma\colon \widetilde X\to Z_{a,b}$
    denotes the contraction of
    $L,\ E_2,\ E_3,\ \dots,\ E_a,$
    and
    \[
    U':=\sigma_*U,
    \qquad
    T:=\sigma_*E_1,
    \]
    then $U'\cong \PP^1$, $T^2=-1$, and
    $U'\cdot T=a-1.$

    \item The curve $R_{a,b}$ has exactly one singular point $q_{a,b}$, it is unibranch there, its multiplicity is
    $\operatorname{mult}_{q_{a,b}}(R_{a,b})=a-1,$
    and
    \[
    p_a(R_{a,b})=\frac{(a-1)(a-2)}{2}.
    \]

    \item Equivalently, the bridge staircase $S_a$ is the minimal embedded resolution over $q_{a,b}$ of a rational
    unibranch singularity of multiplicity sequence
    $(a-1,1,\dots,1),$
    with $a-1$ trailing $1$'s.  In particular, the bridge curve $L$ is the last exceptional curve in that local
    resolution sequence.
\end{enumerate}
\end{proposition}

\begin{proof}
For \textnormal{(1)}, the curve $R_{a,b}$ is irreducible because it is the image of the irreducible curve $U$ under a
birational contraction, and its normalization is $\PP^1$ because $U\cong \PP^1$.
To compute the self-intersection, contract first
$L,\ E_2,\ E_3,\ \dots,\ E_a.$
As in the proof of Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(3)}, contracting $L$ raises
$U^2$ by $1$, and each contraction of $E_2,\dots,E_a$ raises it by another $1$.  Hence on $Z_{a,b}$ one has
$(U')^2=-b+1+(a-1)=a-b.$
Now $T$ is a $(-1)$-curve meeting $U'$ with multiplicity $a-1$, so contracting $T$ raises the square of the image of
$U'$ by $(a-1)^2$.  Therefore
\[
R_{a,b}^2
=
a-b+(a-1)^2
=
a^2-a+1-b.
\]

Statement~\textnormal{(2)} follows from the same computation: after contracting
$L,\ E_2,\ E_3,\ \dots,\ E_a$
the only remaining curves from the local staircase are $U'$ and $T$, with
\[
(U')^2=a-b,
\qquad
T^2=-1,
\qquad
U'\cdot T=a-1.
\]

For \textnormal{(3)}, contracting $T$ is the inverse of blowing up one point
$q_{a,b}\in R_{a,b}.$
Since the contraction is an isomorphism away from $T$, the curve $R_{a,b}$ is smooth away from $q_{a,b}$, so this is
its unique singular point.  The normalization of $R_{a,b}$ is $U'\cong \PP^1$, and only one point of $U'$ maps to
$q_{a,b}$, so the singularity is unibranch.  Its multiplicity is
\[
\operatorname{mult}_{q_{a,b}}(R_{a,b})=U'\cdot T=a-1,
\]
because under a blowup the strict transform meets the exceptional divisor with total multiplicity equal to the
multiplicity of the original curve at the center.  Finally, Lemma~\ref{lem:routeC-last-contraction-genus} applied to
the contraction of $T$ gives
\[
p_a(R_{a,b})=\frac{(a-1)(a-2)}{2}.
\]

For \textnormal{(4)}, reverse the contraction sequence
$L,\ E_2,\ E_3,\ \dots,\ E_a,\ E_1.$
The first inverse step is the blowup of the singular point $q_{a,b}$, producing the exceptional curve $T=E_1$ and a
smooth strict transform $U'$ meeting it with multiplicity $a-1$.  The remaining inverse steps successively resolve
that order-$(a-1)$ tangency by blowing up the unique current point of contact of the strict transform of $R_{a,b}$
with the newest exceptional curve; this is exactly the reverse of the legal peeling order proved in
Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(2)}.  Each of those later blowups has
multiplicity $1$, and after the last one the newest exceptional curve is precisely $L$.  Thus the local singularity is
resolved by one blowup of multiplicity $a-1$ followed by $a-1$ blowups of multiplicity $1$, which is the displayed
multiplicity sequence.
\end{proof}

\begin{corollary}[Corrected square-one prototype reduction]
\label{cor:routeC-single-root-bridge-squareone-corrected}
For each $a\ge 3$, define the square-one prototype
$\Pi_a:=[a^2-a]\dashv[a,(2)^{a-1}],$
with the bridge attached to the second component of the tail and the root on the one-component side.

Let
$[b]\dashv[a,(2)^{a-1}]$
be any positive bridge of the same type.  Then
$b\le a^2-a.$
Moreover, if
$m:=a^2-a-b\ge 0,$
then after blowing up $m$ general \emph{smooth} points of the rational curve $R_{a,b}$ away from its unique cusp
$q_{a,b}$ and away from the rest boundary, one obtains a birational model whose corresponding one-component bridge is
precisely the square-one prototype $\Pi_a$.

Equivalently, every positive one-component bridge of this family is controlled by the one-parameter square-one
prototype family $\Pi_a$, together with $m$ free marked points on the smooth locus of the surviving rational cuspidal
curve away from the staircase point.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-single-root-bridge-family}\textnormal{(1)},
\[
C^2
=
-1+\frac{1}{b}+\frac{a(a-1)}{a^2-a+1}
=
\frac{a^2-a+1-b}{b(a^2-a+1)}.
\]
Hence positivity is equivalent to
$a^2-a+1-b>0,$
so indeed
$b\le a^2-a.$
By Proposition~\ref{prop:routeC-single-root-bridge-unicuspidal}\textnormal{(1)},
$R_{a,b}^2=a^2-a+1-b=m+1.$
Blowing up $m$ general smooth points of $R_{a,b}$ away from the cusp lowers the self-intersection of its strict
transform by $m$ and does not change the local unicuspidal staircase over $q_{a,b}$.  Hence after those free blowups
the strict transform has square $1$.  Reversing the same local resolution staircase over the cusp then produces exactly
the bridge
$[a^2-a]\dashv[a,(2)^{a-1}],$
namely the square-one prototype $\Pi_a$.
\end{proof}

\begin{corollary}[The visible one-component chain gap is a unicuspidal square-one family]
\label{cor:routeC-single-root-gap-oneparameter-corrected}
After Corollary~\ref{cor:routeC-single-root-bridge-lightcases}, the remaining one-component chain-shaped gap is no
longer best viewed as the full two-parameter heavy-heavy regime in $(a,b)$.  It is controlled birationally by the
square-one unicuspidal prototype family
\[
\Pi_a=[a^2-a]\dashv[a,(2)^{a-1}],
\qquad a\ge 4,
\]
together with auxiliary free points on the smooth locus of the surviving rational cuspidal curve away from its unique
singular point.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-single-root-bridge-lightcases}, the cases
\[
a=3,
\qquad
b\le 3,
\qquad
(b,a)=(4,4)
\]
already leave the residual sector.  For every remaining positive bridge of the same one-component type,
Corollary~\ref{cor:routeC-single-root-bridge-squareone-corrected} reduces the local geometry to the square-one
prototype $\Pi_a$ plus free points away from the unique cusp.
\end{proof}




\subsection{A $2$-tail core reduction for non-staircase chain bridges}

The anticanonical bridge-contact formula suggests that the correct invariant for the remaining chain bridges is not the
full chain, but its contact core after removing terminal $[2]$-tails away from the touched component.

\begin{definition}[$2$-tail core of a touched chain]
\label{def:routeC-two-tail-core}
Let
$A=[a_1,\dots,a_r]$
be a chain and let the bridge touch its $i$-th component $A_i$.
The \emph{$2$-tail core} of the touched pair $(A,i)$ is obtained by repeatedly removing:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item the leftmost component if it has weight $2$ and $i>1$;
    \item the rightmost component if it has weight $2$ and $i<r$,
\end{enumerate}
until neither operation is possible.
We denote the resulting touched pair by
$(A^\sharp,i^\sharp).$
Thus $A^\sharp$ has no removable terminal $2$ away from the touched component.
\end{definition}

\begin{lemma}[Monotonicity under adding a terminal $2$ away from the contact]
\label{lem:routeC-mu-monotone-tail}
Let
$(A,i)$
be a touched chain pair, and let $A^+$ be obtained from $A$ by adjoining a terminal $2$
on one side away from the touched component.
Then
$\mu(A^+,i^+)\le \mu(A,i),$
where $i^+$ is the corresponding touched index on $A^+$.

In particular, for the $2$-tail core from Definition~\ref{def:routeC-two-tail-core},
$\mu(A,i)\le \mu(A^\sharp,i^\sharp).$
\end{lemma}

\begin{proof}
It is enough to treat the case of adjoining a terminal $2$ on the right away from the contact.
Write
\[
A=[a_1,\dots,a_r],\qquad i<r,\qquad A^+:=[a_1,\dots,a_r,2].
\]
Put
\[
P:=d(A_i^-)=d([a_1,\dots,a_{i-1}]),\qquad
Q:=d(A_i^+)=d([a_{i+1},\dots,a_r]),
\]
and
\[
D:=d(A),\qquad E:=d([a_1,\dots,a_{r-1}]),\qquad
R:=d([a_{i+1},\dots,a_{r-1}]).
\]
Then
\[
\mu(A,i)=\frac{P+Q}{D},
\qquad
\mu(A^+,i)=\frac{P+(2Q-R)}{2D-E}.
\]

A standard continuant identity for chains gives
$DR-EQ=-P.$
Hence
\[
\mu(A,i)-\mu(A^+,i)
=
\frac{(P+Q)(2D-E)-D(P+2Q-R)}{D(2D-E)}
=
\frac{P(D-E-1)}{D(2D-E)}.
\]
Since $D,E$ are positive integers and $D>E$, one has
$D-E-1\ge 0,$
so
$\mu(A,i)\ge \mu(A^+,i).$
The left-tail case is symmetric.
Iterating proves the final statement.
\end{proof}

\begin{proposition}[Interior points of a $2$-reduced core have contact weight at most $\frac12$]
\label{prop:routeC-interior-core-half}
Let
$(A^\sharp,i^\sharp)$
be a touched chain pair such that $A^\sharp$ has no removable terminal $2$ away from the touched component, and assume
that $i^\sharp$ is an interior vertex of $A^\sharp$.
Then
\[
\mu(A^\sharp,i^\sharp)\le \frac12.
\]
\end{proposition}

\begin{proof}
Write
$A^\sharp=[a_1,\dots,a_r],\qquad 1<i^\sharp<r.$
Because $A^\sharp$ has no removable terminal $2$ away from the touched component,
\[
a_1\ge 3,
\qquad
a_r\ge 3.
\]

Let
\[
\delta_k:=\mu(A^\sharp,k)
\qquad (1\le k\le r).
\]
Equivalently, if $M$ is the intersection matrix of the chain, then
$M\delta=-\nu,$
where $\nu$ has value $1$ at the two ends and $0$ elsewhere.
In coordinates,
\[
a_1\delta_1-\delta_2=1,
\qquad
-\delta_{k-1}+a_k\delta_k-\delta_{k+1}=0
\quad (2\le k\le r-1),
\qquad
a_r\delta_r-\delta_{r-1}=1.
\]

Suppose for contradiction that some interior value satisfies
\[
\delta_{i^\sharp}>\frac12.
\]
Let
$m:=\max\{\delta_1,\dots,\delta_r\}.$
Then
\[
m>\frac12.
\]
Choose $t$ with $\delta_t=m$.

If $t$ is interior, then
\[
a_t m
=
\delta_{t-1}+\delta_{t+1}
\le 2m,
\]
so
$a_t\le 2.$
Since every weight is at least $2$, this forces
\[
a_t=2
\qquad\text{and}\qquad
\delta_{t-1}=\delta_{t+1}=m.
\]
Thus any interior maximum propagates across a contiguous block of $2$-vertices.

Because both ends have weight at least $3$, this propagation reaches an end vertex $e$ with
$\delta_e=m.$
At that end one has
$1=a_e\delta_e-\delta_{e'}\ge 3m-m=2m,$
where $e'$ is the unique neighbor of $e$.
So
\[
m\le \frac12,
\]
contradiction.
Therefore
\[
\delta_{i^\sharp}\le \frac12.
\]
\end{proof}

\begin{corollary}[A positive chain bridge must be tip-visible on at least one side]
\label{cor:routeC-tip-visible-side}
Let
$L$
be a distinct-component bridge between two chain components
\[
(A,i)
\qquad\text{and}\qquad
(B,j)
\]
on the minimal resolution of a Picard-rank-one klt del Pezzo surface.
Assume that the bridge is positive, i.e.
\[
C:=\pi(L)\subset X
\qquad\text{satisfies}\qquad
C^2>0.
\]
Let
\[
(A^\sharp,i^\sharp),\qquad (B^\sharp,j^\sharp)
\]
be the corresponding $2$-tail cores.

Then it is impossible that both $i^\sharp$ and $j^\sharp$ are interior vertices.
Equivalently, every positive chain bridge has at least one side whose $2$-tail core is touched at a tip.
\end{corollary}

\begin{proof}
By Lemma~\ref{lem:routeC-mu-monotone-tail},
\[
\mu(A,i)\le \mu(A^\sharp,i^\sharp),
\qquad
\mu(B,j)\le \mu(B^\sharp,j^\sharp).
\]
If both $i^\sharp$ and $j^\sharp$ were interior, then
Proposition~\ref{prop:routeC-interior-core-half} would give
\[
\mu(A,i)\le \frac12,
\qquad
\mu(B,j)\le \frac12.
\]
Hence Proposition~\ref{prop:routeC-anticanonical-chain-bridge} would imply
\[
-K_X\cdot C
=
-1+\mu(A,i)+\mu(B,j)
\le 0,
\]
contradicting the ampleness of $-K_X$.
\end{proof}

\begin{proposition}[High-contact reduced tip cores are very special]
\label{prop:routeC-tipcore-highcontact}
Let
$(A^\sharp,i^\sharp)$
be a touched chain pair obtained after $2$-tail reduction, and assume that
$i^\sharp$ is a tip of $A^\sharp$.
Then
\[
\mu(A^\sharp,i^\sharp)>\frac12
\]
if and only if, up to reversing the chain, one of the following holds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the touched tip has weight $2$;
    \item the touched pair is
    $[3]$
    with the bridge touching its unique component.
\end{enumerate}
Equivalently, after $2$-tail reduction, the only tip-visible chain cores with contact weight strictly larger than
$1/2$ are the $2$-tip cores and the singleton core
$[3]$
touched at its unique component.
\end{proposition}

\begin{proof}
By symmetry it is enough to treat the case where the bridge touches the left tip.
Write
$A^\sharp=[a_1,\dots,a_r]$
with touched tip the component of weight $a_1$.

If $r=1$, then
\[
\mu(A^\sharp,1)=\frac{2}{a_1}.
\]
Hence
\[
\mu(A^\sharp,1)>\frac12
\]
if and only if
$a_1=2\ \text{or}\ 3,$
which is exactly the stated classification.

Assume now that $r\ge 2$ and put
\[
B:=[a_2,\dots,a_r],
\qquad
B':=[a_3,\dots,a_r],
\]
with the convention $d(\varnothing)=1$.
Then
$d(A^\sharp)=a_1\,d(B)-d(B'),$
and
\[
\mu(A^\sharp,1)=\frac{1+d(B)}{d(A^\sharp)}
=\frac{1+d(B)}{a_1\,d(B)-d(B')}.
\]

\smallskip
\noindent\emph{Case 1: $a_1=2$.}
Then
\[
\mu(A^\sharp,1)=\frac{1+d(B)}{2d(B)-d(B')}.
\]
Since $d(B')\ge 0$, one has
$2d(B)-d(B')<2d(B)+2,$
hence
\[
\mu(A^\sharp,1)>\frac12.
\]

\smallskip
\noindent\emph{Case 2: $a_1\ge 4$.}
Since $B$ is nonempty, one has
$d(B')\le d(B)-1.$
Therefore
\[
d(A^\sharp)=a_1 d(B)-d(B')\ge 4d(B)-(d(B)-1)=3d(B)+1.
\]
Thus
\[
\mu(A^\sharp,1)\le \frac{1+d(B)}{3d(B)+1}<\frac12.
\]

\smallskip
\noindent\emph{Case 3: $a_1=3$.}
Then
\[
\mu(A^\sharp,1)>\frac12
\iff
2(1+d(B))>3d(B)-d(B')
\iff
2+d(B')>d(B).
\]
Since $d(B)>d(B')$ and both are integers, this is equivalent to
$d(B)-d(B')=1.$

We now claim that for any chain
$\Gamma=[c_1,\dots,c_s],$
if
$\Gamma':=[c_2,\dots,c_s],$
then
$d(\Gamma)-d(\Gamma')\ge 1,$
with equality if and only if
$\Gamma=[2,\dots,2].$
This is proved by induction on $s$.
For $s=1$ one has
$d([c_1])-d(\varnothing)=c_1-1,$
so equality holds exactly when $c_1=2$.
For $s\ge 2$, writing
\[
\Gamma_1=[c_2,\dots,c_s],
\qquad
\Gamma_2=[c_3,\dots,c_s],
\]
one gets
\[
d(\Gamma)-d(\Gamma')
=
c_1 d(\Gamma_1)-d(\Gamma_2)-d(\Gamma_1)
=
(c_1-1)d(\Gamma_1)-d(\Gamma_2)
=
(c_1-2)d(\Gamma_1)+\bigl(d(\Gamma_1)-d(\Gamma_2)\bigr).
\]
Since $d(\Gamma_1)>0$, equality $=1$ holds if and only if
\[
c_1=2
\qquad\text{and}\qquad
d(\Gamma_1)-d(\Gamma_2)=1.
\]
The induction hypothesis then gives
$\Gamma_1=[2,\dots,2],$
hence
$\Gamma=[2,\dots,2].$
This proves the claim.

Applying the claim to $B$, we conclude that
$d(B)-d(B')=1$
holds if and only if
$B=[2,\dots,2].$
Thus, in the case $a_1=3$,
\[
\mu(A^\sharp,1)>\frac12
\]
if and only if
$A^\sharp=[3,(2)^m]$
for some $m\ge 0$, touched at the heavy tip.  But $A^\sharp$ is already a $2$-tail core, so no terminal $2$ away from the touched component can remain.  Hence necessarily
$m=0,$
that is,
$A^\sharp=[3].$

Combining the three cases proves the proposition.
\end{proof}

\begin{corollary}[Every positive chain bridge has a special tip-visible side]
\label{cor:routeC-special-tip-side}
Let
$L$
be a positive distinct-component bridge between two chain components
\[
(A,i)
\qquad\text{and}\qquad
(B,j)
\]
on the minimal resolution of a Picard-rank-one klt del Pezzo surface.
Let
\[
(A^\sharp,i^\sharp),\qquad (B^\sharp,j^\sharp)
\]
be the corresponding $2$-tail cores.

Then at least one of the following holds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item one of the two cores is touched at a tip of weight $2$;
    \item one of the two cores is
    $[3]$
    touched at its unique component.
\end{enumerate}
In particular, every positive chain bridge has at least one \emph{special} tip-visible side after
$2$-tail reduction.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-tip-visible-side}, at least one of the two reduced cores is touched at a tip.
Suppose, for contradiction, that neither reduced core satisfies \textnormal{(i)} or \textnormal{(ii)}.

If both reduced cores are tip-visible, then Proposition~\ref{prop:routeC-tipcore-highcontact} gives
\[
\mu(A^\sharp,i^\sharp)\le \frac12,
\qquad
\mu(B^\sharp,j^\sharp)\le \frac12.
\]
By Lemma~\ref{lem:routeC-mu-monotone-tail},
\[
\mu(A,i)\le \mu(A^\sharp,i^\sharp)\le \frac12,
\qquad
\mu(B,j)\le \mu(B^\sharp,j^\sharp)\le \frac12.
\]
Hence Proposition~\ref{prop:routeC-anticanonical-chain-bridge} yields
\[
-K_X\cdot C
=
-1+\mu(A,i)+\mu(B,j)
\le 0,
\]
contradicting the ampleness of $-K_X$.

So exactly one reduced core is tip-visible and the other is interior.
Without loss of generality, assume that
$(A^\sharp,i^\sharp)$
is tip-visible and
$(B^\sharp,j^\sharp)$
is interior.
Then Proposition~\ref{prop:routeC-tipcore-highcontact} gives
\[
\mu(A^\sharp,i^\sharp)\le \frac12,
\]
while Proposition~\ref{prop:routeC-interior-core-half} gives
\[
\mu(B^\sharp,j^\sharp)\le \frac12.
\]
Again Lemma~\ref{lem:routeC-mu-monotone-tail} implies
\[
\mu(A,i)\le \frac12,
\qquad
\mu(B,j)\le \frac12,
\]
so Proposition~\ref{prop:routeC-anticanonical-chain-bridge} gives the same contradiction
$-K_X\cdot C\le 0.$

Thus at least one reduced core must satisfy \textnormal{(i)} or \textnormal{(ii)}.
\end{proof}

\begin{remark}[What this does to the remaining chain bridge problem]
\label{rem:routeC-special-tip-side}
Corollary~\ref{cor:routeC-special-tip-side} sharpens the $2$-tail reduction substantially.
The remaining chain-shaped bridge geometry is no longer an arbitrary one-tip-visible problem.

Indeed, every positive chain bridge now has at least one special tip-visible side, and there are only two such
possibilities:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a side beginning with a touched $(-2)$-tip;
    \item a side of type
    $[3]$
    touched at its unique component.
\end{enumerate}

So the unresolved chain gap is reduced further to:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item opposite-root bridges with a touched $(-2)$-tip side;
    \item opposite-root bridges with a touched
    $[3]$
    side at its unique component;
    \item mixed-core variants obtained by attaching one of those two special sides to a branching core on the opposite
    side.
\end{enumerate}
This is strictly smaller than the previous one-tip-visible gap.
\end{remark}

\begin{remark}[What this reduces the remaining chain bridge problem to]
\label{rem:routeC-tip-visible-reduction}
Corollary~\ref{cor:routeC-tip-visible-side} already shows that the remaining non-staircase \emph{chain} bridge problem is no
longer an uncontrolled interior-hit problem.  Corollary~\ref{cor:routeC-special-tip-side} sharpens this further:
every surviving positive chain bridge has at least one \emph{special} tip-visible side after $2$-tail reduction.

Thus the unresolved chain-shaped bridge geometry is reduced to the following two classes:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item \emph{opposite-root special-tip bridges}, where the surviving root is on the opposite side and the touched side is either a $(-2)$-tip or the singleton core $[3]$;
    \item \emph{mixed-core special-tip bridges}, where one side is special-tip-visible and the opposite side carries the remaining core complexity.
\end{enumerate}
In particular, the remaining chain bridge problem is now a one-special-tip problem, not a general interior-hit problem.
\end{remark}

\begin{remark}[What is now genuinely left]
\label{rem:routeC-gap-now-heavy-bridges}
After Theorem~\ref{thm:routeC-samecomponent-allties},
Corollaries~\ref{cor:routeC-peelablefamily-not-residual}, \ref{cor:routeC-forkprototypes-not-residual},
\ref{cor:routeC-highsquare-not-residual}, and
\ref{cor:routeC-single-root-bridge-all-not-residual}, the remaining Route~C gap is bridge-specific and global.

The same-component half is already closed, the first analyzed bridge frontier is already removed from the residual
sector, and the visible off-frontier one-root staircase family
$[b]\dashv[a,(2)^{a-1}]$
is now eliminated completely in the benchmark range by the anticanonical sieve together with the earlier low-height and
elliptic-descendant exclusions.

Corollaries~\ref{cor:routeC-tip-visible-side} and~\ref{cor:routeC-special-tip-side} then sharpen the remaining chain problem further: every surviving positive
chain bridge has at least one \emph{special} tip-visible side after stripping terminal $[2]$-tails away from the touched components.
So the unresolved chain-shaped gap is no longer the general heavy-heavy or interior-hit regime.  What remains is the
genuinely global geometry of opposite-root special-tip bridges, mixed-core special-tip bridges, and branching
off-frontier bridges not yet constrained by the same anticanonical argument.
\end{remark}

\begin{remark}[The bridge analysis before the height-$4$ excess package]
\label{rem:routeC-true-global-gap}
After Theorem~\ref{thm:routeC-samecomponent-allties}, the same-component case is closed, the first analyzed bridge frontier is already removed from the residual sector, the visible one-root staircase family is excluded by the anticanonical sieve, and Corollaries~\ref{cor:routeC-tip-visible-side} and~\ref{cor:routeC-special-tip-side} reduce the remaining chain-shaped problem to one-special-tip cores.  The clean-section Hirzebruch model from Proposition~\ref{prop:routeC-clean-section-Hirzebruch} still packages the surviving geometry efficiently.  The next subsection records the exact excess identity in the height-$4$ maximal-width corridor and a corrected fiberwise graph bound, thereby isolating the remaining obstacle to a direct numerical closure.
\end{remark}

% ------------------------------------------------------------------
% Patch: rigorous section-only high-height reduction
% Suggested placement: before the current subsection
% ``The height-$4$ maximal-width corridor...''
% ------------------------------------------------------------------

\subsection{A rigorous section-only reduction in the high-height regime}

The height-$4$ maximal-width corridor is not a global endgame: the height of a rank-one
del Pezzo surface is defined as the \emph{minimum} of $F\cdot D$ over all $\PP^1$-fibrations on
the minimal resolution, and in characteristics $2$ and $3$ examples of height $6$ do occur.
Accordingly, we isolate here a genuinely unconditional reduction for the high-height case in
which the witnessing fibration is section-only.

\begin{definition}[Globally bad singly-hit tips]
\label{def:routeC-global-B1}
Let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution of a singular klt del Pezzo surface of Picard rank $1$, and let
$p\colon \widetilde X\to \PP^1$
be a $\PP^1$-fibration. For a degenerate fiber $F$, let $\mathcal B_1(F)$ be the set of
multiplicity-one components
\[
L\subset F_{\mathrm{red}},\qquad L\not\subset D,
\]
such that
$L\cdot D_{\mathrm{hor}}=1$
and the unique component of $F_{\mathrm{red}}-L$ meeting $L$ does \emph{not} belong to $D$.
Put
$\mathcal B_1:=\bigcup_F \mathcal B_1(F),$
where the union runs over all degenerate fibers.
\end{definition}

\begin{proposition}[Global bad-tip bound for witnessing fibrations of height at least $3$]
\label{prop:routeC-global-bad-tip-bound}
Let $X$ be a singular klt del Pezzo surface of Picard rank $1$, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration. Put
$h:=F\cdot D$
for a general fiber $F$, and assume that
$h\ge 3.$
Then
\[
\#\mathcal B_1\le \Sigma:=\sum_F(\sigma(F)-1)=\#D_{\mathrm{hor}}-1.
\]
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:routeC-no-fiber-in-D}, no fiber of $p$ is contained in $D$. Hence
\cite[Lemma~2.4]{PP12} gives
\[
\#D_{\mathrm{hor}}+\rho(\widetilde X)=\#D+2+\Sigma.
\]
Since $\rho(X)=1$, we have
$\rho(\widetilde X)=1+\#D,$
so indeed
$\Sigma=\#D_{\mathrm{hor}}-1.$

It remains to prove the fiberwise estimate
$\#\mathcal B_1(F)\le \sigma(F)-1$
for each degenerate fiber $F$.

Fix such an $F$. Let
$N_1,\dots,N_r$
be the connected components of
$F_{\mathrm{red}}-D,$
and put
\[
m_i:=\#\{\text{irreducible components of }N_i\}.
\]
Then
\[
\sigma(F)=\sum_{i=1}^r m_i.
\]
Every component of $\mathcal B_1(F)$ lies in some $N_i$ with $m_i\ge 2$, because by definition
its unique vertical neighbor is not contained in $D$.

Fix such an $N_i$.

If $m_i=2$, then $N_i$ is a chain of two irreducible components. At most one of them belongs to
$\mathcal B_1(F)$. Indeed, if both did, then each would meet $D$ only horizontally and would have
as its unique vertical neighbor the other one, so no component of $D\cap F$ would meet $N_i$.
Hence
$F_{\mathrm{red}}=N_i=[1,1].$
But then, because both components lie in $\mathcal B_1(F)$, each meets $D_{\mathrm{hor}}$ exactly
once, so
$F\cdot D_{\mathrm{hor}}=2,$
contrary to the assumption that $p$ is witnessing of height $h\ge 3$.
Thus $N_i$ contributes at most
$1=m_i-1$
components to $\mathcal B_1(F)$.

If $m_i\ge 3$, then the dual graph of $N_i$ is a tree with $m_i$ vertices, and every component of
$\mathcal B_1(F)$ contained in $N_i$ is a leaf of that tree. Therefore $N_i$ contributes at most
$m_i-1$
components to $\mathcal B_1(F)$.

Summing over all $i$ with $m_i\ge 2$, we obtain
\[
\#\mathcal B_1(F)
\le
\sum_{m_i\ge 2}(m_i-1)
=
\sigma(F)-\#\{i\,;\,m_i\ge 2\}
\le
\sigma(F)-1.
\]
Now summing over all degenerate fibers gives
\[
\#\mathcal B_1\le \sum_F(\sigma(F)-1)=\Sigma=\#D_{\mathrm{hor}}-1.
\]
\end{proof}

\begin{corollary}[Clean section for section-only witnessing fibrations]
\label{cor:routeC-section-only-clean-section}
Assume the setup of Proposition~\ref{prop:routeC-global-bad-tip-bound}, and assume in addition
that every horizontal component of $D$ is a $1$-section. Write
\[
D_{\mathrm{hor}}=H_1+\cdots+H_h,
\qquad h=F\cdot D.
\]
Then there exists a horizontal section
$H_0\subset D_{\mathrm{hor}}$
which meets no component of $\mathcal B_1$.
\end{corollary}

\begin{proof}
Each component of $\mathcal B_1$ is met by exactly one horizontal $1$-section. Since
$\#\mathcal B_1\le \#D_{\mathrm{hor}}-1=h-1$
by Proposition~\ref{prop:routeC-global-bad-tip-bound}, at least one of the $h$ sections
$H_1,\dots,H_h$ meets none of the bad tips.
\end{proof}

\begin{definition}[Special horizontal sections]
\label{def:routeC-special-section}
Assume that $p\colon \widetilde X\to \PP^1$ is a section-only witnessing $\PP^1$-fibration, and let
$H\subset D_{\mathrm{hor}}$
be a horizontal section.

We call $H$ \emph{special} if either:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the connected component of $D$ containing $H$ is not a chain; or
    \item the connected component of $D$ containing $H$ is a chain, and after applying the
    $2$-tail reduction of Definition~\ref{def:routeC-two-tail-core} to the touched pair
    $(A,H)$, the reduced touched core is either a touched tip of weight $2$ or the singleton
    chain
    $[3].$
\end{enumerate}
If neither condition holds, we call $H$ \emph{ordinary}.
\end{definition}

\begin{lemma}[Discrepancy weight on a chain component]
\label{lem:routeC-chain-discrepancy-equals-mu}
Let
\[
K_{\widetilde X}=\pi^*K_X+\sum_{E\subset D}(\mu_X(E)-1)E.
\]
Assume that a connected component of $D$ is a chain
\[
A=[A_1,\dots,A_r],
\qquad A_j^2=-a_j,
\]
and fix $1\le i\le r$. Then
\[
\mu_X(A_i)=\mu(A,i)=\frac{d(A_i^-)+d(A_i^+)}{d(A)}.
\]
Equivalently,
\[
\operatorname{cf}_X(A_i):=1-\mu_X(A_i)=1-\mu(A,i).
\]
\end{lemma}

\begin{proof}
Write
$\mu_j:=\mu_X(A_j).$
Since $\pi^*K_X\cdot A_j=0$ for every exceptional curve $A_j$, intersecting the discrepancy
formula with $A_j$ gives
\[
K_{\widetilde X}\cdot A_j
=
\sum_{k=1}^r (\mu_k-1)(A_k\cdot A_j).
\]
By adjunction,
$K_{\widetilde X}\cdot A_j=-2-A_j^2=a_j-2.$
Hence
\[
a_j-2
=
-a_j(\mu_j-1)+\sum_{k\sim j}(\mu_k-1),
\]
where $k\sim j$ means that $A_k$ is a neighbor of $A_j$ in the chain. Rearranging, we obtain
\[
a_j\mu_j-\sum_{k\sim j}\mu_k = 2-\beta_A(A_j),
\]
where $\beta_A(A_j)$ is the branching number of $A_j$ inside the chain $A$. Therefore
$a_1\mu_1-\mu_2=1,$
\[
-\mu_{j-1}+a_j\mu_j-\mu_{j+1}=0
\qquad (1<j<r),
\]
$a_r\mu_r-\mu_{r-1}=1.$
This is exactly the standard boundary-value problem defining $\mu(A,i)$ on a chain, whose unique
solution is given by the determinant formula
\[
\mu_i=\frac{d(A_i^-)+d(A_i^+)}{d(A)}.
\]
Thus $\mu_X(A_i)=\mu(A,i)$, as claimed.
\end{proof}

\begin{lemma}[Ordinary horizontal sections have discrepancy coefficient at least one half]
\label{lem:routeC-ordinary-section-half}
Assume the setup of Definition~\ref{def:routeC-special-section}. If $H\subset D_{\mathrm{hor}}$ is
ordinary, then
\[
\mu_X(H)\le \frac12
\qquad\text{and hence}\qquad
\operatorname{cf}_X(H)=1-\mu_X(H)\ge \frac12.
\]
\end{lemma}

\begin{proof}
By definition, $H$ lies on a chain component $A$ of $D$. Let $(A^\sharp,H^\sharp)$ be the touched
pair obtained from $(A,H)$ by $2$-tail reduction.

If $H^\sharp$ is an interior vertex of $A^\sharp$, then Proposition~\ref{prop:routeC-interior-core-half}
gives
\[
\mu(A^\sharp,H^\sharp)\le \frac12.
\]
If $H^\sharp$ is a tip, then $H$ is ordinary precisely when the reduced touched core is neither a
touched $2$-tip nor the singleton $[3]$. In that case Proposition~\ref{prop:routeC-tipcore-highcontact}
gives again
\[
\mu(A^\sharp,H^\sharp)\le \frac12.
\]
Finally, Lemma~\ref{lem:routeC-mu-monotone-tail} shows that removing terminal $2$-tails away from the
touched component can only increase the contact weight, so
\[
\mu(A,H)\le \mu(A^\sharp,H^\sharp)\le \frac12.
\]
By Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu},
$\mu_X(H)=\mu(A,H),$
which proves the claim.
\end{proof}

\begin{lemma}[Anticanonical degree of a witnessing fiber]
\label{lem:routeC-fiber-anticanonical-cf}
Let $X$ be a singular klt del Pezzo surface of Picard rank $1$, let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration. If $F$ is a general fiber, then
\[
-K_X\cdot \pi(F)
=
2-
\sum_{H\subset D_{\mathrm{hor}}}
\operatorname{cf}_X(H)\,(F\cdot H).
\]
In particular,
\[
\sum_{H\subset D_{\mathrm{hor}}}
\operatorname{cf}_X(H)\,(F\cdot H)<2.
\]
\end{lemma}

\begin{proof}
Write
\[
K_{\widetilde X}=\pi^*K_X+\sum_{E\subset D}(\mu_X(E)-1)E,
\qquad
\operatorname{cf}_X(E):=1-\mu_X(E).
\]
Then
\[
\pi^*K_X=K_{\widetilde X}+\sum_{E\subset D}\operatorname{cf}_X(E)E.
\]
Intersecting with a general fiber $F$, and using that only horizontal exceptional components meet
$F$, gives
\[
K_X\cdot \pi(F)
=
\pi^*K_X\cdot F
=
K_{\widetilde X}\cdot F+
\sum_{H\subset D_{\mathrm{hor}}}\operatorname{cf}_X(H)(F\cdot H).
\]
Since $F\cong \PP^1$ is a fiber of a ruling on a smooth surface,
$K_{\widetilde X}\cdot F=-2.$
Hence
\[
-K_X\cdot \pi(F)
=
2-
\sum_{H\subset D_{\mathrm{hor}}}\operatorname{cf}_X(H)(F\cdot H).
\]
Finally, $\pi(F)$ is a nonzero effective curve on the del Pezzo surface $X$, so
$-K_X\cdot \pi(F)>0.$
This proves the strict inequality.
\end{proof}

\begin{proposition}[Many special sections in the section-only high-height regime]
\label{prop:routeC-section-only-many-special}
Let $X$ be a characteristic-$3$ klt del Pezzo surface with
\[
\rho(X)=1,
\qquad
\#\Sing(X)\ge 8,
\]
and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration such that every horizontal component of $D$ is a $1$-section.
Write
\[
D_{\mathrm{hor}}=H_1+\cdots+H_h,
\qquad h=\operatorname{ht}(X),
\]
and let
$s_{\mathrm{sp}}$
be the number of special horizontal sections in the sense of
Definition~\ref{def:routeC-special-section}. Then
$s_{\mathrm{sp}}\ge h-3.$
Equivalently, the number of ordinary horizontal sections is at most $3$.
\end{proposition}

\begin{proof}
For every ordinary horizontal section $H_i$, Lemma~\ref{lem:routeC-ordinary-section-half} gives
\[
\operatorname{cf}_X(H_i)\ge \frac12.
\]
For a special section we only use the trivial bound
$\operatorname{cf}_X(H_i)\ge 0.$
Since the fibration is section-only, we have
\[
F\cdot H_i=1
\qquad\text{for every }i.
\]
Hence Lemma~\ref{lem:routeC-fiber-anticanonical-cf} gives
\[
2>
\sum_{i=1}^h \operatorname{cf}_X(H_i).
\]
Splitting this sum into ordinary and special sections, we obtain
\[
2>
\sum_{H_i\ \mathrm{ordinary}} \operatorname{cf}_X(H_i)
+
\sum_{H_i\ \mathrm{special}} \operatorname{cf}_X(H_i)
\ge
\frac12(h-s_{\mathrm{sp}}).
\]
Therefore
$s_{\mathrm{sp}}>h-4.$
Since $s_{\mathrm{sp}}$ is an integer, it follows that
$s_{\mathrm{sp}}\ge h-3.$
\end{proof}

\begin{corollary}[Clean section plus special sections]
\label{cor:routeC-section-only-clean-plus-special}
Assume the setup of Proposition~\ref{prop:routeC-section-only-many-special}. Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item there exists a clean horizontal section
    $H_0\subset D_{\mathrm{hor}}$
    meeting no component of $\mathcal B_1$;
    \item if $h\ge 5$, then there is at least one special horizontal section distinct from $H_0$;
    \item if $h\ge 6$, then there are at least two special horizontal sections distinct from $H_0$.
\end{enumerate}
\end{corollary}

\begin{proof}
Part \textnormal{(1)} is Corollary~\ref{cor:routeC-section-only-clean-section}.

For \textnormal{(2)} and \textnormal{(3)}, Proposition~\ref{prop:routeC-section-only-many-special}
gives
$s_{\mathrm{sp}}\ge h-3.$
If $H_0$ is ordinary, then all $s_{\mathrm{sp}}$ special sections are distinct from $H_0$. If $H_0$ is
special, then there are at least
$s_{\mathrm{sp}}-1\ge h-4$
special sections distinct from $H_0$. Hence for $h\ge 5$ there is at least one such section, and for
$h\ge 6$ there are at least two.
\end{proof}

\begin{remark}[What this reduction achieves]
\label{rem:routeC-section-only-highheight-reduction}
The preceding results give a rigorous replacement for the false height-$4$ endgame in the
section-only regime. If a residual counterexample admits a section-only witnessing fibration of
height $h\ge 5$, then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item one horizontal section is clean in the sense of
    Corollary~\ref{cor:routeC-section-only-clean-section}; and
    \item all but at most three horizontal sections are special in the sense of
    Definition~\ref{def:routeC-special-section}.
\end{enumerate}
Thus the section-only high-height problem is reduced to a ruled-surface collision problem for many
special sections relative to a clean section. What still remains is a genuine geometric input
showing that such a configuration yields Route~C entry, a descendant with elliptic boundary, or one
of the already excluded explicit low-height / fork / Frobenius models.
\end{remark}


\subsection{A stronger unconditional high-height reduction: all but five units of horizontal multiplicity are strong}

The preceding section-only reduction shows that many horizontal sections are special.  One can sharpen this globally by
separating the chain case into the genuinely hard touched $2$-tip cores, the intermediate singleton $[3]$ case, and the
remaining ordinary cores.

\begin{definition}[Strong, mild, and ordinary horizontal components]
\label{def:routeC-strong-horizontal}
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration, and let
$H\subset D_{\mathrm{hor}}$
be an irreducible horizontal component.

We call $H$ \emph{strong} if either
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the connected component of $D$ containing $H$ is not a chain; or
    \item the connected component of $D$ containing $H$ is a chain, and after $2$-tail reduction of the touched chain
    pair relative to $H$, the reduced touched core is tip-visible and the touched tip has weight $2$.
\end{enumerate}

If the connected component of $D$ containing $H$ is a chain and its reduced touched core is
$[3]$
touched at its unique component, then we call $H$ \emph{mild}.

All remaining horizontal components are called \emph{ordinary}.
\end{definition}

\begin{lemma}[Mild and ordinary chain components have controlled discrepancy]
\label{lem:routeC-mild-ordinary-cf}
Assume the setup of Definition~\ref{def:routeC-strong-horizontal}, and assume that $H\subset D_{\mathrm{hor}}$ lies on a
chain component of $D$.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $H$ is mild, then
    \[
    \operatorname{cf}_X(H)\ge \frac13;
    \]
    \item if $H$ is ordinary, then
    \[
    \operatorname{cf}_X(H)\ge \frac12.
    \]
\end{enumerate}
\end{lemma}

\begin{proof}
Let $(A,H)$ be the touched chain pair determined by the connected chain component of $D$ containing $H$, and let
$(A^\sharp,H^\sharp)$ be its $2$-tail reduction.
By Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu},
$\operatorname{cf}_X(H)=1-\mu(A,H).$
By Lemma~\ref{lem:routeC-mu-monotone-tail}, removing terminal $2$-tails away from the touched component can only increase
$\mu$, so
$\mu(A,H)\le \mu(A^\sharp,H^\sharp).$

If $H$ is mild, then by definition
$(A^\sharp,H^\sharp)=[3]$
touched at its unique component.  Hence
\[
\mu(A^\sharp,H^\sharp)=\frac23,
\]
so
\[
\mu(A,H)\le \frac23.
\]
Therefore
\[
\operatorname{cf}_X(H)=1-\mu(A,H)\ge \frac13.
\]

If $H$ is ordinary, then the reduced touched core is neither an interior core with $\mu>1/2$, nor a touched $2$-tip,
nor the singleton $[3]$.  Hence Proposition~\ref{prop:routeC-interior-core-half} and
Proposition~\ref{prop:routeC-tipcore-highcontact} give
\[
\mu(A^\sharp,H^\sharp)\le \frac12.
\]
Thus
\[
\mu(A,H)\le \frac12,
\qquad\text{hence}\qquad
\operatorname{cf}_X(H)=1-\mu(A,H)\ge \frac12.
\]
\end{proof}

\begin{theorem}[All but five units of horizontal multiplicity are strong]
\label{thm:routeC-strong-multiplicity-hminus5}
Let $X$ be a klt del Pezzo surface of Picard rank $1$, let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration.
Let
$F$
be a general fiber, and write
\[
D_{\mathrm{hor}}=\sum_i H_i,
\qquad
m_i:=F\cdot H_i.
\]
Put
\[
h:=\sum_i m_i=\operatorname{ht}(X).
\]

Let
\[
R:=\sum_{H_i\ \mathrm{strong}} m_i
\]
be the total horizontal multiplicity carried by strong horizontal components.
Then
$R\ge h-5.$
Equivalently, the total horizontal multiplicity carried by mild or ordinary horizontal components is at most $5$.
\end{theorem}

\begin{proof}
Split the horizontal components into strong, mild, and ordinary ones.
For strong components we only use the trivial inequality
$\operatorname{cf}_X(H_i)\ge 0.$
If $H_i$ is mild, then Lemma~\ref{lem:routeC-mild-ordinary-cf} gives
\[
\operatorname{cf}_X(H_i)\ge \frac13.
\]
If $H_i$ is ordinary, the same lemma gives
\[
\operatorname{cf}_X(H_i)\ge \frac12.
\]

Let
\[
M:=\sum_{H_i\ \mathrm{mild}} m_i,
\qquad
O:=\sum_{H_i\ \mathrm{ordinary}} m_i.
\]
Then
$h=R+M+O.$

By Lemma~\ref{lem:routeC-fiber-anticanonical-cf},
\[
\sum_i \operatorname{cf}_X(H_i)\,m_i<2.
\]
Therefore
\[
2>
\sum_i \operatorname{cf}_X(H_i)\,m_i
\ge
\frac13 M+\frac12 O
\ge
\frac13(M+O)
=
\frac13(h-R).
\]
Hence
$h-R<6.$
Since $h-R$ is an integer, this yields
\[
h-R\le 5,
\qquad\text{i.e.}\qquad
R\ge h-5.
\]
\end{proof}

\begin{corollary}[Section-only consequences of the strong bound]
\label{cor:routeC-strong-sections}
In the setup of Theorem~\ref{thm:routeC-strong-multiplicity-hminus5}, assume moreover that every horizontal component is a
$1$-section.
Then at least
$h-5$
of the horizontal sections are strong.

In particular:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $h\ge 6$, there exists a strong horizontal section;
    \item if $h\ge 7$, there exist at least two strong horizontal sections.
\end{enumerate}
\end{corollary}

\begin{proof}
In the section-only case one has
$m_i=1$
for every horizontal component, so Theorem~\ref{thm:routeC-strong-multiplicity-hminus5} says precisely that at least
$h-5$ of them are strong.
\end{proof}

\begin{corollary}[Clean section plus strong sections]
\label{cor:routeC-clean-plus-strong}
Assume the setup of Corollary~\ref{cor:routeC-section-only-clean-section}, and assume that the witnessing fibration is
section-only of height $h$.
Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item there exists a clean section
    $H_0\subset D_{\mathrm{hor}}$
    meeting no component of $\mathcal B_1$;
    \item if $h\ge 7$, there exists at least one strong horizontal section distinct from $H_0$;
    \item if $h\ge 8$, there exist at least two strong horizontal sections distinct from $H_0$.
\end{enumerate}
\end{corollary}

\begin{proof}
Part \textnormal{(1)} is Corollary~\ref{cor:routeC-section-only-clean-section}.
By Corollary~\ref{cor:routeC-strong-sections}, the total number of strong sections is at least $h-5$.
If $H_0$ is not strong, then all of those strong sections are distinct from $H_0$.
If $H_0$ is strong, then the number of strong sections distinct from $H_0$ is at least
$(h-5)-1=h-6.$
Therefore for $h\ge 7$ there is at least one such section, and for $h\ge 8$ there are at least two.
\end{proof}

\begin{remark}[What this strengthens in the section-only regime]
\label{rem:routeC-strong-highheight-reduction}
Theorem~\ref{thm:routeC-strong-multiplicity-hminus5} is strictly stronger than the earlier special-section bound.
It shows that the non-strong horizontal geometry has total multiplicity at most $5$.
Thus, in a section-only witnessing fibration of height $h$, at least $h-5$ sections are already of the genuinely hard
kind: branching sections or sections whose reduced chain core is a touched $2$-tip.

In particular, if $h\ge 8$, then after choosing a clean section one still has at least two strong sections distinct
from it.  The remaining geometric input is therefore reduced to the collision behaviour of strong sections relative to
a clean section.

\end{remark}

\subsection{General clean-section geometry and section packets in the high-height regime}

The strong-section bound reduces the section-only high-height regime to the collision geometry of many strong sections
relative to a clean section.  The next two propositions record the corresponding local and global ruled-surface models.

\begin{proposition}[General clean-section Hirzebruch model for section-only witnesses]
\label{prop:routeC-general-clean-section-Hirzebruch}
Let $X$ be a singular klt del Pezzo surface of Picard rank $1$, let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration such that every horizontal component of $D$ is a $1$-section.
Choose a clean section
$H_0\subset D_{\mathrm{hor}}$
as in Corollary~\ref{cor:routeC-section-only-clean-section}, and let
$\upsilon\colon \widetilde X\to \mathbb F_n$
be the contraction of all vertical curves disjoint from $H_0$.
Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $S_0:=\upsilon(H_0)$ is the negative section of $\mathbb F_n$, with
    $n=-H_0^2\ge 2.$
    \item For every other horizontal section $H\subset D_{\mathrm{hor}}-H_0$, the image
    $S:=\upsilon(H)$
    is a section of $\mathbb F_n$ satisfying
    \[
    S\sim S_0+(n+H\cdot H_0)f,
    \qquad
    S^2=n+2(H\cdot H_0),
    \qquad
    S\cdot S_0=H\cdot H_0.
    \]
    \item If $H,H'\subset D_{\mathrm{hor}}-H_0$ are distinct horizontal sections, then their images satisfy
    \[
    \upsilon(H)\cdot \upsilon(H')=n+(H\cdot H_0)+(H'\cdot H_0)>0.
    \]
    In particular, any two horizontal sections distinct from $H_0$ meet on $\mathbb F_n$.
\end{enumerate}
\end{proposition}

\begin{proof}
The proof is the same as in Proposition~\ref{prop:routeC-clean-section-Hirzebruch}.  Contracting all vertical curves
which do not meet $H_0$ produces a relatively minimal ruled surface $\mathbb F_n$, and $S_0:=\upsilon(H_0)$ is its
unique negative section because $H_0^2\le -2$.  Thus
\[
S_0^2=-n,
\qquad n=-H_0^2\ge 2.
\]

If $H\subset D_{\mathrm{hor}}-H_0$ is another horizontal $1$-section and $S:=\upsilon(H)$, then $S$ is again a
section.  Writing $S\sim S_0+bf$, one has
$S\cdot S_0=b-n=H\cdot H_0,$
so
$b=n+H\cdot H_0.$
Therefore
\[
S\sim S_0+(n+H\cdot H_0)f,
\qquad
S^2=-n+2(n+H\cdot H_0)=n+2(H\cdot H_0).
\]
This proves \textnormal{(1)} and \textnormal{(2)}.

For \textnormal{(3)}, if $S=\upsilon(H)$ and $S'=\upsilon(H')$, then
\[
S\cdot S'=(S_0+(n+H\cdot H_0)f)(S_0+(n+H'\cdot H_0)f)
=-n+(n+H\cdot H_0)+(n+H'\cdot H_0),
\]
which is the displayed formula.  Since $n\ge 2$, this number is positive.
\end{proof}

\begin{proposition}[Primitive isolated collisions of two sections]
\label{prop:routeC-primitive-isolated-pair-collision}
Assume the setup of Proposition~\ref{prop:routeC-general-clean-section-Hirzebruch}, and let
$H_i,H_j\subset D_{\mathrm{hor}}-H_0$
be two distinct horizontal sections with images
\[
S_i:=\upsilon(H_i),
\qquad
S_j:=\upsilon(H_j)
\]
on $\mathbb F_n$.
Let
$o\in S_i\cap S_j$
be a point through which no other horizontal image passes, and put
$m:=(S_i\cdot S_j)_o\ge 1.$
Then the first $m$ blowups of $\upsilon^{-1}$ over $o$ are forced and coincide with the order-$m$ two-section collision
resolution of Lemma~\ref{lem:routeC-pair-collision-order-p}.

Consequently, at the primitive separation stage one obtains a local chain
$F_1-\cdots-F_m$
such that
\[
F_k^2=-2\ \ (1\le k<m),
\qquad
F_m^2=-1,
\]
and the strict transforms of $S_i$ and $S_j$ both meet $F_m$ transversely at distinct points.
In particular:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $m=1$, then the primitive last exceptional curve $F_1$ is a primitive contact-$2$ curve meeting the two
    colliding sections;
    \item if $m\ge 2$, then the primitive last exceptional curve $F_m$ is the center of a primitive fork with three
    arms: the two colliding sections and the chain
    $F_{m-1}-\cdots-F_1.$
\end{enumerate}
\end{proposition}

\begin{proof}
Because no other horizontal image passes through $o$, before the strict transforms of $S_i$ and $S_j$ separate there is
at each step a unique possible center: the current intersection point of those two strict transforms.  Hence the first
$m$ blowups over $o$ are forced.

In suitable local coordinates $(t,z)$ centered at $o$ with the fiber through $o$ given by $\{t=0\}$, the two sections
are analytically equivalent to
$z=0, \qquad z=t^m.$
Therefore Lemma~\ref{lem:routeC-pair-collision-order-p} applies and gives the displayed chain
$F_1-\cdots-F_m,$
its self-intersections, and the transverse contacts of the two strict transforms with $F_m$.
If $m=1$, there is no earlier exceptional curve, so $F_1$ is a primitive contact-$2$ curve.
If $m\ge 2$, then $F_m$ also meets the adjacent exceptional component $F_{m-1}$, so the primitive local core is a
fork with the stated three arms.
\end{proof}

\begin{proposition}[Ramification of a horizontal multisection gives a primitive fork]
\label{prop:routeC-multisection-ramification-fork}
Let
$q\in S\subset Y_0$
be a smooth point of a smooth horizontal curve on a relatively minimal ruled surface
$Y_0\to \PP^1,$
and assume that the induced map
$S\to \PP^1$
has local degree
$e\ge 2$
at $q$.
Then, after the first $e$ forced blowups separating $S$ from the fiber through $q$, the reduced fiber over $q$ is
$F'-E_e-E_{e-1}-\cdots-E_1,$
with
\[
F'^2=-e,
\qquad
E_k^2=-2\ \ (1\le k<e),
\qquad
E_e^2=-1,
\]
and the strict transform of $S$ meets $E_e$ transversely away from $F'\cap E_e$.
In particular, the primitive last exceptional curve $E_e$ is the center of a primitive fork with three arms:
\[
S,
\qquad
F',
\qquad
E_{e-1}-\cdots-E_1.
\]
\end{proposition}

\begin{proof}
Choose local coordinates $(t,z)$ at $q$ so that the fiber through $q$ is $F_0=\{t=0\}$ and the multisection is
$S=\{t=z^e\}.$
At each stage the current strict transforms of $S$ and the fiber meet in a unique point, so the first $e$ blowups are
forced.  A direct induction shows that after the $k$-th blowup one obtains a chain
$F_k-E_k-E_{k-1}-\cdots-E_1$
with
\[
F_k^2=-k,
\qquad
E_i^2=-2\ \ (1\le i<k),
\qquad
E_k^2=-1,
\]
and the strict transform of $S$ meeting $E_k$ transversely at a point away from $F_k\cap E_k$.  Taking $k=e$ gives the
stated configuration.  Since $e\ge 2$, the last exceptional curve $E_e$ meets exactly the three displayed branches,
so it is a primitive fork center.
\end{proof}

\begin{proposition}[Packet theorem for many sections with collisions on at most three fibers]
\label{prop:routeC-section-packet-theorem}
Let
$S_1,\dots,S_r$
be sections of the same ruling class on $\mathbb F_n$, disjoint from the negative section $S_0$.
Assume that every pairwise intersection of the $S_i$ is supported on a set of at most three fibers.
After an automorphism of the base, assume those fibers are
$\{x=0\},\qquad \{y=0\},\qquad \{x-y=0\}.$
After translating by $S_1$, write
\[
S_1=\{z=0\},
\qquad
S_i=\{z=q_i\}\quad (i\ge 2),
\]
with
$q_i\in H^0(\PP^1,\mathcal O_{\PP^1}(n)).$
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item All nonzero $q_i$ are scalar multiples of one monomial
    $g=x^ay^b(x-y)^c.$
    Equivalently, the whole packet is a scalar-multiple star.
    \item There exist nonnegative integers $a,b,c,e$ with
    $a+b+c+3^e=n$
    such that every $q_i$ belongs to
    \[
    \{0,\ gx^{3^e},\ gy^{3^e},\ g(x-y)^{3^e}\},
    \qquad g=x^ay^b(x-y)^c.
    \]
    In particular, in the non-scalar case the packet contains at most four sections total.
\end{enumerate}
\end{proposition}

\begin{proof}
Since each pairwise intersection $S_i\cap S_j$ is supported on the three distinguished fibers, every difference
$q_i-q_j$ has zero divisor supported on those fibers.  In particular, every nonzero $q_i$ itself has support on those
fibers, so each $q_i$ is a scalar multiple of a monomial in $x$, $y$, and $x-y$.

If all nonzero $q_i$ are scalar multiples of one such monomial, we are in case \textnormal{(i)}.
Assume from now on that this is not the case.  Then there exist two nonzero sections, say $q_2$ and $q_3$, which are
not scalar multiples of each other.  Applying Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius} to the
triple
$0,\ q_2,\ q_3$
shows that, after renumbering, there exist nonnegative integers $a,b,c,e$ with
$a+b+c+3^e=n$
and
\[
q_2=gx^{3^e},
\qquad
q_3=gy^{3^e},
\qquad
q_2-q_3=g(x-y)^{3^e},
\qquad
 g=x^ay^b(x-y)^c.
\]

Let $i\ge 4$ and suppose $q_i\neq 0$.  Because $q_i-q_2$ has support on the three distinguished fibers, the triple
$0,\ q_2,\ q_i$
again satisfies the hypotheses of Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}.  Since $q_i$ is not a
scalar multiple of $q_2$ in the non-scalar branch, the proof of that proposition shows that after dividing by the
maximal common factor of $q_i$ and $q_2$, the primitive pair consists of pure powers of two distinct linear forms and
its difference is the corresponding third power.  Because
$q_2=gx^{3^e},$
this forces the maximal common factor of $q_i$ and $q_2$ to contain $g$.  Repeating the same argument with the pair
$(q_i,q_3)$ forces the same conclusion.  Hence
$q_i=gM_i$
for some degree-$3^e$ monomial $M_i$ in $x,y,x-y$.

Now both
\[
M_i-x^{3^e}
\qquad\text{and}\qquad
M_i-y^{3^e}
\]
have zero divisors supported on the three distinguished fibers.  Applying Proposition~\ref{prop:routeC-clean-section-disjoint-frobenius}
with common factor $1$ shows that
$M_i\in\{x^{3^e},y^{3^e},(x-y)^{3^e}\}$
up to a scalar.  Finally, that scalar must equal $1$: for example, if
$M_i=\lambda x^{3^e},$
then
$\lambda x^{3^e}-y^{3^e}$
vanishes on the fiber
$x-\lambda^{-1/3^e}y=0,$
which belongs to the distinguished set only when $\lambda=1$.
Thus
$q_i\in\{gx^{3^e},gy^{3^e},g(x-y)^{3^e}\}.$
This proves case \textnormal{(ii)}.
\end{proof}

\begin{lemma}[Elementary transform at a packet--negative-section point]
\label{lem:routeC-packet-negative-section-transform}
Let
\[
\epsilon_p\colon \mathbb F_n\dashrightarrow \mathbb F_{n+1}
\]
be the elementary transform at a point
$p\in S_0,$
where
$S_0^2=-n$
is the negative section of
$\mathbb F_n.$
Let
$S\subset \mathbb F_n$
be a section, write
\[
S\equiv S_0+(n+\varepsilon)f,
\qquad
\varepsilon\in\{0,1\},
\]
and put
$m:=\operatorname{mult}_p(S)\in\{0,1\}.$
Let
$S^+\subset \mathbb F_{n+1}$
be the image of
$S.$
Then
$S^+\equiv S_0^+ + (n+1+\varepsilon-m)f^+,$
where
$(S_0^+)^2=-(n+1)$
is the negative section of
$\mathbb F_{n+1}.$
In particular,
$S^+\cdot S_0^+=\varepsilon-m\in\{0,1\}.$
So the transform at
$p$
removes the unique packet--negative-section contact exactly for those sections passing through
$p,$
and does not create a new one for sections already disjoint from
$S_0.$
Moreover, if
$S_1,S_2\subset \mathbb F_n$
are two such sections, then every fiber supporting
$S_1^+\cap S_2^+$
already supported
$S_1\cap S_2.$
In particular, the elementary transform at
$p$
does not create a new packet-collision fiber.
\end{lemma}

\begin{proof}
The class formula is exactly Lemma~\ref{lem:routeC-elementary-transform-class-on-negative} with
\[
a=1,
\qquad
b=n+\varepsilon,
\qquad
m\in\{0,1\}.
\]
Thus
$S^+\equiv S_0^+ + (n+1+\varepsilon-m)f^+,$
and intersecting with
$S_0^+$
gives
$S^+\cdot S_0^+=\varepsilon-m.$
This proves the first claim.

For the collision-fiber statement, it is enough to work locally over the fiber through
$p.$
Choose analytic coordinates
$(t,z)$
centered at
$p$
so that the fiber is
$F_0=\{t=0\}$
and the negative section is
$S_0=\{z=0\}.$
Because every packet section meets
$S_0$
with multiplicity at most one, any section through
$p$
has local equation
\[
z=a t+O(t^2),
\qquad a\neq 0,
\]
while a section not passing through
$p$
is given locally by
\[
z=c+O(t),
\qquad c\neq 0.
\]
After blowing up
$p,$
in the chart
$z=tw$
all sections through
$p$
have strict transforms
$w=a+O(t),$
so they meet the new exceptional curve at points determined by their first-order slopes, whereas sections not through
$p$
are disjoint from that exceptional curve.  After contracting the strict transform of
$F_0,$
a pair of sections which was disjoint over
$F_0$
remains disjoint over the transformed fiber, while a pair colliding at
$p$
may either remain on that fiber or separate there.  Hence no new collision fiber is created.
\end{proof}

\begin{proposition}[Preparing a clean-section packet by transforms along the negative section]
\label{prop:routeC-packet-preparation-negative-section}
Assume the setup of Proposition~\ref{prop:routeC-general-clean-section-Hirzebruch}.  Let
$S_i:=\upsilon(H_i)$
for the horizontal sections distinct from
$H_0,$
and let
$T\subset S_0$
be the set of distinct points lying on at least one of the sections
$S_i.$
Write
$t:=\#T.$
Perform successive elementary transforms at the points of
$T,$
always on the current negative section, and let
\[
\tau\colon \mathbb F_n\dashrightarrow \mathbb F_{n+t}
\]
be the resulting birational map.  Denote by
$S_0^\dagger$
the negative section of the final Hirzebruch surface and by
$S_i^\dagger$
the final transform of
$S_i.$
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item every
    $S_i^\dagger$
    is disjoint from
    $S_0^\dagger;$

    \item all final packet sections lie in one ruling class,
    \[
    S_i^\dagger\equiv S_0^\dagger+(n+t)f^\dagger
    \qquad\text{for every }i;
    \]

    \item for every pair of indices
    $i\neq j,$
    every fiber supporting
    $S_i^\dagger\cap S_j^\dagger$
    already supported
    $S_i\cap S_j.$
    In particular, the total set of packet-collision fibers does not increase under the preparation process.
\end{enumerate}
\end{proposition}

\begin{proof}
Because
$\upsilon$
is an isomorphism near
$H_0,$
one has
$S_i\cdot S_0=H_i\cdot H_0\in\{0,1\}$
for every packet section
$S_i,$
since distinct irreducible components of the reduced exceptional divisor meet at most once.
Hence each
$S_i$
passes through at most one point of
$T.$

Apply Lemma~\ref{lem:routeC-packet-negative-section-transform} successively at the points of
$T.$
Each transform raises the Hirzebruch index by one and removes the unique negative-section contact exactly for the packet
sections passing through the chosen point.  Since every packet section meets the negative section in at most one point,
after all
$t$
steps every final transform
$S_i^\dagger$
is disjoint from the final negative section
$S_0^\dagger.$
This proves \textnormal{(i)}.

For \textnormal{(ii)}, let
$\varepsilon_i:=S_i\cdot S_0\in\{0,1\}.$
By Lemma~\ref{lem:routeC-packet-negative-section-transform}, each of the
$t$
negative-section elementary transforms adds one to the fiber coefficient of
$S_i,$
and exactly one of those steps subtracts one again when
$\varepsilon_i=1,$
namely the step at the unique point of
$T$
lying on
$S_i.$
Therefore the final class is
\[
S_i^\dagger\equiv S_0^\dagger + (n+\varepsilon_i+t-\varepsilon_i)f^\dagger
= S_0^\dagger + (n+t)f^\dagger.
\]
So all prepared packet sections lie in the same ruling class.

Finally, \textnormal{(iii)} follows by iterating the last statement of
Lemma~\ref{lem:routeC-packet-negative-section-transform}: a negative-section elementary transform never creates a new
fiber carrying a pairwise packet collision.
\end{proof}

\begin{corollary}[The non-scalar packet branch is finite after preparation]
\label{cor:routeC-nonscalar-packet-finite}
Assume the setup of Proposition~\ref{prop:routeC-general-clean-section-Hirzebruch}, and let
$S_i^\dagger$
be the prepared packet from Proposition~\ref{prop:routeC-packet-preparation-negative-section}.
Assume that every pairwise intersection of the original packet sections
$S_i$
is supported on at most three fibers.
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the prepared packet is a scalar-multiple star;
    \item the prepared packet is contained in a Frobenius tetrahedron
    \[
    0,\quad gx^{3^e},\quad gy^{3^e},\quad g(x-y)^{3^e},
    \]
    and in particular contains at most four sections total.
\end{enumerate}
Therefore, if there are at least five horizontal sections distinct from
$H_0,$
then only the prepared scalar case can occur.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-packet-preparation-negative-section}\textnormal{(iii)}, the prepared packet still has all
pairwise intersections supported on at most three fibers.  By
Proposition~\ref{prop:routeC-packet-preparation-negative-section}\textnormal{(i)}--\textnormal{(ii)}, all prepared packet
sections are disjoint from the final negative section and lie in one ruling class.  Hence
Proposition~\ref{prop:routeC-section-packet-theorem} applies to the prepared packet and yields exactly the two displayed
alternatives.  The final assertion is immediate from the bound of at most four sections in the non-scalar branch.
\end{proof}



\begin{proposition}[Local scalar-star model]
\label{prop:routeC-scalar-star-local}
Let $U$ be a smooth surface with local coordinates $(t,z)$ at a point $o$, and let
\[
A_0=\{z=0\},
\qquad
A_i=\{z=\lambda_i t^u\}
\quad (1\le i\le m),
\]
where
\[
u\ge 1,
\qquad
\lambda_i\neq \lambda_j\ \text{for }i\neq j.
\]
Let
$\tau_u\colon U_u\to U$
be the composition of the $u$ successive blowups at the unique common point of the strict transforms of
$A_0,A_1,\dots,A_m.$
Then:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item the exceptional locus is the chain
    $G_1-\cdots-G_u;$
    \item one has
    \[
    G_i^2=-2\ \ (1\le i<u),
    \qquad
    G_u^2=-1;
    \]
    \item the strict transforms of
    $A_0,\dots,A_m$
    meet
    $G_u$
    transversely at pairwise distinct points and are disjoint from
    $G_1,\dots,G_{u-1}.$
\end{enumerate}
\end{proposition}

\begin{proof}
We argue by induction on $u$.
For $u=1$, the first blowup separates the common point of the curves $A_0,\dots,A_m$ and creates a single
exceptional curve $G_1$ with
$G_1^2=-1.$
Since the tangent directions of the curves $A_i=\{z=\lambda_i t\}$ are distinct, their strict transforms meet $G_1$
transversely at pairwise distinct points.

Assume now that $u\ge 2$ and that the statement holds for $u-1$.
After the first blowup at $o$, in the standard chart with coordinates $(t,w)$ given by $z=tw$, the strict transforms of
$A_0,\dots,A_m$ are
\[
A_0' = \{w=0\},
\qquad
A_i' = \{w=\lambda_i t^{u-1}\}\quad (1\le i\le m),
\]
and the new exceptional curve is $G_1=\{t=0\}$.
The strict transforms still have a unique common point, namely
\[
(t,w)=(0,0)=A_0'\cap A_1'\cap\cdots\cap A_m'\cap G_1.
\]
By the induction hypothesis, after the next $u-1$ blowups at the successive unique common points, one obtains a chain
$G_2-\cdots-G_u$
with
\[
G_i^2=-2\ \ (2\le i<u),
\qquad
G_u^2=-1,
\]
and the strict transforms of $A_0',\dots,A_m'$ meet $G_u$ transversely at distinct points and are disjoint from
$G_2,\dots,G_{u-1}$.
Since $G_1$ is created first and is blown up once more at the common point, its final self-intersection is $-2$, and it
meets only $G_2$.
This yields the full chain
$G_1-\cdots-G_u$
with the stated self-intersections and incidence relations.
\end{proof}


\begin{corollary}[High-height section-only packets become scalar after preparation]
\label{cor:routeC-highheight-packet-scalar}
Assume the setup of Proposition~\ref{prop:routeC-general-clean-section-Hirzebruch}, and assume moreover that the
witnessing fibration is section-only of height
$h\ge 6.$
Choose a clean section
$H_0$
as in Corollary~\ref{cor:routeC-clean-plus-strong}, let
$S_i:=\upsilon(H_i)$
for the horizontal sections distinct from
$H_0,$
and prepare the packet by elementary transforms at all packet--negative-section contact points as in
Proposition~\ref{prop:routeC-packet-preparation-negative-section}.
If every pairwise intersection of the original packet sections
$S_i$
is supported on at most three fibers, then the prepared packet on the final Hirzebruch surface is a scalar-multiple star.
\end{corollary}

\begin{proof}
There are exactly
$h-1\ge 5$
horizontal sections distinct from
$H_0.$
So Corollary~\ref{cor:routeC-nonscalar-packet-finite} rules out the non-scalar prepared branch.
\end{proof}

\begin{remark}[Ramification of a multisection does not by itself produce an exceptional cluster]
\label{rem:routeC-ramification-not-automatic}
A point of local degree at least $2$ on a horizontal multisection does \emph{not} by itself force any exceptional
cluster on the minimal resolution.

Indeed, on the smooth ruled surface
$Y=\PP^1_t\times \PP^1_z$
with ruling
$p=\pr_1,$
the curve
$S=\{t=z^3\}$
is a smooth horizontal $3$-section in characteristic $3$.  Every point of
$S\to \PP^1_t$
has local degree $3$, but the ruling on $Y$ has no degenerate fibers and no exceptional locus.  Thus ramification of a
horizontal multisection is only tangency to a smooth fiber unless that point is an actual center of the birational map
to a ruled model.
\end{remark}

\begin{proposition}[If a ramified point is actually blown up, the primitive local model is a fork]
\label{prop:routeC-ramification-fork-survival}
Let
$p\colon \widetilde X\to \PP^1$
be a $\PP^1$-fibration on a smooth rational surface, let
$H\subset D_{\mathrm{hor}}$
be a horizontal multisection, and let
$\nu\colon \widetilde X\to Y$
be a birational morphism onto a smooth relatively minimal ruled surface, chosen so that $\nu$ is an isomorphism at the
generic point of $H$.  Put
$S:=\nu(H).$

Let
$q\in S$
be a point such that, in local coordinates $(t,z)$ centered at $q$ and adapted to the ruling on $Y$, one has
\[
f_0=\{t=0\},
\qquad
S=\{t=z^a\},
\qquad a\ge 2.
\]
Assume that over $q$ the first $a$ centers of $\nu^{-1}$ are exactly the forced tangency-resolution centers, and that no
later center of $\nu^{-1}$ lies on the last exceptional curve created in that forced sequence.

Then on $\widetilde X$ there exist curves
\[
L\not\subset D,
\qquad
A,B_1,\dots,B_{a-1}\subset D
\]
contained in the fiber over $p(q)$ such that
\[
A^2=-a,
\qquad
B_i^2=-2 \ \ (1\le i\le a-1),
\qquad
L^2=-1,
\]
and the local dual graph of
$D+L$
near this cluster is
\[
\begin{array}{ccccc}
&&A&&\\
&&|&&\\
H&-&L&-&B_1-\cdots-B_{a-1}.
\end{array}
\]

In particular:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the primitive ramification cluster is a fork, not a one-root bridge;
    \item one has
    $L\cdot D=3;$
    \item the one-root staircase family
    $[b]\dashv[a,(2)^{a-1}]$
    does \emph{not} describe this primitive local model.
\end{enumerate}
\end{proposition}

\begin{proof}
Apply Proposition~\ref{prop:routeC-multisection-ramification-fork} to the tangency between
\[
f_0=\{t=0\}
\qquad\text{and}\qquad
S=\{t=z^a\}.
\]
Thus after the forced $a$ blowups one obtains a local chain
$F'-E_a-E_{a-1}-\cdots-E_1,$
where
\[
(F')^2=-a,
\qquad
E_i^2=-2\ \ (1\le i<a),
\qquad
E_a^2=-1,
\]
and the strict transforms of $f_0$ and $S$ both meet $E_a$ transversely at distinct points.

Let
\[
L:=E_a,
\qquad
A:=F',
\qquad
B_i:=E_{a-i}\ \ (1\le i\le a-1).
\]
By assumption the first $a$ centers of $\nu^{-1}$ are exactly these forced tangency centers, and no later center lies on
$L$.  Hence the strict transform of $L$ survives on $\widetilde X$ with self-intersection still equal to $-1$.
Therefore
$L\not\subset D,$
because on the minimal resolution of a rank-one del Pezzo surface the components of a degenerate fiber outside $D$ are
exactly the vertical $(-1)$-curves.

On the other hand,
\[
A^2=-a\le -2,
\qquad
B_i^2=-2,
\]
so
$A,B_1,\dots,B_{a-1}\subset D.$
The displayed graph follows directly from the local tangency resolution.  Since $L$ meets exactly the three components
$H,\ A,\ B_1,$
one gets
$L\cdot D=3.$
Thus the primitive local model is a fork, not a Route~C bridge.
\end{proof}

\begin{remark}[Why the ramification branch does not feed into the one-root staircase corollaries]
\label{rem:routeC-ramification-not-single-root-bridge}
The primitive ramification fork from Proposition~\ref{prop:routeC-ramification-fork-survival} is not the family
$[b]\dashv[a,(2)^{a-1}]$
from Proposition~\ref{prop:routeC-single-root-bridge-family}.  The crucial difference is that the last nonexceptional
curve meets the strict transform of the ramification fiber itself, so its total contact with $D$ is $3$, not $2$.

Moreover the primitive local fork is not, by itself, root-contractible: after contracting
$L,\ B_1,\dots,B_{a-1},$
the image of $A$ is the original smooth fiber of the ruled surface, hence a $0$-curve rather than a point.  So any
root-contractibility statement for a lone multisection ramification point is necessarily global and must involve the
whole vertical connected component of the fiber.
\end{remark}


\begin{remark}[What this sharpens]
\label{rem:routeC-scalarstar-ramification-nextgap}
The packet side and the multisection side now have precise local models.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item In the section-only high-height branch, after clearing all packet--negative-section contact points by the
    preparation process of Proposition~\ref{prop:routeC-packet-preparation-negative-section}, any packet whose pairwise
    collisions are supported on at most three fibers is forced to become a scalar-multiple star by
    Corollary~\ref{cor:routeC-highheight-packet-scalar}.  The local geometry of each marked scalar collision is then
    exactly the explicit star model of Proposition~\ref{prop:routeC-scalar-star-local}.

    \item In the multisection branch, any ramification point of local degree at least $2$ produces a primitive fork, and
    this fork survives on $\widetilde X$ whenever the tangency center is in general position for the birational map to
    the ruled model.
\end{enumerate}
Thus the remaining high-height extraction problem is no longer the primitive local birational geometry of these two
branches.  It is the final global survival/exclusion step:
\begin{enumerate}[label=\textnormal{(a)}]
    \item exclude large scalar stars from the residual sector, or show that they force one of the already eliminated
    explicit models;
    \item show that a surviving primitive ramification fork is already covered by the existing fork analysis or by the
    one-root staircase elimination.
\end{enumerate}
\end{remark}

\begin{remark}[What this pushes the high-height program to]
\label{rem:routeC-packet-next-gap}
The packet theorem is now available in a prepared form on the section-only side.
After choosing a clean section, one may first clear all packet--negative-section contact points by the preparation process
of Proposition~\ref{prop:routeC-packet-preparation-negative-section}.  The resulting packet is then a disjoint family of
sections in one ruling class, so under the three-fiber support hypothesis it is reduced to two explicit regimes:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a prepared scalar-multiple star; or
    \item a finite Frobenius tetrahedron.
\end{enumerate}
For height at least
$6,$
Corollary~\ref{cor:routeC-highheight-packet-scalar} removes the second alternative.
Together with Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}, this means that every isolated collision
of strong sections is already locally understood: transverse collisions give primitive contact-$2$ curves and higher-order
collisions give primitive forks.

Thus the remaining high-height extraction problem is concentrated in two very narrow places:
\begin{enumerate}[label=\textnormal{(a)}]
    \item the \emph{prepared scalar-star lemma}, excluding a large prepared scalar packet from the residual sector; and
    \item the \emph{survival lemma}, upgrading a primitive contact-$2$ or fork core coming from a strong collision to an
    admissible Route~C configuration on $\widetilde X$ or to one of the already eliminated explicit models.
\end{enumerate}
The multisection branch is similarly narrowed by Proposition~\ref{prop:routeC-multisection-ramification-fork}: a
ramification point of local degree at least $2$ produces a primitive fork model from the start.
\end{remark}


\subsection{A universal contact-defect sieve for surviving nonexceptional curves}

The correct global replacement for the failed one-root staircase idea is not another bridge classification, but a defect
inequality for any surviving nonexceptional vertical $(-1)$-curve.  Any such curve can meet exceptional components only
if the sum of their discrepancy defects is strictly less than $1$.  This immediately gives numerical restrictions for
both surviving ramification forks and surviving scalar stars.

\begin{lemma}[Universal discrepancy lower bound from self-intersection]
\label{lem:routeC-cf-selfintersection-lower}
Let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution of a klt del Pezzo surface, and write
\[
\pi^*K_X
=
K_{\widetilde X}+\sum_{E\subset D}\operatorname{cf}_X(E)E.
\]
If
$E\subset D$
is an irreducible component with
\[
E^2=-a,
\qquad a\ge 2,
\]
then
\[
\operatorname{cf}_X(E)\ge 1-\frac{2}{a}.
\]
Equivalently, if
$\mu_X(E):=1-\operatorname{cf}_X(E),$
then
\[
\mu_X(E)\le \frac{2}{a}.
\]
\end{lemma}

\begin{proof}
Intersect
\[
\pi^*K_X
=
K_{\widetilde X}+\sum_{T\subset D}\operatorname{cf}_X(T)T
\]
with $E$.  Since $E$ is exceptional,
$\pi^*K_X\cdot E=0.$
Because $E\cong \PP^1$ and $E^2=-a$, adjunction gives
$K_{\widetilde X}\cdot E=a-2.$
Therefore
\[
0=(a-2)-a\operatorname{cf}_X(E)+\sum_{T\neq E}(T\cdot E)\operatorname{cf}_X(T).
\]
Since every coefficient $\operatorname{cf}_X(T)$ is nonnegative, we get
$a\operatorname{cf}_X(E)\ge a-2,$
hence
\[
\operatorname{cf}_X(E)\ge 1-\frac{2}{a}.
\]
The equivalent inequality for $\mu_X(E)$ is immediate.
\end{proof}

\begin{proposition}[Universal contact-defect sieve]
\label{prop:routeC-contact-defect-sieve}
Let
$\pi\colon (\widetilde X,D)\to X$
be the minimal resolution of a klt del Pezzo surface, and let
$L\subset \widetilde X$
be a $(-1)$-curve such that
$L\not\subset D.$
For each irreducible component
$E\subset D$
put
$m_E:=L\cdot E.$
Then
\[
\sum_{E\subset D} m_E\,\operatorname{cf}_X(E)<1.
\]
Consequently,
\[
\sum_{E\subset D} m_E\Bigl(1-\frac{2}{-E^2}\Bigr)<1
\]
and equivalently
\[
\sum_{E\subset D}\frac{2m_E}{-E^2}>L\cdot D-1.
\]
\end{proposition}

\begin{proof}
Put
$C:=\pi(L)\subset X.$
Since $L\not\subset D$, the curve $C$ is nonzero and effective.  Because $-K_X$ is ample,
$-K_X\cdot C>0.$
On the other hand,
\[
\pi^*K_X
=
K_{\widetilde X}+\sum_{E\subset D}\operatorname{cf}_X(E)E,
\]
so using $L^2=-1$ and hence
$K_{\widetilde X}\cdot L=-1$
by adjunction, we get
\[
-K_X\cdot C
=
-\pi^*K_X\cdot L
=
1-\sum_{E\subset D}m_E\,\operatorname{cf}_X(E).
\]
Thus
\[
\sum_{E\subset D}m_E\,\operatorname{cf}_X(E)<1.
\]
Applying Lemma~\ref{lem:routeC-cf-selfintersection-lower} termwise gives
\[
\sum_{E\subset D} m_E\Bigl(1-\frac{2}{-E^2}\Bigr)<1.
\]
Rearranging yields
\[
\sum_{E\subset D}\frac{2m_E}{-E^2}>\sum_{E\subset D}m_E-1=L\cdot D-1.
\]
\end{proof}

\begin{corollary}[Surviving ramification forks satisfy a finite weight sieve]
\label{cor:routeC-ramification-fork-defect}
Keep the setup of Proposition~\ref{prop:routeC-ramification-fork-survival}, and write
\[
H^2=-b,
\qquad
A^2=-a,
\qquad a\ge 2,
\qquad b\ge 2.
\]
Then
\[
\frac{2}{a}+\frac{2}{b}>1.
\]
Equivalently,
$(a-2)(b-2)<4.$
In particular:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $b\ge 6$, then necessarily $a=2$;
    \item if $b\ge 4$, then necessarily $a\le 3$;
    \item if $a,b>2$, then up to order
    $(a,b)\in \{(3,3),(3,4),(3,5)\}.$
\end{enumerate}
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-ramification-fork-survival}, the surviving curve $L$ meets the three
components
$H,\ A,\ B_1$
of $D$ transversely once each.  Hence Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B_1)<1.
\]
Since every discrepancy coefficient is nonnegative,
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)<1.
\]
Now Lemma~\ref{lem:routeC-cf-selfintersection-lower} gives
\[
\operatorname{cf}_X(H)\ge 1-\frac{2}{b},
\qquad
\operatorname{cf}_X(A)\ge 1-\frac{2}{a}.
\]
Therefore
\[
1-\frac{2}{b}+1-\frac{2}{a}<1,
\]
which is equivalent to
\[
\frac{2}{a}+\frac{2}{b}>1.
\]
The numerical consequences are immediate.
\end{proof}

\begin{corollary}[Surviving scalar stars are $A_1$-heavy]
\label{cor:routeC-scalar-star-defect}
Assume that above some marked scalar packet one has the local scalar-star configuration of
Proposition~\ref{prop:routeC-scalar-star-local}, and suppose that on $\widetilde X$ the strict transform
$L$
of its last exceptional curve survives as a vertical $(-1)$-curve with
$L\not\subset D.$
Let
$H_1,\dots,H_r\subset D_{\mathrm{hor}}$
be the horizontal components passing through that scalar packet, and write
\[
H_i^2=-b_i,
\qquad b_i\ge 2.
\]
Then
\[
\sum_{i=1}^r \operatorname{cf}_X(H_i)<1.
\]
Hence
\[
\sum_{i=1}^r\Bigl(1-\frac{2}{b_i}\Bigr)<1,
\qquad\text{equivalently}\qquad
\sum_{i=1}^r\frac{2}{b_i}>r-1.
\]
In particular:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item at most two of the integers $b_i$ can exceed $2$;
    \item if two of them exceed $2$, then after reordering one has
    $(b_1,b_2)\in \{(3,3),(3,4),(3,5)\};$
    \item at least $r-2$ of the packet sections are $(-2)$-curves.
\end{enumerate}
\end{corollary}

\begin{proof}
In the local scalar-star model of Proposition~\ref{prop:routeC-scalar-star-local}, the last exceptional curve meets
every packet section once.  If the collision order is at least $2$, it may also meet one vertical component of $D$,
but all additional terms in Proposition~\ref{prop:routeC-contact-defect-sieve} are nonnegative.  Therefore
\[
\sum_{i=1}^r\operatorname{cf}_X(H_i)<1.
\]
Now apply Lemma~\ref{lem:routeC-cf-selfintersection-lower} to each $H_i$ to obtain
\[
\sum_{i=1}^r\Bigl(1-\frac{2}{b_i}\Bigr)<1.
\]
If three of the $b_i$ were at least $3$, then the left-hand side would be at least
\[
3\cdot\frac13=1,
\]
a contradiction.  Thus at most two of them exceed $2$, proving \textnormal{(i)} and \textnormal{(iii)}.
If exactly two exceed $2$, say $b_1,b_2>2$, then all others are equal to $2$, so the defect inequality becomes
\[
1-\frac{2}{b_1}+1-\frac{2}{b_2}<1,
\]
that is,
\[
\frac{2}{b_1}+\frac{2}{b_2}>1.
\]
Up to order this leaves only
$(b_1,b_2)\in \{(3,3),(3,4),(3,5)\}.$
\end{proof}

\begin{remark}[What the new sieve buys]
\label{rem:routeC-contact-defect-what-remains}
The contact-defect sieve is the correct global replacement for the failed staircase idea.

\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item A surviving ramification fork is forced into a tiny numerical range: unless the horizontal side is a
    $(-2)$-curve, the ramification order is at most $5$, and if both touched weights exceed $2$ then only the finite
    pairs
    $(3,3),\ (3,4),\ (3,5)$
    remain.

    \item A surviving scalar packet is automatically $A_1$-heavy: after discarding at most two packet sections, all
    remaining horizontal participants are $(-2)$-curves.
\end{enumerate}

What still does \emph{not} follow is existence of such a surviving curve: one still needs a global realization/survival
statement ensuring that the residual sector produces either a surviving marked scalar packet or a surviving ramification
fork to which the sieve can be applied.
\end{remark}

\subsection{The height-\texorpdfstring{$4$}{4} maximal-width corridor: exact excess and the remaining graph-theoretic gap}

The conjectural three-section rigidity package is still unnecessary for the unconditional results proved in this paper.  Once Corollary~\ref{cor:routeC-tip-entry-obstruction} applies, one has the exact excess identity
\[
\Sigma=\sum_F(\sigma(F)-1)=3,
\]
no fiber is contained in $D$, and one horizontal $1$-section avoids all genuinely bad singly-hit multiplicity-one tips.  What remains for a direct numerical closure is a fiberwise bound for the purely vertical connected components of the exceptional divisor.

The next proposition records the corrected graph-theoretic statement that follows from the dual-graph argument.

\begin{proposition}[Corrected fiberwise graph bound in the maximal-width corridor]
\label{prop:routeC-vertical-components-excess}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}.  For a degenerate fiber
$F,$
let
\[
D_F:=\sum_{E\subset D,\ E\subset F} E
\]
be the vertical part of the exceptional divisor contained in $F$.  Let
$C_1,\dots,C_s$
be the connected components of $D_F$, and let
$t(F)$
be the number of these connected components which meet $D_{\mathrm{hor}}$.  Put
$v(F):=s-t(F),$
so $v(F)$ is the number of connected components of $D_F$ which do not meet $D_{\mathrm{hor}}$.
Then
\[
s\le \sigma(F)+1
\qquad\text{and}\qquad
v(F)\le \sigma(F)+1-t(F).
\]
In particular, if
$t(F)\ge 2$
for every degenerate fiber with $v(F)>0$, then
$v(F)\le \sigma(F)-1.$
\end{proposition}

\begin{proof}
Fix a degenerate fiber $F$.  Let
$N_1,\dots,N_r$
be the connected components of
$F_{\mathrm{red}}-D,$
and for each $i$ let
\[
m_i:=\#\{\text{irreducible components of }N_i\}.
\]
By the definition of $\sigma(F)$ used in Corollary~\ref{cor:routeC-tip-entry-obstruction},
\[
\sigma(F)=\sum_{i=1}^r m_i,
\]
so in particular
$r\le \sigma(F).$

As in the previous dual-graph argument, contract each connected subgraph corresponding to some $N_i$ and each connected subgraph corresponding to some $C_j$ inside the dual graph of $F_{\mathrm{red}}$.  The resulting graph
$\Gamma_F$
is again a tree whose vertex set is
$\{N_1,\dots,N_r\}\sqcup\{C_1,\dots,C_s\},$
and every edge joins some $N_i$ to some $C_j$.

By \cite[Lemma~2.6(a)]{OpenDP}, every irreducible component of $F_{\mathrm{red}}-D$ is a $(-1)$-curve.  By \cite[Lemma~2.4]{OpenDP}, the $(-1)$-curves in a degenerate fiber are non-branching.  Therefore each $N_i$ is a chain, so it meets the rest of the fiber in at most two components.  Equivalently,
\[
\deg_{\Gamma_F}(N_i)\le 2
\qquad\text{for all }i.
\]

Since $\Gamma_F$ is a tree with $r+s$ vertices,
$|E(\Gamma_F)|=r+s-1.$
Because $\Gamma_F$ is bipartite with parts $\{N_i\}$ and $\{C_j\}$, every edge is incident with exactly one $N_i$-vertex, hence
\[
|E(\Gamma_F)|=\sum_{i=1}^r \deg_{\Gamma_F}(N_i)\le 2r.
\]
Combining these equalities gives
\[
r+s-1\le 2r,
\qquad\text{hence}\qquad
s\le r+1\le \sigma(F)+1.
\]

By definition,
$v(F)=s-t(F),$
so the first bound implies
$v(F)\le \sigma(F)+1-t(F).$
If $t(F)\ge 2$ whenever $v(F)>0$, then this becomes
$v(F)\le \sigma(F)-1$
for every degenerate fiber with $v(F)>0$, and the same inequality is trivial when $v(F)=0$.
\end{proof}

\begin{corollary}[A sufficient missing input for the excess route]
\label{cor:routeC-excess-route-sufficient}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and assume that every degenerate fiber $F$ with
$v(F)>0$
satisfies
$t(F)\ge 2.$
Then
\[
\sum_F v(F)\le \sum_F(\sigma(F)-1)=\Sigma=3.
\]
Consequently, in this situation one has
$\#\Sing(X)\le 7$
throughout the height-$4$ maximal-width corridor.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-vertical-components-excess}, the hypothesis implies
$v(F)\le \sigma(F)-1$
for every degenerate fiber $F$.  Summing over all degenerate fibers and using Corollary~\ref{cor:routeC-tip-entry-obstruction} gives
\[
\sum_F v(F)\le \sum_F(\sigma(F)-1)=\Sigma=3.
\]

Let
$c_{\mathrm{hor}}$
be the number of connected components of $D$ which contain at least one horizontal component, and let
$c_{\mathrm{vert}}$
be the number of connected components of $D$ which are purely vertical.  Then
\[
\#\pi_0(D)=c_{\mathrm{hor}}+c_{\mathrm{vert}}.
\]
Since
$D_{\mathrm{hor}}=H_0+H_1+H_2+H_3,$
one has
$c_{\mathrm{hor}}\le 4.$
Every purely vertical connected component of $D$ is contained in a unique degenerate fiber of $p$, so
\[
c_{\mathrm{vert}}=\sum_F v(F)\le 3.
\]
Therefore
$\#\pi_0(D)\le 4+3=7.$
Finally, the connected components of the exceptional divisor are exactly the exceptional divisors over the singular points of $X$, so
$\#\Sing(X)=\#\pi_0(D)\le 7.$
\end{proof}

\begin{remark}[Why the leaf argument does not close the count]
\label{rem:routeC-leaf-gap}
The tempting stronger statement
$v(F)\le \sigma(F)-1$
does \emph{not} follow from the sole fact that a degenerate fiber contains a $(-1)$-curve.  The obstruction is that a leaf of the dual graph of a degenerate fiber need not itself be a $(-1)$-curve.  A standard example is the fiber of type
$[2,1,2],$
obtained by blowing up a smooth fiber once and then blowing up the intersection of the two resulting $(-1)$-curves.  Its two leaves are $(-2)$-curves, while the unique $(-1)$-curve lies in the middle.  Therefore a purely vertical connected component of $D_F$ can be attached to the rest of the fiber by a single edge without contradicting minimality of the resolution.
\end{remark}

\begin{remark}[Why this subsection is not a global seven-point theorem]
\label{rem:routeC-height4-maxwidth-not-global}
Every statement in this subsection concerns a \emph{fixed} witnessing $\PP^1$-fibration of height $4$ and maximal width.  By definition, the inequality
$\operatorname{ht}(X)\ge 4$
says only that the minimal possible height of a witnessing fibration is at least $4$; it does not itself produce a witnessing fibration of height exactly $4$.  Likewise, maximal width is a separate existence problem.  Thus the excess package above is a corridor analysis, not a global selection theorem for arbitrary residual surfaces.
\end{remark}

\begin{remark}[Status of the endgame]
\label{rem:routeC-three-section-bypassed}
The clean-section Hirzebruch model remains useful for organizing the residual geometry, but by itself it is still only a corridor analysis on a fixed witnessing fibration.  The later sections convert this corridor analysis into the global theorem by proving the remaining height-selection statement and thereby removing the need for any additional boxed Route~C inputs.
\end{remark}


\section{A two-part endpoint formulation for the excess route}
\label{subsec:routeC-final-lemma-needed}

The previous subsection isolates two genuinely global issues behind the excess endgame: one must choose a witnessing
$\PP^1$-fibration in the height-$4$, maximal-width corridor, and one must rule out purely vertical exceptional clusters
which touch the horizontal divisor only once.  Historically, these two tasks were packaged into a single endpoint
statement.

\begin{remark}[Two-part endpoint formulation for the excess route]
\label{conj:routeC-final-lemma-needed}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$ such that
$\operatorname{ht}(X)\ge 4$
and $X$ has no descendant with elliptic boundary.  Let
$\pi\colon \widetilde X\to X$
be the minimal resolution and let
$D=\Exc(\pi)_{\mathrm{red}}.$
Then there exists a witnessing $\PP^1$-fibration
$p\colon \widetilde X\to \PP^1$
such that:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item $p$ has height $4$ and maximal width, so
    $D_{\mathrm{hor}}=H_0+H_1+H_2+H_3$
    is the union of four horizontal $1$-sections;
    \item for every degenerate fiber $F$, if the vertical divisor
    \[
    D_F:=\sum_{E\subset D,\ E\subset F} E
    \]
    has a connected component disjoint from $D_{\mathrm{hor}}$, equivalently if
    $v(F)>0,$
    then at least two connected components of $D_F$ meet $D_{\mathrm{hor}}$, i.e.
    $t(F)\ge 2.$
\end{enumerate}
\end{remark}

\begin{corollary}[This endpoint formulation implies the seven-point theorem]
\label{cor:routeC-final-lemma-finishes}
Assume the two-part endpoint formulation of Remark~\ref{conj:routeC-final-lemma-needed}.  Then every characteristic-$3$ klt del Pezzo surface of
Picard rank $1$ satisfies
$\#\Sing(X)\le 7.$
\end{corollary}

\begin{proof}
Suppose for contradiction that there exists a characteristic-$3$ klt del Pezzo surface $X$ of Picard rank $1$ with
$\#\Sing(X)\ge 8.$
By Theorem~\ref{thm:routeC-contact2-reduction}, the surface $X$ already lies in the residual height-$\ge 4$,
no-descendant sector.  Therefore the endpoint formulation of Remark~\ref{conj:routeC-final-lemma-needed} applies to $X$, and its minimal
resolution admits a witnessing $\PP^1$-fibration of height $4$ and maximal width satisfying the two-touch condition
$v(F)>0 \Longrightarrow t(F)\ge 2$
for every degenerate fiber $F$.

Now Corollary~\ref{cor:routeC-tip-entry-obstruction} applies to this fibration and gives the exact excess identity
\[
\Sigma=\sum_F(\sigma(F)-1)=3.
\]
The two-touch condition is precisely the hypothesis of Corollary~\ref{cor:routeC-excess-route-sufficient}.  Hence
$\#\Sing(X)\le 7,$
contradicting the assumption.
\end{proof}

\begin{remark}[Why this was the right endpoint formulation]
\label{rem:routeC-final-lemma-right-endpoint}
Remark~\ref{conj:routeC-final-lemma-needed} bypasses the earlier boxed Route~C entry and frontier-completeness
inputs.  It does not ask for a surviving contact-$2$ bridge, a frontier classification beyond what had already been
proved, or a surviving scalar packet / ramification fork to which the contact-defect sieve applies.  It asks only for a
global choice of witnessing fibration in the exact corridor where the excess count is sharp, together with the
fiberwise two-touch property that converts Proposition~\ref{prop:routeC-vertical-components-excess} into the numerical
bound
\[
\sum_F v(F)\le \Sigma=3.
\]
This was the cleanest two-part formulation of the last missing step before the final closure.
\end{remark}


\section{Reduction of the endpoint formulation}
\label{subsec:routeC-final-lemma-proof-program}

The endpoint formulation of Remark~\ref{conj:routeC-final-lemma-needed} has two logically separate parts: a \emph{height-selection} problem
(producing a witnessing fibration of height $4$ and maximal width) and the \emph{two-touch} problem inside that
maximal-width corridor.  The next results record the reductions that isolate the height-selection side before the final
closure in Section~\ref{sec:routeC-height-selection}.

\begin{corollary}[First sharp reduction of the height-selection problem]
\label{cor:routeC-height-selection-two-global-branches}
Let $X$ lie in the residual sector of Theorem~\ref{thm:routeC-contact2-reduction}.
To prove the height-selection statement
$\operatorname{ht}(X)\ge 4$, no elliptic descendant $\Longrightarrow$ existence of a witnessing fibration of height $4$
and maximal width, it is enough to rule out the following two global branches.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the \emph{multi-special-horizontal branch}: some witnessing fibration carries several special horizontal
    components, already reduced on the section-only side to packet / isolated-collision geometry by
    Remarks~\ref{rem:routeC-scalarstar-ramification-nextgap} and~\ref{rem:routeC-packet-next-gap};

    \item the \emph{special-multisection branch}: some witnessing fibration carries a special horizontal multisection of
    degree at least $2$.  On the purely inseparable side, Corollary~\ref{cor:routeC-final-lemma-pureinseparable-fork}
    yields a surviving primitive ramification fork, while Corollary~\ref{cor:routeC-ordinary-ramification-cubic} and
    Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers} force the ordinary singleton $[2]$-root branch into the
    cubic/two-fiber corridor.  On the separable one-section side, Proposition~\ref{prop:routeC-one-section-image-class-genus}
    and Corollary~\ref{cor:routeC-lowdegree-mixed-no-new-local} show that no separate low-degree mixed local corridor
    remains.
\end{enumerate}
At this stage of the argument, the unresolved part was therefore concentrated in the global survival / realization of
several special horizontals or of a special multisection.
\end{corollary}

\begin{proof}
If a residual witness has height $h\ge 5$, then Corollary~\ref{cor:routeC-final-lemma-highheight-dichotomy} reduces it
to either several special horizontal components or a special multisection.

Assume next that $h=4$ and let $p\colon \widetilde X\to \PP^1$ be a witnessing fibration of height $4$.
If $p$ has maximal width, then Theorem~\ref{thm:routeC-height4-maxwidth-corridor-closed} already removes this case.
Assume therefore that $p$ is not of maximal width.
By Corollary~\ref{cor:routeC-final-lemma-height4-nonmax-special}, some horizontal component is special.

If every horizontal component is a multisection, then one is immediately on the special-multisection branch.
If the degree partition is $3+1$ or $2+1+1$, then Proposition~\ref{prop:routeC-one-section-image-class-genus} shows
that the multisection image on the ruled model is singular, so this case lies in the low-degree mixed corridor.
Corollary~\ref{cor:routeC-lowdegree-mixed-no-new-local} then shows that this corridor contributes no separate local
branch beyond the several-special-horizontal and special-multisection alternatives already listed.
\end{proof}

\begin{definition}[Special horizontal components for arbitrary witnessing fibrations]
\label{def:routeC-global-special-horizontal}
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration on the minimal resolution of a singular klt del Pezzo surface, and let
$H\subset D_{\mathrm{hor}}$
be a horizontal component.

We call $H$ \emph{ordinary} if the connected component of $D$ containing $H$ is a chain and, after applying the
$2$-tail reduction of Definition~\ref{def:routeC-two-tail-core} to the touched pair relative to $H$, the reduced
 touched core is neither a touched tip of weight $2$ nor the singleton chain
$[3].$
Otherwise we call $H$ \emph{special}.
\end{definition}

\begin{lemma}[Ordinary horizontal components contribute at least one half per horizontal degree]
\label{lem:routeC-global-ordinary-horizontal-half}
Assume the setup of Definition~\ref{def:routeC-global-special-horizontal}.  If
$H\subset D_{\mathrm{hor}}$
is ordinary, then
\[
\mu_X(H)\le \frac12
\qquad\text{and hence}\qquad
\operatorname{cf}_X(H)=1-\mu_X(H)\ge \frac12.
\]
\end{lemma}

\begin{proof}
Because $H$ is ordinary, the connected component of $D$ containing $H$ is a chain
$A=[A_1,\dots,A_r],$
and $H=A_i$ for some index $i$.  Let $(A^\sharp,H^\sharp)$ be the corresponding $2$-tail reduction.

If $H^\sharp$ is an interior vertex of $A^\sharp$, then Proposition~\ref{prop:routeC-interior-core-half} gives
\[
\mu(A^\sharp,H^\sharp)\le \frac12.
\]
If $H^\sharp$ is a tip, then ordinariness means precisely that the reduced touched core is neither a touched $2$-tip
nor the singleton chain $[3]$.  Hence Proposition~\ref{prop:routeC-tipcore-highcontact} again gives
\[
\mu(A^\sharp,H^\sharp)\le \frac12.
\]
By Lemma~\ref{lem:routeC-mu-monotone-tail}, removing terminal $2$-tails away from the touched component can only
increase the local weight, so
\[
\mu(A,H)\le \mu(A^\sharp,H^\sharp)\le \frac12.
\]
Finally, Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu} identifies
$\mu_X(H)=\mu(A,H),$
which proves the claim.
\end{proof}

\begin{proposition}[Global special-horizontal multiplicity bound]
\label{prop:routeC-global-special-horizontal-multiplicity}
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration of height
$h:=F\cdot D,$
where $F$ is a general fiber.  Let
\[
S:=\sum_{H\ \mathrm{special}} (F\cdot H)
\]
be the total horizontal multiplicity carried by special horizontal components in the sense of
Definition~\ref{def:routeC-global-special-horizontal}.  Then
$S\ge h-3.$
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:routeC-fiber-anticanonical-cf},
\[
\sum_{H\subset D_{\mathrm{hor}}}\operatorname{cf}_X(H)\,(F\cdot H)<2.
\]
Split the horizontal components into special and ordinary ones.  For special components we only use the trivial bound
$\operatorname{cf}_X(H)\ge 0.$
For ordinary components, Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
Therefore
\[
2>
\sum_{H\ \mathrm{special}}\operatorname{cf}_X(H)\,(F\cdot H)
+
\sum_{H\ \mathrm{ordinary}}\operatorname{cf}_X(H)\,(F\cdot H)
\ge
0+\frac12(h-S).
\]
Hence
$S>h-4.$
Since $S$ and $h$ are integers, this implies
$S\ge h-3.$
\end{proof}

\begin{corollary}[Residual high-height dichotomy for the endpoint formulation]
\label{cor:routeC-final-lemma-highheight-dichotomy}
Let $X$ lie in the residual sector of Theorem~\ref{thm:routeC-contact2-reduction}, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration of height
$h\ge 5.$
Then at least one of the following holds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item there are at least two distinct special horizontal components;
    \item there exists a special horizontal multisection of degree at least $2$.
\end{enumerate}
In particular, the high-height part of the endpoint formulation of Remark~\ref{conj:routeC-final-lemma-needed} is already reduced to a
special-horizontal problem.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-global-special-horizontal-multiplicity}, the total special horizontal multiplicity
satisfies
$S\ge h-3\ge 2.$
If all special horizontal components were $1$-sections and there were at most one of them, then one would have
$S\le 1,$
a contradiction.  Thus either there are at least two distinct special horizontal components, or some special
horizontal component has degree at least $2$ over the base.
\end{proof}

\begin{corollary}[Height-$4$ non-maximal-width witnesses already contain a special horizontal component]
\label{cor:routeC-final-lemma-height4-nonmax-special}
Assume that
$\operatorname{ht}(X)=4$
and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration of height $4$.  If $p$ is not of maximal width, then at least one horizontal
component is special in the sense of Definition~\ref{def:routeC-global-special-horizontal}.

In particular, the only height-$4$ non-maximal-width partitions in which the special component need not itself be a
multisection are
\[
3+1
\qquad\text{and}\qquad
2+1+1.
\]
\end{corollary}

\begin{proof}
If $p$ is not of maximal width, then some horizontal component has degree greater than $1$, but in every case the height
is still
$h=4.$
Applying Proposition~\ref{prop:routeC-global-special-horizontal-multiplicity} gives
$S\ge 1,$
so at least one horizontal component is special.

If the degree partition is $4$ or $2+2$, then every horizontal component is a multisection, so any special horizontal
component is automatically a special multisection.  The only remaining non-maximal-width partitions are therefore
\[
3+1
\qquad\text{and}\qquad
2+1+1,
\]
in which the special component could in principle be the section side.
\end{proof}

The multisection branch can also be pushed further on the purely inseparable side.

\begin{corollary}[Ramification forks survive away from finitely many bad points]
\label{cor:routeC-final-lemma-ramification-finite-bad}
Let
$p\colon \widetilde X\to \PP^1$
be a $\PP^1$-fibration on a smooth rational surface, let
$H\subset D_{\mathrm{hor}}$
be a horizontal multisection, and let
$\nu\colon \widetilde X\to Y$
be a birational morphism onto a smooth relatively minimal ruled surface, chosen so that $\nu$ is an isomorphism at the
generic point of $H$.  Put
$S:=\nu(H).$
Fix a factorization of
$\nu^{-1}$
into finitely many blowups.  Then there exists a finite subset
$B_\nu\subset S$
with the following property: if
$q\in S\setminus B_\nu$
is a ramification point of local degree at least $2$ for
$S\to \PP^1,$
then the primitive ramification fork of Proposition~\ref{prop:routeC-ramification-fork-survival} survives on
$\widetilde X$ as a vertical curve
\[
L\not\subset D,
\qquad
L^2=-1.
\]
\end{corollary}

\begin{proof}
For each ramification point
$q\in S$
of local degree at least $2$, let
$a(q)$
be that local degree.  Proposition~\ref{prop:routeC-ramification-fork-survival} identifies the forced initial
order-$a(q)$ tangency-resolution subcluster over $q$ and its last exceptional curve.  Declare $q$ to lie in
$B_\nu$
if, after that forced tangency-resolution stage, some later blowup center in the chosen factorization lies on the then-
current transform of the last exceptional curve.

Because the factorization uses only finitely many blowups, only finitely many points $q$ can be bad in this sense.
Hence
$B_\nu$
is finite.  For
$q\notin B_\nu,$
all the hypotheses of Proposition~\ref{prop:routeC-ramification-fork-survival} are satisfied by construction, so the
primitive ramification fork survives on $\widetilde X$.
\end{proof}

\begin{corollary}[Purely inseparable multisections automatically yield surviving primitive ramification forks]
\label{cor:routeC-final-lemma-pureinseparable-fork}
Keep the setup of Corollary~\ref{cor:routeC-final-lemma-ramification-finite-bad}.  Assume in addition that
$S\to \PP^1$
is purely inseparable of degree greater than $1$.  Then there exists a point
$q\in S$
and a vertical curve
\[
L\subset \widetilde X,
\qquad
L\not\subset D,
\]
such that the primitive ramification fork over $q$ survives on $\widetilde X$.
\end{corollary}

\begin{proof}
For a purely inseparable map of degree greater than $1$, every point of
$S\cong \PP^1$
has local degree at least $2$.  Since the bad set
$B_\nu$
from Corollary~\ref{cor:routeC-final-lemma-ramification-finite-bad} is finite, one can choose
$q\in S\setminus B_\nu.$
Applying that corollary to $q$ gives the required surviving primitive ramification fork.
\end{proof}

\begin{corollary}[Ordinary surviving ramification forks are cubic in characteristic $3$]
\label{cor:routeC-ordinary-ramification-cubic}
Keep the setup of Proposition~\ref{prop:routeC-ramification-fork-survival}, and let
$a$
be the local ramification degree, so that
$A^2=-a.$
Assume that the horizontal multisection
$H$
is ordinary in the sense of Definition~\ref{def:routeC-global-special-horizontal}.  Then
$a\le 3.$
If moreover
$\operatorname{char}k=3$
and the map
$H\to \PP^1$
is purely inseparable of degree greater than $1$, then
$a=3.$

If in addition the connected component of
$D$
containing
$H$
is the one-component chain
$[b],$
then
\[
\frac1a+\frac1b>\frac12.
\]
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-ramification-fork-survival}, the surviving curve
$L$
meets
$H,\ A,\ B_1$
transversely once each.  Proposition~\ref{prop:routeC-contact-defect-sieve} therefore gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B_1)<1.
\]
Since
$H$
is ordinary, Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} yields
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
Hence
\[
\operatorname{cf}_X(A)<\frac12.
\]
By Lemma~\ref{lem:routeC-cf-selfintersection-lower},
\[
\operatorname{cf}_X(A)\ge 1-\frac{2}{a}.
\]
Therefore
\[
1-\frac{2}{a}<\frac12,
\]
so
$a<4.$
Because
$a\ge 2$
is an integer, it follows that
$a\le 3.$

If the map
$H\to \PP^1$
is purely inseparable in characteristic $3$, then its local degree is a positive power of $3$.  Since
\[
a>1
\qquad\text{and}\qquad
a\le 3,
\]
one gets
$a=3.$

Finally, when the connected component containing
$H$
is the singleton chain
$[b],$
the last inequality is exactly the reformulation of
Corollary~\ref{cor:routeC-ramification-fork-defect}.
\end{proof}

\begin{proposition}[Smooth rational cubic multisections on Hirzebruch surfaces]
\label{prop:routeC-cubic-multisection-backbone}
Let
$Y=\mathbb F_n$
with negative section
$S_0$
and fiber class
$f.$
If
$M\equiv 3S_0+mf$
is a smooth rational multisection not containing
$S_0,$
then
\[
m=\frac{3n+2}{2},
\qquad
M^2=6.
\]
In particular,
$n$
is even.
\end{proposition}

\begin{proof}
Using
\[
S_0^2=-n,
\qquad
S_0\cdot f=1,
\qquad
f^2=0,
\qquad
K_{\mathbb F_n}=-2S_0-(n+2)f,
\]
one computes
\[
M^2=-9n+6m,
\qquad
M\cdot K_{\mathbb F_n}=3(n-2)-2m.
\]
Since
$M\cong \PP^1,$
adjunction gives
\[
\frac{M^2+M\cdot K_{\mathbb F_n}}{2}+1=0.
\]
Substituting the above formulas yields
$-9n+6m+3(n-2)-2m=-2,$
hence
\[
4m=6n+4,
\qquad
m=\frac{3n+2}{2}.
\]
Then
\[
M^2=-9n+6\cdot \frac{3n+2}{2}=6.
\]
Thus
$3n+2$
is even, so
$n$
is even.
\end{proof}

\begin{remark}[The cubic backbone and a supplementary search reduction]
\label{rem:routeC-cubic-backbone-search}
Corollary~\ref{cor:routeC-ordinary-ramification-cubic} and
Proposition~\ref{prop:routeC-cubic-multisection-backbone} show that the purely inseparable ordinary-root branch is
governed by a rigid ruled-model backbone: after passing to a relatively minimal ruled surface, the horizontal image is a
smooth rational cubic multisection of self-intersection
$6.$
A supplementary finite ordinary-fiber search on that backbone shows that over an ordinary fiber no valid
interesting local module of Picard defect
$0$
appears through eight additional blowups, and defects add across disjoint ordinary fibers.
\end{remark}

\begin{proposition}[Blowup budget in the clean threaded \texorpdfstring{$[2]$}{[2]}-root model]
\label{prop:routeC-root-two-clean-threaded}
Let
$Y\to \PP^1$
be a relatively minimal ruled surface, and let
$S\subset Y$
be a smooth rational multisection of degree
$a\ge 2.$
Then
$S^2=2a.$
Consequently, if a birational morphism
$\nu^{-1}\colon Y\dashrightarrow \widetilde X$
is obtained by blowing up finitely many points on successive strict transforms of
$S,$
and the final strict transform
$H\subset \widetilde X$
satisfies
$H^2=-2,$
then exactly
$2a+2$
of the blowup centers lie on successive strict transforms of
$S.$

Assume now that
\[
\operatorname{char}k=3,
\qquad
a=3^e\ge 3,
\]
and that every blowup centered on a transform of
$S$
is either:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item one of the $a$ forced centers resolving a primitive order-$a$ tangency over one of
    $r$
    distinct fibers of the ruling; or
    \item a \emph{thread blowup} at the unique point where the current strict transform of
    $H$
    meets the current unique nonexceptional $(-1)$-curve over one of those same
    $r$
    fibers.
\end{enumerate}
Then
$r\le 2.$
Moreover every extra thread blowup merely lengthens the detached chain over that fiber by one terminal $(-2)$-curve and
does not create any new connected component of
$D.$
In particular, this clean threaded branch contributes at most three connected components to
$D:$
the root component containing
$H,$
and at most two detached ramification chains.
\end{proposition}

\begin{proof}
Write
\[
Y=\mathbb F_n,
\qquad
S\sim aS_0+bf.
\]
Since
$p_a(S)=0,$
adjunction gives
$(a-1)(2b-na-2)=0.$
Because
$a\ge 2,$
one gets
$2b=na+2.$
Therefore
$S^2=-na^2+2ab=2a.$
Each blowup centered on the current strict transform of
$S$
lowers its self-intersection by
$1.$
So changing
$2a$
to
$-2$
requires exactly
$2a+2$
such centers.

Now consider one primitive order-$a$ tangency
$t=z^a.$
Its full forced tangency resolution consists of
$a$
successive blowups at the unique current point where the strict transforms of the multisection and the fiber meet.  The
resulting local configuration is the primitive fork
\[
H-L-F,
\qquad
L-E_{a-1}-\cdots-E_1,
\]
where
\[
L^2=-1,
\qquad
F^2=-a,
\qquad
E_i^2=-2
\quad (1\le i\le a-1).
\]
Thus each primitive ramification fiber consumes exactly
$a$
of the
$2a+2$
available centers on successive transforms of
$S.$
Hence
$ra\le 2a+2,$
which implies
$r\le 2$
for every
$a\ge 3.$

Finally, a thread blowup is centered at the unique point where the current transform of
$H$
meets the current nonexceptional $(-1)$-curve over one of those fibers.  The old $(-1)$-curve becomes a new $(-2)$-curve
in
$D,$
while a new $(-1)$-curve appears at the
$H$
-end of the same local branch.  So the detached chain over that fiber simply gains one more terminal $(-2)$-curve and
no new connected component of
$D$
is created.  Since each primitive ramification fiber contributes one detached chain, the total number of connected
components of
$D$
coming from this branch is at most
$1+r\le 3.$
\end{proof}

\begin{remark}[Meaning for the remaining special singleton \texorpdfstring{$[2]$}{[2]} branch]
\label{rem:routeC-root-two-clean-threaded}
Proposition~\ref{prop:routeC-root-two-clean-threaded} shows that the clean threaded purely inseparable
$[2]$
-root model is too small to serve as a stand-alone source of a Picard-rank-one surface with at least eight singular
points.  Any remaining candidate in that branch must therefore use non-threaded service patterns or interact with other
residual geometry.
\end{remark}

\begin{corollary}[Non-clean purely inseparable service patterns still live on at most two fibers]
\label{cor:routeC-root-two-allpatterns-twofibers}
Keep the setup of the first paragraph of Proposition~\ref{prop:routeC-root-two-clean-threaded}.  Assume only that
\[
\operatorname{char}k=3,
\qquad
 a=3^e\ge 3,
\]
and that the final strict transform
$H\subset \widetilde X$
satisfies
$H^2=-2.$
Do \emph{not} assume the clean threaded hypotheses \textnormal{(i)} and \textnormal{(ii)} of that proposition.
Let
$\mathcal T$
be the set of fibers of
$Y\to \PP^1$
whose total transforms contain at least one blowup center lying on a successive strict transform of
$S.$
Then
$\#\mathcal T\le 2.$
In particular, even a non-clean purely inseparable ordinary singleton
$[2]$
-root service pattern can involve the multisection side over at most two fibers of the ruled model.
\end{corollary}

\begin{proof}
The first part of Proposition~\ref{prop:routeC-root-two-clean-threaded} is independent of the clean threaded
assumptions and shows that exactly
$2a+2$
of the blowup centers lie on successive strict transforms of
$S.$
Now fix a fiber in
$\mathcal T.$
Because the map
$S\to \PP^1$
is purely inseparable of degree
$a,$
every point of
$S$
has local degree
$a.$
Therefore, once a blowup center lies on the current transform of
$S$
over that fiber, the first
$a$
such centers over that fiber are the forced order-$a$ tangency-resolution centers.  Distinct fibers of the ruling are
disjoint on the ruled surface, so the centers lying on transforms of
$S$
over different fibers are disjoint groups.  Hence
$a\,\#\mathcal T\le 2a+2.$
Since
$a\ge 3,$
this implies
$\#\mathcal T\le 2.$
\end{proof}


\begin{remark}[What was genuinely left at this stage]
\label{rem:routeC-final-lemma-what-remains-after-proof-program}
At this stage, the proved reductions above sharpened the endpoint formulation of Remark~\ref{conj:routeC-final-lemma-needed} in four ways.
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item In the high-height regime
    $h\ge 5,$
    one is no longer facing an arbitrary witnessing fibration.  Corollary~\ref{cor:routeC-final-lemma-highheight-dichotomy}
    reduces the problem to either several special horizontal components or a special multisection.

    \item On the multisection side, the purely inseparable branch is now split into explicit subcorridors.
    Corollary~\ref{cor:routeC-final-lemma-pureinseparable-fork} produces a surviving primitive ramification fork;
    Corollary~\ref{cor:routeC-ordinary-ramification-cubic} and
    Proposition~\ref{prop:routeC-cubic-multisection-backbone} show that every ordinary surviving fork is cubic in
    characteristic $3$ and comes from a rigid smooth rational cubic backbone of self-intersection
    $6;$
    and Proposition~\ref{prop:routeC-root-two-clean-threaded} shows that the clean threaded special singleton
    $[2]$
    branch has at most two primitive ramification fibers and at most three connected components of
    $D$
    coming from that branch.

    \item A supplementary finite ordinary-fiber search on the cubic backbone shows that over an ordinary fiber no
    valid interesting local module of Picard defect
    $0$
    appears through eight additional blowups, and defects add across disjoint ordinary fibers.  So coupled ordinary-fiber
    constructions in that searched range cannot preserve Picard rank
    $1.$

    \item In height $4$, a non-maximal-width witness already contains a special horizontal component by
    Corollary~\ref{cor:routeC-final-lemma-height4-nonmax-special}.  Thus the height-selection part of the conjecture is
    reduced to understanding how special horizontal components survive in the residual sector, while the two-touch part is
    concentrated entirely in the height-$4$, maximal-width corridor of
    Section~\ref{subsec:routeC-final-lemma-needed}.
\end{enumerate}

So at this stage the remaining proof was no longer a single amorphous global selection problem.  It was concentrated in
the following explicit residual branches:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item separable special multisections of degree at least $2$, together with the residual global survival / exclusion
    problem for the non-clean purely inseparable service patterns not covered by
    Proposition~\ref{prop:routeC-root-two-clean-threaded} and the supplementary cubic-backbone search described above;
    \item the section-only high-height packet / isolated-collision branch already reduced in
    Remarks~\ref{rem:routeC-scalarstar-ramification-nextgap} and~\ref{rem:routeC-packet-next-gap};
    \item the height-$4$, maximal-width single-attached-fiber problem
    $v(F)>0\Longrightarrow t(F)\ge 2.$
\end{enumerate}
This was the sharpest honest reduction available at this stage of the argument.

\end{remark}

\subsection{The one-section separable branch: every higher-degree image is singular}
\label{subsec:routeC-one-section-separable-branch}

A further unconditional reduction becomes available as soon as one horizontal $1$-section is present.  After
contracting the vertical curves disjoint from that section, every other horizontal component of degree greater than $1$
acquires positive arithmetic genus on the ruled model.  Since its normalization is still $\PP^1$, such an image is
automatically singular.  This applies in particular to the height-$4$ partitions
\[
3+1
\qquad\text{and}\qquad
2+1+1
\]
isolated in Corollary~\ref{cor:routeC-final-lemma-height4-nonmax-special}.

\begin{proposition}[One-section ruled model: class and arithmetic genus of every other horizontal image]
\label{prop:routeC-one-section-image-class-genus}
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration, and assume that
$H_0\subset D_{\mathrm{hor}}$
is a horizontal $1$-section.  Let
$\nu\colon \widetilde X\to \mathbb F_n$
be the contraction of all vertical curves disjoint from $H_0$, and put
$S_0:=\nu(H_0).$
Then
$S_0^2=-n,\qquad n=-H_0^2\ge 2,$
so $S_0$ is the negative section of $\mathbb F_n$.

Let
$H\subset D_{\mathrm{hor}}-H_0$
be any other horizontal component, and put
\[
d:=F\cdot H,
\qquad
C:=\nu(H),
\qquad
s:=H\cdot H_0.
\]
Then
\[
s\in\{0,1\},
\qquad
C\equiv dS_0+(nd+s)f,
\]
where $f$ denotes a fiber of the ruling on $\mathbb F_n$, and
\[
p_a(C)=\frac{(d-1)(nd+2s-2)}{2}.
\]
\end{proposition}

\begin{proof}
Because $\nu$ contracts only vertical curves, it is a birational morphism over the same base
$\PP^1,$
and its target is a relatively minimal ruled surface.  Since the contracted curves are disjoint from $H_0$, the morphism
$\nu$ is an isomorphism in a neighborhood of $H_0$.  Hence
$S_0^2=H_0^2<0.$
Therefore $S_0$ is the negative section of some Hirzebruch surface $\mathbb F_n$, and
$n=-H_0^2\ge 2$
because $H_0\subset D$ is not a $(-1)$-curve.

Now let $H\subset D_{\mathrm{hor}}-H_0$ be another horizontal component.  Since $\nu$ is over the base,
$C\cdot f=d.$
Write
$C\equiv dS_0+bf$
for some integer $b$.  Because $\nu$ is an isomorphism near $H_0$, one has
$C\cdot S_0 = H\cdot H_0 = s.$
Distinct irreducible components of the reduced exceptional divisor meet at most once, so
$s\in\{0,1\}.$
On the other hand,
$(dS_0+bf)\cdot S_0 = -nd+b,$
hence
$b=nd+s.$
This proves the class formula
$C\equiv dS_0+(nd+s)f.$

Finally, on $\mathbb F_n$ one has
$K_{\mathbb F_n}=-2S_0-(n+2)f.$
Therefore
\[
C^2
=
(dS_0+(nd+s)f)^2
=
-nd^2+2d(nd+s)
=
nd^2+2ds,
\]
and
\[
K_{\mathbb F_n}\cdot C
=
(-2S_0-(n+2)f)\cdot(dS_0+(nd+s)f)
=
-dn-2d-2s.
\]
Adjunction for the arithmetic genus gives
\[
p_a(C)
=
1+\frac{C^2+K_{\mathbb F_n}\cdot C}{2}
=
1+\frac{nd^2+2ds-dn-2d-2s}{2}
=
\frac{(d-1)(nd+2s-2)}{2}.
\]
\end{proof}

\begin{corollary}[With one section fixed, every genuine multisection image is singular]
\label{cor:routeC-one-section-higherdegree-singular}
Keep the setup of Proposition~\ref{prop:routeC-one-section-image-class-genus}.  If
$d\ge 2,$
then the image
$C=\nu(H)$
is singular.

Equivalently, once a horizontal section is fixed, every smooth horizontal image on the ruled model is again a section.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-one-section-image-class-genus},
\[
p_a(C)=\frac{(d-1)(nd+2s-2)}{2}.
\]
Since
\[
n\ge 2,
\qquad
d\ge 2,
\qquad
s\ge 0,
\]
one has
$nd+2s-2\ge 2d-2>0,$
and therefore
$p_a(C)>0.$
On the other hand, $H\cong \PP^1$ and $\nu|_H\colon H\to C$ is birational, so the normalization of $C$ is rational.
Hence $C$ cannot be smooth.
\end{proof}

\begin{lemma}[A singular horizontal point is either a branch-collision point or a ramification point]
\label{lem:routeC-singular-horizontal-branch-or-ramification}
Let
$q\in C\subset Y$
be a singular point of an irreducible horizontal curve on a smooth ruled surface
$Y\to \PP^1,$
and let
$\eta\colon \overline C\to C$
be the normalization.  Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the fiber
    $\eta^{-1}(q)$
    contains at least two points, so $q$ is a collision point of at least two local branches of $C$;
    \item
    $\eta^{-1}(q)=\{x\}$
    and the map
    $\overline C\to \PP^1$
    has local degree at least $2$ at $x$.
\end{enumerate}
\end{lemma}

\begin{proof}
If
$\eta^{-1}(q)$
contains at least two points, then $q$ has at least two branches by definition.

Assume now that
$\eta^{-1}(q)=\{x\}.$
If the induced map
$\overline C\to \PP^1$
had local degree $1$ at $x$, then in local coordinates
$(t,z)$
on the ruled surface adapted to the ruling and a local parameter
$u$
on $\overline C$ at $x$, one could write
\[
t=u,
\qquad
z=\varphi(u)
\]
for some convergent power series $\varphi$.  The image would then be the graph
$z=\varphi(t),$
which is smooth at $q$, contradicting the assumption that $q$ is singular.
Therefore the local degree at $x$ must be at least $2$.
\end{proof}

\begin{corollary}[The one-section height-$4$ partitions are singular-image cases]
\label{cor:routeC-one-section-height4-partitions-singular}
Assume that
$\operatorname{ht}(X)=4$
and that a witnessing fibration has degree partition either
\[
3+1
\qquad\text{or}\qquad
2+1+1.
\]
Choose one of the horizontal sections
$H_0\subset D_{\mathrm{hor}}$
and form
$\nu\colon \widetilde X\to \mathbb F_n$
as in Proposition~\ref{prop:routeC-one-section-image-class-genus}.  Let
$H\subset D_{\mathrm{hor}}-H_0$
be the unique multisection, and put
\[
C:=\nu(H),
\qquad
d:=F\cdot H\in\{2,3\},
\qquad
s:=H\cdot H_0\in\{0,1\}.
\]
Then
$C$
is a singular rational curve.  More precisely:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if
    $d=2,$
    then
    \[
    C\equiv 2S_0+(2n+s)f
    \qquad\text{and}\qquad
    \delta(C)=p_a(C)=n+s-1\ge 1;
    \]
    \item if
    $d=3,$
    then
    \[
    C\equiv 3S_0+(3n+s)f
    \qquad\text{and}\qquad
    \delta(C)=p_a(C)=3n+2s-2\ge 4.
    \]
\end{enumerate}
In particular, every singular point of
$C$
is locally either a collision of horizontal branches or a unibranch ramification point of local degree at least $2$.
\end{corollary}

\begin{proof}
The singularity statement is Corollary~\ref{cor:routeC-one-section-higherdegree-singular}.  Since
$H\cong \PP^1$
and $\nu|_H$ is birational, the normalization of
$C$
is rational, so
$\delta(C)=p_a(C).$
Applying Proposition~\ref{prop:routeC-one-section-image-class-genus} with
\[
d=2
\qquad\text{or}\qquad
d=3
\]
gives the displayed formulas.

The final local dichotomy is exactly Lemma~\ref{lem:routeC-singular-horizontal-branch-or-ramification}.
\end{proof}

\begin{remark}[What this does to the separable special-multisection branch]
\label{rem:routeC-one-section-separable-branch-what-remains}
The one-section part of the separable special-multisection branch is no longer a smooth-multisection problem.

Indeed, once one section is fixed, every horizontal component of degree greater than $1$ is forced by
Corollary~\ref{cor:routeC-one-section-higherdegree-singular} to have a singular ruled-model image.  So in the
partitions
\[
3+1
\qquad\text{and}\qquad
2+1+1,
\]
the residual geometry is concentrated entirely in singular rational $3$-sections and $2$-sections on
$\mathbb F_n,\qquad n\ge 2.$
By Lemma~\ref{lem:routeC-singular-horizontal-branch-or-ramification}, every singular point of such a curve is locally of
one of two broad types:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item a collision of at least two horizontal branches, with the same forced initial blowup pattern as the local
    two-branch collision model of Proposition~\ref{prop:routeC-primitive-isolated-pair-collision};
    \item a unibranch point of local degree at least $2$.
\end{enumerate}
The next subsection pushes the degree-two case further by a ruled-surface obstruction and a marked plane conic/cubic
reduction.  The following subsection shows that in the low-degree partitions
\[
3+1
\qquad\text{and}\qquad
2+1+1
\]
the remaining unibranch side is still a finite low-order problem: after a finite preliminary Euclidean singular phase,
the only remaining local steps are corner blowups and order-$2$ or order-$3$ tangency blocks.

So the one-section separable branch is reduced to a genuinely global survival / root-contractibility problem for these
explicit singular-image models.  In particular, item~\textnormal{(i)} of
Remark~\ref{rem:routeC-final-lemma-what-remains-after-proof-program} can now be sharpened: in the one-section partitions,
only \emph{singular-image} separable multisections remain.
\end{remark}


\subsection{The bisection case: a ruled-surface obstruction and a plane reduction}
\label{subsec:routeC-bisection-plane-reduction}

The degree-two part of the one-section branch can be pushed further.  The first observation is a general ruled-surface
fact: an irreducible multisection cannot avoid a section of nonnegative self-intersection.

\begin{lemma}[Generalized Hirzebruch obstruction]
\label{lem:routeC-generalized-hirzebruch-obstruction}
Let
$C\subset \mathbb F_n$
be a section with
$C^2\ge 0,$
and let
$B\subset \mathbb F_n$
be an irreducible $d$-section with
$d\ge 2.$
Then
$B\cdot C>0.$
In particular,
$B\cap C\neq \varnothing.$
\end{lemma}

\begin{proof}
Write
\[
C\equiv S_0+af,
\qquad
B\equiv dS_0+bf,
\]
where
\[
S_0^2=-n,
\qquad
S_0\cdot f=1,
\qquad
f^2=0.
\]
Since
$C^2=-n+2a\ge 0,$
one has
\[
a>0
\qquad\text{if }n>0,
\]
and
\[
a\ge 0
\qquad\text{if }n=0.
\]
Because $B$ is irreducible and $d\ge 2$, it is not equal to the negative section.  Hence
$B\cdot S_0=b-dn\ge 0,$
so
$b\ge dn.$
Therefore
\[
B\cdot C
=(dS_0+bf)\cdot(S_0+af)
=d(-n+a)+b
\ge da.
\]
If $n>0$, then $a>0$, so $B\cdot C>0$.  If $n=0$ and $a>0$, the same inequality gives $B\cdot C>0$.

It remains to consider the case
$n=0, \qquad a=0.$
Then $\mathbb F_0\cong \PP^1\times \PP^1$ and $C$ is a fiber of one ruling.  In this case
$B\cdot C=b.$
If $b=0$, then $B\equiv dS_0$, but every effective divisor of class $(d,0)$ on $\PP^1\times \PP^1$ is a sum of $d$
fibers and is reducible for $d\ge 2$, contradicting the irreducibility of $B$.  Hence $b>0$, so again
$B\cdot C>0.$
\end{proof}

\begin{corollary}[In the $2+1+1$ partition, the bisection meets the second section on the ruled model]
\label{cor:routeC-one-section-bisection-meets-second-section}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and assume
$d=2.$
Let
$H_1\subset D_{\mathrm{hor}}-H_0-H$
be the remaining horizontal section, and put
$S_1:=\nu(H_1).$
Then
$C\cdot S_1>0.$
In particular, the ruled-model bisection is never disjoint from the second section.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-one-section-image-class-genus}, the curve
$S_1$
is a section on $\mathbb F_n$, and
$S_1^2=n+2(H_0\cdot H_1)\ge n\ge 2.$
Therefore $S_1$ is a section of nonnegative self-intersection.  Since
$C$
is an irreducible $2$-section, Lemma~\ref{lem:routeC-generalized-hirzebruch-obstruction} gives
$C\cdot S_1>0.$
\end{proof}


\begin{lemma}[Local normal forms for singular points of a bisection]
\label{lem:routeC-bisection-local-normal-forms}
Let
$Y\to \PP^1$
be a smooth ruled surface, let
$C\subset Y$
be an irreducible bisection, and let
$p\in C$
be a singular point.  Choose formal local coordinates
$(t,z)$
at
$p$
adapted to the ruling, so that
$t=0$
is the local fiber through
$p.$
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item \textbf{Two-branch collision.}  The completed local equation of
    $C$
    at
    $p$
    is
    $z(z-t^m)=0$
    for some integer
    $m\ge 1.$
    Equivalently,
    $C$
    has two smooth branches through
    $p$
    whose local intersection multiplicity is
    $m.$

    \item \textbf{Unibranch ramification type.}  The normalization of
    $C$
    at the unique preimage of
    $p$
    admits a formal parameter
    $u$
    for which
    \[
    t=u^2,
    \qquad
    z=u^{2m+1}
    \]
    for some integer
    $m\ge 1.$
    Equivalently, the completed local equation of
    $C$
    at
    $p$
    is
    $z^2=t^{2m+1}.$
\end{enumerate}
In case \textnormal{(ii)}, the map from the normalization of
$C$
to the base has ramification index
$2$
at the unique preimage of
$p.$
\end{lemma}

\begin{proof}
Because
$C\cdot F_p=2$
for the fiber
$F_p$
through
$p,$
the singularity multiplicity satisfies
$\operatorname{mult}_p(C)\le 2.$
Since
$p$
is singular, one has
$\operatorname{mult}_p(C)=2.$

Assume first that
$p$
has two branches.  Since the total degree over the base is
$2,$
each branch has local degree
$1$
over the base.  After a formal fiberwise coordinate change, the two branches can therefore be written as
\[
z=\varphi_1(t),
\qquad
z=\varphi_2(t).
\]
Replacing
$z$
by
$z-\varphi_1(t)$
makes the first branch equal to
$z=0.$
Then the second branch is
$z=\psi(t)$
for some nonzero series
$\psi(t).$
Writing
$\psi(t)=u(t)t^m$
with
$u(0)\ne 0$
and replacing
$z$
by
$u(t)^{-1}z$
gives the local equation
$z(z-t^m)=0.$
This is case \textnormal{(i)}.

Assume now that
$p$
is unibranch.  Let
$u$
be a formal parameter on the normalization of
$C$
at the unique preimage of
$p.$
Because the local degree to the base is
$2,$
and
$\operatorname{char}k=3\ne 2,$
one may choose the base coordinate so that
$t=u^2.$
Write
$z=\phi(u).$
Subtracting the even part of
$\phi(u)$
by a formal fiberwise coordinate change
$z\mapsto z-g(t)$
leaves only odd powers of
$u.$
Since
$C$
is singular, the first nonzero odd term has order at least
$3,$
say
$2m+1.$
Thus one can write
$z=u^{2m+1}h(u^2)$
with
$h(0)\ne 0.$
Replacing
$z$
by
$h(t)^{-1}z$
gives
\[
t=u^2,
\qquad
z=u^{2m+1}.
\]
Eliminating
$u$
gives the equation
$z^2=t^{2m+1}.$
This is case \textnormal{(ii)}.

In case \textnormal{(ii)}, the parameterization
$t=u^2$
shows directly that the map from the normalization to the base has ramification index
$2$
at the corresponding point.
\end{proof}

\begin{lemma}[A singular-point elementary transform peels one layer from a bisection singularity]
\label{lem:routeC-bisection-elementary-peel}
Keep the setup of Lemma~\ref{lem:routeC-bisection-local-normal-forms}, and perform the elementary transform at
$p.$
Then the image of
$C$
has the following local form.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item If
    $C$
    is of collision type
    $z(z-t^m)=0,$
    then the image is smooth if
    $m=1,$
    and otherwise it is again of collision type
    $z(z-t^{m-1})=0.$

    \item If
    $C$
    is of unibranch type
    $t=u^2,\ z=u^{2m+1},$
    then the image is smooth if
    $m=1,$
    and otherwise it is again of unibranch type
    $t=u^2,\ z=u^{2m-1}.$
\end{enumerate}
\end{lemma}

\begin{proof}
In formal coordinates adapted to the ruling, the elementary transform is given by
\[
(t,z)\longmapsto (t,w),
\qquad
w=\frac{z}{t},
\]
which is the blowup of
$p$
followed by the contraction of the strict transform of the fiber
$t=0.$

For collision type
$z(z-t^m)=0,$
the two branches become
\[
w=0
\qquad\text{and}\qquad
w=t^{m-1}.
\]
If
$m=1,$
these are two distinct smooth branches through different points, so the image is smooth.  If
$m\ge 2,$
the image is again a collision of order
$m-1.$

For unibranch type
\[
t=u^2,
\qquad
z=u^{2m+1},
\]
one gets
\[
w=\frac{z}{t}=u^{2m-1}.
\]
If
$m=1,$
then
$t=u^2, \qquad w=u,$
so the image is smooth.  If
$m\ge 2,$
the image is again of unibranch type with parameterization
\[
t=u^2,
\qquad
w=u^{2m-1}.
\]
\end{proof}

\begin{corollary}[A bisection has at most two unibranch singular points on every ruled model]
\label{cor:routeC-bisection-at-most-two-unibranch}
Let
$C$
be an irreducible rational bisection on a smooth ruled surface.  Then
$C$
has at most two unibranch singular points.  More generally, every curve obtained from
$C$
by a sequence of singular-point elementary transforms has at most two unibranch singular points.
\end{corollary}

\begin{proof}
The normalization of
$C$
is
$\PP^1,$
and the induced map to the base has degree
$2.$
Because
$\operatorname{char}k=3\ne 2,$
this map is separable.  By Riemann--Hurwitz,
$-2=2(-2)+\deg R,$
so its ramification divisor has degree
$2.$
Hence there are exactly two ramification points on the normalization.

By Lemma~\ref{lem:routeC-bisection-local-normal-forms}, a unibranch singular point on any ruled model of the
bisection comes from a ramification point of that normalization map.  Therefore at most two such points can occur on any
model in a singular-point elementary-transform sequence.
\end{proof}

\begin{lemma}[Elementary transform class formula away from the negative section]
\label{lem:routeC-elementary-transform-class}
Let
\[
\epsilon_p\colon \mathbb F_n\dashrightarrow \mathbb F_{n-1}
\]
be the elementary transform at a point
$p\in \mathbb F_n-S_0.$
Let
$C\subset \mathbb F_n$
be an irreducible curve of class
$C\equiv aS_0+bf,$
and write
$m:=\operatorname{mult}_p(C).$
If
$C'\subset \mathbb F_{n-1}$
is the image of $C$, then
$C'\equiv aS_0'+(b-m)f',$
where
$S_0'^2=-(n-1)$
is the negative section of $\mathbb F_{n-1}$ and $f'$ is a fiber of its ruling.
\end{lemma}

\begin{proof}
Let
\[
\sigma\colon Z:=\operatorname{Bl}_p\mathbb F_n\to \mathbb F_n
\]
be the blowup of $p$, let
$E\subset Z$
be the exceptional curve, let
$F$
be the fiber of $\mathbb F_n$ through $p$, and let
$F'\subset Z$
be the strict transform of $F$.  Then $F'^2=-1$, so contracting $F'$ gives a morphism
$\tau\colon Z\to \mathbb F_{n-1}$
realizing the elementary transform.

Because $p\notin S_0$, one has
$\tau^*f'=\sigma^*f$
and
\[
\tau^*S_0'=
\sigma^*S_0+F'
=
\sigma^*S_0+\sigma^*f-E.
\]
The strict transform of $C$ on $Z$ is
$\widetilde C=\sigma^*C-mE.$
Since
$\widetilde C\cdot F'=(aS_0+bf)\cdot f-m=a-m,$
one gets
\[
\tau^*C'=
\widetilde C+(a-m)F'.
\]
Expanding and using the displayed formulas for $\tau^*S_0'$ and $\tau^*f'$ gives
\[
\tau^*C'
=
a\,\tau^*S_0'+(b-m)\,\tau^*f'.
\]
Hence
$C'\equiv aS_0'+(b-m)f',$
as claimed.
\end{proof}

\begin{proposition}[Rational bisections reduce to plane conics or cubics]
\label{prop:routeC-bisection-plane-reduction}
Let
$C\subset \mathbb F_n$
be an irreducible rational curve with
\[
C\equiv 2S_0+(2n+s)f,
\qquad
s\in\{0,1\}.
\]
Then there exists a sequence of $n-1$ elementary transforms
\[
\mathbb F_n\dashrightarrow \mathbb F_{n-1}\dashrightarrow \cdots \dashrightarrow \mathbb F_1
\]
such that each center is a singular point of the current curve and lies off the current negative section.  The final
image
$C_1\subset \mathbb F_1$
satisfies
$C_1\equiv 2S_1+(2+s)f.$
After contracting the negative section $S_1$, the image of $C_1$ is an irreducible rational plane curve of degree
$2+s.$
Equivalently:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $s=0$, the branch reduces birationally to a plane conic;
    \item if $s=1$, the branch reduces birationally to a singular plane cubic.
\end{enumerate}
\end{proposition}

\begin{proof}
We argue by induction on $n$.  If $n=1$, there is nothing to prove.

Assume now that $n\ge 2$.  By Proposition~\ref{prop:routeC-one-section-image-class-genus}, one has
$p_a(C)=n+s-1>0,$
so $C$ is singular.  Let
$p\in C$
be a singular point.  We claim that
$p\notin S_0.$
Indeed, if $p\in S_0$, then because $S_0$ is smooth and $p$ is singular on $C$, one would have
\[
(C\cdot S_0)_p\ge \operatorname{mult}_p(C)\operatorname{mult}_p(S_0)\ge 2,
\]
contradicting the total intersection number
$C\cdot S_0=s\le 1.$
So every singular point of $C$ lies off the negative section.

Let
$F_p$
be the fiber through $p$.  Since $C$ is a bisection,
$C\cdot F_p=2.$
Hence
$\operatorname{mult}_p(C)\le 2.$
Because $p$ is singular, one also has
$\operatorname{mult}_p(C)\ge 2,$
so in fact
$\operatorname{mult}_p(C)=2.$
Perform the elementary transform at $p$.  By Lemma~\ref{lem:routeC-elementary-transform-class} with
$a=2, \qquad m=2,$
the image
$C'\subset \mathbb F_{n-1}$
has class
$C'\equiv 2S_0'+(2(n-1)+s)f'.$
It is again irreducible and rational, being birational to $C$.  Applying the induction hypothesis to $C'$ finishes the
construction of the sequence.

For the final plane statement, let
$\kappa\colon \mathbb F_1\to \PP^2$
be the contraction of $S_1$, and let
$H:=\kappa^*(\text{line})=S_1+f.$
Then
\[
C_1
\equiv
2S_1+(2+s)f
=(2+s)H-sS_1,
\]
so the image
$\Gamma:=\kappa(C_1)$
has plane degree
$\deg \Gamma=2+s.$
It is irreducible and rational because $C_1$ is.  If $s=0$, then $\Gamma$ is an irreducible plane conic, hence smooth.
If $s=1$, then $\Gamma$ is a rational plane cubic and is therefore singular.
\end{proof}

\begin{proposition}[Marked-point plane models for rational bisections]
\label{prop:routeC-bisection-marked-plane-model}
Keep the setup of Proposition~\ref{prop:routeC-bisection-plane-reduction}, and let
$\kappa\colon \mathbb F_1\to \PP^2$
be the contraction of the negative section
$E\subset \mathbb F_1$
to a point
$q\in \PP^2.$
Let
$\Gamma:=\kappa(C_1).$
Then
$\Gamma$
is an irreducible rational plane curve of degree
$2+s,$
and the ruling on
$\mathbb F_1$
restricts to the normalization of the projection of
$\Gamma$
from
$q.$
More precisely:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if
    $s=0,$
    then
    $q\notin \Gamma$
    and
    $\Gamma$
    is a smooth plane conic;

    \item if
    $s=1,$
    then
    $q\in \Gamma_{\mathrm{reg}},$
    the curve
    $\Gamma$
    is a singular plane cubic, and projection from
    $q$
    realizes the original degree-$2$ map from the normalization of the bisection to the base.
\end{enumerate}
In case
$s=1,$
that projection is separable of degree
$2$
and has exactly two ramification points.
\end{proposition}

\begin{proof}
The final class computation in the proof of Proposition~\ref{prop:routeC-bisection-plane-reduction} gives
$C_1\equiv 2E+(2+s)f=(2+s)H-sE,$
where
$H=\kappa^*(\text{line}).$
Hence
$\deg\Gamma=2+s$
and the multiplicity of
$\Gamma$
at
$q$
is exactly
$s.$
If
$s=0,$
then
$q\notin \Gamma,$
and Proposition~\ref{prop:routeC-bisection-plane-reduction} already shows that
$\Gamma$
is a smooth conic.  If
$s=1,$
then
$q\in \Gamma$
with multiplicity
$1,$
so
$q\in \Gamma_{\mathrm{reg}}.$
In that case Proposition~\ref{prop:routeC-bisection-plane-reduction} shows that
$\Gamma$
is a singular irreducible cubic.

The fibers of the ruling on
$\mathbb F_1$
are the strict transforms of the lines through
$q.$
Therefore the restriction of the ruling to
$C_1$
is exactly the normalization of the projection of
$\Gamma$
from
$q.$
Since all elementary transforms in Proposition~\ref{prop:routeC-bisection-plane-reduction} are performed over the same
base, this is the same degree-$2$ map as the original map from the normalization of the bisection to
$\PP^1.$

When
$s=1,$
the source and target are both
$\PP^1,$
and in characteristic
$3$
a degree-$2$ morphism is separable.  Riemann--Hurwitz therefore gives
$-2=2(-2)+\deg R,$
so
$\deg R=2.$
Since every ramification point of a degree-$2$ separable morphism contributes
$1$
to
$\deg R,$
there are exactly two ramification points.
\end{proof}

\begin{corollary}[The touched bisection subcase is a marked singular plane cubic]
\label{cor:routeC-one-section-bisection-cubic}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and assume
$d=2, \qquad s=1.$
Then, after successive singular-point elementary transforms, the ruled-model bisection reduces birationally to an
irreducible singular plane cubic
$\Gamma\subset \PP^2$
together with a smooth marked point
$q\in \Gamma.$
Projection from
$q$
realizes the original degree-$2$ map from the normalization of the bisection to the base.  In particular, every ruled
model obtained along the singular-point elementary-transform sequence has at most two unibranch singular points.
\end{corollary}

\begin{proof}
Combine Proposition~\ref{prop:routeC-bisection-plane-reduction},
Proposition~\ref{prop:routeC-bisection-marked-plane-model}, and
Corollary~\ref{cor:routeC-bisection-at-most-two-unibranch}.
\end{proof}

\begin{remark}[Relation with the unexpected-cubic route]
\label{rem:routeC-one-section-bisection-kettinger}
Corollary~\ref{cor:routeC-one-section-bisection-cubic} places the touched bisection subcase squarely in marked
plane-cubic geometry: one has a singular cubic
$\Gamma\subset \PP^2$
together with a smooth marked point
$q\in \Gamma,$
and projection from
$q$
recovers the original degree-$2$ map to the base.  By Proposition~\ref{prop:routeC-bisection-marked-plane-model}, this
projection has exactly two ramification points, so the residual local freedom is concentrated in collision strings and at
most two unibranch ramification strings.  This is exactly the range where Kettinger proves that characteristic $3$ has
no unexpected plane cubics \cite{Kettinger}.  The marked-cubic model therefore remains a useful global compression of
the touched degree-two branch.  However, the later Corollary~\ref{cor:routeC-bisection-no-new-local} shows that for the
Route~C singularity-count problem one does not need a separate unexpected-cubic closure on the local side: every
surviving singularity of the ordinary bisection is either impossible by the contact-defect sieve or already absorbed by
the old touched-tip / branching geometry.
\end{remark}

\begin{remark}[What this adds to the one-section separable branch]
\label{rem:routeC-one-section-bisection-what-remains}
The degree-two part of the one-section separable branch is no longer an arbitrary ruled-surface singularity problem.  In
the partition
$2+1+1,$
the bisection is forced by Corollary~\ref{cor:routeC-one-section-bisection-meets-second-section} to meet the second
section on the ruled model; Lemma~\ref{lem:routeC-bisection-local-normal-forms} shows that every singular point is an
explicit plane double point; Lemma~\ref{lem:routeC-bisection-elementary-peel} shows that singular-point elementary
transforms shorten those local strings one step at a time; and Proposition~\ref{prop:routeC-bisection-plane-reduction}
reduces the whole branch to a marked plane conic or cubic.  In the touched subcase
$s=1,$
Corollary~\ref{cor:routeC-one-section-bisection-cubic} gives a marked singular plane cubic with exactly two
ramification directions for the projection from the marked point.  At this stage of the paper, the only one-section separable cases still not reduced
to such explicit plane models are the residual conic bookkeeping problem in degree $2$ and the singular trisection
branch in degree $3$.  The later Subsections~\ref{subsec:routeC-trisection-collision-oldside} and~\ref{subsec:routeC-bisection-oldside}
show that neither creates a genuinely new local corridor.
\end{remark}

\subsection{Low-degree unibranch singularities: only corner and order-$2/3$ terminal phases remain}
\label{subsec:routeC-one-section-unibranch-lowdegree}

The previous remark reduces the one-section separable branch to singular ruled-model images.  On the multibranch side,
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} already gives the primitive local collision model.  The
unibranch side still needs a local endgame analysis.  In the one-section partitions
\[
2+1+1
\qquad\text{and}\qquad
3+1,
\]
that endgame is much smaller than an arbitrary plane-branch resolution, simply because the local degree over the base is at
most $3$.

\begin{lemma}[Order pairs follow the Euclidean algorithm under blowup]
\label{lem:routeC-unibranch-euclidean-orders}
Let
\[
\gamma\colon \operatorname{Spec} k[[u]]\to (\mathbb A^2_{x,y},0)
\]
be a primitive formal branch, and write
\[
\operatorname{ord}_u x(\gamma)=r,
\qquad
\operatorname{ord}_u y(\gamma)=s,
\qquad
\gcd(r,s)=1.
\]
Blow up the origin and choose the unique affine chart containing the strict transform of
$\gamma.$
Then the new vanishing-order pair is
\[
(r,s-r)
\qquad\text{if } s>r,
\]
and
\[
(r-s,s)
\qquad\text{if } r>s.
\]
Consequently, repeated blowups at the unique current point of the strict transform realize the Euclidean algorithm on the
pair
$(r,s).$
In particular, if
$r\le 3,$
then after finitely many such blowups one reaches a smooth strict transform with local order pair
\[
(1,a)
\qquad\text{or}\qquad
(a,1)
\]
for some integer
$1\le a\le 3.$
\end{lemma}

\begin{proof}
Assume first that
$s>r.$
In the blowup chart
\[
x=x_1,
\qquad
y=x_1y_1,
\]
which is the unique chart containing the strict transform of
$\gamma,$
one has
\[
\operatorname{ord}_u x_1=r,
\qquad
\operatorname{ord}_u y_1=s-r.
\]
This gives the pair
$(r,s-r).$
The case
$r>s$
is symmetric, using the chart
\[
x=y_1x_1,
\qquad
y=y_1.
\]
Thus each blowup replaces the current order pair by one subtraction step of the Euclidean algorithm.

Because
$\gcd(r,s)=1,$
iterating this process eventually reaches a pair of the form
\[
(1,a)
\qquad\text{or}\qquad
(a,1)
\]
for some
$a\ge 1.$
At that stage the strict transform is smooth.  If initially
$r\le 3,$
then every remainder in the Euclidean algorithm is at most $3$, so
$a\le 3.$
\end{proof}

\begin{corollary}[First smooth stage for local degree $2$ and $3$]
\label{cor:routeC-unibranch-first-smooth-stage}
Let
$q\in C\subset Y$
be a unibranch singular point of an irreducible horizontal curve on a smooth ruled surface
$Y\to \PP^1,$
and let
$\eta\colon \overline C\to C$
be the normalization.  Assume that the induced map
$\overline C\to \PP^1$
has local degree
$e\in\{2,3\}$
at the unique point of
$\eta^{-1}(q).$
Then after finitely many blowups at the unique current point of the strict transform over
$q,$
one reaches a stage where the strict transform
$\Gamma$
is smooth.  At that stage the reduced exceptional divisor over
$q$
is a simple normal-crossings divisor, and the unique point of
$\Gamma$
on it is of one of the following two kinds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a smooth point of one exceptional component, with local contact order
    $a\le e;$
    if the resolution is not yet finished at that stage, then necessarily
    $a\in\{2,3\};$
    more precisely,
    \[
    a=2 \text{ if } e=2,
    \qquad
    a\in\{2,3\} \text{ if } e=3;
    \]
    \item a corner point of two exceptional components.
\end{enumerate}
\end{corollary}

\begin{proof}
Choose local coordinates
$(t,z)$
at
$q$
adapted to the ruling, so that the local degree over the base is
$\operatorname{ord}_u t=e\in\{2,3\}$
on the normalization parameter
$u.$
Applying Lemma~\ref{lem:routeC-unibranch-euclidean-orders} to the successive strict transforms gives a finite blowup
sequence after which the branch becomes smooth.  At that first smooth stage, the total exceptional divisor created by the
previous blowups is a normal-crossings divisor, so the point of the smooth strict transform on that divisor is either a
smooth point of one exceptional component or a corner of two exceptional components.

In the first case, if the local contact order were
$a=1,$
then the branch would already meet the exceptional divisor transversely and the local embedded resolution would be finished.
So whenever the resolution is not yet finished, one must have
$a\ge 2.$
The bound
$a\le e$
comes from the final order pair
\[
(1,a)
\qquad\text{or}\qquad
(a,1)
\]
of Lemma~\ref{lem:routeC-unibranch-euclidean-orders}.  Since
$e\in\{2,3\},$
this leaves only the displayed possibilities.
\end{proof}

\begin{proposition}[Low-degree unibranch points have only corner and order-$2/3$ terminal steps]
\label{prop:routeC-unibranch-lowdegree-terminal-steps}
Keep the setup of Corollary~\ref{cor:routeC-unibranch-first-smooth-stage}.  Consider the remaining blowups after the
first stage where the strict transform becomes smooth.

Then every such remaining blowup is of one of the following two kinds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a \emph{corner blowup}: the smooth strict transform passes through a corner of the current reduced exceptional
    divisor;
    \item a \emph{tangency blowup block of order $a$}: the smooth strict transform is tangent to a single exceptional
    component with contact order
    $a\in\{2,3\}.$
    In this case the next
    $a$
    forced blowups are exactly the order-$a$ two-branch tangency resolution of
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}, applied to the smooth strict transform and the touched
    exceptional component.
\end{enumerate}
In particular, after the branch first becomes smooth, no higher-order terminal behavior can occur in the one-section
partitions
\[
2+1+1
\qquad\text{and}\qquad
3+1.
\]
\end{proposition}

\begin{proof}
By Corollary~\ref{cor:routeC-unibranch-first-smooth-stage}, once the strict transform is smooth the unresolved contact with
the reduced exceptional divisor is supported either at a corner point or at a smooth point of one exceptional component
with contact order
$a\in\{2,3\}.$
This gives the stated dichotomy for every remaining blowup.

In case~\textnormal{(ii)}, both the smooth strict transform and the touched exceptional component are smooth branches
through one point, with intersection multiplicity exactly
$a.$
So the local analytic model is the same as for two smooth branches with order-$a$ tangency, and
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} gives the forced order-$a$ resolution block.
No order larger than $3$ can occur because Corollary~\ref{cor:routeC-unibranch-first-smooth-stage} already bounds
$a\le 3.$
\end{proof}

\begin{corollary}[Finite low-order endgame for the one-section unibranch branch]
\label{cor:routeC-one-section-unibranch-finite-loworder}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}.  Let
$q\in C$
be a unibranch singular point.
Then, after a finite preliminary singular phase, the remaining local embedded resolution over
$q$
is built only from:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item corner blowups of a smooth branch through a corner of the reduced exceptional divisor;
    \item order-$2$ tangency blocks;
    \item order-$3$ tangency blocks, and these can occur only in the degree-$3$ case.
\end{enumerate}
Equivalently, the unibranch part of the one-section separable branch is not an arbitrary high-order ramification problem:
in the partitions
\[
2+1+1
\qquad\text{and}\qquad
3+1
\]
it is a finite low-order local endgame.
\end{corollary}

\begin{proof}
At a unibranch singular point of the image
$C,$
Lemma~\ref{lem:routeC-singular-horizontal-branch-or-ramification} says that the local degree over the base is at least
$2$.  Since
$C\cdot f=d\in\{2,3\},$
that local degree is also at most
$d.$
Thus the only possibilities are
\[
e=2
\qquad\text{or}\qquad
(e,d)=(3,3).
\]
Applying Proposition~\ref{prop:routeC-unibranch-lowdegree-terminal-steps} gives exactly the three displayed terminal step
types, with order-$3$ occurring only when
$d=3.$
\end{proof}

\begin{remark}[What has now been isolated on the one-section separable side]
\label{rem:routeC-one-section-unibranch-next-gap}
Combined with Proposition~\ref{prop:routeC-primitive-isolated-pair-collision},
Corollary~\ref{cor:routeC-one-section-unibranch-finite-loworder} shows that the one-section separable branch has only two
remaining local mechanisms:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item multibranch collisions of local branches of the multisection image;
    \item unibranch singularities whose post-smoothing endgame uses only corner blowups and order-$2/3$ tangency blocks.
\end{enumerate}
So the unibranch side is no longer a vague ``ramification'' corridor.  It is a finite low-order local problem.  What is
still missing is the genuinely global step: turning one of these low-order local blocks into a surviving admissible
Route~C configuration on
$\widetilde X$
or else showing that the branch cannot occur in the residual sector at all.
\end{remark}


\subsection{Terminal corner and order-$2/3$ blocks are low-order forks, not chain bridges}
\label{subsec:routeC-one-section-terminal-forks}

The natural hope would be that the terminal corner and order-$2$ blocks from
Corollary~\ref{cor:routeC-one-section-unibranch-finite-loworder} already fall into the existing contact-$2$
chain corridors.  The local calculation shows that this is not quite what happens.
The exceptional components touched by those terminal blocks are already exceptional before the block begins, so they have
self-intersection at most
$-1.$
After the terminal block they therefore remain inside
$D,$
and the last exceptional curve has total contact
$L\cdot D=3,$
not $2$.  Thus the correct local output is a small fork corridor rather than a new chain bridge.

\begin{proposition}[A surviving terminal corner blowup is a primitive fork]
\label{prop:routeC-terminal-corner-fork}
Let
$\nu\colon \widetilde X\to Y$
be a birational morphism from a smooth surface to a smooth ruled surface, and let
$H\subset \widetilde X$
be a smooth horizontal curve such that
$\nu|_H\colon H\to C:=\nu(H)$
is birational.
Fix a point
$q\in C.$
Assume that in a factorization of
$\nu^{-1}$
over
$q$
one reaches a smooth intermediate surface on which the strict transform
$\Gamma$
of
$C$
is smooth and passes through a corner
$p=A_0\cap B_0$
of the reduced exceptional divisor over
$q,$
where
$A_0,B_0$
are smooth exceptional curves meeting transversely.
Assume that the next blowup is exactly the blowup of
$p,$
and that no later center of
$\nu^{-1}$
lies on the exceptional curve
$L$
created by that blowup.

Then on
$\widetilde X$
there are vertical curves
\[
L\not\subset D,
\qquad
A,B\subset D
\]
contained in the fiber through
$q$
with local dual graph
\[
\begin{array}{ccccc}
&&A&&\\
&&|&&\\
H&-&L&-&B
\end{array}
\]
and
\[
L^2=-1,
\qquad
A^2\le -2,
\qquad
B^2\le -2.
\]
In particular,
$L\cdot D=3.$
\end{proposition}

\begin{proof}
Choose local coordinates
$(x,y)$
at
$p$
so that
\[
A_0=\{x=0\},
\qquad
B_0=\{y=0\},
\qquad
\Gamma=\{y=x\}.
\]
Blowing up
$p$
creates one exceptional curve
$L$
with
$L^2=-1,$
and the strict transforms of
\[
\Gamma,
\qquad
A_0,
\qquad
B_0
\]
meet
$L$
transversely at three distinct points.
So after that blowup the local graph is exactly the displayed fork.

Because
\[
A_0
\quad\text{and}\quad
B_0
\]
are components of the exceptional divisor already present before the terminal blowup, one has
\[
A_0^2\le -1,
\qquad
B_0^2\le -1.
\]
The corner blowup lowers each self-intersection by $1$, so immediately after that blowup,
\[
(A_0')^2\le -2,
\qquad
(B_0')^2\le -2.
\]
Any later blowup can only decrease those numbers further.  Hence the final strict transforms
$A,B$
on
$\widetilde X$
still satisfy
\[
A^2\le -2,
\qquad
B^2\le -2.
\]
By \cite[Lemma~2.8(b)]{PP12}, every vertical component of a degenerate fiber outside
$D$
is a $(-1)$-curve.  Therefore
$A,B\subset D.$
By assumption no later center lies on
$L,$
so its strict transform on
$\widetilde X$
still has self-intersection
$-1.$
Hence
$L\not\subset D.$
Since
$L$
meets exactly the three displayed components of
$D,$
one gets
$L\cdot D=3.$
\end{proof}

\begin{proposition}[Surviving terminal order-$a$ tangency blocks are primitive forks]
\label{prop:routeC-terminal-tangency-fork}
Keep the setup of Proposition~\ref{prop:routeC-terminal-corner-fork}.
Assume instead that in a factorization of
$\nu^{-1}$
over
$q$
one reaches a smooth intermediate surface on which the strict transform
$\Gamma$
of
$C$
is smooth and tangent of order
$a\in\{2,3\}$
to a smooth exceptional curve
$A_0$
at a smooth point
$p$
of the reduced exceptional divisor over
$q.$
Assume that the next
$a$
blowups are exactly the forced tangency-resolution centers, and that no later center of
$\nu^{-1}$
lies on the last exceptional curve
$L$
created in that forced block.

Then on
$\widetilde X$
there exist vertical curves
\[
L\not\subset D,
\qquad
A,B_1,\dots,B_{a-1}\subset D
\]
contained in the fiber through
$q$
with local dual graph
\[
\begin{array}{cccccc}
&&A&&&\\
&&|&&&\\
H&-&L&-&B_{a-1}&-\cdots-B_1
\end{array}
\]
and
\[
L^2=-1,
\qquad
B_i^2\le -2\ \ (1\le i\le a-1),
\qquad
A^2\le -(a+1).
\]
In particular,
$L\cdot D=3.$
For
$a=2$
one gets the fork
$H-L-A-B_1,$
while for
$a=3$
one gets the fork
$H-L-A-B_2-B_1.$
\end{proposition}

\begin{proof}
Choose local coordinates
$(t,z)$
at
$p$
so that
\[
A_0=\{z=0\},
\qquad
\Gamma=\{z=t^a\}.
\]
Applying Lemma~\ref{lem:routeC-pair-collision-order-p} to the pair
$A_0, \Gamma$
shows that after the forced
$a$
blowups one obtains a chain
$F_1-\cdots-F_a$
with
\[
F_i^2=-2\ \ (1\le i<a),
\qquad
F_a^2=-1,
\]
and the strict transforms of
\[
A_0
\qquad\text{and}\qquad
\Gamma
\]
both meeting
$F_a$
transversely at distinct points.
Set
\[
L:=F_a,
\qquad
B_i:=F_{a-i}\ \ (1\le i\le a-1),
\]
and let
$A$
denote the final strict transform of
$A_0.$

Since
$A_0$
is already exceptional before this terminal block, one has
$A_0^2\le -1.$
Each of the
$a$
forced blowups is centered on the current strict transform of
$A_0,$
so immediately after the block,
$A^2\le -1-a=-(a+1).$
Any later blowup can only decrease that self-intersection further.
Likewise the curves
$B_1,\dots,B_{a-1}$
have self-intersection
$-2$
immediately after the forced block, and later centers can only decrease those values.
Therefore
$A,B_1,\dots,B_{a-1}\subset D$
by \cite[Lemma~2.8(b)]{PP12}.

By assumption no later center lies on
$L,$
so its strict transform on
$\widetilde X$
still satisfies
$L^2=-1.$
Hence
$L\not\subset D.$
The displayed local graph is exactly the one given by Lemma~\ref{lem:routeC-pair-collision-order-p}.  Since
$L$
meets precisely the three components
\[
H,
\qquad
A,
\qquad
B_{a-1}
\]
of
$D,$
one gets
$L\cdot D=3.$
\end{proof}

\begin{corollary}[Finite defect sieve for the terminal low-order forks]
\label{cor:routeC-terminal-loworder-fork-sieve}
The surviving forks from Propositions~\ref{prop:routeC-terminal-corner-fork} and
\ref{prop:routeC-terminal-tangency-fork} satisfy the following numerical restrictions.

\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item In the tangency case, write
    \[
    H^2=-\beta,
    \qquad
    A^2=-\alpha,
    \qquad
    \alpha,\beta\ge 2.
    \]
    Then
    \[
    \frac{2}{\alpha}+\frac{2}{\beta}>1.
    \]
    Equivalently,
    $(\alpha-2)(\beta-2)<4.$
    In particular, if
    $\alpha,\beta>2,$
    then up to order
    $(\alpha,\beta)\in\{(3,3),(3,4),(3,5)\}.$
    Since Proposition~\ref{prop:routeC-terminal-tangency-fork} also gives
    $\alpha\ge a+1,$
    the order-$3$ case forces either
    $\beta=2,$
    or else
    $(\alpha,\beta)\in\{(4,3),(5,3)\}.$

    \item In the corner case, write
    \[
    H^2=-\beta,
    \qquad
    A^2=-\alpha,
    \qquad
    B^2=-\gamma,
    \qquad
    \alpha,\beta,\gamma\ge 2.
    \]
    Then
    \[
    \frac{2}{\alpha}+\frac{2}{\beta}+\frac{2}{\gamma}>2,
    \qquad\text{equivalently}\qquad
    \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>1.
    \]
    After reordering so that
    $\alpha\le \beta\le \gamma,$
    this leaves only the spherical triples
    \[
    (\alpha,\beta,\gamma)=(2,2,n)\ \ (n\ge 2),
    \qquad
    (2,3,3),
    \qquad
    (2,3,4),
    \qquad
    (2,3,5).
    \]
\end{enumerate}
\end{corollary}

\begin{proof}
In the tangency case, Proposition~\ref{prop:routeC-terminal-tangency-fork} shows that the surviving curve
$L$
meets the three components
\[
H,
\qquad
A,
\qquad
B_{a-1}
\]
of
$D$
transversely once each.  Thus Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B_{a-1})<1.
\]
Discarding the nonnegative last term and using Lemma~\ref{lem:routeC-cf-selfintersection-lower}, we obtain
\[
1-\frac{2}{\beta}+1-\frac{2}{\alpha}<1,
\]
which is the displayed inequality.
The finite list for
$\alpha,\beta>2$
is exactly the same elementary consequence as in
Corollary~\ref{cor:routeC-ramification-fork-defect}.
Since Proposition~\ref{prop:routeC-terminal-tangency-fork} also gives
$\alpha\ge a+1,$
in the order-$3$ case one has
$\alpha\ge 4.$
So if
$\beta>2,$
then the only possibilities are
$(\alpha,\beta)\in\{(4,3),(5,3)\}.$

In the corner case, Proposition~\ref{prop:routeC-terminal-corner-fork} gives
$L\cdot H=L\cdot A=L\cdot B=1.$
Hence Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<1.
\]
Applying Lemma~\ref{lem:routeC-cf-selfintersection-lower} to all three terms yields
\[
1-\frac{2}{\beta}+1-\frac{2}{\alpha}+1-\frac{2}{\gamma}<1,
\]
which is equivalent to
\[
\frac{2}{\alpha}+\frac{2}{\beta}+\frac{2}{\gamma}>2.
\]
Now reorder so that
$\alpha\le \beta\le \gamma.$
If
$\alpha\ge 3,$
then
\[
\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\le 1,
\]
so necessarily
$\alpha=2.$
If also
$\beta\ge 4,$
then
\[
\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\le \frac12+\frac14+\frac14=1,
\]
again impossible.  Thus
$\beta\in\{2,3\}.$
If
$\beta=2,$
then any
$\gamma\ge 2$
works.  If
$\beta=3,$
then
\[
\frac12+\frac13+\frac{1}{\gamma}>1
\]
forces
$\gamma<6,$
so
$\gamma\in\{3,4,5\}.$
This gives exactly the displayed list.
\end{proof}

\begin{corollary}[The surviving one-section unibranch endgame is a finite fork corridor]
\label{cor:routeC-one-section-unibranch-finite-fork-corridor}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and let
$q\in C$
be a unibranch singular point.
Assume that, in the factorization of
$\nu^{-1}$
over
$q,$
the last exceptional curve created in the terminal block survives on
$\widetilde X.$
Then the local configuration on
$\widetilde X$
is one of exactly the following three types:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a corner fork
    \[
    \begin{array}{ccccc}
    &&A&&\\
    &&|&&\\
    H&-&L&-&B;
    \end{array}
    \]
    \item an order-$2$ tangency fork
    $H-L-A-B_1;$
    \item an order-$3$ tangency fork
    $H-L-A-B_2-B_1.$
\end{enumerate}
In every case
$L\cdot D=3,$
so the terminal one-section unibranch mechanisms do \emph{not} directly enter the contact-$2$ chain-bridge corridor.
Instead they lie in the explicit low-order fork corridor of
Corollary~\ref{cor:routeC-terminal-loworder-fork-sieve}.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-one-section-unibranch-finite-loworder}, after the preliminary singular phase the terminal
part of the local embedded resolution is either a corner blowup, an order-$2$ tangency block, or an order-$3$ tangency
block.
If the last exceptional curve of that terminal block survives on
$\widetilde X,$
then Proposition~\ref{prop:routeC-terminal-corner-fork} handles the corner case and
Proposition~\ref{prop:routeC-terminal-tangency-fork} handles the order-$2$ and order-$3$ cases.
The final statement follows from those propositions together with
Corollary~\ref{cor:routeC-terminal-loworder-fork-sieve}.
\end{proof}

\begin{remark}[What this honestly leaves]
\label{rem:routeC-one-section-unibranch-fork-corridor}
The attempted direct reduction of the one-section unibranch branch to the existing short-bridge / special-tip
\emph{chain} corridors is therefore too optimistic.
The terminal corner and order-$2/3$ blocks do not create a contact-$2$ bridge: they create a surviving contact-$3$ fork,
with only three possible low-order combinatorial shapes.

So the one-section separable branch is now narrowed as follows.
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item multibranch collisions are governed by Proposition~\ref{prop:routeC-primitive-isolated-pair-collision};
    \item unibranch singularities, if their terminal block survives, land in the finite low-order fork corridor of
    Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor}.
\end{enumerate}
What remains is the genuinely global step: either show that these low-order forks are already absorbed by the existing
fork/frontier analysis, or prove that they cannot survive in the residual one-section separable sector.
\end{remark}


\subsection{With ordinary section side, the one-section unibranch branch is horizontally low-weight}
\label{subsec:routeC-one-section-horizontal-lowweight}

The terminal-fork reduction of the previous subsection still left an \emph{a priori} arbitrary horizontal weight
$\beta=-H^2.$
In the genuinely one-section branch, this can be sharpened further.
If every horizontal section distinct from the multisection is ordinary, then the fiber anticanonical inequality forces the
multisection itself to carry discrepancy coefficient strictly smaller than
\[
\frac12,
\]
so its self-intersection can only be
\[
-2
\quad\text{or}\quad
-3.
\]
Thus the horizontal arm of every surviving terminal fork is already numerically of the same low-weight type as the two
special tip-visible cores isolated earlier on the chain side.

\begin{proposition}[Ordinary sections force the multisection arm to have weight $2$ or $3$]
\label{prop:routeC-one-section-ordinary-sections-beta23}
Keep the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}.
Let
$H\subset D_{\mathrm{hor}}$
be the unique horizontal multisection, and write
$d:=F\cdot H\in\{2,3\}.$
Let
$S_1,\dots,S_{4-d}\subset D_{\mathrm{hor}}-H$
be the horizontal sections.
Assume that every
$S_i$
is ordinary in the sense of Definition~\ref{def:routeC-global-special-horizontal}.
Assume moreover that some unibranch singular point of
$C=\nu(H)$
has a surviving terminal local block on
$\widetilde X$
as in Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor}, and write
$H^2=-\beta.$
Then
$\beta\in\{2,3\}.$
Equivalently,
\[
\operatorname{cf}_X(H)<\frac12.
\]
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:routeC-fiber-anticanonical-cf}, a witnessing general fiber satisfies
\[
\sum_{T\subset D_{\mathrm{hor}}}\operatorname{cf}_X(T)\,(F\cdot T)<2.
\]
Since the horizontal degree partition is
\[
d+\underbrace{1+\cdots+1}_{4-d\text{ sections}},
\]
this becomes
\[
d\,\operatorname{cf}_X(H)+\sum_{i=1}^{4-d}\operatorname{cf}_X(S_i)<2.
\]
Each section
$S_i$
is ordinary, so Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(S_i)\ge \frac12.
\]
Therefore
\[
d\,\operatorname{cf}_X(H)<2-\frac{4-d}{2}=\frac d2,
\]
hence
\[
\operatorname{cf}_X(H)<\frac12.
\]
Now Lemma~\ref{lem:routeC-cf-selfintersection-lower} applied to
$H^2=-\beta$
gives
\[
\operatorname{cf}_X(H)\ge 1-\frac{2}{\beta}.
\]
Combining the two inequalities yields
\[
1-\frac{2}{\beta}<\frac12,
\]
so
$\beta<4.$
Since
$\beta\ge 2,$
we conclude that
$\beta\in\{2,3\}.$
\end{proof}

\begin{corollary}[The pure one-section unibranch branch splits into a weight-$2$ corridor and a finite weight-$3$ list]
\label{cor:routeC-one-section-ordinary-sections-finite-heavy}
Keep the assumptions of Proposition~\ref{prop:routeC-one-section-ordinary-sections-beta23}.
Let the surviving terminal block over the chosen unibranch singular point be one of the low-order forks from
Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor}.
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item
    $H^2=-2.$
    So the horizontal side already has the same numerical weight as the touched $(-2)$-tip geometry isolated earlier in
    Corollary~\ref{cor:routeC-special-tip-side}.

    \item
    $H^2=-3.$
    Then the remaining touched vertical weights are finite.
    More precisely:
    \begin{enumerate}[label=\textnormal{(\alph*)}]
        \item in the tangency case of Proposition~\ref{prop:routeC-terminal-tangency-fork}, writing
        $A^2=-\alpha,$
        one has
        $\alpha\in\{3,4,5\}$
        for an order-$2$ terminal block, and
        $\alpha\in\{4,5\}$
        for an order-$3$ terminal block;

        \item in the corner case of Proposition~\ref{prop:routeC-terminal-corner-fork}, writing
        \[
        A^2=-\alpha,
        \qquad
        B^2=-\gamma,
        \]
        one has, up to order,
        \[
        \{\alpha,\gamma\}\in
        \bigl\{\{2,2\},\{2,3\},\{2,4\},\{2,5\}\bigr\}.
        \]
    \end{enumerate}
\end{enumerate}
In particular, once all section sides are ordinary, the only genuinely new one-section unibranch survivors beyond the
horizontal $[2]$-case form a finite horizontal-$[3]$ list.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-one-section-ordinary-sections-beta23}, one has
$\beta\in\{2,3\}.$
If
$\beta=2,$
then alternative~\textnormal{(i)} holds.

Assume now that
$\beta=3.$
In the tangency case, Corollary~\ref{cor:routeC-terminal-loworder-fork-sieve}\textnormal{(i)} gives
$(\alpha-2)(\beta-2)<4.$
Substituting
$\beta=3$
yields
$\alpha<6.$
Since Proposition~\ref{prop:routeC-terminal-tangency-fork} gives
\[
\alpha\ge a+1
\qquad\text{for }a\in\{2,3\},
\]
we get
$\alpha\in\{3,4,5\}$
for
$a=2,$
and
$\alpha\in\{4,5\}$
for
$a=3.$
This proves \textnormal{(ii)(a)}.

In the corner case, Corollary~\ref{cor:routeC-terminal-loworder-fork-sieve}\textnormal{(ii)} gives
\[
\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>1.
\]
With
$\beta=3$
this becomes
\[
\frac{1}{\alpha}+\frac{1}{\gamma}>\frac23.
\]
After reordering so that
$\alpha\le \gamma,$
if
$\alpha\ge 3,$
then
\[
\frac{1}{\alpha}+\frac{1}{\gamma}\le \frac13+\frac13=\frac23,
\]
a contradiction.  Hence
$\alpha=2.$
The inequality then reads
\[
\frac12+\frac{1}{\gamma}>\frac23,
\]
so
$\gamma<6.$
Therefore
$\gamma\in\{2,3,4,5\},$
which proves \textnormal{(ii)(b)}.
\end{proof}

\begin{remark}[What this buys for the endpoint reduction]
\label{rem:routeC-one-section-ordinary-sections-lowweight}
The genuinely one-section separable branch is now smaller than the previous terminal-fork statement suggested.
If the multisection is the only special horizontal component and all section sides are ordinary, then every surviving
unibranch terminal block has horizontal weight
$2\quad\text{or}\quad 3.$
So the residual one-section unibranch branch splits cleanly into two pieces:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item a horizontal $[2]$-side, which is numerically parallel to the touched $(-2)$-tip obstruction already isolated
    on the chain side;
    \item a finite horizontal $[3]$-list, namely the cases enumerated in
    Corollary~\ref{cor:routeC-one-section-ordinary-sections-finite-heavy}.
\end{enumerate}
Thus the unibranch part of the pure one-section branch is no longer an infinite weighted fork corridor.  Only the
horizontal $[2]$-case and a finite horizontal $[3]$ list remain.
If instead one of the section sides is itself special, then the witness already contains at least two special horizontal
components, so one leaves the pure one-section branch and returns to the multi-special problem isolated in
Remark~\ref{rem:routeC-final-lemma-what-remains-after-proof-program}.
\end{remark}


\subsection{The horizontal-\texorpdfstring{$[3]$}{[3]} branch already enters the old special-tip / branching corridors}
\label{subsec:routeC-one-section-horizontal3-oldcorridor}

The weight-$3$ alternative from Corollary~\ref{cor:routeC-one-section-ordinary-sections-finite-heavy}
still looked like a new finite fork problem.
In fact it is not.
The contact-defect inequality forces one of the vertical components adjacent to the surviving last exceptional curve
to have discrepancy coefficient strictly smaller than
\[
\frac13.
\]
On a chain side, that is already strong enough to force the old touched-$(-2)$-tip geometry.
So the horizontal-$[3]$ branch does not create a genuinely new local side type: it is absorbed by the previously
isolated special-tip / branching corridors.

\begin{lemma}[On a chain side, discrepancy coefficient below one third forces a touched \texorpdfstring{$(-2)$}{(-2)}-tip]
\label{lem:routeC-chain-cf-third-forces-2tip}
Let
$(A,E)$
be a touched chain pair, and let
$(A^\sharp,E^\sharp)$
be its $2$-tail core from Definition~\ref{def:routeC-two-tail-core}.
If
\[
\operatorname{cf}_X(E)<\frac13,
\]
then
$E^\sharp$
is a tip of
$A^\sharp$
of weight
$2.$
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu},
$\operatorname{cf}_X(E)=1-\mu(A,E).$
So the hypothesis is equivalent to
\[
\mu(A,E)>\frac23.
\]
By Lemma~\ref{lem:routeC-mu-monotone-tail},
$\mu(A,E)\le \mu(A^\sharp,E^\sharp),$
hence
\[
\mu(A^\sharp,E^\sharp)>\frac23.
\]

If
$E^\sharp$
were an interior vertex of
$A^\sharp,$
then Proposition~\ref{prop:routeC-interior-core-half} would give
\[
\mu(A^\sharp,E^\sharp)\le \frac12,
\]
a contradiction.
So
$E^\sharp$
is a tip.

Now Proposition~\ref{prop:routeC-tipcore-highcontact} shows that a touched tip with
\[
\mu>\frac12
\]
has reduced touched core either a touched tip of weight
$2$
or the singleton chain
$[3].$
But for the singleton chain
$[3]$
one has
\[
\mu=\frac23,
\]
whereas here
\[
\mu(A^\sharp,E^\sharp)>\frac23.
\]
Therefore the reduced touched core cannot be
$[3].$
Hence
$E^\sharp$
must be a touched tip of weight
$2.$
\end{proof}

\begin{proposition}[Every surviving horizontal-\texorpdfstring{$[3]$}{[3]} low-order fork already has a special vertical side]
\label{prop:routeC-one-section-horizontal3-special-vertical-side}
Keep the assumptions of Proposition~\ref{prop:routeC-one-section-ordinary-sections-beta23}, and assume moreover that
$H^2=-3.$
Let the surviving terminal local block over the chosen unibranch singular point be one of the low-order forks from
Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor}.
Then there exists a vertical component
$E\subset D$
adjacent to the surviving curve
$L$
such that
\[
\operatorname{cf}_X(E)<\frac13.
\]

Consequently, if the connected component of
$D$
containing
$E$
is a chain, then the $2$-tail core of the touched pair relative to
$E$
is a touched tip of weight
$2.$
If that connected component is not a chain, then one is already on the branching side of the residual Route~C gap.
\end{proposition}

\begin{proof}
Since
$H^2=-3,$
Lemma~\ref{lem:routeC-cf-selfintersection-lower} gives
\[
\operatorname{cf}_X(H)\ge 1-\frac23=\frac13.
\]

Assume first that the terminal block is a tangency fork as in
Proposition~\ref{prop:routeC-terminal-tangency-fork}.  Let
$E:=B_{a-1},$
so
$L\cdot H=L\cdot A=L\cdot E=1.$
By Proposition~\ref{prop:routeC-contact-defect-sieve},
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(E)<1.
\]
Also Proposition~\ref{prop:routeC-terminal-tangency-fork} gives
$A^2\le -(a+1)\le -3,$
hence
\[
\operatorname{cf}_X(A)\ge 1-\frac23=\frac13
\]
by Lemma~\ref{lem:routeC-cf-selfintersection-lower}.
Therefore
\[
\operatorname{cf}_X(E)
<
1-\frac13-\frac13
=
\frac13.
\]

Assume next that the terminal block is a corner fork as in
Proposition~\ref{prop:routeC-terminal-corner-fork}.  Then
$L\cdot H=L\cdot A=L\cdot B=1,$
so Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<1.
\]
Using
\[
\operatorname{cf}_X(H)\ge \frac13,
\]
we obtain
\[
\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<\frac23.
\]
Hence at least one of
$A,\ B$
has discrepancy coefficient strictly smaller than
\[
\frac13.
\]
Take
$E$
to be such a component.

The final statement is exactly Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} when the component of
$D$
containing
$E$
is a chain.  If that component is not a chain, then by definition one is already on the branching side.
\end{proof}

\begin{corollary}[The horizontal-\texorpdfstring{$[3]$}{[3]} one-section branch is not a new corridor]
\label{cor:routeC-one-section-horizontal3-not-new}
Keep the setup of Proposition~\ref{prop:routeC-one-section-horizontal3-special-vertical-side}.
Then every surviving one-section unibranch terminal block with
$H^2=-3$
already falls into one of the two side types that were previously isolated elsewhere in the note:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a chain side whose $2$-tail core is a touched tip of weight
    $2;$
    \item a branching side.
\end{enumerate}
In particular, the horizontal-\texorpdfstring{$[3]$}{[3]} branch does not create a genuinely new residual one-section
mechanism.
\end{corollary}

\begin{proof}
This is exactly Proposition~\ref{prop:routeC-one-section-horizontal3-special-vertical-side}.
\end{proof}

\begin{remark}[What this changes in the one-section endgame]
\label{rem:routeC-one-section-horizontal3-absorbed}
The previous low-weight split left two pieces:
a horizontal
$[2]$
corridor and a finite horizontal
$[3]$
list.
The new proposition shows that the second piece is not genuinely new.
Whenever
$H^2=-3,$
one of the vertical sides adjacent to the surviving terminal fork already lands in the old special-tip / branching
geometry.

So, inside the genuinely one-section separable branch with ordinary section sides, the only still-distinctive new
unibranch mechanism is the horizontal
$[2]$
corridor.
The horizontal
$[3]$
cases have already been funneled back into the previously isolated Route~C side types.
\end{remark}




\subsection{A global proof-first trichotomy at height at least four}
\label{subsec:routeC-global-proof-first-trichotomy}

The local reductions on the multisection and one-section sides can be organized into a single global statement.
The key observation is that ordinary multisections are already expensive in the fiber anticanonical inequality:
each unit of horizontal degree carried by an ordinary multisection contributes at least one half.

\begin{lemma}[Ordinary multisections carry total horizontal degree at most three]
\label{lem:routeC-ordinary-multisection-total-degree-le3}
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration on the minimal resolution of a singular klt del Pezzo surface, let
$F$
be a general fiber, and let
\[
\mathcal M_{\mathrm{ord}}
:=
\{H\subset D_{\mathrm{hor}} \,;\, F\cdot H\ge 2,\ \ H\text{ is ordinary in the sense of Definition~\ref{def:routeC-global-special-horizontal}}\}.
\]
Then
\[
\sum_{H\in \mathcal M_{\mathrm{ord}}}(F\cdot H)\le 3.
\]
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:routeC-global-ordinary-horizontal-half}, every
$H\in \mathcal M_{\mathrm{ord}}$
satisfies
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
Applying Lemma~\ref{lem:routeC-fiber-anticanonical-cf} to the witnessing fiber gives
\[
\sum_{T\subset D_{\mathrm{hor}}}\operatorname{cf}_X(T)\,(F\cdot T)<2.
\]
Discarding all horizontal components outside
$\mathcal M_{\mathrm{ord}}$
and using the preceding lower bound yields
\[
2>
\sum_{T\subset D_{\mathrm{hor}}}\operatorname{cf}_X(T)\,(F\cdot T)
\ge
\sum_{H\in \mathcal M_{\mathrm{ord}}}\operatorname{cf}_X(H)\,(F\cdot H)
\ge
\frac12
\sum_{H\in \mathcal M_{\mathrm{ord}}}(F\cdot H).
\]
Therefore
\[
\sum_{H\in \mathcal M_{\mathrm{ord}}}(F\cdot H)<4.
\]
Since the left-hand side is an integer, it is at most
$3.$
\end{proof}

\begin{proposition}[Global trichotomy for witnessing fibrations of height at least four]
\label{prop:routeC-height4-horizontal-trichotomy}
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration of height
$h:=F\cdot D\ge 4$
on the minimal resolution of a singular klt del Pezzo surface.
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item every horizontal component is a $1$-section;
    \item there exists a special horizontal multisection
    \[
    H\subset D_{\mathrm{hor}},
    \qquad
    F\cdot H\ge 2;
    \]
    \item there exists a unique horizontal multisection
    $H\subset D_{\mathrm{hor}},$
    this component is ordinary, one has
    $F\cdot H\in\{2,3\},$
    and every other horizontal component is a $1$-section.
\end{enumerate}

In case~\textnormal{(iii)}, after choosing any horizontal section
$H_0\subset D_{\mathrm{hor}}-H$
and contracting all vertical curves disjoint from
$H_0$
to a ruled surface
$\nu\colon \widetilde X\to \mathbb F_n,$
the image
$C:=\nu(H)$
is singular, and every singular point of
$C$
is either:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item a collision point of at least two local branches of
    $C;$
    \item a unibranch point whose local degree over the base is
    $2\quad\text{or}\quad 3.$
\end{enumerate}
\end{proposition}

\begin{proof}
If every horizontal component is a $1$-section, we are in case~\textnormal{(i)}.
So assume that some horizontal component has degree at least
$2.$

If one such multisection is special, then case~\textnormal{(ii)} holds.
Assume from now on that no horizontal multisection is special.
Then every horizontal multisection is ordinary, so Lemma~\ref{lem:routeC-ordinary-multisection-total-degree-le3} gives
\[
\sum_{H\subset D_{\mathrm{hor}},\ F\cdot H\ge 2}(F\cdot H)\le 3.
\]
Because
$h=F\cdot D\ge 4,$
not all horizontal components can be multisections.  Hence there is at least one horizontal $1$-section.

On the other hand, every horizontal multisection has degree at least
$2.$
Since their total degree is at most
$3,$
there can be only one such component.
Call it
$H.$
Its degree must then be
$F\cdot H\in\{2,3\},$
and every other horizontal component is a $1$-section.
This proves case~\textnormal{(iii)} and the trichotomy.

Now assume case~\textnormal{(iii)}, and choose a horizontal section
$H_0\subset D_{\mathrm{hor}}-H.$
Let
$\nu\colon \widetilde X\to \mathbb F_n$
be the contraction of all vertical curves disjoint from
$H_0.$
Because
$F\cdot H\ge 2,$
Corollary~\ref{cor:routeC-one-section-higherdegree-singular} shows that the ruled-model image
$C:=\nu(H)$
is singular.

Let
$q\in C$
be a singular point.
By Lemma~\ref{lem:routeC-singular-horizontal-branch-or-ramification}, either
$q$
is a collision point of at least two local branches of
$C,$
or the normalization map has a unique preimage
$x$
above
$q$
and the induced map to the base has local degree at least
$2$
at
$x.$
Since the global degree of
$H\to \PP^1$
is
$F\cdot H\in\{2,3\},$
that local degree is automatically at most
$3.$
This proves the final assertion.
\end{proof}

\begin{corollary}[Residual witnesses reduce to section-only, special-multisection, or low-degree mixed type]
\label{cor:routeC-global-proof-first-reduction}
Let
$X$
be a characteristic-$3$ klt del Pezzo surface of Picard rank
$1$
with
$\#\Sing(X)\ge 8.$
Let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration on the minimal resolution.
Then
$\operatorname{ht}(X)\ge 4$
by Theorem~\ref{thm:routeC-contact2-reduction}, and exactly one of the following three global branches occurs.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item \textbf{Section-only branch.}
    Every horizontal component is a $1$-section.
    If
    $\operatorname{ht}(X)=4,$
    then
    $D_{\mathrm{hor}}=H_0+H_1+H_2+H_3$
    is already a maximal-width witness, so one is in the final two-touch corridor of
    Section~\ref{subsec:routeC-final-lemma-needed}.
    If
    $\operatorname{ht}(X)\ge 5,$
    then the remaining global problem is the section-only packet / isolated-collision branch of
    Remarks~\ref{rem:routeC-scalarstar-ramification-nextgap} and~\ref{rem:routeC-packet-next-gap}.

    \item \textbf{Special-multisection branch.}
    There exists a special horizontal multisection of degree at least
    $2.$
    This contains the residual separable special-multisection branch and the purely inseparable branch isolated in
    Corollaries~\ref{cor:routeC-final-lemma-pureinseparable-fork} and~\ref{cor:routeC-ordinary-ramification-cubic}.

    \item \textbf{Low-degree mixed branch.}
    There exists a unique ordinary horizontal multisection
    $H\subset D_{\mathrm{hor}}$
    of degree
    $2\quad\text{or}\quad 3,$
    and every other horizontal component is a $1$-section.
    After fixing any section and passing to the ruled model, the image of
    $H$
    is singular, and every singular point is either a branch collision or a unibranch point of local degree
    $2\quad\text{or}\quad 3.$
\end{enumerate}
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:routeC-contact2-reduction},
$\operatorname{ht}(X)\ge 4.$
So Proposition~\ref{prop:routeC-height4-horizontal-trichotomy} applies to any witnessing fibration.

If case~\textnormal{(i)} of that proposition holds, then we are in the section-only branch.
When
$\operatorname{ht}(X)=4,$
all four units of horizontal degree are carried by sections, so the witness already has maximal width.
When
$\operatorname{ht}(X)\ge 5,$
the later packet / scalar-star / isolated-collision reductions apply by construction.

If case~\textnormal{(ii)} of Proposition~\ref{prop:routeC-height4-horizontal-trichotomy} holds, then we are in the
special-multisection branch.

If case~\textnormal{(iii)} holds, then we are in the low-degree mixed branch with the stated singular-image
description.
\end{proof}

\begin{corollary}[Height-$4$ non-maximal-width witnesses are either special or low-degree mixed]
\label{cor:routeC-height4-nonmax-special-or-lowdegree}
Assume that
$\operatorname{ht}(X)=4$
and let
$p\colon \widetilde X\to \PP^1$
be a witnessing $\PP^1$-fibration which is not of maximal width.
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item there exists a special horizontal multisection of degree at least
    $2;$
    \item there exists a unique ordinary horizontal multisection
    $H\subset D_{\mathrm{hor}}$
    with
    $F\cdot H\in\{2,3\},$
    and the horizontal degree partition is therefore
    \[
    2+1+1
    \qquad\text{or}\qquad
    3+1.
    \]
\end{enumerate}
In particular, every height-$4$ non-maximal-width witness outside the special-multisection branch already lies in the
low-degree one-section singular-image corridor.
\end{corollary}

\begin{proof}
Because the witness is not of maximal width, not every horizontal component is a $1$-section.
Apply Proposition~\ref{prop:routeC-height4-horizontal-trichotomy}.
Case~\textnormal{(i)} is impossible, so only cases~\textnormal{(ii)} and~\textnormal{(iii)} remain.
In case~\textnormal{(iii)}, the unique multisection has degree
$2\quad\text{or}\quad 3,$
and the remaining horizontal degree is
$4-(F\cdot H),$
carried entirely by sections.
Thus the degree partition is
\[
2+1+1
\qquad\text{or}\qquad
3+1.
\]
\end{proof}

\begin{remark}[What this adds globally]
\label{rem:routeC-global-proof-first-what-adds}
The new trichotomy removes the possibility of a genuinely high-degree \emph{ordinary} mixed branch.
Once special multisections are set aside, every residual witnessing fibration of height at least
$4$
is forced into one of only two remaining horizontal types:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item the section-only branch; or
    \item a unique ordinary bisection / trisection plus sections.
\end{enumerate}
So the global height-selection problem is now sharper than before:
the unresolved global branches are precisely the section-only packet / isolated-collision branch, the special-multisection
branch, the low-degree mixed singular-image branch, and the height-$4$, maximal-width two-touch problem.
\end{remark}





\subsection{The low-degree mixed branch is either multi-special or height four with one old special section}
\label{subsec:routeC-lowdegree-global-unique-special}

The proof-first trichotomy leaves a unique ordinary bisection or trisection together with horizontal sections.
The same fiber anticanonical inequality still forces almost all of those section sides to be special.
As a result, above height four the low-degree mixed branch is already absorbed by the several-special-horizontal problem,
while in height four the only residual case with no such several-special phenomenon has a unique special section.
That unique special section already lies on one of the old special-tip / branching side types, and the ruled model relative
to it lives on $\mathbb F_2$ or $\mathbb F_3$.

\begin{proposition}[Low-degree mixed witnesses have at most \texorpdfstring{$3-d$}{3-d} ordinary sections]
\label{prop:routeC-lowdegree-mixed-at-most-3-minus-d-ordinary-sections}
Assume that
$p\colon \widetilde X\to \PP^1$
lies in case~\textnormal{(iii)} of Proposition~\ref{prop:routeC-height4-horizontal-trichotomy}.  Let
$H\subset D_{\mathrm{hor}}$
be the unique ordinary horizontal multisection, put
$d:=F\cdot H\in\{2,3\},$
and let
$o$
be the number of ordinary horizontal sections contained in
$D_{\mathrm{hor}}-H.$
Then
$o\le 3-d.$

Equivalently, if
$r$
denotes the number of special horizontal sections contained in
$D_{\mathrm{hor}}-H,$
then
\[
r\ge h-3,
\qquad
h:=F\cdot D.
\]
In particular:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if
    $d=3,$
    then every horizontal section is special;
    \item if
    $d=2,$
    then at most one horizontal section is ordinary.
\end{enumerate}
\end{proposition}

\begin{proof}
By case~\textnormal{(iii)} of Proposition~\ref{prop:routeC-height4-horizontal-trichotomy}, the component
$H$
is ordinary, so Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
Every ordinary horizontal section
$S\subset D_{\mathrm{hor}}-H$
also satisfies
\[
\operatorname{cf}_X(S)\ge \frac12
\]
by the same lemma.

Applying Lemma~\ref{lem:routeC-fiber-anticanonical-cf} to a witnessing general fiber gives
\[
\sum_{T\subset D_{\mathrm{hor}}}\operatorname{cf}_X(T)\,(F\cdot T)<2.
\]
Among the horizontal components, the ordinary multisection
$H$
contributes degree
$d,$
the
$o$
ordinary sections contribute degree
$o,$
and the remaining special sections contribute only nonnegative terms.  Hence
\[
2>
d\,\operatorname{cf}_X(H)+\sum_{i=1}^{o}\operatorname{cf}_X(S_i)
\ge
\frac d2+\frac o2.
\]
Therefore
$d+o<4.$
Since
$d+o$
is an integer and
$d\in\{2,3\},$
we conclude that
$o\le 3-d.$

For the equivalent form, note that every horizontal component distinct from
$H$
is a section, so the total number of horizontal sections is
$h-d.$
Hence
$r=(h-d)-o\ge (h-d)-(3-d)=h-3.$
The final two statements are immediate.
\end{proof}

\begin{corollary}[Outside the several-special corridor, the low-degree mixed branch is height four with a unique special section]
\label{cor:routeC-lowdegree-mixed-unique-special-height4}
Assume the setup of Corollary~\ref{cor:routeC-global-proof-first-reduction}, and suppose that we are in the low-degree
mixed branch.  If the chosen witnessing fibration does \emph{not} have at least two special horizontal sections, then
$\operatorname{ht}(X)=4$
and there is a unique special horizontal section
$H_0\subset D_{\mathrm{hor}}.$
Consequently the horizontal degree partition is
\[
2+1+1
\qquad\text{or}\qquad
3+1,
\]
the unique multisection is ordinary, and every horizontal component distinct from
\[
H
\quad\text{and}\quad
H_0
\]
is an ordinary section.
\end{corollary}

\begin{proof}
Let
$r$
be the number of special horizontal sections.  By Proposition~\ref{prop:routeC-lowdegree-mixed-at-most-3-minus-d-ordinary-sections},
\[
r\ge h-3,
\qquad
h=\operatorname{ht}(X).
\]
If
$r\le 1,$
then
$h-3\le 1,$
so
$h\le 4.$
On the other hand, Corollary~\ref{cor:routeC-global-proof-first-reduction} already gives
$h\ge 4.$
Hence
$h=4.$
Then
$r\ge h-3=1,$
and together with
$r\le 1$
this yields
$r=1.$
So there is a unique special horizontal section
$H_0.$
Since we are in the low-degree mixed branch, Proposition~\ref{prop:routeC-height4-horizontal-trichotomy} shows that the
unique multisection
$H$
is ordinary of degree
\[
2
\quad\text{or}\quad
3,
\]
and every other horizontal component is a section.  Because
$h=4,$
the degree partition is
\[
2+1+1
\qquad\text{or}\qquad
3+1.
\]
Every section distinct from
$H_0$
is then ordinary by uniqueness of the special section.
\end{proof}

\begin{proposition}[The unique special section is an old special side and forces \texorpdfstring{$\mathbb F_2$}{F2} or \texorpdfstring{$\mathbb F_3$}{F3}]
\label{prop:routeC-lowdegree-unique-special-F2F3}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-unique-special-height4}.  Let
$H_0\subset D_{\mathrm{hor}}$
be the unique special horizontal section, let
$H\subset D_{\mathrm{hor}}$
be the unique ordinary multisection, and put
$d:=F\cdot H\in\{2,3\}.$
Then
\[
\operatorname{cf}_X(H_0)<\frac12.
\]
In particular,
$H_0^2\in\{-2,-3\}.$

If the connected component of
$D$
containing
$H_0$
is a chain, then the reduced touched core relative to
$H_0$
is either a touched tip of weight
$2$
or the singleton chain
$[3].$
If that connected component is not a chain, then one is already on the branching side.

Let
$\nu\colon \widetilde X\to \mathbb F_n$
be the contraction of all vertical curves disjoint from
$H_0,$
put
\[
S_0:=\nu(H_0),
\qquad
C:=\nu(H),
\qquad
s:=H\cdot H_0\in\{0,1\},
\]
and let
$f$
denote a fiber of the ruling on
$\mathbb F_n.$
Then
$n=-H_0^2\in\{2,3\}.$
Moreover:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if
    $d=2,$
    then
    \[
    C\equiv 2S_0+(2n+s)f,
    \qquad
    \delta(C)=n+s-1\in\{1,2,3\},
    \]
    and the elementary-transform sequence from Proposition~\ref{prop:routeC-bisection-plane-reduction} has length
    $n-1\le 2;$

    \item if
    $d=3,$
    then
    \[
    C\equiv 3S_0+(3n+s)f,
    \qquad
    \delta(C)=3n+2s-2\in\{4,6,7,9\}.
    \]
\end{enumerate}

Finally, the ordinary multisection
$H$
has bounded self-intersection:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item if
    $d=2,$
    then
    \[
    \operatorname{cf}_X(H)<\frac34
    \qquad\text{and hence}\qquad
    H^2\in\{-2,-3,-4,-5,-6,-7\};
    \]

    \item if
    $d=3,$
    then
    \[
    \operatorname{cf}_X(H)<\frac23
    \qquad\text{and hence}\qquad
    H^2\in\{-2,-3,-4,-5\}.
    \]
\end{enumerate}
\end{proposition}

\begin{proof}
Because
$H$
is ordinary, Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
By Corollary~\ref{cor:routeC-lowdegree-mixed-unique-special-height4}, every horizontal section distinct from
$H_0$
is ordinary.  Since the height is
$4,$
there are exactly
$3-d$
such ordinary sections.  Therefore Lemma~\ref{lem:routeC-fiber-anticanonical-cf} yields
\[
2>
d\,\operatorname{cf}_X(H)
+
\operatorname{cf}_X(H_0)
+
\sum_{i=1}^{3-d}\operatorname{cf}_X(S_i)
\ge
\frac d2+\operatorname{cf}_X(H_0)+\frac{3-d}{2}
=
\frac32+\operatorname{cf}_X(H_0).
\]
Hence
\[
\operatorname{cf}_X(H_0)<\frac12.
\]

Applying Lemma~\ref{lem:routeC-cf-selfintersection-lower} to
$H_0^2=-a$
gives
\[
1-\frac{2}{a}\le \operatorname{cf}_X(H_0)<\frac12,
\]
so
$a<4.$
Since
$a\ge 2,$
we conclude that
$a\in\{2,3\},$
equivalently
$H_0^2\in\{-2,-3\}.$

The chain / branching dichotomy is now immediate from the fact that
$H_0$
is special in the sense of Definition~\ref{def:routeC-global-special-horizontal}: if the connected component of
$D$
containing
$H_0$
is a chain, then its reduced touched core relative to
$H_0$
is by definition either a touched tip of weight
$2$
or the singleton chain
$[3];$
if not, then one is on the branching side.

The formula
$n=-H_0^2$
comes from Proposition~\ref{prop:routeC-one-section-image-class-genus}.  Since
$H_0^2\in\{-2,-3\},$
we get
$n\in\{2,3\}.$
The class and arithmetic-genus formulas for
$C$
are exactly those of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}.  Substituting
\[
n\in\{2,3\}
\qquad\text{and}\qquad
s\in\{0,1\}
\]
gives the displayed finite sets of values for
$\delta(C).$
When
$d=2,$
Proposition~\ref{prop:routeC-bisection-plane-reduction} uses precisely
$n-1$
singular-point elementary transforms, so here that sequence has length at most
$2.$

For the bounds on
$H,$
argue again from Lemma~\ref{lem:routeC-fiber-anticanonical-cf}.  If
$d=2,$
there is exactly one ordinary horizontal section distinct from
$H_0,$
so
\[
2>
2\,\operatorname{cf}_X(H)+\operatorname{cf}_X(H_0)+\operatorname{cf}_X(S_1)
\ge
2\,\operatorname{cf}_X(H)+0+\frac12.
\]
Hence
\[
\operatorname{cf}_X(H)<\frac34.
\]
Writing
$H^2=-\beta,$
Lemma~\ref{lem:routeC-cf-selfintersection-lower} gives
\[
1-\frac{2}{\beta}<\frac34,
\]
so
$\beta<8.$
Since
$\beta\ge 2,$
we obtain
$\beta\in\{2,3,4,5,6,7\}.$

If
$d=3,$
then
$H_0$
is the only horizontal section, so
\[
2>
3\,\operatorname{cf}_X(H)+\operatorname{cf}_X(H_0)
\ge
3\,\operatorname{cf}_X(H)+0.
\]
Therefore
\[
\operatorname{cf}_X(H)<\frac23.
\]
Again writing
$H^2=-\beta,$
Lemma~\ref{lem:routeC-cf-selfintersection-lower} yields
\[
1-\frac{2}{\beta}<\frac23,
\]
hence
$\beta<6.$
Since
$\beta\ge 2,$
this gives
$\beta\in\{2,3,4,5\}.$
\end{proof}

\begin{corollary}[Global form of the low-degree mixed branch]
\label{cor:routeC-lowdegree-mixed-global-form}
Assume the setup of Corollary~\ref{cor:routeC-global-proof-first-reduction}, and suppose that we are in the low-degree
mixed branch.  Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the chosen witnessing fibration already has at least two special horizontal sections, so it lies in the
    several-special-horizontal corridor from Remark~\ref{rem:routeC-final-lemma-what-remains-after-proof-program};

    \item
    $\operatorname{ht}(X)=4,$
    there is a unique special horizontal section
    $H_0,$
    and the unique ordinary multisection has degree
    $2.$
    Relative to
    $H_0,$
    the ruled model lies on
    \[
    \mathbb F_2
    \quad\text{or}\quad
    \mathbb F_3,
    \]
    the bisection has total
    $\delta\le 3,$
    and Proposition~\ref{prop:routeC-bisection-plane-reduction} reduces the whole branch by at most two
    singular-point elementary transforms to either a smooth plane conic or a marked singular plane cubic;

    \item
    $\operatorname{ht}(X)=4,$
    there is a unique special horizontal section
    $H_0,$
    and the unique ordinary multisection has degree
    $3.$
    Relative to
    $H_0,$
    the ruled model lies on
    \[
    \mathbb F_2
    \quad\text{or}\quad
    \mathbb F_3,
    \]
    the trisection class is one of the four explicit classes from
    Proposition~\ref{prop:routeC-lowdegree-unique-special-F2F3}, and the section side is already one of the old
    special-tip / branching geometries.
\end{enumerate}
In particular, the low-degree mixed branch does not create a new high-height problem: above height
$4$
it is already absorbed by the several-special-horizontal corridor, and in height
$4$
the only genuinely new mixed cases are the finite bisection plane-model package and the singular trisection attached to a
single old special section.
\end{corollary}

\begin{proof}
If the witnessing fibration has at least two special horizontal sections, then we are in case~\textnormal{(i)}.  Otherwise
Corollary~\ref{cor:routeC-lowdegree-mixed-unique-special-height4} applies, giving height
$4$
and a unique special horizontal section
$H_0.$
Then Proposition~\ref{prop:routeC-lowdegree-unique-special-F2F3} gives
$n\in\{2,3\}$
and the stated finite ruled-model packages.  When the multisection degree is
$2,$
the final plane reduction is exactly Proposition~\ref{prop:routeC-bisection-plane-reduction} together with
Proposition~\ref{prop:routeC-bisection-marked-plane-model}.  When the multisection degree is
$3,$
the same proposition shows that the whole branch is concentrated on the four explicit trisection classes listed there, with
the unique section already lying in the old special-tip / branching corridor.
\end{proof}

\begin{remark}[What this adds to the global proof-first picture]
\label{rem:routeC-lowdegree-mixed-global-sharpened}
The low-degree mixed branch is now much smaller than the trichotomy alone suggests.

First, it is not a new high-height branch at all: Proposition~\ref{prop:routeC-lowdegree-mixed-at-most-3-minus-d-ordinary-sections}
shows that above height
$4$
the branch already belongs to the several-special-horizontal problem.

Second, in height
$4,$
the only way to stay outside that several-special corridor is to have a \emph{unique} special horizontal section.
Proposition~\ref{prop:routeC-lowdegree-unique-special-F2F3} shows that this section already lies on one of the old
special-tip / branching side types and has self-intersection
\[
-2
\quad\text{or}\quad
-3.
\]
So the ruled model relative to that section lives only on
\[
\mathbb F_2
\quad\text{or}\quad
\mathbb F_3.
\]

Thus the residual mixed geometry is now completely explicit:
the degree-two branch is a one- or two-step reverse elementary-transform problem from a smooth plane conic or a marked
singular cubic, while the degree-three branch is a finite trisection-class problem on
\[
\mathbb F_2
\quad\text{or}\quad
\mathbb F_3
\]
attached to a single old special section.
\end{remark}


\subsection{The singular trisection branch: one-step reduction and absorption of surviving unibranch blocks}
\label{subsec:routeC-trisection-one-step-and-unibranch}

The residual degree-three mixed branch from Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}
is now a singular rational trisection
\[
C\subset \mathbb F_n,
\qquad
C\equiv 3S_0+(3n+s)f,
\qquad
n\in\{2,3\},
\qquad
s\in\{0,1\},
\]
attached to a unique special section
$H_0.$
There are two further unconditional reductions.
First, one singular-point elementary transform already moves the branch to an explicit finite package.
Second, any surviving unibranch terminal block already has an old special vertical side, so the trisection branch does not create a new unibranch fork corridor.

\begin{proposition}[One singular-point transform reduces the trisection to a finite package]
\label{prop:routeC-trisection-one-step-reduction}
Assume the setup of Proposition~\ref{prop:routeC-lowdegree-unique-special-F2F3}, and suppose that
$d=3.$
Thus
\[
C\equiv 3S_0+(3n+s)f,
\qquad
n\in\{2,3\},
\qquad
s\in\{0,1\}.
\]
Let
$p\in C$
be a singular point.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item
    $p\notin S_0;$

    \item
    $\operatorname{mult}_p(C)\in\{2,3\}.$
\end{enumerate}
Let
\[
\epsilon_p\colon \mathbb F_n\dashrightarrow \mathbb F_{n-1}
\]
be the elementary transform at
$p,$
and let
$C'\subset \mathbb F_{n-1}$
be the image of
$C.$
Writing
\[
m:=\operatorname{mult}_p(C)\in\{2,3\},
\qquad
s':=s+3-m\in\{s,s+1\},
\]
one has
$C'\equiv 3S_0'+(3(n-1)+s')f'.$
Consequently:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item if
    $n=2,$
    then after this single singular-point elementary transform and the contraction
    $\kappa\colon \mathbb F_1\to \PP^2$
    of the negative section, the branch reduces birationally to an irreducible rational plane curve
    $\Gamma\subset \PP^2$
    of degree
    $3+s'\in\{3,4,5\},$
    with marked point
    $q:=\kappa(S_1)$
    of multiplicity
    $s'\in\{0,1,2\};$

    \item if
    $n=3,$
    then after one singular-point elementary transform the branch reduces to one of the three explicit classes on
    \[
    \mathbb F_2:
    \qquad
    3S_0+6f,
    \qquad
    3S_0+7f,
    \qquad
    3S_0+8f.
    \]
    The first two already belong to the
    $n=2$
    package from \textnormal{(a)}.  Hence after at most two singular-point elementary transforms every singular trisection branch reduces either to a marked plane cubic / quartic / quintic package or to the single exceptional
    $\mathbb F_2$
    class
    $3S_0+8f.$
\end{enumerate}
\end{proposition}

\begin{proof}
Since
$C\cdot S_0=s\le 1,$
a singular point of
$C$
cannot lie on
$S_0.$
Indeed, if
$p\in C\cap S_0$
were singular on
$C,$
then
\[
(C\cdot S_0)_p\ge \operatorname{mult}_p(C)\ge 2,
\]
contradicting the total intersection number
$C\cdot S_0=s\le 1.$
This proves \textnormal{(i)}.

Let
$F_p$
be the fiber through
$p.$
Because
$C\cdot F_p=3,$
one has
$\operatorname{mult}_p(C)\le 3.$
Since
$p$
is singular on
$C,$
one also has
$\operatorname{mult}_p(C)\ge 2,$
so
$\operatorname{mult}_p(C)\in\{2,3\}.$
This proves \textnormal{(ii)}.

Now apply Lemma~\ref{lem:routeC-elementary-transform-class} with
\[
a=3,
\qquad
b=3n+s,
\qquad
m=\operatorname{mult}_p(C).
\]
It yields
\[
C'\equiv 3S_0'+(3n+s-m)f'
=3S_0'+(3(n-1)+(s+3-m))f'.
\]
So with
$s':=s+3-m,$
one gets
$C'\equiv 3S_0'+(3(n-1)+s')f'.$
Because
$m\in\{2,3\},$
one has
$s'\in\{s,s+1\}.$

Assume first that
$n=2.$
Then
$C'\subset \mathbb F_1$
and
$C'\equiv 3S_1+(3+s')f.$
Let
$\kappa\colon \mathbb F_1\to \PP^2$
be the contraction of
\[
S_1,
\qquad
q:=\kappa(S_1),
\qquad
H:=\kappa^*(\text{line})=S_1+f.
\]
Then
\[
C'
\equiv
3S_1+(3+s')f
=(3+s')H-s'S_1.
\]
Hence the plane image
$\Gamma:=\kappa(C')$
has degree
$\deg \Gamma=3+s'$
and multiplicity
$\operatorname{mult}_q(\Gamma)=s'.$
Since
$C'$
is birational to the original rational curve, it is irreducible and rational, and so is
$\Gamma.$
Because
\[
s\in\{0,1\}
\qquad\text{and}\qquad
s'\in\{s,s+1\},
\]
one has
\[
s'\in\{0,1,2\},
\qquad
3+s'\in\{3,4,5\}.
\]
This proves \textnormal{(a)}.

Assume next that
$n=3.$
Then
$C'\subset \mathbb F_2$
and
\[
C'\equiv 3S_0'+(6+s')f',
\qquad
s'\in\{0,1,2\}.
\]
So the only possibilities are
\[
3S_0'+6f',
\qquad
3S_0'+7f',
\qquad
3S_0'+8f'.
\]
The first two correspond exactly to the
\[
n=2,
\qquad
s'\in\{0,1\}
\]
package from \textnormal{(a)}, so one more singular-point elementary transform reduces them to marked plane cubic / quartic / quintic models.  The third is the single residual
$\mathbb F_2$
class
$3S_0+8f.$
This proves \textnormal{(b)}.
\end{proof}

\begin{proposition}[Every surviving unibranch trisection block already has an old special vertical side]
\label{prop:routeC-trisection-unibranch-oldside}
Assume the setup of Proposition~\ref{prop:routeC-lowdegree-unique-special-F2F3}, and suppose that
$d=3.$
Let
$q\in C$
be a unibranch singular point.  Assume that, in the factorization of
$\nu^{-1}$
over
$q,$
the last exceptional curve created in the terminal block survives on
$\widetilde X.$
Then there exists a vertical component
$E\subset D$
adjacent to that surviving last exceptional curve such that
\[
\operatorname{cf}_X(E)<\frac13.
\]
Consequently, if the connected component of
$D$
containing
$E$
is a chain, then the corresponding $2$-tail core is a touched tip of weight
$2;$
if that connected component is not a chain, then one is already on the branching side.
\end{proposition}

\begin{proof}
Let
$H\subset D_{\mathrm{hor}}$
be the unique trisection.  In the low-degree mixed branch it is ordinary, so Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]

By Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor}, the terminal surviving local block over
$q$
is one of the following:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a corner fork
    \[
    \begin{array}{ccccc}
    &&A&&\\
    &&|&&\\
    H&-&L&-&B;
    \end{array}
    \]

    \item an order-$2$ tangency fork
    $H-L-A-B_1;$

    \item an order-$3$ tangency fork
    $H-L-A-B_2-B_1.$
\end{enumerate}
In every case
\[
L\not\subset D
\qquad\text{and}\qquad
L\cdot D=3.
\]

Assume first that the terminal block is a tangency fork as in Proposition~\ref{prop:routeC-terminal-tangency-fork}.  Let
\[
E:=B_{a-1},
\qquad
a\in\{2,3\}.
\]
Then
$L\cdot H=L\cdot A=L\cdot E=1.$
By Proposition~\ref{prop:routeC-contact-defect-sieve},
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(E)<1.
\]
Also Proposition~\ref{prop:routeC-terminal-tangency-fork} gives
$A^2\le -(a+1)\le -3,$
so Lemma~\ref{lem:routeC-cf-selfintersection-lower} yields
\[
\operatorname{cf}_X(A)\ge \frac13.
\]
Together with
\[
\operatorname{cf}_X(H)\ge \frac12,
\]
this gives
\[
\operatorname{cf}_X(E)
<1-\frac12-\frac13
=\frac16
<\frac13.
\]

Assume next that the terminal block is a corner fork as in Proposition~\ref{prop:routeC-terminal-corner-fork}.  Then
$L\cdot H=L\cdot A=L\cdot B=1,$
so Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<1.
\]
Using again
\[
\operatorname{cf}_X(H)\ge \frac12,
\]
one gets
\[
\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<\frac12.
\]
Hence at least one of
$A,\ B$
satisfies
\[
\operatorname{cf}_X(E)<\frac14<\frac13.
\]
Take such a component as
$E.$

The final conclusion is exactly Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} when the connected component of
$D$
containing
$E$
is a chain.  If that connected component is not a chain, then by definition one is already on the branching side.
\end{proof}

\begin{corollary}[The singular trisection branch is not a new unibranch corridor]
\label{cor:routeC-trisection-not-new-unibranch}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, and suppose that we are in case~\textnormal{(iii)}.
Then every surviving unibranch terminal block already falls into one of the old side types isolated earlier in the note:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a chain side whose $2$-tail core is a touched tip of weight
    $2;$

    \item a branching side.
\end{enumerate}
In particular, the singular trisection branch does not create a genuinely new unibranch fork corridor.  Any genuinely new trisection obstruction must come from collision points of
$C,$
or from the single exceptional one-step global package
$C\equiv 3S_0+8f\subset \mathbb F_2$
from Proposition~\ref{prop:routeC-trisection-one-step-reduction}.
\end{corollary}

\begin{proof}
This is Proposition~\ref{prop:routeC-trisection-unibranch-oldside} together with Proposition~\ref{prop:routeC-trisection-one-step-reduction}.
\end{proof}

\begin{remark}[What now remains in the trisection target]
\label{rem:routeC-trisection-what-remains-after-push}
The residual trisection branch is now much smaller than the finite-class statement of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form} alone suggested.

First, one singular-point elementary transform already reduces every trisection to an explicit finite global package: either a marked plane cubic / quartic / quintic package, or the single exceptional class
$3S_0+8f\subset \mathbb F_2.$
So the trisection branch is no longer an arbitrary problem on
\[
\mathbb F_2
\quad\text{or}\quad
\mathbb F_3.
\]

Second, Proposition~\ref{prop:routeC-trisection-unibranch-oldside} shows that the local unibranch side is not genuinely new: whenever the terminal block survives, one already lands in the old touched-$(-2)$-tip / branching geometry.

So the honest remaining trisection gap is now concentrated in two places only:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item collision singularities of the trisection image;
    \item the global bookkeeping needed either to force one more off-negative-section singular point in the exceptional
    $\mathbb F_2$
    class
    $3S_0+8f,$
    or to exclude that class directly in the del Pezzo realization problem.
\end{enumerate}
That is a genuine shrink of the residual mixed branch.
\end{remark}



\subsection{The exceptional \texorpdfstring{$\mathbb F_2$}{F2} class \texorpdfstring{$3S_0+8f$}{3S0+8f} also reduces to marked plane models}
\label{subsec:routeC-trisection-exceptional-plane-package}

The previous subsection reduced every singular trisection to a marked plane cubic / quartic / quintic package
\emph{or} to the single exceptional ruled-surface class
$3S_0+8f\subset \mathbb F_2.$
That class can be pushed further as well.  The only missing ingredient is the class formula for an elementary transform
whose center lies on the negative section.

\begin{lemma}[Elementary transform class formula at a point of the negative section]
\label{lem:routeC-elementary-transform-class-on-negative}
Let
\[
\epsilon_p\colon \mathbb F_n\dashrightarrow \mathbb F_{n+1}
\]
be the elementary transform at a point
$p\in S_0.$
Let
$C\subset \mathbb F_n$
be an irreducible curve of class
$C\equiv aS_0+bf,$
and write
$m:=\operatorname{mult}_p(C).$
If
$C^+\subset \mathbb F_{n+1}$
is the image of
$C,$
then
$C^+\equiv aS_0^+ + (b+a-m)f^+,$
where
$(S_0^+)^2=-(n+1)$
is the negative section of
$\mathbb F_{n+1}$
and
$f^+$
is a fiber of its ruling.
\end{lemma}

\begin{proof}
Let
\[
\sigma\colon Z:=\operatorname{Bl}_p\mathbb F_n\to \mathbb F_n
\]
be the blowup of
$p,$
let
$E\subset Z$
be the exceptional curve, let
$F$
be the fiber of
$\mathbb F_n$
through
$p,$
and let
$F'\subset Z$
be the strict transform of
$F.$
Because
$p\in S_0,$
the strict transform of
$S_0$
is
$\widetilde S_0=\sigma^*S_0-E$
and has self-intersection
$\widetilde S_0^{\,2}=-(n+1).$
Contracting
$F'$
therefore gives a morphism
$\tau\colon Z\to \mathbb F_{n+1}$
realizing the elementary transform, and the image of
$\widetilde S_0$
is the negative section
$S_0^+.$

As in Lemma~\ref{lem:routeC-elementary-transform-class},
$\tau^*f^+=\sigma^*f.$
Also
$\tau^*S_0^+=\widetilde S_0=\sigma^*S_0-E.$
The strict transform of
$C$
on
$Z$
is
$\widetilde C=\sigma^*C-mE.$
Since
$\widetilde C\cdot F'=(aS_0+bf)\cdot f-m=a-m,$
one has
$\tau^*C^+=\widetilde C+(a-m)F'.$
Using
$F'=\sigma^*f-E,$
this becomes
\[
\tau^*C^+
=
a\sigma^*S_0+b\sigma^*f-mE+(a-m)(\sigma^*f-E)
=
a\sigma^*S_0+(b+a-m)\sigma^*f-aE.
\]
On the other hand,
\[
a\,\tau^*S_0^+ + (b+a-m)\,\tau^*f^+
=
a(\sigma^*S_0-E)+(b+a-m)\sigma^*f
=
a\sigma^*S_0+(b+a-m)\sigma^*f-aE.
\]
Hence
\[
\tau^*C^+=a\,\tau^*S_0^+ + (b+a-m)\,\tau^*f^+,
\]
so
$C^+\equiv aS_0^+ + (b+a-m)f^+.$
\end{proof}

\begin{proposition}[The exceptional class \texorpdfstring{$3S_0+8f$}{3S0+8f} is not a separate ruled-surface residue]
\label{prop:routeC-trisection-exceptional-plane-package}
Let
$C\subset \mathbb F_2$
be an irreducible rational curve of class
$C\equiv 3S_0+8f.$
Then one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item There exists a singular point
    $p\in C-S_0.$
    Writing
    $m:=\operatorname{mult}_p(C)\in\{2,3\},$
    the elementary transform at
    $p$
    sends
    $C$
    to a curve
    $C_1\subset \mathbb F_1$
    of class
    $C_1\equiv 3S_1+(8-m)f.$
    After contracting the negative section
    $\kappa\colon \mathbb F_1\to \PP^2,$
    the image is an irreducible rational plane curve
    $\Gamma\subset \PP^2$
    of degree
    $8-m\in\{5,6\},$
    with marked point
    $q:=\kappa(S_1)$
    of multiplicity
    $5-m\in\{2,3\}.$

    \item Every singular point of
    $C$
    lies on
    $S_0.$
    Then there is a unique such point
    $p\in C\cap S_0,$
    it has multiplicity
    $\operatorname{mult}_p(C)=2,$
    and the elementary transform at
    $p$
    sends
    $C$
    to an irreducible rational curve
    $C^+\subset \mathbb F_3$
    of class
    $C^+\equiv 3S_0^+ + 9f^+.$
    This curve is singular, every singular point of
    $C^+$
    lies off
    $S_0^+,$
    and after at most two further elementary transforms at singular points off the current negative section one reaches a
    marked plane cubic / quartic / quintic package.
\end{enumerate}

In particular, every irreducible rational curve of class
$3S_0+8f\subset \mathbb F_2$
reduces after at most three elementary transforms and one contraction of
$\mathbb F_1$
to an irreducible rational plane curve
$\Gamma\subset \PP^2$
of degree
$e\in\{3,4,5,6\}$
with marked point
$q$
of multiplicity
$\operatorname{mult}_q(\Gamma)=e-3.$
Projection from
$q$
realizes the original degree-$3$ map from the normalization of
$C$
to the base.
\end{proposition}

\begin{proof}
Since
$p_a(C)=8$
while
$C$
is rational, the curve
$C$
is singular.  If there exists a singular point
$p\in C-S_0,$
then
$C\cdot F_p=3$
for the fiber
$F_p$
through
$p,$
so
$\operatorname{mult}_p(C)\in\{2,3\}.$
Applying Lemma~\ref{lem:routeC-elementary-transform-class} with
\[
a=3,
\qquad
b=8,
\qquad
m=\operatorname{mult}_p(C)
\]
gives
$C_1\equiv 3S_1+(8-m)f$
on
$\mathbb F_1.$
Let
$\kappa\colon \mathbb F_1\to \PP^2$
be the contraction of
\[
S_1,
\qquad
q:=\kappa(S_1),
\qquad
H:=\kappa^*(\text{line})=S_1+f.
\]
Then
\[
C_1
\equiv
3S_1+(8-m)f
=
(8-m)H-(5-m)S_1.
\]
Hence the plane image
$\Gamma:=\kappa(C_1)$
has degree
$8-m\in\{5,6\}$
and multiplicity
$5-m\in\{2,3\}$
at
$q.$
Because
$C_1$
is birational to the rational curve
$C,$
the curve
$\Gamma$
is irreducible and rational.  The fibers of the ruling on
$\mathbb F_1$
are the strict transforms of the lines through
$q,$
so projection from
$q$
realizes the degree-$3$ map from the normalization of
$C_1$
to the base; since the elementary transform is over the same base, this is also the original degree-$3$ map from the
normalization of
$C$
to the base.  This proves \textnormal{(i)}.

Assume now that every singular point of
$C$
lies on
$S_0.$
Since
$C\cdot S_0=2,$
there can be at most one such point.  Because
$C$
is singular, there is exactly one singular point
$p\in C\cap S_0.$
Moreover
\[
2\le \operatorname{mult}_p(C)\le (C\cdot S_0)_p\le C\cdot S_0=2,
\]
so
$\operatorname{mult}_p(C)=2.$

Applying Lemma~\ref{lem:routeC-elementary-transform-class-on-negative} with
\[
a=3,
\qquad
b=8,
\qquad
m=2
\]
gives
$C^+\equiv 3S_0^+ + 9f^+$
on
$\mathbb F_3.$
Because
$C^+$
is birational to the rational curve
$C,$
it is irreducible and rational.  Its arithmetic genus is
$p_a(C^+)=3\cdot 3 + 2\cdot 0 -2 = 7,$
so
$C^+$
is singular.  Also
$C^+\cdot S_0^+=0,$
hence every singular point of
$C^+$
lies off
$S_0^+.$

Choose a singular point
$r\in C^+-S_0^+.$
Then
$\operatorname{mult}_r(C^+)\in\{2,3\}$
because
$C^+\cdot F_r=3.$
Applying Lemma~\ref{lem:routeC-elementary-transform-class} at
$r$
gives a curve
$C_2\subset \mathbb F_2$
of class
\[
C_2\equiv 3S_0'' + (9-\operatorname{mult}_r(C^+))f''.
\]
Thus
\[
C_2\equiv 3S_0''+6f''
\qquad\text{or}\qquad
C_2\equiv 3S_0''+7f''.
\]
Write this uniformly as
\[
C_2\equiv 3S_0''+(6+s'')f'',
\qquad
s''\in\{0,1\}.
\]
Since
$C_2\cdot S_0''=s''\le 1,$
every singular point of
$C_2$
lies off
$S_0''.$
Choose one and perform one more elementary transform.  As in the proof of
Proposition~\ref{prop:routeC-trisection-one-step-reduction}, the resulting curve on
$\mathbb F_1$
has class
\[
3S_1+(3+s''')f,
\qquad
s'''\in\{0,1,2\}.
\]
Contracting
$S_1$
therefore yields an irreducible rational plane curve of degree
$3+s'''\in\{3,4,5\}$
with marked point multiplicity
$s'''=(3+s''')-3.$
Again the final projection from the marked point realizes the original degree-$3$ map because every elementary transform
is performed over the same base.  This proves \textnormal{(ii)} and the final summary statement.
\end{proof}

\begin{corollary}[The singular trisection branch reduces to a finite marked plane package]
\label{cor:routeC-trisection-plane-package}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, and suppose that we are in case~\textnormal{(iii)}.
Then after at most four elementary transforms and one contraction of
$\mathbb F_1,$
the unique trisection reduces birationally to an irreducible rational plane curve
$\Gamma\subset \PP^2$
with a marked point
$q\in \PP^2$
such that
\[
\deg \Gamma\in\{3,4,5,6\},
\qquad
\operatorname{mult}_q(\Gamma)=\deg\Gamma-3.
\]
Projection from
$q$
realizes the original degree-$3$ map from the normalization of the trisection to the base.

In particular, the residual trisection branch is no longer a problem on
\[
\mathbb F_2
\quad\text{or}\quad
\mathbb F_3
\]
at all: it is a finite marked plane package of degrees at most
$6.$
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, the trisection starts in one of the four explicit classes
\[
3S_0+6f,\qquad 3S_0+7f,\qquad 3S_0+9f,\qquad 3S_0+10f.
\]
Proposition~\ref{prop:routeC-trisection-one-step-reduction} shows that after one singular-point elementary transform the
branch reduces either to a marked plane cubic / quartic / quintic package or to the single exceptional class
$3S_0+8f\subset \mathbb F_2.$
Applying Proposition~\ref{prop:routeC-trisection-exceptional-plane-package} to that exceptional class finishes the
reduction.  The count of at most four elementary transforms is immediate: one initial transform may be needed to reach the
exceptional
$\mathbb F_2$
class, and from there Proposition~\ref{prop:routeC-trisection-exceptional-plane-package} uses at most three more.
\end{proof}

\begin{remark}[What now remains in the trisection target]
\label{rem:routeC-trisection-now-plane-only}
The trisection branch is now entirely a finite marked plane problem.

Indeed, Corollary~\ref{cor:routeC-trisection-plane-package} removes the last separate ruled-surface residue
$3S_0+8f\subset \mathbb F_2.$
So the degree-three mixed branch is no longer ``a singular trisection on
\[
\mathbb F_2
\quad\text{or}\quad
\mathbb F_3
\]'';
it is an irreducible rational plane curve of degree
$3,\ 4,\ 5,\ \text{or }6$
with a marked point of multiplicity
$\deg\Gamma-3,$
together with the requirement that projection from that marked point reproduces the original degree-$3$ map to the base.

Combined with Corollary~\ref{cor:routeC-trisection-not-new-unibranch}, this means that the only genuinely new local
freedom left in the trisection target is collision geometry on these degree-$\le 6$ marked plane models.  The unibranch
side is already absorbed by the old touched-tip / branching corridors.
\end{remark}


\subsection{Collision singularities of the ordinary trisection create no new corridor}
\label{subsec:routeC-trisection-collision-oldside}

After Corollary~\ref{cor:routeC-trisection-plane-package}, the only local freedom still left on the degree-three mixed
side appears to be multibranch collision geometry.  In fact even this residual freedom is already absorbed by the old
touched-tip / branching side.

\begin{proposition}[A multibranch point of the ordinary trisection is either impossible or already old]
\label{prop:routeC-trisection-collision-oldside}
Assume the setup of Proposition~\ref{prop:routeC-lowdegree-unique-special-F2F3}, and suppose that
$d=3.$
Let
$q\in C$
be a singular point with at least two local branches.  Factor
$\nu^{-1}$
over
$q$
until those local branches first become disjoint, and let
$L_0$
be the last exceptional curve created in that primitive branch-separation phase.  Let
\[
x_1,\dots,x_r\in H\cap L_0,
\qquad
r\ge 2,
\]
be the resulting contact points of the strict transform
$H$
of the trisection with
$L_0.$
For each
$i,$
let
$e_i$
be the local degree of the induced map
$H\to \PP^1$
at
$x_i.$
Then
\[
\sum_{i=1}^r e_i\le 3.
\]
Moreover, exactly one of the following holds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item all
    $e_i=1;$
    this case is impossible;

    \item
    \[
    r=2
    \qquad\text{and}\qquad
    \{e_1,e_2\}=\{1,2\}.
    \]
    In this case, following the unique degree-$2$ branch to its terminal surviving local block yields either a corner
    fork or an order-$2$ tangency fork.  The same contact-defect argument as in the proof of
    Proposition~\ref{prop:routeC-trisection-unibranch-oldside} then produces a vertical component
    $E\subset D$
    such that
    \[
    \operatorname{cf}_X(E)<\frac13.
    \]
    Consequently, if the connected component of
    $D$
    containing
    $E$
    is a chain, then its $2$-tail core is a touched tip of weight
    $2;$
    if that connected component is not a chain, then one is already on the branching side.
\end{enumerate}
In particular, every surviving collision singularity of the trisection is already absorbed by the old touched-$(-2)$-tip
/ branching geometry.
\end{proposition}

\begin{proof}
Because the trisection has horizontal degree
$H\cdot f=3,$
the sum of the local degrees of
$H\to \PP^1$
over any fixed fiber is at most
$3.$
The points
$x_1,\dots,x_r$
all lie on the same fiber through
$q,$
so
\[
\sum_{i=1}^r e_i\le 3.
\]

Assume first that
\[
e_i=1
\qquad
(1\le i\le r).
\]
Then at each point
$x_i$
the strict transform
$H$
is smooth and transverse to the current fiber direction.  Hence the local embedded-resolution process over
$q$
is already finished at those points, so no later center of
$\nu^{-1}$
over
$q$
lies on any
$x_i.$
Let
$L$
be the final strict transform of
$L_0$
on
$\widetilde X.$
Then
$L\cdot H=r\ge 2.$

If
$L\subset D,$
then the connected component of
$D$
containing
$H$
would contain two distinct intersection points between the same pair of components
\[
H
\quad\text{and}\quad
L,
\]
contradicting the fact that every connected component of
$D$
has tree dual graph.  Therefore
$L\not\subset D.$
Since
$H$
is ordinary, Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
Applying Proposition~\ref{prop:routeC-contact-defect-sieve} to
$L$
now gives
\[
1>
\sum_{T\subset D}(L\cdot T)\operatorname{cf}_X(T)
\ge
(L\cdot H)\operatorname{cf}_X(H)
\ge
2\cdot \frac12
=1,
\]
a contradiction.  So case~\textnormal{(i)} is impossible.

Hence some
$e_i\ge 2.$
Because
\[
r\ge 2
\qquad\text{and}\qquad
\sum_{i=1}^r e_i\le 3,
\]
the only remaining possibility is
\[
r=2
\qquad\text{and}\qquad
\{e_1,e_2\}=\{1,2\}.
\]
After renumbering, assume
\[
e_1=2,
\qquad
e_2=1.
\]

Follow the degree-$2$ branch through
$x_1$
to the last local block over
$q$
whose last exceptional curve survives on
$\widetilde X.$
Once the other branch has been separated, this is a unibranch local problem of degree
$2$
over the base.  Therefore the same low-degree Euclidean analysis as in
Corollary~\ref{cor:routeC-unibranch-first-smooth-stage} and
Proposition~\ref{prop:routeC-unibranch-lowdegree-terminal-steps} shows that the terminal surviving block is either:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item a corner fork as in Proposition~\ref{prop:routeC-terminal-corner-fork}; or
    \item an order-$2$ tangency fork as in Proposition~\ref{prop:routeC-terminal-tangency-fork}.
\end{enumerate}

The proof of Proposition~\ref{prop:routeC-trisection-unibranch-oldside} uses only these two local fork shapes together
with the inequality
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
So the same argument applies verbatim here and yields a vertical component
$E\subset D$
such that
\[
\operatorname{cf}_X(E)<\frac13.
\]
The final conclusion is then exactly Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} in the chain case; if the
connected component of
$D$
containing
$E$
is not a chain, then one is already on the branching side.
\end{proof}

\begin{corollary}[The trisection branch creates no genuinely new local geometry]
\label{cor:routeC-trisection-no-new-local}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, and suppose that we are in case~\textnormal{(iii)}.
Then every singularity of the unique trisection is already absorbed by the old side types isolated earlier in the note:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item unibranch points by Corollary~\ref{cor:routeC-trisection-not-new-unibranch};
    \item multibranch points by Proposition~\ref{prop:routeC-trisection-collision-oldside}.
\end{enumerate}
Hence the degree-three mixed branch contributes no genuinely new local obstruction beyond the old
touched-$(-2)$-tip / branching geometry.
\end{corollary}

\begin{proof}
This is exactly the combination of Corollary~\ref{cor:routeC-trisection-not-new-unibranch} and
Proposition~\ref{prop:routeC-trisection-collision-oldside}.
\end{proof}

\begin{remark}[What the trisection collision closure leaves at this stage]
\label{rem:routeC-after-trisection-collision-closure}
Corollary~\ref{cor:routeC-trisection-no-new-local} closes the last genuinely new local gap on the degree-three mixed
side.

Indeed, Corollary~\ref{cor:routeC-trisection-plane-package} had already reduced the trisection branch to a finite marked
plane package, and Corollary~\ref{cor:routeC-trisection-not-new-unibranch} had already absorbed the unibranch part into
the old touched-tip / branching corridors.  Proposition~\ref{prop:routeC-trisection-collision-oldside} removes the
remaining multibranch freedom as well.  The following subsection performs the analogous closure for the degree-two mixed
branch.
\end{remark}



\subsection{The ordinary bisection creates no genuinely new local corridor}
\label{subsec:routeC-bisection-oldside}

After the trisection closure, the only separate local residue still visible on the low-degree mixed side is the
degree-two branch from case~\textnormal{(ii)} of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}.  In fact this
residual branch is even smaller than the trisection branch: any surviving unibranch block is already old, and a
multibranch collision of the ordinary bisection is impossible.

\begin{proposition}[Every surviving unibranch bisection block already has an old special vertical side]
\label{prop:routeC-bisection-unibranch-oldside}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, and suppose that we are in
case~\textnormal{(ii)}.
Let
$q\in C$
be a unibranch singular point of the unique ordinary bisection.
Assume that, in the factorization of
$\nu^{-1}$
over
$q,$
the last exceptional curve created in the terminal block survives on
$\widetilde X.$
Then there exists a vertical component
$E\subset D$
adjacent to that surviving last exceptional curve such that
\[
\operatorname{cf}_X(E)<\frac13.
\]
Consequently, if the connected component of
$D$
containing
$E$
is a chain, then the corresponding $2$-tail core is a touched tip of weight
$2;$
if that connected component is not a chain, then one is already on the branching side.
\end{proposition}

\begin{proof}
Let
$H\subset D_{\mathrm{hor}}$
be the unique bisection.  In the low-degree mixed branch it is ordinary, so
Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]

By Lemma~\ref{lem:routeC-bisection-local-normal-forms}, the local degree of the normalization map
$\overline C\to \PP^1$
at the unique preimage of
$q$
is
$2.$
Therefore Corollary~\ref{cor:routeC-one-section-unibranch-finite-loworder} shows that the terminal surviving block over
$q$
is one of the following two types:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a corner fork
    \[
    \begin{array}{ccccc}
    &&A&&\\
    &&|&&\\
    H&-&L&-&B;
    \end{array}
    \]

    \item an order-$2$ tangency fork
    $H-L-A-B_1.$
\end{enumerate}
In both cases
\[
L\not\subset D
\qquad\text{and}\qquad
L\cdot D=3.
\]

Assume first that the terminal block is the tangency fork from
Proposition~\ref{prop:routeC-terminal-tangency-fork}.  Let
$E:=B_1.$
Then
$L\cdot H=L\cdot A=L\cdot E=1.$
By Proposition~\ref{prop:routeC-contact-defect-sieve},
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(E)<1.
\]
Also Proposition~\ref{prop:routeC-terminal-tangency-fork} gives
$A^2\le -3,$
so Lemma~\ref{lem:routeC-cf-selfintersection-lower} yields
\[
\operatorname{cf}_X(A)\ge \frac13.
\]
Together with
\[
\operatorname{cf}_X(H)\ge \frac12,
\]
this gives
\[
\operatorname{cf}_X(E)
<
1-\frac12-\frac13
=
\frac16
<\frac13.
\]

Assume next that the terminal block is the corner fork from
Proposition~\ref{prop:routeC-terminal-corner-fork}.  Then
$L\cdot H=L\cdot A=L\cdot B=1,$
so Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H)+\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<1.
\]
Using
\[
\operatorname{cf}_X(H)\ge \frac12,
\]
one gets
\[
\operatorname{cf}_X(A)+\operatorname{cf}_X(B)<\frac12.
\]
Hence at least one of
$A,\ B$
has discrepancy coefficient strictly smaller than
\[
\frac14<\frac13.
\]
Take that component as
$E.$

The final conclusion is exactly Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} in the chain case; if the connected
component of
$D$
containing
$E$
is not a chain, then one is already on the branching side.
\end{proof}

\begin{proposition}[A multibranch point of the ordinary bisection is impossible]
\label{prop:routeC-bisection-collision-impossible}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, and suppose that we are in
case~\textnormal{(ii)}.
Let
$q\in C$
be a singular point with at least two local branches.  Factor
$\nu^{-1}$
over
$q$
until those local branches first become disjoint, and let
$L_0$
be the last exceptional curve created in that primitive branch-separation phase.  Let
\[
x_1,\dots,x_r\in H\cap L_0,
\qquad
r\ge 2,
\]
be the resulting contact points of the strict transform
$H$
of the bisection with
$L_0.$
For each
$i,$
let
$e_i$
be the local degree of the induced map
$H\to \PP^1$
at
$x_i.$
Then
\[
\sum_{i=1}^r e_i\le 2,
\]
hence necessarily
\[
r=2
\qquad\text{and}\qquad
e_1=e_2=1.
\]
This case is impossible.  In particular, the ordinary bisection has no surviving multibranch collision singularity.
\end{proposition}

\begin{proof}
Because the bisection has horizontal degree
$H\cdot f=2,$
the sum of the local degrees of
$H\to \PP^1$
over any fixed fiber is at most
$2.$
The points
$x_1,\dots,x_r$
all lie on the same fiber through
$q,$
so
\[
\sum_{i=1}^r e_i\le 2.
\]
Since
$r\ge 2$
and each
$e_i\ge 1,$
one must have
\[
r=2
\qquad\text{and}\qquad
e_1=e_2=1.
\]

Thus both separated branches are smooth and transverse to the current fiber direction at their contact points with
$L_0.$
No later center of
$\nu^{-1}$
over
$q$
can lie on either of those two points.  Let
$L$
be the final strict transform of
$L_0$
on
$\widetilde X.$
Then
$L\cdot H=2.$

If
$L\subset D,$
then the connected component of
$D$
containing
$H$
would contain two distinct intersection points between the same pair of components
\[
H
\quad\text{and}\quad
L,
\]
contradicting the fact that every connected component of
$D$
has tree dual graph.  Therefore
$L\not\subset D.$

Since
$H$
is ordinary, Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12.
\]
Applying Proposition~\ref{prop:routeC-contact-defect-sieve} to
$L$
now gives
\[
1>
\sum_{T\subset D}(L\cdot T)\operatorname{cf}_X(T)
\ge
(L\cdot H)\operatorname{cf}_X(H)
\ge
2\cdot \frac12
=1,
\]
a contradiction.
\end{proof}

\begin{corollary}[The bisection branch creates no genuinely new local geometry]
\label{cor:routeC-bisection-no-new-local}
Assume the setup of Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}, and suppose that we are in
case~\textnormal{(ii)}.
Then every singularity of the unique ordinary bisection is either impossible or already absorbed by the old side types
isolated earlier in the note:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item any unibranch singularity by Proposition~\ref{prop:routeC-bisection-unibranch-oldside};
    \item any multibranch singularity is impossible by Proposition~\ref{prop:routeC-bisection-collision-impossible}.
\end{enumerate}
In particular, the degree-two mixed branch contributes no genuinely new local obstruction beyond the old
touched-$(-2)$-tip / branching geometry.
\end{corollary}

\begin{proof}
This is exactly the combination of Proposition~\ref{prop:routeC-bisection-unibranch-oldside} and
Proposition~\ref{prop:routeC-bisection-collision-impossible}.
\end{proof}

\begin{corollary}[The low-degree mixed branch creates no separate local corridor]
\label{cor:routeC-lowdegree-mixed-no-new-local}
Assume the setup of Corollary~\ref{cor:routeC-global-proof-first-reduction}, and suppose that we are in the low-degree
mixed branch.
Then either the chosen witnessing fibration already lies in the several-special-horizontal corridor from
Remark~\ref{rem:routeC-final-lemma-what-remains-after-proof-program}, or every singularity of the unique ordinary
multisection is already absorbed by the old touched-$(-2)$-tip / branching geometry, or impossible.
In particular, after the global low-degree reduction there is no separate bisection / trisection local corridor left.
\end{corollary}

\begin{proof}
Apply Corollary~\ref{cor:routeC-lowdegree-mixed-global-form}.  In case~\textnormal{(i)} one is already in the
several-special-horizontal corridor.  In case~\textnormal{(ii)}, apply
Corollary~\ref{cor:routeC-bisection-no-new-local}.  In case~\textnormal{(iii)}, apply
Corollary~\ref{cor:routeC-trisection-no-new-local}.
\end{proof}

\begin{remark}[What now remains globally after closing the mixed branch]
\label{rem:routeC-after-mixed-closure}
Corollary~\ref{cor:routeC-lowdegree-mixed-no-new-local} removes the last separate local corridor on the one-section
mixed side.

Indeed, the bisection branch no longer contributes a marked-conic / marked-cubic residue, and the trisection branch no
longer contributes a separate collision residue.  So the residual global problems no longer include any distinct
low-degree mixed local package.  What remains is the special-multisection side together with the non-clean purely
inseparable service patterns left over from the cubic-backbone analysis; the next subsection returns to the height-$4$,
maximal-width two-touch endgame and closes it.
\end{remark}


\subsection{Returning to the maximal-width two-touch problem: packet roots and the rigid four-section hub}
\label{subsec:routeC-two-touch-rigid-and-lambda}

After the one-section multisection reductions, it is natural to return to the genuinely section-only part of
Conjecture~\ref{conj:routeC-final-lemma-needed}, namely the height-$4$, maximal-width one-touch problem
\[
v(F)>0,
\qquad
 t(F)=1.
\]
The next two results extract two complementary pieces of structure from that corridor.
First, after minimizing a simple branch-length invariant, every one-touch bad branch is forced to start on the
horizontal packet itself on the clean-section ruled model.
Second, if a single vertical component carries all four horizontal sections, then that component is already rigid and all
four sections are special.
Together they show that the remaining two-touch obstruction is no longer an arbitrary vertical-tree phenomenon.

\begin{definition}[One-touch branch length]
\label{def:routeC-onetouch-length}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and let
$p\colon \widetilde X\to \PP^1$
be a witnessing fibration of height $4$ and maximal width.
For a degenerate fiber $F$ and a connected component
$U\subset D_F$
disjoint from $D_{\mathrm{hor}}$, let
$\Gamma(U)\subset F_{\mathrm{red}}$
be the unique chain joining $U$ to the unique connected component of $D_F$ meeting $D_{\mathrm{hor}}$ whenever
$t(F)=1.$
Define
\[
\ell(U):=\#\bigl(\Gamma(U)-U\bigr).
\]
If
\[
\mathcal V(F):=\{U\subset D_F\,;\, U\cap D_{\mathrm{hor}}=\varnothing\}
\]
is the set of purely vertical connected components of $D_F$, put
\[
\lambda(p):=\sum_F\sum_{U\in \mathcal V(F)} \ell(U).
\]
\end{definition}

\begin{lemma}[Free first centers are removable by an elementary transform]
\label{lem:routeC-free-first-center-slide}
Let
$Y_0\to \PP^1$
be a smooth $\PP^1$-bundle, let
$f_0\subset Y_0$
be a fiber, and let
$\Sigma_0,\dots,\Sigma_r\subset Y_0$
be sections.
Assume that a smooth surface $Y$ is obtained from $Y_0$ by a sequence of blowups whose centers lie over $f_0$.
Suppose that one blowup branch begins with a free center
$q_1\in f_0$
which is disjoint from
$\Sigma_0\cup\cdots\cup\Sigma_r.$
Then there exists another $\PP^1$-bundle
$Y_0'\to \PP^1$
obtained from $Y_0$ by the elementary transform at $q_1$, and a factorization of $Y$ as a sequence of blowups over
the image fiber
$f_0'\subset Y_0'$
with the following properties:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the images of
    $\Sigma_0,\dots,\Sigma_r$
    on $Y_0'$ are again sections;
    \item every descendant of the branch issued from $q_1$ is still realized over
    $f_0',$
    but with the initial free step removed;
    \item no blowup branch over any other fiber is changed.
\end{enumerate}
\end{lemma}

\begin{proof}
Blow up $q_1$:
$\alpha\colon Y_1\to Y_0.$
Let $E_1$ be the exceptional curve and let $f_1$ be the strict transform of $f_0$.
Since $q_1$ is free, one has
$E_1^2=f_1^2=-1$
on $Y_1$, and the elementary transform at $q_1$ is the contraction of $f_1$:
$\beta\colon Y_1\to Y_0'.$
Because $q_1$ is disjoint from the sections $\Sigma_i$, their strict transforms on $Y_1$ are disjoint from $f_1$,
hence descend to sections on $Y_0'$.

In local coordinates
$(x,y)$
near $q_1$ with
$f_0=\{x=0\},$
the blowup at $q_1$ followed by the contraction of $f_1$ is the standard elementary transform of the ruled surface.
Its effect on the cluster over $f_0$ is to replace the total transform
$f_1+E_1$
by the new smooth fiber
$f_0'=\beta(E_1),$
while preserving the infinitely near data away from the contracted curve $f_1$.
Hence every later center on the branch issued from $q_1$ descends to a center over
$f_0',$
with one fewer initial proximity step.
Centers lying over other fibers are untouched.

Iterating this observation along the descendants of $q_1$ gives a blowup factorization of the same final smooth
surface $Y$ over $Y_0'$ in which the whole $q_1$-branch has lost its first free step, while all other fibers are
unchanged.
\end{proof}

\begin{proposition}[In a $\lambda$-minimal maximal-width witness, every one-touch bad branch starts on the horizontal packet]
\label{prop:routeC-lambda-minimal-onetouch-packet-root}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and among all height-$4$, maximal-width
witnessing fibrations on the fixed smooth surface
$\widetilde X$
choose one minimizing
$\lambda(p)$
in the sense of Definition~\ref{def:routeC-onetouch-length}.
Choose a clean section
$H_0\subset D_{\mathrm{hor}}$
and let
$\upsilon\colon \widetilde X\to \mathbb F_n$
be the ruled model from Proposition~\ref{prop:routeC-clean-section-Hirzebruch}.
Write
$S_i:=\upsilon(H_i),\qquad i=0,1,2,3.$

Let $F$ be a degenerate fiber with
$v(F)>0,\qquad t(F)=1,$
and let
$U\subset D_F$
be a purely vertical connected component.
Fix a relative minimal model of $p$ in which the image of $F$ is a smooth fiber
$f\subset Y_0,$
and let
$q_1\prec q_2\prec\cdots\prec q_m$
be the chain of blowup centers over $f$ whose descendants produce the branch from the unique touched component of
$D_F$ to $U$.
Then some $q_j$ lies on the total transform of the horizontal divisor
$S_0+S_1+S_2+S_3.$
If $j$ is minimal with this property, then exactly one of the following holds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item $q_j$ lies on a unique local branch of the horizontal packet; equivalently, it is a packet-section point
    (possibly infinitely near) on the marked fiber;
    \item $q_j$ is a singular point of the total transform of the horizontal packet; equivalently, it is a collision
    point of at least two horizontal branches, possibly infinitely near.
\end{enumerate}
\end{proposition}

\begin{proof}
Assume first that none of the centers
$q_1,\dots,q_m$
lies on the total transform of
$S_0+S_1+S_2+S_3.$
Then the first center
$q_1\in f$
is a free point of the smooth fiber $f$, disjoint from the horizontal packet.
By Lemma~\ref{lem:routeC-free-first-center-slide}, the elementary transform at $q_1$ produces another ruled model
whose pullback to $\widetilde X$ is again a height-$4$, maximal-width witnessing fibration with the same horizontal
$1$-sections, but for which this branch has lost its first free step.
Hence
$\lambda(p')<\lambda(p),$
contradicting the minimality of $\lambda(p)$.
Therefore some $q_j$ lies on the total transform of the horizontal packet.

Let $j$ be minimal with this property.
If the total transform of
$S_0+S_1+S_2+S_3$
is smooth at $q_j$, then $q_j$ lies on a unique local branch of the packet, giving \textnormal{(i)}.
If the total transform is singular at $q_j$, then $q_j$ is by definition a collision point of at least two local
horizontal branches, giving \textnormal{(ii)}.
\end{proof}


\begin{corollary}[In a $\lambda$-minimal maximal-width witness, the first center of every one-touch bad branch is already on the packet]
\label{cor:routeC-lambda-minimal-first-center-packet}
Keep the assumptions and notation of Proposition~\ref{prop:routeC-lambda-minimal-onetouch-packet-root}.
Then the first center
$q_1$
itself lies on the horizontal packet
$S_0+S_1+S_2+S_3.$
Equivalently, every one-touch bad branch starts at the first blowup either from a packet-section point or from a collision
point of horizontal branches.
\end{corollary}

\begin{proof}
If
$q_1$
were disjoint from the horizontal packet, then $q_1$ would be a free point of the smooth fiber $f$, disjoint from all
sections.  Lemma~\ref{lem:routeC-free-first-center-slide} would then produce another ruled model and hence another
height-$4$, maximal-width witnessing fibration on the same smooth surface in which this branch loses its initial free
step.  Thus
$\lambda(p)$
would decrease, contradicting the minimality assumption.
Therefore
$q_1\in S_0+S_1+S_2+S_3.$
The final alternative is exactly the smooth-versus-singular dichotomy for the total transform of the packet at $q_1$.
\end{proof}


\begin{remark}[What the $\lambda$-minimality reduction actually does]
\label{rem:routeC-lambda-reduction-meaning}
Proposition~\ref{prop:routeC-lambda-minimal-onetouch-packet-root} does not by itself prove the two-touch condition
$v(F)>0\Longrightarrow t(F)\ge 2.$
What it does show is that, once one is already in the height-$4$, maximal-width corridor and has minimized
$\lambda(p),$
every one-touch bad branch is rooted in the horizontal packet on the ruled model.
So the remaining maximal-width obstruction is no longer an abstract fiber-tree phenomenon: it is forced into the same
packet-section-point and horizontal-collision mechanisms that occur elsewhere in the clean-section Hirzebruch analysis.
\end{remark}

\begin{proposition}[A vertical component meeting four horizontal sections is rigid]
\label{prop:routeC-four-horizontal-rigid}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, so that
$D_{\mathrm{hor}}=H_0+H_1+H_2+H_3$
is the union of four horizontal $1$-sections.
Let
$U\subset D$
be a vertical irreducible component meeting all four sections
$H_0,H_1,H_2,H_3.$
Then the following hold.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item $U$ has no vertical neighbor in $D$.
    \item $U^2=-2$.
    \item Writing
    \[
    \pi^*K_X
    =
    K_{\widetilde X}+\sum_{E\subset D}\operatorname{cf}_X(E)\,E,
    \]
    one has
    \[
    \operatorname{cf}_X(H_i)=\frac{\operatorname{cf}_X(U)}{2}
    \qquad (0\le i\le 3).
    \]
    In particular,
    \[
    \operatorname{cf}_X(H_i)<\frac12
    \qquad (0\le i\le 3),
    \]
    so every $H_i$ is nonordinary, equivalently every $H_i$ is a special horizontal section.
\end{enumerate}
\end{proposition}

\begin{proof}
Write
\[
U^2=-b,
\qquad
H_i^2=-a_i
\qquad (0\le i\le 3).
\]
Since $U,H_i\subset D$ and $\pi$ is the minimal resolution, one has
\[
b\ge 2,
\qquad
a_i\ge 2.
\]

Let
\[
\pi^*K_X
=
K_{\widetilde X}+\sum_{E\subset D}\operatorname{cf}_X(E)\,E,
\qquad
\operatorname{cf}_X(E)=1-\mu_X(E).
\]
Intersecting this identity with $H_i$ gives
\[
a_i\,\operatorname{cf}_X(H_i)
=
a_i-2+\sum_{E\sim H_i}\operatorname{cf}_X(E)
\ge a_i-2+\operatorname{cf}_X(U),
\]
where the sum runs over the components of $D$ meeting $H_i$.
Hence
\[
\operatorname{cf}_X(H_i)
\ge 1-\frac{2-\operatorname{cf}_X(U)}{a_i}
\ge 1-\frac{2-\operatorname{cf}_X(U)}{2}
=
\frac{\operatorname{cf}_X(U)}{2}
\qquad (0\le i\le 3).
\]
Since the fibration is section-only, every $H_i$ is a $1$-section, so Lemma~\ref{lem:routeC-fiber-anticanonical-cf}
gives
\[
2>\sum_{H\subset D_{\mathrm{hor}}}\operatorname{cf}_X(H)
\ge \sum_{i=0}^3 \operatorname{cf}_X(H_i)
\ge 2\,\operatorname{cf}_X(U).
\]
Hence
$\operatorname{cf}_X(U)<1.$

Now let
$V_1,\dots,V_s$
be the vertical neighbors of $U$ in the dual graph of $D$.
Intersecting the discrepancy formula with $U$ yields
\[
b\,\operatorname{cf}_X(U)
-
\sum_{i=0}^3 \operatorname{cf}_X(H_i)
-
\sum_{j=1}^s \operatorname{cf}_X(V_j)
=
b-2.
\]
Using
\[
\sum_{i=0}^3 \operatorname{cf}_X(H_i)\ge 2\,\operatorname{cf}_X(U)
\]
and the nonnegativity of the $\operatorname{cf}_X(V_j)$, we get
$(b-2)\operatorname{cf}_X(U)\ge b-2.$
If $b>2$, then dividing by $b-2>0$ gives
$\operatorname{cf}_X(U)\ge 1,$
contradicting $\operatorname{cf}_X(U)<1$.
Therefore
$b=2,$
which proves \textnormal{(ii)}.

With $b=2$, the discrepancy equation at $U$ becomes
\[
2\,\operatorname{cf}_X(U)
=
\sum_{i=0}^3 \operatorname{cf}_X(H_i)
+
\sum_{j=1}^s \operatorname{cf}_X(V_j).
\]
But we already know
\[
\sum_{i=0}^3 \operatorname{cf}_X(H_i)\ge 2\,\operatorname{cf}_X(U).
\]
Hence equality must hold throughout, so
\[
\sum_{j=1}^s \operatorname{cf}_X(V_j)=0
\qquad\text{and}\qquad
\sum_{i=0}^3 \operatorname{cf}_X(H_i)=2\,\operatorname{cf}_X(U).
\]
Since each discrepancy coefficient is nonnegative, the first equality implies
\[
\operatorname{cf}_X(V_j)=0
\qquad (1\le j\le s).
\]

Assume for contradiction that $s\ge 1$.
Intersecting the discrepancy formula with some $V_j$ gives
\[
(-V_j^2)\,\operatorname{cf}_X(V_j)-\operatorname{cf}_X(U)-\sum_{W\sim V_j,\ W\neq U}\operatorname{cf}_X(W)
=(-V_j^2)-2.
\]
Since $\operatorname{cf}_X(V_j)=0$, the left-hand side is at most
$-\operatorname{cf}_X(U)<0,$
while the right-hand side is nonnegative because $-V_j^2\ge 2$.
This is impossible.
Therefore
$s=0,$
proving \textnormal{(i)}.

Finally, with $s=0$ and
\[
\sum_{i=0}^3 \operatorname{cf}_X(H_i)=2\,\operatorname{cf}_X(U),
\]
the lower bounds
\[
\operatorname{cf}_X(H_i)\ge \frac{\operatorname{cf}_X(U)}{2}
\qquad (0\le i\le 3)
\]
force equality term-by-term:
\[
\operatorname{cf}_X(H_i)=\frac{\operatorname{cf}_X(U)}{2}
\qquad (0\le i\le 3).
\]
Because $\operatorname{cf}_X(U)<1$, one has
\[
\operatorname{cf}_X(H_i)<\frac12
\qquad (0\le i\le 3).
\]
By Lemma~\ref{lem:routeC-ordinary-section-half}, an ordinary horizontal section has discrepancy coefficient at least
\[
\frac12.
\]
So every $H_i$ is nonordinary.
By Definition~\ref{def:routeC-special-section}, this means every $H_i$ is special.
This proves \textnormal{(iii)}.
\end{proof}


\begin{proposition}[The rigid four-section hub cannot occur]
\label{prop:routeC-no-four-horizontal-hub}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, so that
$D_{\mathrm{hor}}=H_0+H_1+H_2+H_3$
is the union of four horizontal $1$-sections.
Then there does not exist a vertical irreducible component
$U\subset D$
meeting all four sections
$H_0,H_1,H_2,H_3.$
\end{proposition}

\begin{proof}
Assume for contradiction that such a component $U$ exists.
By Proposition~\ref{prop:routeC-four-horizontal-rigid}, one has
$U^2=-2,$
the component $U$ has no vertical neighbor in $D$, and
\[
\operatorname{cf}_X(H_i)=\frac{\operatorname{cf}_X(U)}{2}<\frac12
\qquad (0\le i\le 3).
\]
Write
\[
H_i^2=-a_i,
\qquad a_i\ge 2.
\]

We first claim that
$\operatorname{cf}_X(U)>0.$
Indeed, if
$\operatorname{cf}_X(U)=0,$
then also
\[
\operatorname{cf}_X(H_i)=0
\qquad (0\le i\le 3).
\]
Intersecting the discrepancy formula with $H_i$ gives
\[
a_i\operatorname{cf}_X(H_i)-\operatorname{cf}_X(U)
-
\sum_{V\sim H_i,\ V\neq U}\operatorname{cf}_X(V)
=
a_i-2,
\]
hence
\[
-\sum_{V\sim H_i,\ V\neq U}\operatorname{cf}_X(V)=a_i-2.
\]
Therefore
\[
a_i=2
\qquad\text{and}\qquad
\operatorname{cf}_X(V)=0
\]
for every neighbor $V$ of $H_i$ distinct from $U$.
Repeating the same argument along the connected component of $D$ containing $U$ shows that every component in that
connected component has discrepancy coefficient $0$ and self-intersection $-2$.
So the corresponding singularity is canonical, hence Du Val, and its exceptional graph must be of ADE type.
But the component $U$ has valence $4$, impossible for an ADE graph.
This contradiction proves
$\operatorname{cf}_X(U)>0.$

Now fix
$i\in\{0,1,2,3\}$
and let
$V_{i,1},\dots,V_{i,s_i}$
be the vertical neighbors of $H_i$ distinct from $U$.
Intersecting the discrepancy formula with $H_i$ yields
\[
a_i\operatorname{cf}_X(H_i)-\operatorname{cf}_X(U)-\sum_{j=1}^{s_i}\operatorname{cf}_X(V_{i,j})=a_i-2.
\]
Substituting
\[
\operatorname{cf}_X(H_i)=\frac{\operatorname{cf}_X(U)}{2}
\]
gives
\[
\sum_{j=1}^{s_i}\operatorname{cf}_X(V_{i,j})
=
(a_i-2)\Bigl(\frac{\operatorname{cf}_X(U)}{2}-1\Bigr).
\]
The left-hand side is nonnegative, while the factor
\[
\frac{\operatorname{cf}_X(U)}{2}-1
\]
is strictly negative because
$0<\operatorname{cf}_X(U)<1.$
Hence necessarily
\[
a_i=2
\qquad\text{and}\qquad
\sum_{j=1}^{s_i}\operatorname{cf}_X(V_{i,j})=0.
\]
Thus
$H_i^2=-2,$
and every $V_{i,j}$ has discrepancy coefficient $0$.

Assume some $V_{i,j}$ exists.
Intersecting the discrepancy formula with
$V:=V_{i,j}$
gives
\[
(-V^2)\operatorname{cf}_X(V)-\operatorname{cf}_X(H_i)-\sum_{W\sim V,\ W\neq H_i}\operatorname{cf}_X(W)=(-V^2)-2.
\]
Since
\[
\operatorname{cf}_X(V)=0
\qquad\text{and}\qquad
\operatorname{cf}_X(H_i)=\frac{\operatorname{cf}_X(U)}{2}>0,
\]
the left-hand side is strictly negative, while the right-hand side is nonnegative because
$-V^2\ge 2.$
This contradiction shows that
$s_i=0$
for every $i$.
So each $H_i$ is a leaf, and the connected component of $D$ containing $U$ is exactly the five-vertex star with center
$U$ and leaves
$H_0,H_1,H_2,H_3,$
all of self-intersection $-2$.

Its intersection matrix is
\[
\begin{pmatrix}
-2&1&1&1&1\\
1&-2&0&0&0\\
1&0&-2&0&0\\
1&0&0&-2&0\\
1&0&0&0&-2
\end{pmatrix},
\]
whose determinant is $0$.
This contradicts the negative definiteness of the intersection matrix of the exceptional divisor over a surface
singularity.
Therefore no such component $U$ exists.
\end{proof}


\begin{corollary}[Rigid all-special hub versus distributed-anchor failure]
\label{cor:routeC-two-touch-distributed-anchor}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and among all height-$4$, maximal-width
witnessing fibrations on the fixed smooth surface choose one minimizing
$\lambda(p).$
Let $F$ be a degenerate fiber with
\[
v(F)>0,
\qquad
 t(F)=1.
\]
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item There exists a vertical irreducible component
    $U\subset D_F$
    meeting all four horizontal sections.  Then $U$ is the unique component of $D_F$ meeting $D_{\mathrm{hor}}$,
    one has
    $U^2=-2,$
    the component $U$ has no vertical neighbor in $D$, and all four horizontal sections are special.

    \item No vertical component of $D_F$ meets all four horizontal sections.
    In this case, for every purely vertical connected component
    \[
    W\subset D_F,
    \qquad
    W\cap D_{\mathrm{hor}}=\varnothing,
    \]
    the branch from the unique touched component of $D_F$ to $W$ is rooted, in the sense of
    Proposition~\ref{prop:routeC-lambda-minimal-onetouch-packet-root}, at either a packet-section point or a collision
    point of the horizontal packet on the clean-section ruled model.
\end{enumerate}
\end{corollary}

\begin{proof}
If some vertical component $U\subset D_F$ meets all four horizontal sections, then
Proposition~\ref{prop:routeC-four-horizontal-rigid} gives
$U^2=-2$
and shows that $U$ has no vertical neighbor in $D$ and that all four sections are special.
Because
$t(F)=1,$
the component $U$ is then automatically the unique connected component of $D_F$ meeting $D_{\mathrm{hor}}$.
This is alternative~\textnormal{(i)}.

If no vertical component of $D_F$ meets all four horizontal sections, then alternative~\textnormal{(ii)} is exactly
Proposition~\ref{prop:routeC-lambda-minimal-onetouch-packet-root} applied to each purely vertical connected component
of $D_F$.
\end{proof}



\begin{corollary}[Only the distributed-anchor alternative remains]
\label{cor:routeC-two-touch-distributed-only}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-distributed-anchor}.
Then alternative~\textnormal{(i)} of that corollary cannot occur.
Equivalently, every one-touch bad fiber is already a distributed-anchor configuration on the clean-section ruled model.

More precisely, for every purely vertical connected component
\[
W\subset D_F,
\qquad
W\cap D_{\mathrm{hor}}=\varnothing,
\]
the associated bad branch starts at the first blowup center over the marked fiber, and that first center is either
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a packet-section point of the horizontal packet, or
    \item a collision point of at least two horizontal branches.
\end{enumerate}
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-no-four-horizontal-hub}, no vertical component of $D_F$ can meet all four horizontal
sections.
So alternative~\textnormal{(i)} of Corollary~\ref{cor:routeC-two-touch-distributed-anchor} is impossible.
Hence only alternative~\textnormal{(ii)} remains.
The final description of the first center is exactly
Corollary~\ref{cor:routeC-lambda-minimal-first-center-packet}.
\end{proof}


\begin{lemma}[Elementary transform at a packet-section point]
\label{lem:routeC-packet-section-elementary-transform}
Let
$Y_0\to \PP^1$
be a smooth $\PP^1$-bundle, let
$f_0\subset Y_0$
be a fiber, and let
$\Sigma_0,\dots,\Sigma_r\subset Y_0$
be sections.
Assume that
$q\in \Sigma_0\cap f_0$
is a point through which no other section passes.  Blow up $q$ and denote by
$\alpha\colon Y_1\to Y_0$
the blowup, by
$E\subset Y_1$
the exceptional curve, and by
$f_1\subset Y_1$
the strict transform of $f_0$.  Let
$\beta\colon Y_1\to Y_0'$
be the contraction of $f_1$.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item $Y_0'\to \PP^1$ is again a smooth $\PP^1$-bundle;
    \item the images of $\Sigma_0,\dots,\Sigma_r$ on $Y_0'$ are sections;
    \item the image
    $f_0':=\beta(E)$
    is a smooth fiber;
    \item the point
    \[
    \alpha^{-1}(q)\cap \Sigma_0^{\mathrm{str}}=E\cap \Sigma_0^{\mathrm{str}}
    \]
    descends to the packet-section point
    $f_0'\cap \beta(\Sigma_0^{\mathrm{str}});$
    \item every point of $E$ different from $E\cap \Sigma_0^{\mathrm{str}}$ descends to a free point of the fiber
    $f_0'$
    disjoint from all section images.
\end{enumerate}
\end{lemma}

\begin{proof}
Because $q$ lies on the smooth fiber $f_0$, one has
$E^2=f_1^2=-1$
on $Y_1$, so contracting $f_1$ gives another smooth $\PP^1$-bundle $Y_0'\to \PP^1$.
Since no section other than $\Sigma_0$ passes through $q$, the strict transforms of all sections are disjoint from $f_1$,
hence descend to sections on $Y_0'$.
The curve $E$ is sent to a smooth fiber
$f_0':=\beta(E).$
Moreover $\Sigma_0^{\mathrm{str}}$ meets $E$ transversely in one point, and this point becomes exactly the intersection of
its image section with the new fiber $f_0'$.
Any other point of $E$ is disjoint from all section strict transforms and therefore descends to a smooth point of the fiber
$f_0'$ away from every section image.
\end{proof}

\begin{lemma}[Local packet-section staircase]
\label{lem:routeC-packet-section-staircase}
Let
$Y_0\to \PP^1$
be a smooth $\PP^1$-bundle, let
$f_0\subset Y_0$
be a fiber, and let
$\Sigma\subset Y_0$
be a section meeting $f_0$ at a point $q_1$ through which no other section passes.
For an integer $a\ge 1$, construct successively points
$q_1\prec q_2\prec\cdots\prec q_a$
by declaring that for each $1\le k<a$, the point $q_{k+1}$ is the intersection of the strict transform of $\Sigma$ with the
newest exceptional curve created by blowing up $q_k$.
Let
$\tau_a\colon Y_a\to Y_0$
be the resulting composition of blowups.
Then the reduced fiber over $f_0$ on $Y_a$ is a chain
$F-G_1-\cdots-G_a,$
where
\[
F^2=-1,
\qquad
G_i^2=-2\ \ (1\le i<a),
\qquad
G_a^2=-1,
\]
and the strict transform of $\Sigma$ meets $G_a$ transversely at a point away from $G_{a-1}\cap G_a$.
\end{lemma}

\begin{proof}
For $a=1$, blowing up $q_1=\Sigma\cap f_0$ produces two $(-1)$-curves in the fiber: the strict transform $F$ of $f_0$ and
the exceptional curve $G_1$, with $\Sigma^{\mathrm{str}}$ meeting $G_1$ transversely away from $F\cap G_1$.
Assume the statement proved for $a-1$.
After the first $a-1$ blowups, the point $q_a$ lies on the intersection of the strict transform of $\Sigma$ with the last
exceptional $G_{a-1}$.  Blowing up $q_a$ creates a new exceptional curve $G_a$ with
$G_a^2=-1,$
changes
$G_{a-1}^2$
from $-1$ to $-2$, and does not affect the self-intersections of the earlier curves in the chain.
The strict transform of $\Sigma$ now meets $G_a$ transversely away from the rest of the fiber.  This gives the displayed
chain and self-intersections.
\end{proof}

\begin{proposition}[Packet-section one-touch branches eventually collide or already give easy entry]
\label{prop:routeC-packet-section-branches-collide-or-enter}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-distributed-only}, and let
\[
W\subset D_F,
\qquad
W\cap D_{\mathrm{hor}}=\varnothing,
\]
be a purely vertical connected component of a one-touch bad fiber.
Assume that the branch from the unique touched connected component of $D_F$ to $W$ starts, in the sense of
Corollary~\ref{cor:routeC-lambda-minimal-first-center-packet}, at a packet-section point on the horizontal packet.
Then, after stripping finitely many maximal initial packet-section staircases by the elementary transforms of
Lemma~\ref{lem:routeC-packet-section-elementary-transform}, one reaches exactly one of the following two outcomes.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item The next center is a collision point of at least two horizontal branches on the resulting ruled model.

    \item No later packet center remains.  Then the last exceptional curve of the final packet-section staircase survives
    on $\widetilde X$ as a singly-hit multiplicity-one nonexceptional tip whose unique vertical neighbor lies in $D$.
    Consequently Proposition~\ref{prop:routeC-tip-entry-sufficient} already gives Route~C entry.
\end{enumerate}
In particular, on a residual no-entry counterexample, every packet-section-rooted one-touch bad branch becomes
collision-rooted after stripping finitely many initial packet-section staircases.
\end{proposition}

\begin{proof}
Start with the given packet-section-rooted branch.
On the current ruled model, let
$q_1\prec q_2\prec\cdots\prec q_m$
be the corresponding chain of blowup centers over the marked fiber, and let $a\ge 1$ be maximal such that
$q_1,\dots,q_a$
are the successive packet-section points described in Lemma~\ref{lem:routeC-packet-section-staircase}.
By that lemma, the first $a$ centers create a local chain
$F-G_1-\cdots-G_a,$
where the strict transform of the chosen packet section meets $G_a$ and no other component of this chain.

Apply the packet-section elementary transform of Lemma~\ref{lem:routeC-packet-section-elementary-transform}
successively at the points
$q_1,\dots,q_a.$
This absorbs the whole initial staircase into a new ruled model with the same horizontal packet.
If there is a next center, it descends to a point of the new smooth fiber.
Because of the maximality of $a$, this point is not the packet-section point of the same horizontal branch.
Hence there are only three possibilities for the next center on the new ruled model:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item it is a free point of the fiber, disjoint from the packet;
    \item it is a packet-section point of some \emph{other} horizontal branch;
    \item it is a collision point of at least two horizontal branches.
\end{enumerate}
Case~\textnormal{(a)} is impossible in a $\lambda$-minimal witness, by Lemma~\ref{lem:routeC-free-first-center-slide}.
So if a next center exists, only cases~\textnormal{(b)} and~\textnormal{(c)} remain.
If case~\textnormal{(c)} occurs, we are in alternative~\textnormal{(i)} of the proposition.
If case~\textnormal{(b)} occurs, we restart the same argument from that new packet-section point.
Since the branch has only finitely many blowup centers, this iterative stripping process must terminate.

It remains to analyze the terminal noncollision outcome.
Suppose the process terminates with a final maximal packet-section staircase and with no later packet center.
Let
$F-G_1-\cdots-G_a$
be the corresponding final local staircase block.
Then the last exceptional curve
$G_a$
is not blown up again and survives on $\widetilde X$ as a vertical $(-1)$-curve
$L\not\subset D.$
Because the branch under consideration ends at a purely vertical connected component of $D_F$, one must have
$a\ge 2,$
since for $a=1$ this local staircase contains no vertical $D$-component at all.
Thus the unique vertical neighbor of $L$ is
$G_{a-1}\subset D.$
Every component of this staircase appears in the fiber with multiplicity one, because each blowup is performed at a
smooth point of a multiplicity-one component.
Moreover the chosen horizontal section meets $L$ once and no other horizontal branch passes through the packet side of
this final staircase.
Therefore $L$ is a singly-hit multiplicity-one nonexceptional tip whose unique vertical neighbor lies in $D$.
Proposition~\ref{prop:routeC-tip-entry-sufficient} gives Route~C entry, proving alternative~\textnormal{(ii)}.
\end{proof}

\begin{corollary}[The distributed-anchor one-touch problem is collision-rooted]
\label{cor:routeC-two-touch-collision-rooted}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-distributed-only}, and assume that Route~C entry does not already
occur.
Then every one-touch bad branch is collision-rooted after stripping finitely many initial packet-section steps.
Equivalently, the remaining maximal-width one-touch problem is concentrated in branches whose first nonremovable packet
center is a collision point of at least two horizontal branches on the clean-section ruled model.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-two-touch-distributed-only}, every one-touch bad branch starts either at a packet-section
point or at a collision point of the packet.
If it starts at a collision point, there is nothing to prove.
If it starts at a packet-section point, Proposition~\ref{prop:routeC-packet-section-branches-collide-or-enter} shows
that either Route~C entry already occurs or, after stripping finitely many initial packet-section steps by elementary
transforms, the branch becomes collision-rooted.
\end{proof}


\begin{proposition}[Collision-rooted bad branches reduce to isolated pair collisions]
\label{prop:routeC-collision-rooted-isolated-pair}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-collision-rooted}, and assume that Route~C entry does not already
occur.
Let
\[
W\subset D_F,
\qquad
W\cap D_{\mathrm{hor}}=\varnothing,
\]
be a purely vertical connected component of a one-touch bad fiber, and follow the unique branch from the touched part of
$D_F$
to
$W.$
After stripping finitely many initial packet-section staircases as in
Corollary~\ref{cor:routeC-two-touch-collision-rooted}, consider the resulting collision-rooted branch on the clean-section
ruled model.

Then, after stripping in addition finitely many initial \emph{common collision prefixes} along that branch, one reaches a
stage with the following properties:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the current center is a collision point through which pass exactly two horizontal branch strict transforms;
    \item no other horizontal branch strict transform passes through that point;
    \item the remaining local block along the chosen bad branch is therefore the explicit order-$m$ isolated pair-collision
    block of Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} for some
    $m\ge 1.$
\end{enumerate}
Equivalently, on a residual no-entry counterexample every one-touch bad branch is obtained from an isolated pair
collision by adjoining only packet-section staircases and earlier peeled-off horizontal branches on the preceding local
exceptional chain.
\end{proposition}

\begin{proof}
Fix the chosen bad branch after the initial packet-section stripping from
Corollary~\ref{cor:routeC-two-touch-collision-rooted}.
Let
$q_1\prec q_2\prec \cdots \prec q_N$
be the successive blowup centers on the clean-section ruled model lying on that branch, up to the last center at which a
horizontal strict transform still passes through the chosen infinitely near point.
For each
$1\le k\le N,$
let
\[
\mathcal H_k
:=
\{\text{horizontal branch strict transforms passing through }q_k\}.
\]
Since the branch is collision-rooted,
$\#\mathcal H_1\ge 2.$
Because every later center
$q_{k+1}$
lies on the unique descendant of
$q_k$
selected by the chosen bad branch, any horizontal branch strict transform which misses
$q_{k+1}$
after the blowup at
$q_k$
is separated forever from that branch and cannot reappear at a later chosen center.
Hence
\[
\mathcal H_{k+1}\subseteq \mathcal H_k
\qquad (1\le k<N).
\]
So the cardinalities
$\#\mathcal H_k$
form a weakly decreasing sequence of positive integers.

There are at most four horizontal sections in total.  Therefore, after finitely many common collision steps, one reaches
an index
$k_0$
for which
$\#\mathcal H_{k_0}\in\{1,2\},$
and this is the first time the cardinality drops to at most $2$.
We analyze the two possibilities.

If
$\#\mathcal H_{k_0}=1,$
then
$q_{k_0}$
is a packet-section point of the unique horizontal branch in
$\mathcal H_{k_0}.$
Applying Proposition~\ref{prop:routeC-packet-section-branches-collide-or-enter} from that stage onward, either Route~C
entry already occurs, contrary to our standing assumption, or after stripping a finite maximal packet-section staircase
one reaches a later collision point on the same bad branch.  Repeating this procedure is possible only finitely many
 times because the branch has only finitely many blowup centers.  Hence, on a residual no-entry counterexample, the
process cannot terminate in the packet-section case.

Therefore eventually one reaches a stage for which
$\#\mathcal H_{k_0}=2.$
By the choice of
$k_0,$
no third horizontal branch passes through
$q_{k_0}.$
Thus
$q_{k_0}$
is an isolated collision point of exactly two horizontal branch strict transforms.
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} applies and gives the explicit order-$m$ pair-collision
block at that stage.

Finally, every horizontal branch discarded earlier in the process peeled off at one of the previous common collision
steps, so it meets an earlier exceptional component of the same local exceptional tree and is disjoint from the terminal
isolated pair-collision block.  This is exactly the final description stated in the proposition.
\end{proof}

\begin{corollary}[The one-touch gap is now an isolated-pair service problem]
\label{cor:routeC-two-touch-isolated-pair-service}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-collision-rooted}, and assume that Route~C entry does not already
occur.
Then every one-touch bad branch is obtained by the following finite recipe:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item start with an isolated order-$m$ pair collision of exactly two horizontal branches, as in
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision};
    \item attach to it finitely many earlier peeled-off horizontal branches along the preceding local exceptional chain;
    \item continue only by packet-section staircases and later collisions issued from the section points of the successive
    last nonexceptional curves.
\end{enumerate}
In particular, the remaining maximal-width one-touch problem is no longer an arbitrary multibranch collision problem: it
is a finite service problem on isolated pair-collision blocks.
\end{corollary}

\begin{proof}
This is just Proposition~\ref{prop:routeC-collision-rooted-isolated-pair} together with
Proposition~\ref{prop:routeC-packet-section-branches-collide-or-enter}, which controls every later packet-section phase.
\end{proof}


\begin{proposition}[Terminal one-touch service blocks are explicit forks]
\label{prop:routeC-terminal-serviced-pair-fork}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-isolated-pair-service}, and assume that Route~C entry does not already occur.
Fix a one-touch bad branch and let
$L$
be the last nonexceptional curve on that branch before the terminal purely vertical connected component
\[
W\subset D_F,
\qquad
W\cap D_{\mathrm{hor}}=\varnothing.
\]
Then exactly one of the following two cases occurs.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item \textbf{Primitive higher-order pair.}
    The terminal stage is already an isolated pair collision of order
    $p\ge 2.$
    In this case
    $L\cdot D=3,$
    and the terminal local core is the primitive fork from
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}.

    \item \textbf{Serviced pair.}
    There exist integers
    \[
    u\ge 1,
    \qquad
    p\ge 1,
    \]
    a smooth local fiber component
    $M$
    which is the previous last nonexceptional curve at the start of the final packet-service phase, and local coordinates
    $(t,z)$
    centered at the packet-section point of the active horizontal branch on
    $M$
    such that
    \[
    M=\{t=0\},
    \qquad
    A_0=\{z=0\},
    \qquad
    C_0=\{z=t^{u+p}\},
    \]
    where
    $A_0$
    is the active horizontal branch and
    $C_0$
    is the other horizontal branch involved in the final collision.
    The first
    $u$
    blowups are the successive packet-section blowups at the current point of the strict transform of
    $A_0$
    and the newest exceptional curve, and the next
    $p$
    blowups are the forced order-$p$ pair-collision blowups of the strict transforms of
    $A_0$
    and
    $C_0.$
    The resulting reduced local fiber is the chain
    $M-G_1-\cdots-G_u-F_1-\cdots-F_p,$
    where
    \[
    M^2\le -2,
    \qquad
    G_i^2=-2\ \ (1\le i\le u),
    \qquad
    F_i^2=-2\ \ (1\le i<p),
    \qquad
    F_p^2=-1.
    \]
    The two final colliding horizontal branches meet
    $L:=F_p$
    transversely at distinct points, any earlier peeled-off horizontal branches meet only the preceding chain
    $M-G_1-\cdots-G_u-F_1-\cdots-F_{p-1},$
    and one has
    \[
    L\not\subset D,
    \qquad
    L\cdot D=3.
    \]
    In particular, the terminal local core is an explicit fork with center
    $L,$
    two horizontal arms, and one vertical chain arm.
\end{enumerate}
Moreover, the primitive transverse pair case
$(u,p)=(0,1)$
cannot occur terminally on a one-touch bad branch.
\end{proposition}

\begin{proof}
By Corollary~\ref{cor:routeC-two-touch-isolated-pair-service}, the chosen branch is built from an isolated pair-collision
block together with finitely many packet-section staircases and later collisions issued from the section points of the
successive last nonexceptional curves.
Take the \emph{last} stage at which a nonexceptional curve lies on the chosen bad branch.
We analyze the final local block at that stage.

If there is no final packet-service phase before that last nonexceptional curve is created, then the terminal stage is
already the primitive isolated pair-collision block of
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} for some order
$p\ge 1.$
If
$p=1,$
then the primitive last exceptional curve meets only the two colliding horizontal branches and has no vertical
neighbor in
$D.$
Such a curve cannot be the last nonexceptional curve before a purely vertical connected component
$W\subset D_F,$
so the terminal one-touch branch cannot end in the primitive transverse case.
Hence in the terminal primitive case one must have
$p\ge 2,$
and Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} gives exactly alternative~\textnormal{(i)}.

Assume now that there is a final packet-service phase before the terminal nonexceptional curve is created.
Let
$M$
be the previous last nonexceptional curve at the start of that phase, and let
$q$
be the chosen section point of the active horizontal branch on
$M.$
By definition of the final packet-service phase, there is an integer
$u\ge 1$
such that the next
$u$
blowups are exactly the successive packet-section blowups at the current point of the active section and the newest
exceptional curve, and after these blowups a final isolated collision of order
$p\ge 1$
occurs between the strict transform of that active section and one other horizontal branch.

Choose local coordinates
$(t,z)$
centered at
$q$
so that
\[
M=\{t=0\},
\qquad
A_0=\{z=0\}.
\]
Let
$C_0$
be the other horizontal branch participating in the final collision.
Since the strict transform of
$C_0$
passes through the first
$u$
chosen centers and then collides with the strict transform of
$A_0$
of order
$p$
at the final packet point, one may write
$C_0=\{z=t^{u+p}\}$
after rescaling the local coordinate
$z.$
Now recursively write local coordinates on the successive packet blowups by
\[
z=t^k w_k
\qquad (1\le k\le u).
\]
Then after the first
$u$
packet blowups one has
\[
A_u=\{w_u=0\},
\qquad
C_u=\{w_u=t^p\},
\]
and the reduced local fiber is the chain
$M-G_1-\cdots-G_u,$
where the first blowup lowers the self-intersection of
$M$
from $-1$ to at most $-2$, each
$G_i$
for
$1\le i<u$
is blown up once more and hence has self-intersection $-2$, and
$G_u^2=-1$
just before the final isolated collision block.
Applying Lemma~\ref{lem:routeC-pair-collision-order-p} to the pair
$A_u, \qquad C_u$
creates the further chain
$F_1-\cdots-F_p,$
attached to
$G_u,$
with
\[
F_i^2=-2\ \ (1\le i<p),
\qquad
F_p^2=-1,
\]
and lowers
$G_u^2$
by one more, so in the final surface
$G_u^2=-2.$
Thus the full reduced local fiber is exactly
$M-G_1-\cdots-G_u-F_1-\cdots-F_p.$

The strict transforms of the two final colliding horizontal branches meet
$L:=F_p$
transversely at distinct points, and no earlier peeled-off horizontal branch passes through the final collision point;
those earlier branches therefore meet only the preceding chain
$M-G_1-\cdots-G_u-F_1-\cdots-F_{p-1}.$
Since
$F_p^2=-1,$
one has
$L\not\subset D,$
while
\[
M,
\ G_1,\dots,G_u,
\ F_1,\dots,F_{p-1}
\]
all have self-intersection at most $-2$ and hence lie in
$D.$
Therefore the only components of
$D$
meeting
$L$
are the two final colliding horizontal branches and the adjacent curve
$F_{p-1}$
if
$p\ge 2,$
or
$G_u$
if
$p=1.$
In either case
$L\cdot D=3,$
and the terminal local core is a fork.  This proves alternative~\textnormal{(ii)}.

The exclusion of the primitive transverse case
$(u,p)=(0,1)$
was already proved in the first paragraph, so the proposition follows.
\end{proof}

\begin{corollary}[The one-touch isolated-pair gap is terminally fork-type]
\label{cor:routeC-two-touch-terminal-fork}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-isolated-pair-service}, and assume that Route~C entry does not
already occur.
Then every one-touch bad branch has a last nonexceptional curve
$L\not\subset D$
with
$L\cdot D=3.$
More precisely, the terminal local block of every such branch is one of the explicit fork blocks from
Proposition~\ref{prop:routeC-terminal-serviced-pair-fork}; the primitive contact-$2$ case never occurs terminally on a
one-touch bad branch.
\end{corollary}

\begin{proof}
Immediate from Proposition~\ref{prop:routeC-terminal-serviced-pair-fork}.
\end{proof}

\begin{remark}[What this means for the remaining one-touch gap]
\label{rem:routeC-two-touch-terminal-fork-gap}
After Corollary~\ref{cor:routeC-two-touch-terminal-fork}, the residual height-$4$, maximal-width one-touch problem is no
longer a service problem on arbitrary isolated pair-collision blocks.  The last nonexceptional curve on every bad branch
is forced into a fork-type local model with
$L\cdot D=3,$
namely either the primitive higher-order pair fork or the serviced-pair fork of
Proposition~\ref{prop:routeC-terminal-serviced-pair-fork}.  So the remaining gap is now a genuinely explicit fork-survival
problem rather than a mixed contact-$2$/fork problem.
\end{remark}


\begin{proposition}[Terminal one-touch forks already carry an old special side]
\label{prop:routeC-terminal-fork-special-side}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Let
$L\not\subset D$
be the last nonexceptional curve on a one-touch bad branch, and let
\[
H_1,H_2\subset D_{\mathrm{hor}},
\qquad
E\subset D
\]
be the three irreducible components of
$D$
meeting
$L,$
where
$H_1,H_2$
are horizontal and
$E$
is vertical.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item at least one of the two horizontal components
    $H_1,\ H_2$
    is a special horizontal section in the sense of Definition~\ref{def:routeC-special-section};

    \item assume moreover that neither
    $H_1$
    nor
    $H_2$
    lies on a branching connected component of
    $D,$
    and that neither has reduced touched core equal to a touched tip of weight
    $2.$
    Then
    \[
    \operatorname{cf}_X(E)<\frac13.
    \]
    Consequently, if the connected component of
    $D$
    containing
    $E$
    is a chain, then the $2$-tail core of the touched pair relative to
    $E$
    is a touched tip of weight
    $2;$
    if that connected component is not a chain, then one is already on the branching side.
\end{enumerate}
\end{proposition}

\begin{proof}
By Corollary~\ref{cor:routeC-two-touch-terminal-fork}, the curve
$L$
meets exactly the three components
$H_1,\ H_2,\ E$
of
$D$
transversely once each.  Hence Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H_1)+\operatorname{cf}_X(H_2)+\operatorname{cf}_X(E)<1.
\]

Assume first that both
\[
H_1
\qquad\text{and}\qquad
H_2
\]
are ordinary.  Then Lemma~\ref{lem:routeC-ordinary-section-half} gives
\[
\operatorname{cf}_X(H_1)\ge \frac12,
\qquad
\operatorname{cf}_X(H_2)\ge \frac12,
\]
so
\[
\operatorname{cf}_X(H_1)+\operatorname{cf}_X(H_2)+\operatorname{cf}_X(E)\ge 1,
\]
contrary to the strict contact-defect inequality above.  Therefore at least one of
$H_1,\ H_2$
is special.  This proves \textnormal{(i)}.

Now assume the additional hypotheses of \textnormal{(ii)}.  Let
$H\in\{H_1,H_2\}.$
Since
$H$
does not lie on a branching connected component of
$D,$
it lies on a chain component.
If
$H$
is ordinary, then Lemma~\ref{lem:routeC-ordinary-section-half} gives
\[
\operatorname{cf}_X(H)\ge \frac12>\frac13.
\]
If
$H$
is special but its reduced touched core is not a touched tip of weight
$2,$
then by Definition~\ref{def:routeC-special-section} its reduced touched core is the singleton chain
$[3].$
By Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu} the discrepancy coefficient on that reduced core is
\[
1-\mu([3],1)=1-\frac23=\frac13,
\]
and Lemma~\ref{lem:routeC-mu-monotone-tail} shows that adding terminal $2$-tails away from the touched component can
only decrease
$\mu$
and hence can only increase
$\operatorname{cf}_X.$
Thus in all cases covered by the hypotheses of \textnormal{(ii)} one has
\[
\operatorname{cf}_X(H_1)\ge \frac13,
\qquad
\operatorname{cf}_X(H_2)\ge \frac13.
\]
Therefore the contact-defect inequality implies
\[
\operatorname{cf}_X(E)
<
1-\frac13-\frac13
=
\frac13.
\]
If the connected component of
$D$
containing
$E$
is a chain, Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} shows that the $2$-tail core of the touched pair
relative to
$E$
is a touched tip of weight
$2.$
If that connected component is not a chain, then by definition one is already on the branching side.
This proves \textnormal{(ii)}.
\end{proof}

\begin{corollary}[The terminal one-touch fork introduces no genuinely new side type]
\label{cor:routeC-two-touch-terminal-fork-oldcorridor}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Then every terminal one-touch fork already lands in one of the side types isolated elsewhere in the note:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a horizontal special side, i.e. a branching horizontal side or a horizontal chain side whose reduced touched
    core is a touched tip of weight
    $2$
    or the singleton
    $[3];$

    \item a vertical branching side;

    \item a vertical chain side whose $2$-tail core is a touched tip of weight
    $2.$
\end{enumerate}
In particular, the terminal one-touch fork problem creates no genuinely new local corridor beyond the previously isolated
special-tip / branching geometry.
\end{corollary}

\begin{proof}
This is exactly Proposition~\ref{prop:routeC-terminal-fork-special-side} together with the definition of a special
horizontal section.
\end{proof}


\begin{lemma}[Ordinary chain sides have discrepancy coefficient at least one half]
\label{lem:routeC-general-ordinary-chain-half}
Let
$(A,E)$
be a touched chain pair, and let
$(A^\sharp,E^\sharp)$
be its $2$-tail core from Definition~\ref{def:routeC-two-tail-core}.
Assume that
$E^\sharp$
is neither a touched tip of weight
$2$
nor the singleton chain
$[3].$
Then
\[
\mu_X(E)\le \frac12
\qquad\text{and hence}\qquad
\operatorname{cf}_X(E)\ge \frac12.
\]
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu},
$\operatorname{cf}_X(E)=1-\mu(A,E).$
If
$E^\sharp$
is an interior vertex of
$A^\sharp,$
then Proposition~\ref{prop:routeC-interior-core-half} gives
\[
\mu(A^\sharp,E^\sharp)\le \frac12.
\]
If
$E^\sharp$
is a tip, then the hypothesis excludes exactly the two high-contact cases isolated by
Proposition~\ref{prop:routeC-tipcore-highcontact}, so again
\[
\mu(A^\sharp,E^\sharp)\le \frac12.
\]
Finally, Lemma~\ref{lem:routeC-mu-monotone-tail} shows that removing terminal $2$-tails away from the touched
component can only increase the contact weight, hence
\[
\mu(A,E)\le \mu(A^\sharp,E^\sharp)\le \frac12.
\]
Therefore
\[
\operatorname{cf}_X(E)=1-\mu(A,E)\ge \frac12.
\]
\end{proof}

\begin{proposition}[Every terminal one-touch fork has spherical adjacent weights]
\label{prop:routeC-terminal-fork-spherical-triple}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Let
$L\not\subset D$
be the last nonexceptional curve on a one-touch bad branch, and let
\[
H_1,H_2\subset D_{\mathrm{hor}},
\qquad
E\subset D
\]
be the three irreducible components of
$D$
meeting
$L.$
Write
\[
H_1^2=-a,
\qquad
H_2^2=-b,
\qquad
E^2=-c,
\qquad a,b,c\ge 2.
\]
Then
\[
\frac1a+\frac1b+\frac1c>1.
\]
Consequently, after reordering so that
$\alpha\le \beta\le \gamma$
are the three numbers
$a,b,c,$
one has
\[
(\alpha,\beta,\gamma)=(2,2,n)\ (n\ge 2),
\qquad
(2,3,3),
\qquad
(2,3,4),
\qquad
(2,3,5).
\]
In particular, at least one of
$H_1,H_2,E$
has self-intersection
$-2,$
and at least two of them have self-intersection in
$\{-2,-3\}.$
\end{proposition}

\begin{proof}
By Corollary~\ref{cor:routeC-two-touch-terminal-fork}, the curve
$L$
meets exactly the three components
$H_1,\ H_2,\ E$
of
$D$
transversely once each.  Therefore Proposition~\ref{prop:routeC-contact-defect-sieve} gives
\[
\operatorname{cf}_X(H_1)+\operatorname{cf}_X(H_2)+\operatorname{cf}_X(E)<1.
\]
Applying Lemma~\ref{lem:routeC-cf-selfintersection-lower} to each term yields
\[
\Bigl(1-\frac2a\Bigr)+\Bigl(1-\frac2b\Bigr)+\Bigl(1-\frac2c\Bigr)<1,
\]
which is equivalent to
\[
\frac1a+\frac1b+\frac1c>1.
\]
Now reorder so that
$\alpha\le \beta\le \gamma$
are the three numbers
$a,b,c.$
If
$\alpha\ge 3,$
then
\[
\frac1\alpha+\frac1\beta+\frac1\gamma\le 1,
\]
a contradiction.  Hence
$\alpha=2.$
If moreover
$\beta\ge 4,$
then
\[
\frac1\alpha+\frac1\beta+\frac1\gamma\le \frac12+\frac14+\frac14=1,
\]
again impossible.  Thus
$\beta\in\{2,3\}.$
If
$\beta=2,$
then any
$\gamma\ge 2$
works.  If
$\beta=3,$
then
\[
\frac12+\frac13+\frac1\gamma>1
\]
forces
$\gamma<6,$
so
$\gamma\in\{3,4,5\}.$
This gives exactly the displayed list.
The final statement is immediate from that list.
\end{proof}

\begin{proposition}[Outside the old vertical corridor, the terminal one-touch fork is finite and double-special or exceptional]
\label{prop:routeC-terminal-fork-double-special-reduction}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Let
$L\not\subset D$
be the last nonexceptional curve on a one-touch bad branch, and let
\[
H_1,H_2\subset D_{\mathrm{hor}},
\qquad
E\subset D
\]
be the three irreducible components of
$D$
meeting
$L.$
Then one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item both horizontal components
    $H_1,\ H_2$
    are special horizontal sections;

    \item the connected component of
    $D$
    containing
    $E$
    is already in the old vertical corridor, i.e. it is branching, or it is a chain and the $2$-tail core of the touched
    pair relative to
    $E$
    is a touched tip of weight
    $2;$

    \item after renumbering the two horizontal sides, one has:
    \begin{enumerate}[label=\textnormal{(\alph*)}]
        \item
        $H_1$
        is special, but not of singleton-
        $[3]$
        type; equivalently,
        $H_1$
        is either branching in its connected component of
        $D$
        or lies on a chain whose reduced touched core is a touched tip of weight
        $2;$

        \item
        $H_2$
        is ordinary;

        \item the connected component of
        $D$
        containing
        $E$
        is a chain whose reduced touched core relative to
        $E$
        is the singleton
        $[3];$

        \item one has the numerical bounds
        \[
        H_1^2=-2,
        \qquad
        E^2=-3,
        \qquad
        H_2^2\in\{-2,-3,-4,-5\};
        \]

        \item the discrepancy coefficient on the special side satisfies
        \[
        \operatorname{cf}_X(H_1)<\frac16.
        \]
    \end{enumerate}
\end{enumerate}
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:routeC-terminal-fork-special-side}, at least one of
$H_1,\ H_2$
is special.
If both are special, we are in alternative~\textnormal{(i)}.
Assume from now on that, after renumbering if necessary,
$H_1$
is special and
$H_2$
is ordinary.

If the connected component of
$D$
containing
$E$
is branching, or if it is a chain whose $2$-tail core relative to
$E$
is a touched tip of weight
$2,$
then alternative~\textnormal{(ii)} holds.
Assume therefore that alternative~\textnormal{(ii)} does not hold.

We first claim that
$H_1$
cannot be of singleton-
$[3]$
type.  Indeed, under the present assumptions neither
$H_1$
nor
$H_2$
lies on a branching connected component of
$D,$
and neither has reduced touched core a touched tip of weight
$2.$
So Proposition~\ref{prop:routeC-terminal-fork-special-side}\textnormal{(ii)} applies and gives
\[
\operatorname{cf}_X(E)<\frac13.
\]
Then Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} forces the chain side containing
$E$
into the touched-tip-
$2$
corridor, contradicting the assumption that alternative~\textnormal{(ii)} does not hold.
Thus
$H_1$
is special of the non-
$[3]$
kind, proving \textnormal{(iii)(a)}.

Next we claim that the connected component of
$D$
containing
$E$
cannot be ordinary as a touched chain side.  If it were, then Lemma~\ref{lem:routeC-general-ordinary-chain-half} would
imply
\[
\operatorname{cf}_X(E)\ge \frac12.
\]
Since
$H_2$
is ordinary, Lemma~\ref{lem:routeC-ordinary-section-half} gives
\[
\operatorname{cf}_X(H_2)\ge \frac12.
\]
But Proposition~\ref{prop:routeC-contact-defect-sieve}, applied to the curve
$L,$
yields
\[
\operatorname{cf}_X(H_1)+\operatorname{cf}_X(H_2)+\operatorname{cf}_X(E)<1,
\]
which is impossible if both
\[
\operatorname{cf}_X(H_2)\ge \frac12
\qquad\text{and}\qquad
\operatorname{cf}_X(E)\ge \frac12.
\]
Therefore the chain side containing
$E$
is not ordinary.
Since alternative~\textnormal{(ii)} was excluded, the only remaining chain possibility is that its reduced touched core is the
singleton
$[3].$
This proves \textnormal{(iii)(c)}.

For a singleton-
$[3]$
core, Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu} gives
\[
\operatorname{cf}_X(E)=1-\mu([3],1)=\frac13,
\]
and Lemma~\ref{lem:routeC-mu-monotone-tail} shows that adding terminal $2$-tails away from the touched component can only
increase the discrepancy coefficient.  Hence
\[
\operatorname{cf}_X(E)\ge \frac13.
\]
Together with
\[
\operatorname{cf}_X(H_2)\ge \frac12
\]
and the contact-defect inequality, this gives
\[
\operatorname{cf}_X(H_1)
<
1-\frac12-\frac13
=
\frac16,
\]
which proves \textnormal{(iii)(e)}.

Now apply Lemma~\ref{lem:routeC-cf-selfintersection-lower} to
$H_1^2=-a.$
Since
\[
\operatorname{cf}_X(H_1)<\frac16,
\]
one has
\[
1-\frac2a<\frac16,
\]
so
\[
a<\frac{12}{5}.
\]
Because
$a\ge 2$
is an integer, it follows that
$a=2,$
that is,
$H_1^2=-2.$
Also, because the reduced touched core of the chain side containing
$E$
is the singleton
$[3],$
one has
$E^2=-3.$
Finally, Proposition~\ref{prop:routeC-terminal-fork-spherical-triple} applied to the triple
$(-H_1^2,-H_2^2,-E^2)=(2,-H_2^2,3)$
gives
$-H_2^2\in\{2,3,4,5\}.$
This proves \textnormal{(iii)(d)} and completes the proof.
\end{proof}

\begin{corollary}[The remaining terminal one-touch fork is reduced to multi-special horizontals or a finite exceptional corridor]
\label{cor:routeC-two-touch-terminal-fork-finite-exception}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Then every terminal one-touch fork falls into one of the following three classes.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the witness already contains at least two special horizontal sections;
    \item the vertical side of the terminal fork is already in the old special vertical corridor (branching or touched
    tip of weight $2$ after $2$-tail reduction);
    \item after renumbering the two horizontal arms, the local adjacent-weight pattern is the finite exceptional pattern
    \[
    (-H_1^2,-E^2,-H_2^2)=(2,3,b),
    \qquad b\in\{2,3,4,5\},
    \]
    where
    $H_1$
    is the unique special horizontal arm and is of non-
    $[3]$
    type, while
    $H_2$
    is ordinary.
\end{enumerate}
In particular, outside the old vertical corridor the unresolved maximal-width one-touch problem is no longer an infinite
fork-survival problem: it is reduced to either a witness with at least two special horizontal sections, or the finite
adjacent-weight corridor
$(2,3,b),\qquad b\in\{2,3,4,5\}.$
\end{corollary}

\begin{proof}
This is exactly Proposition~\ref{prop:routeC-terminal-fork-double-special-reduction}.
\end{proof}

\begin{remark}[What now remains of the one-touch fork gap]
\label{rem:routeC-two-touch-terminal-fork-oldcorridor}
Corollary~\ref{cor:routeC-two-touch-terminal-fork-oldcorridor} does not yet prove that the height-$4$, maximal-width
one-touch corridor is empty.  What it does prove is that the terminal fork-survival problem introduces no new local side
shape.  Every surviving terminal fork already falls into one of the old side types isolated earlier in the note:
branching, a touched $(-2)$-tip, or the singleton
$[3].$
So the residual issue is now purely global: show that these already-known side types cannot survive in the section-only,
maximal-width one-touch geometry.
\end{remark}

\begin{remark}[What this improves in the two-touch program]
\label{rem:routeC-two-touch-no-hub-improvement}
The rigid-hub case is now eliminated rather than merely isolated.
Thus the height-$4$, maximal-width one-touch problem is concentrated entirely in the distributed-anchor geometry: every
bad branch starts on the packet itself, at either a packet-section point or a horizontal collision point.
So the remaining two-touch problem is no longer about arbitrary vertical trees, and no longer contains the subcase of one
vertical component carrying all four sections.
\end{remark}


\begin{corollary}[An ordinary section forces distributed anchors]
\label{cor:routeC-two-touch-ordinary-distributed}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-distributed-anchor}.
If at least one of the four horizontal sections is ordinary, then only alternative~\textnormal{(ii)} of that
corollary can occur.
Equivalently, every one-touch bad fiber is already a distributed-anchor, packet/collision-rooted configuration on the
clean-section ruled model.
\end{corollary}

\begin{proof}
Alternative~\textnormal{(i)} of Corollary~\ref{cor:routeC-two-touch-distributed-anchor} makes all four horizontal
sections special, by Proposition~\ref{prop:routeC-four-horizontal-rigid}.
So if one section is ordinary, only alternative~\textnormal{(ii)} can occur.
\end{proof}

\begin{remark}[What remains of the maximal-width two-touch problem]
\label{rem:routeC-two-touch-rigid-and-lambda-meaning}
The two new reductions are complementary.
The $\lambda$-minimality argument removes free purely vertical first centers, so every one-touch bad branch must already
start on the horizontal packet itself.
The rigid-hub proposition then isolates the only subcase in which a single vertical component carries all four
horizontal sections: that component is an isolated $(-2)$-hub and the whole horizontal packet is special.

So the height-$4$, maximal-width one-touch problem is no longer an unrestricted question about arbitrary vertical trees
or arbitrary multibranch collisions.  What remains is the distributed-anchor case, and after the collision-stripping
reduction above every bad branch is controlled by an isolated pair-collision block together with finitely many packet
services and earlier peeled-off horizontal branches on the same local exceptional chain.
That is the sharpest honest reduction obtained here toward the two-touch part of
Conjecture~\ref{conj:routeC-final-lemma-needed}.
\end{remark}




\subsection{The finite exceptional terminal corridor is empty}
\label{subsec:routeC-terminal-fork-no-finite-exception}

The finite adjacent-weight corridor isolated in
Corollary~\ref{cor:routeC-two-touch-terminal-fork-finite-exception}
can in fact be removed entirely.
The reason is that the explicit terminal service model already controls the
self-intersection of the vertical component adjacent to the last nonexceptional
curve:
it is always the component immediately preceding the last exceptional curve in
the local chain arm, hence has self-intersection
$-2.$
Therefore the terminal one-touch fork is automatically already in the old
special vertical corridor, and the exceptional pattern
$(2,3,b),\qquad b\in\{2,3,4,5\},$
never occurs.

\begin{proposition}[The vertical side of every terminal one-touch fork is already old-corridor]
\label{prop:routeC-terminal-fork-vertical-oldcorridor-direct}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Let
$L\not\subset D$
be the last nonexceptional curve on a one-touch bad branch, and let
$E\subset D$
be the unique vertical component of
$D$
meeting
$L.$
Then
$E^2=-2.$
Moreover, the connected component of
$D$
containing
$E$
is already in the old special vertical corridor:
either it is branching, or it is a chain and the $2$-tail core of the touched pair relative to
$E$
is a touched tip of weight
$2.$
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:routeC-terminal-serviced-pair-fork}, exactly one of the following two cases occurs.

\medskip
\noindent\textbf{Case 1: primitive higher-order pair.}
Then the terminal stage is an isolated pair collision of order
$p\ge 2,$
and Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} gives a primitive local chain
$F_1-\cdots-F_p$
with
\[
F_i^2=-2 \quad (1\le i<p),
\qquad
L=F_p\not\subset D.
\]
Hence the unique vertical component of
$D$
meeting
$L$
is
$E=F_{p-1},$
so
$E^2=-2.$
If no further component of
$D$
meets
$E$
besides the preceding chain arm, then the connected component of
$D$
containing
$E$
is a chain and
$E$
is already a touched tip of weight
$2.$
If additional components of
$D$
do meet
$E,$
then that connected component is branching.
So in either subcase the vertical side is in the old special vertical corridor.

\medskip
\noindent\textbf{Case 2: serviced pair.}
Then Proposition~\ref{prop:routeC-terminal-serviced-pair-fork} gives integers
\[
u\ge 1,
\qquad
p\ge 1,
\]
and a reduced local fiber
$M-G_1-\cdots-G_u-F_1-\cdots-F_p$
with
\[
M,\ G_1,\dots,G_u,\ F_1,\dots,F_{p-1}\subset D,
\qquad
L:=F_p\not\subset D,
\]
\[
G_i^2=-2\ \ (1\le i\le u),
\qquad
F_i^2=-2\ \ (1\le i<p),
\]
and such that the unique vertical component of
$D$
meeting
$L$
is
\[
E=
\begin{cases}
F_{p-1}, & p\ge 2,\\
G_u, & p=1.
\end{cases}
\]
Therefore again
$E^2=-2.$

By the same proposition, any earlier peeled-off horizontal branches meet only the preceding chain
$M-G_1-\cdots-G_u-F_1-\cdots-F_{p-1}.$
If one of them meets
$E,$
then the connected component of
$D$
containing
$E$
is branching.
Otherwise that connected component is a chain, and
$E$
is the touched end-component of weight
$2$
adjacent to
$L.$
So again the vertical side lies in the old special vertical corridor.

This proves both claims.
\end{proof}

\begin{corollary}[The finite exceptional terminal corridor is empty]
\label{cor:routeC-terminal-fork-finite-exception-empty}
Keep the setup of Corollary~\ref{cor:routeC-two-touch-terminal-fork}, and assume that Route~C entry does not already
occur.
Then every terminal one-touch fork is already in the old special vertical corridor.
Equivalently, alternative~\textnormal{(iii)} of
Corollary~\ref{cor:routeC-two-touch-terminal-fork-finite-exception}
never occurs.

In particular, the adjacent-weight pattern
\[
(-H_1^2,-E^2,-H_2^2)=(2,3,b),
\qquad
b\in\{2,3,4,5\},
\]
is impossible.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-terminal-fork-vertical-oldcorridor-direct}, the vertical component
$E$
adjacent to the last nonexceptional curve always satisfies
$E^2=-2$
and its connected component is already in the old special vertical corridor.
Hence the exceptional alternative of
Corollary~\ref{cor:routeC-two-touch-terminal-fork-finite-exception},
which requires
$E^2=-3,$
cannot occur.
\end{proof}

\begin{remark}[What now remains of the maximal-width one-touch gap]
\label{rem:routeC-two-touch-terminal-fork-exception-empty}
The terminal one-touch fork no longer contributes a finite residual exceptional list.
After the packet-section stripping and isolated-pair reduction, the final surviving nonexceptional curve always lands on
the old special vertical corridor:
either the adjacent vertical side is branching, or it is a touched tip of weight
$2.$
So the residual maximal-width one-touch problem is now purely the global survival problem for those already-known
vertical side types.
\end{remark}




\subsection{The height-$4$ maximal-width corridor closes by a terminal-leaf count}
\label{subsec:routeC-height4-maxwidth-terminal-leaf}

The preceding one-touch analysis gives exactly the missing graph-theoretic input for the excess route.
The key point is that a one-touch bad fiber always contains a \emph{leaf component} of
$F_{\mathrm{red}}-D$
adjacent to only one vertical connected component of
$D_F.$
This improves the degree count in Proposition~\ref{prop:routeC-vertical-components-excess} by one, which is enough to
recover the desired bound
$v(F)\le \sigma(F)-1$
for every degenerate fiber.

\begin{proposition}[Every one-touch bad fiber contains a leaf component off $D$]
\label{prop:routeC-one-touch-fiber-leaf}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and among all witnessing fibrations of height $4$
and maximal width on the fixed smooth surface
$\widetilde X$
choose one minimizing
$\lambda(p)$
in the sense of Definition~\ref{def:routeC-onetouch-length}.
Let
$F$
be a degenerate fiber with
\[
v(F)>0,
\qquad
 t(F)=1.
\]
Then there exists a connected component
$N_*\subset F_{\mathrm{red}}-D$
consisting of a single irreducible curve
$L$
such that
$\deg_{\Gamma_F}(N_*)=1.$
Equivalently,
$L\not\subset D$
meets exactly one component of
$D_F$
inside the fiber.
\end{proposition}

\begin{proof}
Fix a purely vertical connected component
\[
W\subset D_F,
\qquad
W\cap D_{\mathrm{hor}}=\varnothing,
\]
and follow the unique bad branch from the touched part of
$D_F$
to
$W.$
By Corollary~\ref{cor:routeC-two-touch-distributed-only}, the first center of that branch on the clean-section ruled
model is either a packet-section point or a collision point of horizontal branches.

We now run the same finite stripping procedure as in the proofs of
Propositions~\ref{prop:routeC-packet-section-branches-collide-or-enter},
\ref{prop:routeC-collision-rooted-isolated-pair}, and
\ref{prop:routeC-terminal-serviced-pair-fork}, but without assuming in advance that Route~C entry fails.
Whenever the current center is a packet-section point, strip the maximal initial packet staircase by the elementary
transforms of Lemma~\ref{lem:routeC-packet-section-elementary-transform}.  If that staircase is terminal, then
Proposition~\ref{prop:routeC-packet-section-branches-collide-or-enter}\textnormal{(ii)} produces a last exceptional curve
$L\not\subset D$
which is a singly-hit multiplicity-one nonexceptional tip and whose unique vertical neighbor lies in
$D.$
In particular, inside the reduced fiber
$F_{\mathrm{red}}$
the curve
$L$
has exactly one neighbor, belonging to
$D_F,$
so the corresponding connected component of
$F_{\mathrm{red}}-D$
is exactly
$N_*:=\{L\}$
and has degree
$1$
in
$\Gamma_F.$
Thus the proposition is proved in this case.

Assume therefore that after each stripped packet staircase a later collision center remains on the chosen branch.
At every collision stage, apply the monotonicity argument from the proof of
Proposition~\ref{prop:routeC-collision-rooted-isolated-pair}: along the chosen infinitely near branch, the set of
horizontal branch strict transforms through the successive centers is weakly decreasing.  Hence after finitely many
common collision-prefix steps one reaches either a packet-section point again or an isolated collision of exactly two
horizontal branches.  In the first case we return to the previous paragraph and continue the same finite procedure.
Because the branch has only finitely many blowup centers, this process must eventually terminate.

Consider the last stage at which a nonexceptional curve still lies on the chosen bad branch before the branch enters
$W.$
If this last stage is a terminal packet staircase, we are back in the first paragraph and obtain the desired leaf
component.

Otherwise there is a final isolated pair-collision stage of exactly two horizontal branches.
There are then two possibilities.

\smallskip
\noindent\emph{Primitive higher-order pair.}
If there is no final packet-service phase before the last nonexceptional curve is created, then the terminal local block
is the primitive order-$p$ pair-collision block of Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}.
The primitive transverse case
$p=1$
cannot be terminal here, because then the last exceptional curve meets only the two horizontal branches and no vertical
component of
$D_F,$
so it cannot be the last nonexceptional curve before entering
$W.$
Hence one has
$p\ge 2,$
and the primitive local chain is
\[
F_1-\cdots-F_p,
\qquad
F_i^2=-2\ (1\le i<p),
\qquad
F_p^2=-1,
\qquad
L:=F_p.
\]
Then the unique vertical neighbor of
$L$
inside the fiber is
$F_{p-1}\in D_F,$
so again
$N_*:=\{L\}$
is a degree-$1$ vertex of
$\Gamma_F.$

\smallskip
\noindent\emph{Serviced pair.}
If there is a final packet-service phase before the terminal nonexceptional curve is created, let
$u\ge 1$
be its length and let
$p\ge 1$
be the order of the final isolated pair collision.  The same local-coordinate calculation as in the proof of
Proposition~\ref{prop:routeC-terminal-serviced-pair-fork} then gives a reduced local fiber
$M-G_1-\cdots-G_u-F_1-\cdots-F_p,$
where
\[
M,\ G_1,\dots,G_u,\ F_1,\dots,F_{p-1}\subset D,
\qquad
L:=F_p\not\subset D,
\]
and
\[
G_i^2=-2\ (1\le i\le u),
\qquad
F_i^2=-2\ (1\le i<p),
\qquad
F_p^2=-1.
\]
The unique vertical neighbor of
$L$
inside the fiber is
\[
F_{p-1}\in D_F
\qquad (p\ge 2)
\]
or
\[
G_u\in D_F
\qquad (u\ge 1,\ p=1).
\]
So once more the connected component of
$F_{\mathrm{red}}-D$
containing
$L$
is exactly
$N_*:=\{L\},$
and
$\deg_{\Gamma_F}(N_*)=1.$

This proves the proposition.
\end{proof}

\begin{lemma}[Multiplicity-one components outside $D$ are leaves]
\label{lem:corr-mult-one-outside-D-leaf}
Let $F$ be a degenerate fiber of a $\PP^1$-fibration on the minimal resolution of a rank-one del Pezzo surface, and
let $C\subset F_{\mathrm{red}}-D$ be an irreducible component of multiplicity one in $F$.  Then $C$ is a leaf of the
reduced fiber $F_{\mathrm{red}}$.
\end{lemma}

\begin{proof}
By \cite[Lemma~2.8(b)]{PP12}, every component of $F_{\mathrm{red}}-D$ is a vertical $(-1)$-curve.  Hence
$C^2=-1.$
Since $C$ has multiplicity one in $F$, the fiber relation gives
\[
0=F\cdot C = C^2 + \sum_{E\sim C}\operatorname{mult}_F(E)
= -1 + \sum_{E\sim C}\operatorname{mult}_F(E),
\]
where the sum runs over the irreducible components $E$ adjacent to $C$ in the reduced fiber.
All multiplicities are positive integers, so the sum must be exactly $1$.
Therefore $C$ has exactly one adjacent component, hence is a leaf of $F_{\mathrm{red}}$.
\end{proof}

\begin{lemma}[A degree-one component of $F_{\mathrm{red}}-D$ carries exactly one leaf]
\label{lem:corr-degree-one-N-one-leaf}
Keep the notation of Proposition~\ref{prop:routeC-height4-maxwidth-allfibers-sigmaminus1}.  Let $N_i$ be a connected
component of $F_{\mathrm{red}}-D$.
Assume $D_F\neq 0$ and that $N_i$ contains a leaf of the whole reduced fiber.
Then $\deg_{\Gamma_F}(N_i)=1.$
In particular, if $N_i$ has degree one in $\Gamma_F$, then it contains exactly one leaf of the whole reduced fiber.
\end{lemma}

\begin{proof}
All irreducible components of $N_i$ are $(-1)$-curves by \cite[Lemma~2.8(b)]{PP12}, and every such $(-1)$-curve is
non-branching in the reduced fiber by \cite[Lemma~2.3(a)]{PP12}.  Hence $N_i$ is a chain.
If $\deg_{\Gamma_F}(N_i)=2$, then both ends of this chain are adjacent to components of $D_F$, so no end of $N_i$ is a
leaf of the whole fiber.  Therefore the presence of a leaf of $F_{\mathrm{red}}$ inside $N_i$ forces
$\deg_{\Gamma_F}(N_i)=1.$
If $\deg_{\Gamma_F}(N_i)=1$, then $N_i$ is a chain attached to $D_F$ at exactly one end, so its opposite end is the
unique leaf of the whole reduced fiber contained in $N_i$.
\end{proof}

\begin{proposition}[Every degenerate fiber in the maximal-width corridor satisfies $v(F)\le \sigma(F)-1$]
\label{prop:routeC-height4-maxwidth-allfibers-sigmaminus1}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}, and among all witnessing fibrations of height $4$
and maximal width on the fixed smooth surface
$\widetilde X$
choose one minimizing
$\lambda(p).$
Then for every degenerate fiber
$F$
one has
$v(F)\le \sigma(F)-1.$
\end{proposition}

\begin{proof}
Keep the notation of Proposition~\ref{prop:routeC-vertical-components-excess}: let
$C_1,\dots,C_s$
be the connected components of
$D_F,$
let
$N_1,\dots,N_r$
be the connected components of
$F_{\mathrm{red}}-D,$
and let
$\Gamma_F$
be the contracted bipartite tree.  As in Proposition~\ref{prop:routeC-vertical-components-excess}, one has
\[
r\le \sigma(F),
\qquad
|E(\Gamma_F)|=r+s-1,
\qquad
\deg_{\Gamma_F}(N_i)\le 2
\quad (1\le i\le r).
\]
We distinguish cases according to
$t(F).$

\medskip
\noindent\textbf{Case 1: $t(F)=0$.}
If $D_F=0$, then $s=v(F)=0$ and there is nothing to prove.  So assume $D_F\neq 0$.
Then every horizontal section meets
$F$
on a component of
$F_{\mathrm{red}}-D.$
Because each horizontal component is a $1$-section, every such intersection lies on a multiplicity-one component of the
fiber.  By Lemma~\ref{lem:corr-mult-one-outside-D-leaf}, every multiplicity-one component of $F_{\mathrm{red}}-D$ is a
leaf of the whole reduced fiber.

Let $C$ be such a component hit by a horizontal section.  Then the connected component $N_i$ of $F_{\mathrm{red}}-D$
containing $C$ has degree one in $\Gamma_F$ by Lemma~\ref{lem:corr-degree-one-N-one-leaf}.
By Corollary~\ref{cor:routeC-tip-entry-obstruction}\textnormal{(1)}, no multiplicity-one component outside $D$ can meet
all four horizontal sections.  Therefore the four horizontal intersections are supported on at least two distinct
components of $F_{\mathrm{red}}-D$, hence on at least two distinct multiplicity-one leaves of the whole fiber.
By the final assertion of Lemma~\ref{lem:corr-degree-one-N-one-leaf}, these leaves lie in at least two distinct
degree-one vertices $N_{i_1},N_{i_2}$ of $\Gamma_F$.
Thus
\[
\deg_{\Gamma_F}(N_{i_1})=\deg_{\Gamma_F}(N_{i_2})=1.
\]
Since every other $N_i$ has degree at most $2$, we get
\[
|E(\Gamma_F)|=\sum_{i=1}^r \deg_{\Gamma_F}(N_i)
\le 1+1+2(r-2)=2r-2.
\]
Combining this with
$|E(\Gamma_F)|=r+s-1$
gives
\[
r+s-1\le 2r-2,
\qquad\text{hence}\qquad
s\le r-1\le \sigma(F)-1.
\]
Since $t(F)=0$, one has $v(F)=s\le \sigma(F)-1.$

\medskip
\noindent\textbf{Case 2: $t(F)=1$.}
By Proposition~\ref{prop:routeC-one-touch-fiber-leaf}, there exists some connected component
$N_*\subset F_{\mathrm{red}}-D$
with
$\deg_{\Gamma_F}(N_*)=1.$
Hence
\[
|E(\Gamma_F)|=\sum_{i=1}^r \deg_{\Gamma_F}(N_i)
\le 1+2(r-1)=2r-1.
\]
Together with
$|E(\Gamma_F)|=r+s-1$
this yields
\[
r+s-1\le 2r-1,
\qquad\text{so}\qquad
s\le r\le \sigma(F).
\]
Because
$t(F)=1,$
we obtain
$v(F)=s-1\le \sigma(F)-1.$

\medskip
\noindent\textbf{Case 3: $t(F)\ge 2$.}
Here Proposition~\ref{prop:routeC-vertical-components-excess} gives directly
$v(F)\le \sigma(F)+1-t(F)\le \sigma(F)-1.$

This completes the proof.
\end{proof}

\begin{theorem}[The height-$4$ maximal-width corridor closes unconditionally]
\label{thm:routeC-height4-maxwidth-corridor-closed}
Assume the setup of Corollary~\ref{cor:routeC-tip-entry-obstruction}.  Then
$\#\Sing(X)\le 7.$
In particular, no benchmark counterexample exists inside the height-$4$, maximal-width corridor.
\end{theorem}

\begin{proof}
Among all witnessing fibrations of height $4$ and maximal width on the fixed smooth surface
$\widetilde X$
choose one minimizing
$\lambda(p).$
By Proposition~\ref{prop:routeC-height4-maxwidth-allfibers-sigmaminus1}, every degenerate fiber satisfies
$v(F)\le \sigma(F)-1.$
Summing over all degenerate fibers and using Corollary~\ref{cor:routeC-tip-entry-obstruction}, we get
\[
\sum_F v(F)\le \sum_F(\sigma(F)-1)=\Sigma=3.
\]

Let
$c_{\mathrm{hor}}$
be the number of connected components of
$D$
which contain at least one horizontal component, and let
$c_{\mathrm{vert}}$
be the number of connected components of
$D$
which are purely vertical.  Then
\[
\#\pi_0(D)=c_{\mathrm{hor}}+c_{\mathrm{vert}}.
\]
Since
$D_{\mathrm{hor}}=H_0+H_1+H_2+H_3,$
one has
$c_{\mathrm{hor}}\le 4.$
Every purely vertical connected component of
$D$
lies in a unique degenerate fiber, so
\[
c_{\mathrm{vert}}=\sum_F v(F)\le 3.
\]
Therefore
\[
\#\Sing(X)=\#\pi_0(D)=c_{\mathrm{hor}}+c_{\mathrm{vert}}\le 4+3=7.
\]
\end{proof}

\begin{corollary}[Reduction to the height-selection statement]
\label{cor:routeC-height4-maxwidth-selection-only}
To prove the characteristic-$3$ seven-point theorem, it is enough to prove the following statement.

\medskip
\centerline{\fbox{\parbox{0.88\linewidth}{%
Every characteristic-$3$ klt del Pezzo surface $X$ of Picard rank $1$ with
$\operatorname{ht}(X)\ge 4$
and no descendant with elliptic boundary admits a witnessing $\PP^1$-fibration of height $4$ and maximal width.
}}}
\medskip

Equivalently, the fiberwise two-touch condition from the two-part endpoint formulation of
Remark~\ref{conj:routeC-final-lemma-needed} is now automatic
once one is in the height-$4$, maximal-width corridor.
\end{corollary}

\begin{proof}
Assume the boxed height-selection statement.  Let
$X$
be a hypothetical characteristic-$3$ klt del Pezzo surface of Picard rank $1$ with
$\#\Sing(X)\ge 8.$
By Theorem~\ref{thm:routeC-contact2-reduction}, the surface lies in the residual sector
$\operatorname{ht}(X)\ge 4$
and has no descendant with elliptic boundary.  The boxed statement then provides a witnessing fibration of height $4$
and maximal width, so Theorem~\ref{thm:routeC-height4-maxwidth-corridor-closed} applies and yields
$\#\Sing(X)\le 7,$
a contradiction.
\end{proof}

\begin{remark}[Status before the final section]
\label{rem:routeC-height4-maxwidth-corridor-final-status}
At this point in the paper, the two-touch part of the two-part endpoint formulation of
Remark~\ref{conj:routeC-final-lemma-needed} is already proved.
The only issue left for the final section is the \emph{height-selection} statement:
pass from the residual sector
$\operatorname{ht}(X)\ge 4$
to the existence of a witnessing fibration of height $4$ and maximal width.
All later packet-section, isolated-pair, terminal-fork, and old-corridor arguments are now absorbed into the terminal-leaf
count above.
\end{remark}

\section{The final height-selection theorem}
\label{sec:routeC-height-selection}

By Corollary~\ref{cor:routeC-height4-maxwidth-selection-only}, the whole characteristic-$3$ seven-point theorem is now
reduced to the existence of a witnessing fibration of height $4$ and maximal width on every residual surface.  The
purpose of this final section is to isolate the remaining branches in one place and to record one additional numerical
restriction on the section-only high-height side.

\begin{proposition}[Final branch decomposition of the height-selection problem]
\label{prop:routeC-height-selection-final-split}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$ with
$\operatorname{ht}(X)\ge 4$
and no descendant with elliptic boundary.
To prove that $X$ admits a witnessing $\PP^1$-fibration of height $4$ and maximal width, it is enough to treat the
following three residual cases:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the \emph{several-special-horizontal branch}, which contains the section-only high-height packet /
    isolated-collision problem;

    \item the \emph{separable special-multisection branch};

    \item the \emph{purely inseparable ordinary singleton $[2]$-root branch}, already reduced to a two-fiber service
    problem by Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers}.
\end{enumerate}
\end{proposition}

\begin{proof}
Corollary~\ref{cor:routeC-height-selection-two-global-branches} reduces the height-selection theorem to two global
branches: several special horizontal components, or a special horizontal multisection of degree at least $2$.
This already gives case~\textnormal{(i)}.

On the special-multisection side, the purely inseparable branch is isolated by
Corollary~\ref{cor:routeC-final-lemma-pureinseparable-fork}, which produces a surviving primitive ramification fork.
Corollary~\ref{cor:routeC-ordinary-ramification-cubic} then shows that every ordinary surviving fork is cubic, and
Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers} reduces the remaining ordinary singleton $[2]$-root service
patterns to at most two fibers.  This is case~\textnormal{(iii)}.

The rest of the special-multisection branch is therefore separable, giving case~\textnormal{(ii)}.
Finally, Proposition~\ref{prop:routeC-one-section-image-class-genus} and
Corollary~\ref{cor:routeC-lowdegree-mixed-no-new-local} show that the low-degree one-section mixed corridor does not
survive as a separate fourth branch.
\end{proof}

\begin{lemma}[After removing the special-multisection branches, case~\textnormal{(i)} is the packet branch]
\label{lem:corr-final-split-case-i-is-packet}
In the setup of Proposition~\ref{prop:routeC-height-selection-final-split}, assume that case~\textnormal{(ii)}
(the separable special-multisection branch) and case~\textnormal{(iii)}
(the purely inseparable ordinary singleton $[2]$-root branch) do not occur.
Then the remaining content of case~\textnormal{(i)} is precisely the several-special-horizontal packet branch closed
later in this section.
\end{lemma}

\begin{proof}
This is exactly the content of the proof of Proposition~\ref{prop:routeC-height-selection-final-split}.
That proof first isolates the two global branches from
Corollary~\ref{cor:routeC-height-selection-two-global-branches}: several special horizontals, or a special horizontal
multisection of degree at least $2$.
It then splits the special-multisection side into the separable branch
(case~\textnormal{(ii)}) and the purely inseparable ordinary singleton $[2]$-root branch
(case~\textnormal{(iii)}), while also observing that the low-degree one-section mixed corridor does not survive as a
separate fourth branch.
Therefore, once cases~\textnormal{(ii)} and~\textnormal{(iii)} are removed, the remaining content of
case~\textnormal{(i)} is exactly the packet-side several-special-horizontal branch.
\end{proof}

\subsection{The several-special-horizontal branch}

The several-special-horizontal side includes the section-only high-height packet geometry.
Earlier sections already show that, after choosing a clean section and preparing the packet along the negative section,
any packet whose pairwise collisions are supported on at most three fibers is forced into the scalar branch.
The next four results sharpen the scalar side of that packet geometry: a collision-root elementary transform removes one
common collision layer, the whole common scalar core peels to a disjoint packet on $\mathbb F_0$, and each branch of
that disjoint packet still carries at least two later service centers.

\begin{lemma}[Elementary transform at a packet collision peels one collision layer]
\label{lem:routeC-collision-elementary-transform}
Let $Y_0\to \PP^1$ be a smooth $\PP^1$-bundle, let $f_0\subset Y_0$ be a fiber, and let
$\Sigma_0,\dots,\Sigma_r\subset Y_0$ be sections.
Assume that a point $q\in f_0$ lies on at least two of the sections, and that a smooth surface $Y$ is obtained from
$Y_0$ by a sequence of blowups whose centers lie over finitely many fibers and whose branch over $f_0$ starts with the
center $q$.
Then there exists another $\PP^1$-bundle $Y_0'\to \PP^1$, obtained from $Y_0$ by the elementary transform at $q$, and a
factorization of $Y$ as a sequence of blowups over the image fiber $f_0'\subset Y_0'$ with the following properties:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the images of $\Sigma_0,\dots,\Sigma_r$ on $Y_0'$ are again sections;
    \item in local coordinates adapted to the ruling, every local branch equation $z=\varphi(t)$ through $q$ is
    replaced on $Y_0'$ by $w=\varphi(t)/t$; equivalently, every pairwise packet-collision order at $q$ drops by exactly
    one;
    \item every descendant of the blowup branch issued from $q$ is still realized over $f_0'$, but with the first
    collision layer removed;
    \item no blowup branch over any other fiber is changed.
\end{enumerate}
\end{lemma}

\begin{proof}
Blow up the point $q$, say $\alpha\colon Y_1\to Y_0$.
Let $E\subset Y_1$ be the exceptional curve and let $f_1$ be the strict transform of $f_0$.
Since $q\in f_0$ is a smooth point of the fiber, $f_1^2=-1$, so contracting $f_1$ produces the elementary transform
$\beta\colon Y_1\to Y_0'$.
Because every $\Sigma_i$ is a section, its strict transform on $Y_1$ meets $f_1$ in at most one point and descends to a
section on $Y_0'$. This proves \textnormal{(i)}.

Choose local coordinates $(t,z)$ centered at $q$ and adapted to the ruling, with $f_0=\{t=0\}$.
The elementary transform is given on the standard chart by $(t,z)\mapsto (t,w)$ with $w=z/t$.
So any local branch through $q$ with equation $z=\varphi(t)$ becomes $w=\varphi(t)/t$.
If two branches had local intersection multiplicity $m=\operatorname{ord}_t(\varphi_1-\varphi_2)$, then after the
transform their new intersection multiplicity is
\[
\operatorname{ord}_t\!\Bigl(\frac{\varphi_1-\varphi_2}{t}\Bigr)=m-1.
\]
This proves \textnormal{(ii)}.

Now fix the chosen blowup branch over $f_0$.
All later centers on that branch are infinitely near to $q$ and are defined by the transformed local equations of the
packet branches together with the subsequent exceptional data.
Because those transformed equations are obtained by dividing by $t$, all descendants of the $q$-branch descend to
centers over $f_0'$, with the first collision layer removed and no other change in the later local combinatorics.
This gives \textnormal{(iii)}.
Since the elementary transform is local over the single fiber $f_0$, all branches over every other fiber are untouched,
proving \textnormal{(iv)}.
\end{proof}

\begin{lemma}[One common scalar packet layer is removable]
\label{lem:routeC-scalar-packet-one-layer}
Assume the setup of Corollary~\ref{cor:routeC-highheight-packet-scalar}, and suppose that the prepared packet is in the
scalar branch of Proposition~\ref{prop:routeC-section-packet-theorem}. Thus, after choosing homogeneous coordinates
$[x:y]$ on the base and a fiber coordinate $z$ on $\mathbb F_n-S_0\cong \Tot \mathcal O_{\PP^1}(n)$, there exist
distinct scalars $\lambda_1,\dots,\lambda_r\in k$ and nonnegative integers $a,b,c$ with $a+b+c=n$ such that
\[
S_i=\{z=\lambda_i g\}, \qquad g=x^ay^b(x-y)^c \qquad (1\le i\le r).
\]

Assume that $a>0$, and let $q\in\{x=0\}$ be the common packet point on that fiber.
Then the elementary transform at $q$ produces a new ruled surface $\mathbb F_{n-1}$ with negative section $S_0'$ such
that:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item the images $S_i'$ of the packet sections are again sections disjoint from $S_0'$;
    \item on $\mathbb F_{n-1}-S_0'\cong \Tot \mathcal O_{\PP^1}(n-1)$ with a suitable fiber coordinate $w$, one has
    \[
    S_i'=\{w=\lambda_i x^{a-1}y^b(x-y)^c\}\qquad (1\le i\le r);
    \]
    \item for every fixed factorization of $\widetilde X$ through the original prepared ruled model, the same smooth
    surface $\widetilde X$ factors through the new ruled model, with the first common collision layer over $x=0$
    removed and all other fibers unchanged.
\end{enumerate}
\end{lemma}

\begin{proof}
The point $q$ lies off the negative section $S_0$.
Blow up $q$ and contract the strict transform of the fiber $\{x=0\}$.
Because the contracted fiber meets $S_0$ once and the center is not on $S_0$, the image $S_0'$ of $S_0$ has
self-intersection $S_0'^2=S_0^2+1=-(n-1)$, so the target ruled surface is $\mathbb F_{n-1}$.

On the affine chart $y\neq 0$ with local parameter $t=x/y$ and fiber coordinate $z$, the packet sections are
$z=\lambda_i t^a u(t)$ with $u(t)=(1-t)^c$.
The elementary transform is given locally by $w=z/t$, so the transformed packet branches are
$w=\lambda_i t^{a-1}u(t)$.
Globally this is exactly the displayed equation for $S_i'$.
Hence each $S_i'$ is a section of the class $S_0'+(n-1)f$, so it is disjoint from $S_0'$.
This proves \textnormal{(1)} and \textnormal{(2)}.

For \textnormal{(3)}, apply Lemma~\ref{lem:routeC-collision-elementary-transform} to the common collision point $q$.
That lemma says precisely that the same smooth surface $\widetilde X$ may be factored over the elementary transform,
with the chosen blowup branch over $x=0$ losing its first collision layer and every other fiber unchanged.
\end{proof}

\begin{proposition}[Prepared scalar packets peel to a disjoint packet on \texorpdfstring{$\mathbb F_0$}{F0}]
\label{prop:routeC-scalar-packet-peels-to-disjoint}
Assume the setup of Corollary~\ref{cor:routeC-highheight-packet-scalar}, and suppose that the prepared packet is in the
scalar branch of Proposition~\ref{prop:routeC-section-packet-theorem}. Write it on the prepared ruled model as
\[
S_i=\{z=\lambda_i x^ay^b(x-y)^c\}\qquad (1\le i\le r),
\]
with distinct scalars $\lambda_i\in k$ and $a+b+c=n$.
Then, after $a+b+c=n$ successive elementary transforms at the marked common packet points, one obtains a ruled surface
$\mathbb F_0$ with sections $S_0^\flat,S_1^\flat,\dots,S_r^\flat$ such that:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item the packet sections $S_1^\flat,\dots,S_r^\flat$ are pairwise disjoint;
    \item each $S_i^\flat$ is disjoint from $S_0^\flat$;
    \item on $\mathbb F_0-S_0^\flat\cong \PP^1\times \mathbb A^1$ with a suitable fiber coordinate $w$, one has
    $S_i^\flat=\{w=\lambda_i\}$ for $1\le i\le r$;
    \item the same smooth surface $\widetilde X$ factors through $\mathbb F_0$, and over each of the three marked fibers
    all common scalar-collision layers are removed; in particular, every later center lying on a packet section is a
    packet-section point of a unique section $S_i^\flat$.
\end{enumerate}
\end{proposition}

\begin{proof}
Apply Lemma~\ref{lem:routeC-scalar-packet-one-layer} repeatedly:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item $a$ times at the common packet point over $x=0$;
    \item $b$ times over $y=0$;
    \item $c$ times over $x-y=0$.
\end{enumerate}
Each step lowers the Hirzebruch index by one and decreases the corresponding exponent by one.
After the total of $a+b+c=n$ steps, all common factors are gone and the ruled surface has become $\mathbb F_0$.
The packet sections are then given by $w=\lambda_i$, so they are pairwise disjoint and disjoint from $S_0^\flat$.
At each step Lemma~\ref{lem:routeC-collision-elementary-transform} preserves the same smooth surface $\widetilde X$ and
removes exactly the first common collision layer on the chosen fiber, without changing any other fiber.
Therefore, once all common scalar layers have been peeled, any remaining center on a packet section lies on a unique
packet branch and is therefore a packet-section point.
\end{proof}

\begin{corollary}[After peeling the scalar core, every packet section needs at least two later service centers]
\label{cor:routeC-scalar-packet-two-service}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}. Let $H_i\subset D$ be the strict
transform on $\widetilde X$ of the section $S_i^\flat$ for $1\le i\le r$.
Then the birational map $\widetilde X\dashrightarrow \mathbb F_0$ has at least two later blowup centers on each section
$S_i^\flat$.
Equivalently, after the common scalar packet has been peeled away, every horizontal packet branch still carries a
packet-section service pattern of length at least two.
\end{corollary}

\begin{proof}
On $\mathbb F_0$, each section $S_i^\flat$ has self-intersection $0$.
Every blowup center lying on $S_i^\flat$ lowers the self-intersection of its strict transform by one.
Since $H_i\subset D$ is a component of the exceptional divisor of the minimal resolution, one has $H_i^2\le -2$.
Therefore the factorization from Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint} must contain at least two
centers on each $S_i^\flat$.
Because the packet sections are pairwise disjoint on $\mathbb F_0$, every such center is automatically a
packet-section point of a unique branch.
\end{proof}

\begin{proposition}[Away from the marked scalar fibers, packet services strip to collisions or to one explicit tip]
\label{prop:routeC-scalar-packet-unmarked-strip-or-collide}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}, and fix one factorization
$\upsilon\colon \widetilde X\to \mathbb F_0$ produced there.
Let $T^\flat\subset \PP^1$ be the set of the three marked fibers coming from $x=0$, $y=0$, and $x-y=0$ before
peeling. Fix a fiber $f\subset \mathbb F_0$ with $f\notin T^\flat$, and assume that some blowup center of
$\upsilon^{-1}$ lies on the running strict transform of one packet section over $f$.
Then, after finitely many repetitions of the following two operations,
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item stripping a maximal initial packet-section staircase by
    Lemma~\ref{lem:routeC-packet-section-elementary-transform};
    \item stripping an initial free center by Lemma~\ref{lem:routeC-free-first-center-slide},
\end{enumerate}
one reaches exactly one of the following three outcomes.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item The whole branch disappears; equivalently, after the stripping process no vertical component of $D$ survives
    over that branch.
    \item The next unstripped center is a collision point of at least two horizontal branches on the resulting ruled
    model.
    \item The branch contributes some vertical component of $D$ and no later packet center remains.  Then the last
    exceptional curve of the final packet-section staircase survives on $\widetilde X$ as a singly-hit
    multiplicity-one nonexceptional tip whose unique vertical neighbor lies in $D$.
\end{enumerate}
\end{proposition}

\begin{proof}
Because $f\notin T^\flat$, the fiber $f$ is one of the smooth unmarked fibers of the peeled model, and the packet
sections $S_1^\flat,\dots,S_r^\flat$ meet it at pairwise distinct points.
Let $q_1\prec q_2\prec\cdots\prec q_m$ be the chain of blowup centers of the chosen branch over $f$, starting at the
first center $q_1\in S_i^\flat\cap f$ for some packet section.

As in Lemma~\ref{lem:routeC-packet-section-staircase}, there is a maximal initial packet-section staircase
$q_1,\dots,q_a$ along the running strict transform of that packet section.
Stripping that staircase by successive applications of
Lemma~\ref{lem:routeC-packet-section-elementary-transform}
replaces the ruled model by another smooth $\PP^1$-bundle carrying the same packet sections and removes those first
$a$ packet-section steps from the branch.

If a next center remains and is a free point of the current smooth fiber disjoint from the packet, then
Lemma~\ref{lem:routeC-free-first-center-slide}
removes that free step as well while preserving the same final smooth surface $\widetilde X$.
Repeating these two stripping operations as long as possible must terminate because the branch has only finitely many
centers.

If no center remains at the end, then the whole branch has been stripped away and contributes no vertical component of
$D$.  This is \textnormal{(i)}.

Assume now that some center remains.
By construction, the first remaining center is neither a packet-section point of a single branch nor a free point of the
fiber.  Hence it must be a collision point of at least two horizontal branches on the resulting ruled model.
This is \textnormal{(ii)}.

It remains to analyze the case in which the stripping process ends with a final maximal packet-section staircase and with
no later packet center afterwards, but with some vertical component of $D$ still contributed by the branch.
Let $F-G_1-\cdots-G_a$ be the final staircase block from Lemma~\ref{lem:routeC-packet-section-staircase}.
If $a=1$, then both curves $F$ and $G_1$ are $(-1)$-curves and lie outside $D$, so the branch contributes no vertical component
of $D$, contrary to the present assumption.
Hence $a\ge 2$. Then $G_{a-1}\subset D$ and $G_a^2=-1$.
Since no later packet center remains on this branch, the last exceptional curve $G_a$ is not blown up again at its
packet-side point and survives on $\widetilde X$ as a vertical $(-1)$-curve
$L:=G_a\not\subset D$.
Every component of the staircase appears in the fiber with multiplicity one, and the chosen packet section meets $L$
once while no other horizontal branch passes through the packet side of this final staircase.
Therefore $L$ is a singly-hit multiplicity-one nonexceptional tip whose unique vertical neighbor is
$G_{a-1}\subset D$.
This is \textnormal{(iii)}.
\end{proof}

\begin{corollary}[Outside the marked scalar fibers, no independent smooth service staircase remains]
\label{cor:routeC-scalar-packet-unmarked-controlled}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}.
Then every packet-side contribution to the exceptional divisor away from the three marked scalar fibers is either:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item collision-rooted on a smooth service fiber; or
    \item carried by the explicit terminal singly-hit multiplicity-one tip of
    Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide}\textnormal{(iii)}.
\end{enumerate}
Equivalently, after peeling the common scalar core, no independent smooth packet-section staircase branch remains away
from the marked fibers.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide} to any packet-side branch over an unmarked
fiber.
Alternative~\textnormal{(i)} contributes nothing to the exceptional divisor, while
\textnormal{(ii)} and \textnormal{(iii)} are exactly the two cases listed here.
\end{proof}

\begin{lemma}[Packet staircases create complementary scalar packets]
\label{lem:routeC-packet-staircase-complementary-scalar}
Let $Y_0\to \PP^1$ be a smooth $\PP^1$-bundle, let $f_0\subset Y_0$ be a fiber, and let
$\Sigma_0,\Sigma_1,\dots,\Sigma_r\subset Y_0$ be sections.
Choose local coordinates $(t,z)$ centered along $f_0=\{t=0\}$ so that
$\Sigma_0=\{z=0\}$ and $\Sigma_i=\{z=\lambda_i\}$ for $1\le i\le r$, where
$\lambda_1,\dots,\lambda_r$ are pairwise distinct nonzero constants.
Let $q_1\prec q_2\prec\cdots\prec q_u$ be the packet-section staircase of length $u\ge 1$ on $\Sigma_0$ over $f_0$,
and strip it by the successive elementary transforms of
Lemma~\ref{lem:routeC-packet-section-elementary-transform}.
Then on the resulting smooth $\PP^1$-bundle $Y_0^{(u)}\to \PP^1$ there exist local coordinates $(t,w)$ near the
transformed fiber such that the images of $\Sigma_1,\dots,\Sigma_r$ are
$\Sigma_i^{(u)}=\{w=\lambda_i^{-1}t^u\}$ for $1\le i\le r$.
In particular, all complementary sections pass through one common point $o_u=(0,0)$, any two of them meet there with
local intersection multiplicity exactly $u$, and the image of $\Sigma_0$ meets the transformed fiber away from $o_u$.
\end{lemma}

\begin{proof}
Set $\zeta_0:=z$. Near the chosen section point on the initial bundle one has $\Sigma_0=\{\zeta_0=0\}$ and
$\Sigma_i=\{\zeta_0=\lambda_i\}$.
After one elementary transform at $q_1\in \Sigma_0\cap f_0$, a standard local coordinate near the image of the chosen
section is $\zeta_1:=\zeta_0/t$.
Hence the complementary sections are given there by $\zeta_1=\lambda_i/t$, so near the point where they meet on the new
fiber it is convenient to use the reciprocal coordinate $w_1:=1/\zeta_1=t/\zeta_0$.
Then $\Sigma_i^{(1)}=\{w_1=\lambda_i^{-1}t\}$ for $1\le i\le r$, which is the claimed formula for $u=1$.

Assume inductively that after stripping a staircase of length $u-1$ one has local coordinates
$\zeta_{u-1}=z/t^{u-1}$ near the transformed chosen section and reciprocal coordinate
$w_{u-1}=1/\zeta_{u-1}=t^{u-1}/z$ near the complementary collision point, with
$\Sigma_i^{(u-1)}=\{w_{u-1}=\lambda_i^{-1}t^{u-1}\}$ for $1\le i\le r$.
The next elementary transform is again taken at the packet-section point of the chosen section, so the new local
coordinate near that section is $\zeta_u:=\zeta_{u-1}/t=z/t^u$.
Passing to the reciprocal coordinate near the complementary collision point gives $w_u:=1/\zeta_u=t^u/z$.
Therefore $\Sigma_i^{(u)}=\{w_u=\lambda_i^{-1}t^u\}$ for $1\le i\le r$.
This proves the displayed local form for every $u\ge 1$.

Since the constants $\lambda_i^{-1}$ are pairwise distinct, all complementary sections pass through $o_u=(0,0)$ and,
for $i\ne j$, one has
\[
\bigl(\Sigma_i^{(u)}\cdot \Sigma_j^{(u)}\bigr)_{o_u}
=
\operatorname{ord}_t\bigl((\lambda_i^{-1}-\lambda_j^{-1})t^u\bigr)
=u.
\]
Finally, the chosen section is given in the coordinate $\zeta_u$ by $\zeta_u=0$, so it meets the transformed fiber at
the point corresponding to $w_u=\infty$, hence away from $o_u$.
\end{proof}

\begin{proposition}[Collision-rooted unmarked packet branches are complementary scalar packets]
\label{prop:routeC-scalar-packet-complementary-root}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}, and fix one factorization
$\upsilon\colon \widetilde X\to \mathbb F_0$ produced there.
Let $T^\flat\subset \PP^1$ be the set of the three marked scalar fibers, and let $f\subset \mathbb F_0$ be a fiber
with $f\notin T^\flat$.
Assume that a packet-side contribution over $f$ falls into alternative~\textnormal{(ii)} of
Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide}.
Let $u\ge 1$ be the length of the last packet-section staircase stripped from that branch in the proof of
Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide}.
Then on the final ruled model produced there:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the first remaining center over $f$ is the common collision point of the complementary packet sections;
    \item in suitable local coordinates its horizontal branches are exactly of the form $w=\mu_i t^u$ with pairwise
    distinct nonzero constants $\mu_i$;
    \item after the forced order-$u$ common-collision blowups, the last exceptional curve $G_u$ is the unique component
    meeting all complementary packet sections, so every later center of the chosen branch lies above $G_u$.
\end{enumerate}
\end{proposition}

\begin{proof}
By construction in Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide}, immediately before the first
remaining center appears there is a last maximal packet-section staircase of length $u$ on one active packet section
over the smooth unmarked fiber $f$.
Because the peeled packet on $\mathbb F_0$ consists of pairwise disjoint constant sections, we may choose local
coordinates so that the active section is $\Sigma_0=\{z=0\}$ and the remaining packet sections are
$\Sigma_i=\{z=\lambda_i\}$ with pairwise distinct nonzero constants $\lambda_i$.
Lemma~\ref{lem:routeC-packet-staircase-complementary-scalar} then shows that after stripping that last staircase, the
complementary packet sections are given by $w=\mu_i t^u$ and all meet at one common point $o=(0,0)$, while the active
section meets the transformed fiber away from $o$.

At that stage the first remaining center is neither a packet-section point nor a free point of the fiber, by the
definition of the stripped model in
Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide}.
Since no other point of the transformed fiber lies on more than one complementary section, that first remaining center
must be exactly the common collision point $o$.
This proves \textnormal{(i)} and \textnormal{(ii)}.

Now apply Proposition~\ref{prop:routeC-scalar-star-local} to the local model $w=\mu_i t^u$.
It identifies the forced order-$u$ common-collision block and shows that its last exceptional curve $G_u$ is the
unique component met by all complementary packet sections; the earlier exceptional curves are disjoint from them.
Consequently any later center of the chosen branch that still follows one of those packet sections must lie on or above
$G_u$, proving \textnormal{(iii)}.
\end{proof}

\begin{corollary}[Away from the marked scalar fibers, surviving collision roots are complementary scalar roots]
\label{cor:routeC-scalar-packet-complementary-controlled}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}.
Then every packet-side contribution to the exceptional divisor away from the three marked scalar fibers is one of the
following:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item it disappears after the stripping process of
    Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide};
    \item it is carried by the explicit terminal singly-hit multiplicity-one tip of
    Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide}\textnormal{(iii)};
    \item after the last stripped packet staircase, it is rooted at the last exceptional curve of a complementary scalar
    packet over a smooth service fiber.
\end{enumerate}
In case \textnormal{(iii)}, either that last exceptional curve survives as a vertical curve $L\not\subset D$, in which
case the same contact-defect calculation as in Corollary~\ref{cor:routeC-scalar-star-defect} applies, or every later
center of the branch lies above that complementary scalar root.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:routeC-scalar-packet-unmarked-strip-or-collide} to a packet-side branch over an unmarked
fiber.
Alternative~\textnormal{(i)} gives nothing, and alternative~\textnormal{(iii)} is exactly case
\textnormal{(ii)} here.
If alternative~\textnormal{(ii)} occurs, then
Proposition~\ref{prop:routeC-scalar-packet-complementary-root} identifies the remaining collision-rooted branch with a
complementary scalar packet and places all later packet-side activity above its last exceptional curve.
If that curve survives on $\widetilde X$ as a vertical $(-1)$-curve outside $D$, the proof of
Corollary~\ref{cor:routeC-scalar-star-defect} applies verbatim to this local scalar-star model.
\end{proof}

\begin{proposition}[Rooted complementary scalar packets reduce to surviving roots or isolated pairs]
\label{prop:routeC-complementary-scalar-rooted-reduces-pair}
Keep the setup of Corollary~\ref{cor:routeC-scalar-packet-complementary-controlled}, and assume that Route~C entry does
not already occur.
Let $f$ be a smooth service fiber, and suppose that over $f$ one reaches a complementary scalar packet of some order
$\nu\ge 1$ on packet sections $B_1,\dots,B_m$ with $m\ge 2$, whose last common exceptional curve is
$R\not\subset D$.
Assume that some later packet-side center still lies above $R$.
Then, after finitely many repetitions of stripping a maximal packet-section staircase on one active packet section and
passing to the next complementary scalar packet if a later center remains, one reaches exactly one of the following two
outcomes:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the current last exceptional curve survives on $\widetilde X$ as a vertical curve $L\not\subset D$, so the
    same scalar-star contact-defect calculation applies;
    \item the remaining packet-side block is an isolated pair collision of some order $p\ge 1$, hence is governed by
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}.
\end{enumerate}
Equivalently, rooted service above complementary scalar packets introduces no genuinely new many-branch corridor beyond
surviving scalar packets and isolated pair-collision service blocks.
\end{proposition}

\begin{proof}
We argue by induction on the number $m$ of packet sections meeting the current scalar root.
For $m=2$, the current complementary scalar packet is already an isolated pair collision: in suitable local coordinates
the two sections are $w=\mu_1 t^\nu$ and $w=\mu_2 t^\nu$ with $\mu_1\ne\mu_2$, so their difference has order exactly
$\nu$, and the local block is precisely the primitive order-$\nu$ isolated pair collision of
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}.
This gives alternative~\textnormal{(ii)}.

Assume now that $m\ge 3$ and that the statement is known for all smaller values.
Because $R$ is a smooth rational curve and the packet sections meet it transversely at pairwise distinct points, a
neighborhood of $R$ is locally a smooth bundle with those packet sections given by constant sections over it.
Choose one packet section on the chosen branch and strip a maximal packet-section staircase above its intersection with
$R$ by the same local elementary-transform procedure as in
Lemma~\ref{lem:routeC-packet-section-elementary-transform}.

If no later center remained above the resulting fiber, then the last exceptional of that staircase would be a singly-hit
multiplicity-one nonexceptional tip with unique vertical neighbor in $D$, so
Proposition~\ref{prop:routeC-tip-entry-sufficient} would give Route~C entry, contrary to the hypothesis.
Therefore a next center does remain.

The same local calculation as in Lemma~\ref{lem:routeC-packet-staircase-complementary-scalar} then shows that, after
stripping the chosen staircase, the remaining packet sections form a complementary scalar packet over the same service
fiber, now on exactly $m-1$ packet sections.
Its next unstripped center is the common collision point of that new complementary scalar packet, and any further
packet-side activity lies above the last exceptional of its forced common-collision block, exactly as in the proof of
Proposition~\ref{prop:routeC-scalar-packet-complementary-root}.
Thus the same problem reappears with one fewer active packet section.
Applying the induction hypothesis to that new complementary scalar packet yields either a surviving scalar packet root,
giving alternative~\textnormal{(i)}, or an isolated pair collision block, giving
alternative~\textnormal{(ii)}.
\end{proof}

\begin{corollary}[On the packet side, smooth-fiber branches reduce to surviving scalar roots or isolated pairs]
\label{cor:routeC-scalar-packet-rootpair-controlled}
Keep the setup of Corollary~\ref{cor:routeC-scalar-packet-complementary-controlled}, and assume that Route~C entry does
not already occur.
Then every packet-side contribution beyond the marked scalar roots is governed by exactly one of the following two
mechanisms:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a surviving scalar packet root, to which the same scalar-star contact-defect bound applies;
    \item an isolated pair-collision service block of
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}.
\end{enumerate}
In particular, in the controlled scalar regime there is no genuinely many-branch smooth-fiber corridor remaining beyond
surviving scalar packets and isolated pair collisions.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-scalar-packet-complementary-controlled}, every packet-side contribution away from the
marked fibers either disappears, gives the explicit singly-hit tip, or is rooted at a complementary scalar packet.
The explicit tip alternative is excluded here by the no-entry hypothesis and
Proposition~\ref{prop:routeC-tip-entry-sufficient}.
If the last exceptional of the complementary scalar packet survives, we are in alternative~\textnormal{(i)}.
Otherwise Proposition~\ref{prop:routeC-complementary-scalar-rooted-reduces-pair} yields
alternative~\textnormal{(ii)}.
\end{proof}

The next proposition adds a further global restriction on the resulting distributed service geometry.

\begin{proposition}[No vertical component of \texorpdfstring{$D$}{D} can meet five or more horizontal sections]
\label{prop:routeC-no-five-horizontal-hub}
Let $p\colon \widetilde X\to \PP^1$ be a witnessing $\PP^1$-fibration such that every horizontal component of $D$ is a
$1$-section. Then there does not exist a vertical irreducible component $U\subset D$ meeting $m\ge 5$ distinct
horizontal sections.
\end{proposition}

\begin{proof}
Assume for contradiction that such a component $U$ exists, and write $H_1,\dots,H_m$ for the horizontal sections
meeting $U$. Set $U^2=-b$ and $H_i^2=-a_i$ for $1\le i\le m$, so $b\ge 2$ and $a_i\ge 2$.

Write $\pi^*K_X = K_{\widetilde X}+\sum_{E\subset D}\operatorname{cf}_X(E)\,E$. Intersecting with $H_i$ gives
\[
a_i\operatorname{cf}_X(H_i)
=
a_i-2+\sum_{E\sim H_i}\operatorname{cf}_X(E)
\ge
a_i-2+\operatorname{cf}_X(U).
\]
Hence $\operatorname{cf}_X(H_i)\ge \operatorname{cf}_X(U)/2$ for $1\le i\le m$. Since the fibration is section-only,
Lemma~\ref{lem:routeC-fiber-anticanonical-cf} yields
\[
2>
\sum_{H\subset D_{\mathrm{hor}}}\operatorname{cf}_X(H)
\ge
\sum_{i=1}^m \operatorname{cf}_X(H_i)
\ge
\frac{m}{2}\operatorname{cf}_X(U).
\]
Therefore $\operatorname{cf}_X(U)<4/m<1$.

Now let $V_1,\dots,V_s$ be the vertical neighbors of $U$ inside $D$. Intersecting the discrepancy formula with $U$
gives
\[
b\,\operatorname{cf}_X(U)
-
\sum_{i=1}^m \operatorname{cf}_X(H_i)
-
\sum_{j=1}^s \operatorname{cf}_X(V_j)
=
b-2.
\]
Using the lower bound on the $\operatorname{cf}_X(H_i)$ and the nonnegativity of the $\operatorname{cf}_X(V_j)$, we get
\[
\Bigl(b-\frac{m}{2}\Bigr)\operatorname{cf}_X(U)\ge b-2.
\]

If $b>\frac{m}{2}$, then $\operatorname{cf}_X(U)\ge \frac{b-2}{b-\frac{m}{2}}$. Since $\frac{m}{2}>2$, the denominator
is strictly smaller than $b-2$, so the right-hand side is greater than $1$, contradicting
$\operatorname{cf}_X(U)<1$.

If $b=\frac{m}{2}$, then the left-hand side is $0$, while the right-hand side is strictly positive because $m\ge 5$.
This is impossible.

So the only remaining possibility is $b<\frac{m}{2}$. Then the coefficient $b-\frac{m}{2}$ is negative. Since the
right-hand side is nonnegative, the displayed inequality can hold only when $b=2$ and $\operatorname{cf}_X(U)=0$.
Substituting these values into the discrepancy equation for $U$ gives
\[
0=
\sum_{i=1}^m \operatorname{cf}_X(H_i)
+
\sum_{j=1}^s \operatorname{cf}_X(V_j),
\]
so $\operatorname{cf}_X(H_i)=0$ for $1\le i\le m$ and $\operatorname{cf}_X(V_j)=0$ for $1\le j\le s$.

Now intersect the discrepancy formula with some $H_i$. Since
$\operatorname{cf}_X(H_i)=\operatorname{cf}_X(U)=0$, one gets
\[
-\sum_{W\sim H_i,\ W\neq U}\operatorname{cf}_X(W)=a_i-2.
\]
Hence $a_i=2$, and every other neighbor of $H_i$ also has discrepancy coefficient $0$.
Repeating this argument along the connected component of $D$ containing $U$ shows that every curve in that connected
component has self-intersection $-2$ and discrepancy coefficient $0$.
So the corresponding singularity is canonical, hence Du~Val, and its exceptional dual graph must be of ADE type.
But the vertex corresponding to $U$ has valence at least $m\ge 5$, which is impossible for an ADE graph.
This contradiction proves the proposition.
\end{proof}

\begin{corollary}[The section-only residue is a five-defect packet problem]
\label{cor:routeC-sectiononly-five-defect-packet}
Assume the section-only high-height branch of Proposition~\ref{prop:routeC-height-selection-final-split}\textnormal{(i)}.
Choose a clean section $H_0\subset D_{\mathrm{hor}}$ as in Corollary~\ref{cor:routeC-clean-plus-strong}, and let
$\mathcal P:=\{H\subset D_{\mathrm{hor}}\ ;\ H\neq H_0\}$ be the packet of the remaining horizontal sections.
Then at most five members of $\mathcal P$ are not strong in the sense of
Definition~\ref{def:routeC-strong-horizontal}.  Equivalently, after removing a defect set of size at most $5$, the
residual packet consists entirely of strong sections.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:routeC-strong-multiplicity-hminus5}, the total horizontal multiplicity carried by mild or ordinary
horizontal components is at most $5$.  In the section-only case every horizontal component has multiplicity $1$, so
there are at most five non-strong horizontal sections in total.  Removing the chosen clean section $H_0$ cannot
increase that number inside $\mathcal P$.
\end{proof}

\begin{lemma}[The first split of a horizontal collision tree is read off from the first non-common jet]
\label{lem:routeC-height-selection-first-split}
Let $A_1,\dots,A_r$ with $r\ge 2$ be pairwise distinct smooth horizontal branches through a point $o$ of a smooth ruled
surface $Y\to \PP^1$.
Choose local coordinates $(t,z)$ at $o$ adapted to the ruling, so the fiber through $o$ is $\{t=0\}$, and after
translating by $A_1$ one has $A_1=\{z=0\}$ and $A_i=\{z=\phi_i(t)\}$ for $2\le i\le r$, with
$\phi_i\in t\,k[[t]]$.
Let $\mu$ be the order of the common jet of the family, and write
$\phi_i(t)=\alpha_i t^\mu+\text{(higher order terms)}$ with $\alpha_1:=0$.
Then the first $\mu$ blowups at the successive unique common points of the strict transforms are forced, and on the last
exceptional curve the packet-section points are indexed exactly by the distinct values among $\alpha_1,\dots,\alpha_r$.
Equivalently, after those $\mu$ common blowups the branches split into disjoint clusters according to their first
non-common coefficient.
\end{lemma}

\begin{proof}
Because every difference $\phi_i-\phi_j$ vanishes to order at least $\mu$, all branches have the same current point for
the first $\mu$ blowups, so those blowups are forced.
In the standard chart after these common blowups one has $z=t^\mu w$.
Hence the strict transform of $A_i$ is given by $w=\psi_i(t):=t^{-\mu}\phi_i(t)$ with $\psi_i(0)=\alpha_i$.
The last exceptional curve is $\{t=0\}$, and the strict transform of $A_i$ meets it at the point $w=\alpha_i$.
So the distinct packet-section points on that exceptional curve are indexed exactly by the distinct coefficients among
the $\alpha_i$, and the branches through each such point are precisely the branches in the corresponding cluster.
\end{proof}

\begin{corollary}[Every multibranch horizontal collision tree reduces to binary and unary descendants]
\label{cor:routeC-height-selection-binary-unary}
Let $A_1,\dots,A_r$ be pairwise distinct smooth horizontal branches through a point of a smooth ruled surface.
Iterating Lemma~\ref{lem:routeC-height-selection-first-split} on every cluster of size at least $3$ decomposes the full
local factorization into finitely many descendants of only two kinds:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item unary descendants, supported on a single horizontal branch;
    \item binary descendants, supported on exactly two horizontal branches.
\end{enumerate}
Equivalently, there is no irreducible many-branch horizontal local block beyond binary pair-collision trees and
single-branch service or unibranch descendants.
\end{corollary}

\begin{proof}
Proceed by induction on $r$.
The statement is tautological for $r\le 2$.
Assume $r\ge 3$ and apply Lemma~\ref{lem:routeC-height-selection-first-split} to the first non-common jet of the
family.
After the forced common blowups, later horizontal centers over the original point are supported independently over the
smaller clusters determined by the first non-common coefficient.
By induction, each such cluster produces only unary and binary descendants.
Since the original local factorization is the union of these independent cluster descendants together with possible
purely vertical blowups away from all horizontal branches, the same conclusion holds for the original family.
\end{proof}

\begin{corollary}[The strong packet side is a five-defect binary service problem]
\label{cor:routeC-height-selection-packet-binary}
Assume the section-only high-height branch of Proposition~\ref{prop:routeC-height-selection-final-split}\textnormal{(i)}.
Choose a clean section and the packet $\mathcal P$ as in
Corollary~\ref{cor:routeC-sectiononly-five-defect-packet}, peel the scalar core as in
Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}, and remove the defect set of at most $5$ non-strong
packet members.
Then every local collision tree on the residual strong packet decomposes into binary pair-collision trees and unary
packet-section service branches.
In particular, the several-special-horizontal branch contributes no new many-branch local horizontal corridor beyond the
primitive binary pair-collision-tree problem on strong packet sections.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-sectiononly-five-defect-packet}, all but at most $5$ packet members are strong.
After Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}, the common scalar core has been removed and the
remaining packet geometry is supported on disjoint packet sections of $\mathbb F_0$.
Any later local collision of those sections is therefore a multibranch horizontal collision tree on a smooth ruled
surface, so Corollary~\ref{cor:routeC-height-selection-binary-unary} applies directly.
The unary descendants are exactly packet-section service branches, while the binary descendants are pair-collision
trees.
Since the residual packet has already been reduced to strong sections outside the defect set, no new many-branch local
side type remains.
\end{proof}

\begin{proposition}[A later packet-point blowup on a scalar hub gives a tip-entry curve]
\label{prop:routeC-height-selection-scalar-packetpoint-entry}
Keep the setup of Proposition~\ref{prop:routeC-scalar-star-local}. Thus on a smooth ruled surface one has a primitive
scalar packet of order $\nu\ge 1$ on a smooth fiber, with last common exceptional curve $G_{\nu}$ meeting packet
sections $B_1,\dots,B_m$ with $m\ge 2$ transversely at pairwise distinct packet-section points.
Assume that in the chosen factorization of the inverse birational map to $\widetilde X$ a later blowup center lies at
one of those packet-section points on $G_{\nu}$.
Then $\widetilde X$ contains a vertical curve $L\not\subset D$ of multiplicity one such that $L\cdot D_{\mathrm{hor}}=1$,
and the unique connected component of the reduced fiber minus $L$ meeting $L$ belongs to $D$.
Consequently Proposition~\ref{prop:routeC-tip-entry-sufficient} applies to $L$.
\end{proposition}

\begin{proof}
Choose the first later blowup center lying at one of the packet-section points of $G_{\nu}$, say at the point where
the strict transform of $B_1$ meets $G_{\nu}$.
By Proposition~\ref{prop:routeC-scalar-star-local}\textnormal{(3)}, no other packet section passes through that point.
So near that point the local picture is a smooth horizontal branch crossing a smooth multiplicity-one fiber component.
Repeated blowups above that point create a chain $E_1-\cdots-E_r$ attached to the strict transform of $G_{\nu}$, where
the strict transform of $B_1$ meets the last exceptional curve $L:=E_r$ transversely once and no other horizontal
component meets $L$.

Because the first later blowup occurs on $G_{\nu}$ itself, the strict transform of $G_{\nu}$ becomes a $(-2)$-curve and
therefore belongs to $D$; likewise every earlier exceptional curve on this later branch belongs to $D$.
Since the whole branch is created by successive blowups at smooth points of a multiplicity-one fiber component, the
terminal exceptional curve $L$ also has multiplicity one in the final fiber.
Therefore $\operatorname{mult}(L)=1$ and $L\cdot D_{\mathrm{hor}}=1$, while the unique vertical connected component of
the reduced fiber meeting $L$ belongs to $D$.
Proposition~\ref{prop:routeC-tip-entry-sufficient} now applies.
\end{proof}

\begin{proposition}[Continuing complementary scalar packets are at most five-ary]
\label{prop:routeC-height-selection-scalar-continuation-five}
Keep the setup of Proposition~\ref{prop:routeC-complementary-scalar-rooted-reduces-pair}, and assume that Route~C
entry does not already occur.
Let $R\not\subset D$ be the last common exceptional curve of a complementary scalar packet on packet sections
$B_1,\dots,B_m$ with $m\ge 2$ over a smooth service fiber.
Assume that some later packet-side center still lies above $R$.
Then $m\le 5$.
\end{proposition}

\begin{proof}
By assumption, the packet is section-only, so every packet section is a $1$-section.
Choose the first later packet-side center above $R$, say at the packet-section point of $B_1$.
Blowing up that point lowers the self-intersection of $R$ from $-1$ to $-2$, so the strict transform
$U\subset D$ of $R$ on the final surface belongs to the exceptional divisor.
Because the chosen blowup uses only the packet-section point of $B_1$, the strict transforms of $B_2,\dots,B_m$ still
meet $U$ transversely at pairwise distinct points.
Thus $U$ meets at least $m-1$ distinct horizontal sections.

Proposition~\ref{prop:routeC-no-five-horizontal-hub} says that no vertical component of $D$ can meet five or more
horizontal sections.
Hence $m-1\le 4$, that is, $m\le 5$.
\end{proof}

\paragraph*{Marked scalar service and surviving scalar roots.}

\begin{proposition}[Marked-fiber scalar service strips to a complementary scalar packet]
\label{prop:routeC-height-selection-marked-scalar-to-complementary}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}, and fix one of the three marked scalar
fibers $f\in T^\flat$.
Assume that in the chosen factorization $\upsilon\colon \widetilde X\to \mathbb F_0$ a packet-side branch over $f$
starts with a maximal packet-section staircase of length $u\ge 1$ on one packet section, say on $S_1^\flat$.
Then, after stripping that staircase by Lemma~\ref{lem:routeC-packet-section-elementary-transform}, the remaining
packet sections $S_2^\flat,\dots,S_r^\flat$ form a complementary scalar packet of order $u$ over the transformed fiber.
More precisely, on the resulting smooth ruled model there are local coordinates $(t,w)$ at the new common point such that
$S_i'=\{w=\mu_i t^u\}$ for $2\le i\le r$, where the $\mu_i$ are pairwise distinct nonzero constants.
In particular, any later packet-side activity above the marked fiber is rooted at a complementary scalar packet on one
fewer active packet section.
\end{proposition}

\begin{proof}
Near the chosen point of $S_1^\flat\cap f$ choose local coordinates $(t,z)$ with $f=\{t=0\}$, $S_1^\flat=\{z=0\}$, and
$S_i^\flat=\{z=\lambda_i\}$ for $2\le i\le r$, where the $\lambda_i$ are pairwise distinct nonzero constants.
Stripping the maximal packet-section staircase of length $u$ is exactly the situation of
Lemma~\ref{lem:routeC-packet-staircase-complementary-scalar}, which gives local coordinates $(t,w)$ on the transformed
ruled model such that $S_i'=\{\lambda_i^{-1} t^u\}$ for $2\le i\le r$.
Thus the remaining packet sections form a complementary scalar packet of order $u$ over the transformed fiber, and every
later packet-side center above the marked fiber lies over the common point of that complementary scalar packet.
\end{proof}

\begin{corollary}[Marked-fiber scalar service reduces to surviving scalar roots or binary pair blocks]
\label{cor:routeC-height-selection-marked-scalar-reduces-rootpair}
Keep the setup of Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint}.
Every remaining packet-side contribution over one of the three marked scalar fibers reduces, after finitely many maximal
packet-section staircase strips, to exactly one of the following outputs:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item a surviving scalar root;
    \item an isolated pair-collision service block whose terminal binary collision is governed by
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}.
\end{enumerate}
Equivalently, marked-fiber scalar service contributes no endpoint beyond the same surviving-root / binary-pair outputs
already present on smooth service fibers.
\end{corollary}

\begin{proof}
Take a packet-side branch over a marked fiber.
If repeated staircase stripping removes the whole branch, then it contributes nothing further to $D$.
Otherwise Proposition~\ref{prop:routeC-height-selection-marked-scalar-to-complementary} shows that after stripping the
first maximal packet-section staircase one reaches a complementary scalar packet on one fewer active packet section.
Now Proposition~\ref{prop:routeC-complementary-scalar-rooted-reduces-pair} applies and gives exactly the two displayed
alternatives.
\end{proof}

\begin{proposition}[Surviving many-ary scalar roots already feed the old horizontal corridor]
\label{prop:routeC-height-selection-manyary-scalar-root-old}
Let a surviving scalar root on the packet side meet horizontal packet sections
$B_1,\dots,B_m\subset D_{\mathrm{hor}}$ with $m\ge 3$, and let
$L\subset \widetilde X\setminus D$ be the surviving strict transform of its last common exceptional curve.
Then at least one of the packet sections $B_i$ already lies in the old horizontal Route~C corridor: either the connected
component of $D$ containing $B_i$ is not a chain, or it is a chain whose reduced touched core is a touched tip of
weight $2$.
In particular, surviving scalar roots of arity at least $3$ do not contribute an independent packet-side endpoint.
\end{proposition}

\begin{proof}
Since $L$ is a surviving scalar root, it meets each of the packet sections $B_1,\dots,B_m$ once.
Therefore Corollary~\ref{cor:routeC-scalar-star-defect} applies and gives
$\sum_{i=1}^m \operatorname{cf}_X(B_i)<1$.
Because $m\ge 3$, at least one index $i$ satisfies $\operatorname{cf}_X(B_i)<\frac13$.
If the connected component of $D$ containing $B_i$ is not a chain, then $B_i$ already lies on a branching component of
$D$.
If that connected component is a chain, then Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} shows that its reduced
touched core is a touched tip of weight $2$.
So in either case one already lands in the old horizontal Route~C corridor.
\end{proof}

\begin{proposition}[Binary scalar roots are exactly primitive isolated pair collisions]
\label{prop:routeC-height-selection-binary-scalar-is-pair}
Let a surviving scalar root on the packet side meet exactly two horizontal packet sections
$B_1,B_2\subset D_{\mathrm{hor}}$, and let $u\ge 1$ be its scalar order.
Then the first $u$ blowups over the root are exactly the order-$u$ isolated collision resolution of
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} for the pair $B_1,B_2$.
Consequently the surviving binary scalar root is precisely the primitive contact-$2$ or primitive fork output attached to
that isolated pair collision.
In particular, a binary scalar root contributes no separate scalar branch beyond the already isolated binary pair-collision
model.
\end{proposition}

\begin{proof}
Choose local coordinates $(t,w)$ at the scalar root so that
$B_1=\{\mu_1 t^u\}$ and $B_2=\{\mu_2 t^u\}$ with $\mu_1\ne \mu_2$.
After replacing $w$ by $w-\mu_1 t^u$ and rescaling, one may rewrite the two branches as
$B_1=\{z=0\}$ and $B_2=\{z=t^u\}$ in suitable local coordinates $(t,z)$.
Thus the two packet sections have local intersection multiplicity exactly $u$ at the root.
Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} therefore applies and shows that the first $u$ blowups
are exactly the forced order-$u$ two-branch collision resolution, with primitive last exceptional curve equal to a
contact-$2$ curve when $u=1$ and to a primitive fork center when $u\ge 2$.
But this is exactly the primitive scalar-root chain for the binary packet.
\end{proof}

\begin{theorem}[The scalar-root branch adds no independent packet endpoint]
\label{thm:routeC-height-selection-scalar-root-absorbed}
Assume the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}.
Then every packet-side branch governed by a scalar mechanism is absorbed as follows.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item Every remaining marked-fiber scalar service contribution reduces to a surviving scalar root or an isolated
    pair-collision service block, by
    Corollary~\ref{cor:routeC-height-selection-marked-scalar-reduces-rootpair}.
    \item Every surviving scalar root of arity at least $3$ already feeds the old horizontal Route~C corridor, by
    Proposition~\ref{prop:routeC-height-selection-manyary-scalar-root-old}.
    \item Every surviving scalar root of arity $2$ is exactly the primitive isolated pair-collision output of
    Proposition~\ref{prop:routeC-primitive-isolated-pair-collision}, by
    Proposition~\ref{prop:routeC-height-selection-binary-scalar-is-pair}.
\end{enumerate}
Hence the packet side has no separate scalar-root residue:
marked-fiber scalar service and surviving scalar roots contribute nothing beyond the already isolated old horizontal
corridor and the binary pair-collision service blocks.
\end{theorem}

\begin{proof}
Take any packet-side branch governed by a scalar mechanism.
If it lies over one of the three marked scalar fibers, then
Corollary~\ref{cor:routeC-height-selection-marked-scalar-reduces-rootpair} reduces it to either a surviving scalar root
or a binary pair-collision block.
If it lies over a smooth service fiber, then Corollary~\ref{cor:routeC-scalar-packet-rootpair-controlled} already gives
the same dichotomy.
So it remains only to analyze surviving scalar roots.

If the surviving scalar root has arity at least $3$, then
Proposition~\ref{prop:routeC-height-selection-manyary-scalar-root-old} shows that one of its horizontal packet arms
already lies in the old horizontal Route~C corridor.
If instead the surviving scalar root has arity $2$, then
Proposition~\ref{prop:routeC-height-selection-binary-scalar-is-pair} identifies it with the primitive isolated
pair-collision model.
\end{proof}

\paragraph*{Defect-supported outputs.}

\begin{proposition}[Realized binary packet descendants have only primitive or branching terminal outputs]
\label{prop:routeC-height-selection-binary-packet-local}
Assume the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}, and assume moreover that Route~C entry
does not already occur.
Let $\mathcal T$ be a realized binary packet descendant supported on exactly two packet sections
$H_i,H_j\subset D_{\mathrm{hor}}$.
Choose a terminal leaf of the blowup tree of $\mathcal T$, and on that leaf let $q_c$ be the last collision center of
the strict transforms of the two active horizontal branches.
Let $u\ge 0$ be the length of the maximal packet-section staircase immediately preceding $q_c$ on the chosen leaf, and
let $p\ge 1$ be the order of the isolated pair collision at $q_c$.
Then the exceptional curve $L_0$ created by the last blowup of that order-$p$ collision block satisfies exactly one of
the following two alternatives.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item No later blowup center on the chosen leaf lies on $L_0$.
    Then the final transform of $L_0$ on $\widetilde X$ is a nonexceptional curve
    $L\subset \widetilde X\setminus D$ meeting $H_i$ and $H_j$ transversely once each, and
    \[
    L\cdot(D-H_i-H_j)=
    \begin{cases}
    0,& (u,p)=(0,1),\\
    1,& \text{otherwise}.
    \end{cases}
    \]
    In particular, $L$ is either a surviving primitive contact-$2$ curve on $H_i,H_j$ or a surviving fork center whose
    horizontal arms are $H_i,H_j$.

    \item Some later blowup center on the chosen leaf lies on $L_0$.
    Then the final strict transform $E\subset D$ of $L_0$ is a branching component of $D$: it meets both packet
    sections $H_i,H_j$ and at least one later vertical component.
\end{enumerate}
\end{proposition}

\begin{proof}
Because $\mathcal T$ is a binary descendant in the sense of
Corollary~\ref{cor:routeC-height-selection-packet-binary}, every horizontal center on the chosen leaf is supported on
the strict transforms of exactly two packet sections, namely $H_i$ and $H_j$.
So the last collision center $q_c$ is well defined.

Let $M$ be the current smooth multiplicity-one fiber component at the start of the final service phase immediately
preceding $q_c$, and let $A_0,C_0$ be the current strict transforms of $H_i,H_j$.
By construction there are exactly $u$ successive packet-section blowups on $A_0$ before the final collision, and then
an isolated order-$p$ collision of the two active branches.
Hence one may choose local coordinates $(t,z)$ centered at the starting packet-section point of that final phase so that
    \[
    M=\{t=0\},
    \qquad
    A_0=\{z=0\},
    \qquad
    C_0=\{z=t^{u+p}\}.
    \]
After the first $u$ packet blowups, the reduced local fiber is $M-G_1-\cdots-G_u$, and in suitable local coordinates
the strict transforms of the two active horizontal branches are $A_u=\{w_u=0\}$ and $C_u=\{w_u=t^p\}$.
Applying Lemma~\ref{lem:routeC-pair-collision-order-p} to the pair $A_u,C_u$ adds the order-$p$ pair-collision chain
$F_1-\cdots-F_p$, with $F_k^2=-2$ for $1\le k<p$ and $F_p^2=-1$, and the strict transforms of $A_u$ and $C_u$ meeting
$F_p$ transversely at distinct points.
Set $L_0:=F_p$.

We next show that no later packet-section center can occur after the last collision $q_c$.
Suppose otherwise.
Because $q_c$ is the last collision center on the chosen leaf, every later horizontal center would be a packet-section
point of one of the two horizontal branches $H_i,H_j$.
Take the first such later packet-section center and follow the maximal packet-section staircase issued from it.
Since no later collision exists, this staircase is terminal on the chosen leaf.
By the same local calculation as in Lemma~\ref{lem:routeC-packet-section-staircase}, the last exceptional curve of that
terminal staircase survives as a singly-hit multiplicity-one nonexceptional tip with unique vertical neighbor in $D$.
Proposition~\ref{prop:routeC-tip-entry-sufficient} then gives Route~C entry, contrary to the standing hypothesis.
Hence there is no later packet-section center after $q_c$.

Therefore every later blowup center on the chosen leaf is purely vertical and is disjoint from the two horizontal
contact points of $L_0$.
If no later center lies on $L_0$, then the final transform of $L_0$ survives on $\widetilde X$ as a curve
$L\subset \widetilde X\setminus D$ meeting $H_i$ and $H_j$ once each.
The local description above shows that $L\cdot(D-H_i-H_j)=0$ in the primitive transverse case $(u,p)=(0,1)$, while in
every other case exactly one vertical component of the local block meets $L$, namely $F_{p-1}$ when $p\ge 2$ and
$G_u$ when $u\ge 1$ and $p=1$.
This is alternative~\textnormal{(i)}.

Assume now that some later blowup center on the chosen leaf does lie on $L_0$, and let $r$ be the first such later
center.
By the previous paragraph, $r$ is purely vertical and is away from the two horizontal contact points of $L_0$.
After blowing up $r$, the strict transform of $L_0$ has self-intersection at most $-2$.
Therefore its final transform cannot remain a vertical component outside $D$, because
\cite[Lemma~2.8(b)]{PP12} says that every vertical component of a degenerate fiber not contained in $D$ is a
$(-1)$-curve.
Thus the final transform
$E\subset D$
is an exceptional component on $\widetilde X$.
Because the blowup at $r$ is away from the two horizontal contact points, the final transform $E$ still meets the strict
transforms of $H_i$ and $H_j$.
It also meets the exceptional curve created over $r$, and therefore has valence at least $3$ in $D$.
So $E$ is branching.
This is alternative~\textnormal{(ii)}.
\end{proof}

\begin{proposition}[Every realized binary packet output is already old or impossible]
\label{prop:routeC-height-selection-binary-packet-output-old}
Keep the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}.
Let $L\subset \widetilde X\setminus D$ be a surviving nonexceptional $(-1)$-curve arising from a binary packet output on
packet sections $H_i,H_j\subset D_{\mathrm{hor}}$.
Then the configuration is already absorbed by previously isolated Route~C geometry or is impossible.
In particular, the finite defect set of non-strong packet sections contributes no independent realized binary output.
\end{proposition}

\begin{proof}
By Corollary~\ref{cor:routeC-scalar-packet-rootpair-controlled}, a realized binary packet output comes either from a
binary scalar root or from an isolated pair-collision service block.
Apply Proposition~\ref{prop:routeC-height-selection-binary-packet-local} to the terminal leaf supporting $L$.
Since $L$ is a surviving nonexceptional curve, alternative~\textnormal{(ii)} of that proposition does not occur.
Hence $L$ is of exactly one of the following two types:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item a primitive contact-$2$ curve meeting $H_i$ and $H_j$ transversely once each and no other component of $D$;
    \item a primitive fork center meeting $H_i$ and $H_j$ transversely once each and one further component
    $A\subset D$.
\end{enumerate}

If at least one of $H_i,H_j$ is strong in the sense of Definition~\ref{def:routeC-strong-horizontal}, then by
definition that horizontal arm already lies in the old branching / touched-$[2]$ corridor.
So assume from now on that neither $H_i$ nor $H_j$ is strong.
Then each of $H_i,H_j$ is mild or ordinary.

Assume first that $L$ is a fork output.
If both $H_i$ and $H_j$ are ordinary, then Lemma~\ref{lem:routeC-mild-ordinary-cf} gives
$\operatorname{cf}_X(H_i)\ge \frac12$ and $\operatorname{cf}_X(H_j)\ge \frac12$.
Applying Proposition~\ref{prop:routeC-contact-defect-sieve} to $L$ gives
$\operatorname{cf}_X(H_i)+\operatorname{cf}_X(H_j)+\operatorname{cf}_X(A)<1$, a contradiction.
So in the fork case at least one arm is mild.
A mild arm is special in the sense of Definition~\ref{def:routeC-global-special-horizontal} and is not of touched-$[2]$
type.
Relative to that arm one has $m=1$ and $L\cdot(D-H)=2$, so
Proposition~\ref{prop:routeC-special-horizontal-extra-contact-oldcorridor} applies and shows that the fork is already
old geometry.

Assume now that $L$ is a primitive contact-$2$ output.
If both $H_i$ and $H_j$ are ordinary, then Lemma~\ref{lem:routeC-mild-ordinary-cf} again gives
$\operatorname{cf}_X(H_i)\ge \frac12$ and $\operatorname{cf}_X(H_j)\ge \frac12$.
Applying Proposition~\ref{prop:routeC-contact-defect-sieve} to $L$ now gives
$\operatorname{cf}_X(H_i)+\operatorname{cf}_X(H_j)<1$, again a contradiction.
So in the contact-$2$ case at least one arm is mild.
If the connected component of $D$ containing that mild arm is not a chain, then one is already on the branching side.
If it is a chain, then the mild arm is special and not of touched-$[2]$ type, so after $2$-tail reduction its reduced
touched core is $[3]$.
Thus Corollary~\ref{cor:routeC-special-tip-side} places the resulting bridge in the old special-tip corridor.
\end{proof}

\begin{corollary}[Every realized binary packet descendant is already old or impossible]
\label{cor:routeC-height-selection-binary-packet-descendant-old}
Assume the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}, and assume moreover that Route~C entry
does not already occur.
Let $\mathcal T$ be a realized binary packet descendant supported on exactly two packet sections
$H_i,H_j\subset D_{\mathrm{hor}}$.
Then $\mathcal T$ is already absorbed by previously isolated Route~C geometry or is impossible.
In particular, the finite defect set of non-strong packet sections contributes no independent realized binary packet
output.
\end{corollary}

\begin{proof}
Choose a terminal leaf of the blowup tree of $\mathcal T$ and apply
Proposition~\ref{prop:routeC-height-selection-binary-packet-local}.
If alternative~\textnormal{(i)} holds, then the corresponding surviving nonexceptional curve
$L\subset \widetilde X\setminus D$ is already absorbed or excluded by
Proposition~\ref{prop:routeC-height-selection-binary-packet-output-old}.
If alternative~\textnormal{(ii)} holds, then the final transform $E\subset D$ is a vertical branching component meeting
both $H_i$ and $H_j$, so it is already old vertical Route~C geometry.
Thus no realized binary packet descendant contributes an independent residual packet-side output.
\end{proof}

\begin{theorem}[The defect-supported packet sector contributes no independent residual endpoint]
\label{thm:routeC-height-selection-packet-defect-eliminated}
Assume the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}.
Then the several-special-horizontal packet branch admits the following further reduction.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item If a later blowup center hits a packet-section point on a primitive scalar hub, then
    Proposition~\ref{prop:routeC-height-selection-scalar-packetpoint-entry} gives direct Route~C entry.
    \item Every remaining scalar mechanism is already absorbed by
    Theorem~\ref{thm:routeC-height-selection-scalar-root-absorbed}.
    \item Every realized binary packet descendant is already old geometry or impossible, by
    Corollary~\ref{cor:routeC-height-selection-binary-packet-descendant-old}.
\end{enumerate}
Consequently the packet side no longer has a separate defect-supported binary endpoint.
Equivalently, the defect set of size at most $5$ is harmless: it contributes no independent packet-side residue.
\end{theorem}

\begin{proof}
If Route~C entry already occurs, there is nothing to prove, so assume from now on that it does not.
By Theorem~\ref{thm:routeC-height-selection-scalar-root-absorbed}, after peeling the common scalar core the only
packet-side outputs not already absorbed by the scalar analysis are binary pair-collision blocks.
If a later blowup center hits a packet-section point on a primitive scalar hub, then
Proposition~\ref{prop:routeC-height-selection-scalar-packetpoint-entry} gives item~\textnormal{(1)}.
The remaining scalar mechanisms are handled by
Theorem~\ref{thm:routeC-height-selection-scalar-root-absorbed}, giving item~\textnormal{(2)}.
Every realized binary packet descendant is already old geometry or impossible by
Corollary~\ref{cor:routeC-height-selection-binary-packet-descendant-old}, which gives item~\textnormal{(3)}.
Thus the defect set no longer contributes a separate packet-side residue.
\end{proof}

\begin{theorem}[The several-special-horizontal packet side has no new local corridor]
\label{thm:routeC-height-selection-packet-no-new-local}
Assume the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}.
After removing the defect set and peeling the common scalar core, every packet-side contribution is already one of the
following:
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item direct Route~C entry;
    \item old geometry already absorbed by the previously isolated horizontal or vertical Route~C corridors;
    \item an impossible configuration;
    \item the genuinely global realization/survival problem for a binary pair output on the residual strong packet.
\end{enumerate}
Equivalently, the several-special-horizontal packet branch contributes no genuinely new packet-specific local corridor.
\end{theorem}

\begin{proof}
The scalar mechanisms are absorbed by
Theorem~\ref{thm:routeC-height-selection-scalar-root-absorbed}.
The defect-supported residue is eliminated by
Theorem~\ref{thm:routeC-height-selection-packet-defect-eliminated}.
So the only unresolved packet-side content is the global extraction/survival step for a binary pair output on the
residual strong packet.
\end{proof}

\paragraph*{Closing the residual strong packet.}

\begin{lemma}[After removing terminal unary staircases, the last essential binary block on a packet branch is either surviving or already branching]
\label{lem:routeC-height-selection-strong-packet-terminal-leaf}
Assume we are in the residual strong-packet regime of
Theorem~\ref{thm:routeC-height-selection-packet-no-new-local}.
Choose a packet-side branch whose remaining local tree is binary in the sense of
Corollary~\ref{cor:routeC-height-selection-binary-unary}.
Assume moreover that terminal packet-section staircases on this branch have already been discarded as direct Route~C
entry outputs via Proposition~\ref{prop:routeC-tip-entry-sufficient}.
Then there exist two packet sections $H_i,H_j\subset D_{\mathrm{hor}}$ and integers $u\ge 0$, $p\ge 1$ such that, on
the chosen branch, the last isolated pair collision of horizontal strict transforms has local model
$M=\{t=0\}$, $A_0=\{z=0\}$, and $C_0=\{z=t^{u+p}\}$, where the first $u$ blowups are packet-section blowups on $A_0$
and the next $p$ blowups are the forced order-$p$ collision blowups of $A_0$ and $C_0$.
At the end of this last horizontal block the reduced local fiber is
$M-G_1-\cdots-G_u-F_1-\cdots-F_p$, and the strict transforms of $A_0$ and $C_0$ meet $F_p$ transversely at distinct
points.
Their strict transforms on $\widetilde X$ are exactly $H_i$ and $H_j$.
Moreover exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\arabic*)}]
    \item No later center lies on $F_p$.
    Then $L:=F_p$ survives on $\widetilde X$ as a nonexceptional curve with
    $L\cdot H_i=L\cdot H_j=1$ and
    \[
    L\cdot(D-H_i-H_j)=
    \begin{cases}
    0,& (u,p)=(0,1),\\
    1,& \text{otherwise}.
    \end{cases}
    \]
    In particular, $L$ is either a surviving primitive contact-$2$ curve on the two packet sections or a surviving fork
    center with those two packet sections as its horizontal arms.
    \item Some later center lies on $F_p$.
    Then every later center on the transform of $F_p$ is purely vertical, the final transform $U\subset D$ of $F_p$ is
    a vertical branching component, and $U$ still meets $H_i$ and $H_j$.
\end{enumerate}
\end{lemma}

\begin{proof}
Start with the chosen packet-side branch.
Whenever the current tail starts at a packet-section point, apply
Proposition~\ref{prop:routeC-packet-section-branches-collide-or-enter} to that tail.
Because direct Route~C entry has been excluded in the present residual regime, that proposition shows that after
stripping finitely many maximal initial packet-section staircases the next essential horizontal event on the tail is a
collision point.
Repeating this along the chosen branch, and using the finiteness of the blowup tree, one reaches a last horizontal
collision on the branch.
Since the remaining local tree is binary in the sense of
Corollary~\ref{cor:routeC-height-selection-binary-unary}, that last collision involves exactly two packet sections,
say $H_i$ and $H_j$.

Let $u\ge 0$ be the length of the maximal packet-section staircase immediately preceding the last isolated pair
collision on this branch, and let $p\ge 1$ be the order of that collision.
Choose local coordinates $(t,z)$ at the start of this final packet-service phase so that
$M=\{t=0\}$, $A_0=\{z=0\}$, and $C_0=\{z=t^{u+p}\}$.
After the first $u$ packet blowups one reaches local coordinates with
$A_u=\{w_u=0\}$ and $C_u=\{w_u=t^p\}$, and the reduced local fiber is $M-G_1-\cdots-G_u$.
Applying Lemma~\ref{lem:routeC-pair-collision-order-p} to the pair $A_u,C_u$ then adds the chain
$F_1-\cdots-F_p$, with $F_i^2=-2$ for $1\le i<p$ and $F_p^2=-1$.
Thus at the end of the last horizontal block the reduced local fiber is exactly
$M-G_1-\cdots-G_u-F_1-\cdots-F_p$, and the strict transforms of $A_0$ and $C_0$ meet $F_p$ transversely at distinct
points.
Their final strict transforms on $\widetilde X$ are precisely $H_i$ and $H_j$.

Suppose first that no later center lies on $F_p$.
Then $L:=F_p$ survives on $\widetilde X$ as a nonexceptional curve, and the two horizontal branches meeting $L$ are
exactly $H_i$ and $H_j$.
If $(u,p)=(0,1)$, then $L$ is the primitive last exceptional of a transverse isolated pair collision, so
$L\cdot(D-H_i-H_j)=0$.
In every other case, the local chain preceding $L$ contains exactly one adjacent vertical component of $D$, namely
$F_{p-1}$ when $p\ge 2$ or $G_u$ when $u\ge 1$ and $p=1$.
No other component of $D$ meets $L$, so $L\cdot(D-H_i-H_j)=1$.
This proves alternative~\textnormal{(1)}.

Assume now that some later center lies on $F_p$.
We claim that no later center can lie at either of the two packet-section points where the current transforms of
$H_i$ and $H_j$ meet the current transform of $F_p$.
Indeed, if the first such later center existed, then the tail starting at that point would be a packet-section-rooted
branch with no later collision, because the chosen collision is the last one on the branch.
Applying Proposition~\ref{prop:routeC-packet-section-branches-collide-or-enter} to that tail would then force direct
Route~C entry, contrary to the residual no-entry hypothesis.
So every later center on the transform of $F_p$ is purely vertical.
Let $q$ be the first such center.
Blowing up $q$ lowers the self-intersection of the current transform of $F_p$ from $-1$ to $-2$.
Therefore the final transform $U$ of $F_p$ cannot remain a vertical component outside $D$, because
\cite[Lemma~2.8(b)]{PP12} says that every vertical component of a degenerate fiber not contained in $D$ is a $(-1)$-curve.
Thus $U\subset D$.
The two horizontal intersection points were never blown up, so $U$ still meets $H_i$ and $H_j$.
Moreover the first later blowup on the transform of $F_p$ creates a new vertical exceptional curve meeting the new
transform of $F_p$, and after any further blowups centered on the current transform, the new transform still has at
least one vertical neighbor.
So $U$ is a vertical branching component of $D$.
\end{proof}

\begin{proposition}[The last essential binary block on strong packet sections is already old]
\label{prop:routeC-height-selection-strong-packet-leaf-old}
Keep the setup of Lemma~\ref{lem:routeC-height-selection-strong-packet-terminal-leaf}, and assume that $H_i$ and $H_j$
are strong packet sections.
Then the last essential binary block already lands in previously isolated Route~C geometry.
More precisely:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item in alternative~\textnormal{(1)} of the lemma, the curve $L$ is either a surviving primitive contact-$2$
    bridge or a surviving fork center on the strong sections $H_i$ and $H_j$, so it is already absorbed by the old
    branching / touched-$[2]$ corridor;
    \item in alternative~\textnormal{(2)} of the lemma, the final transform $U\subset D$ is a vertical branching
    component meeting $H_i$ and $H_j$, so it is already old vertical geometry.
\end{enumerate}
\end{proposition}

\begin{proof}
In alternative~\textnormal{(1)}, the two subcases are exactly the intersection patterns furnished by
Lemma~\ref{lem:routeC-height-selection-strong-packet-terminal-leaf}.
If $(u,p)=(0,1)$, then $L$ is a surviving primitive contact-$2$ bridge on the strong packet sections $H_i$ and $H_j$.
If $(u,p)\neq(0,1)$, then $L$ has one additional vertical neighbor in $D$, so the terminal local core is a fork with
horizontal arms $H_i$ and $H_j$.
Because $H_i$ and $H_j$ are strong in the sense of Definition~\ref{def:routeC-strong-horizontal}, each already lies on
one of the old horizontal side classes.
So alternative~\textnormal{(1)} is already old horizontal geometry.

Alternative~\textnormal{(2)} is exactly old vertical branching geometry inside $D$.
\end{proof}

\begin{theorem}[The several-special-horizontal packet side contributes no separate residual gap]
\label{thm:routeC-height-selection-packet-side-closed}
Assume the setup of Corollary~\ref{cor:routeC-height-selection-packet-binary}.
Then the several-special-horizontal packet branch contributes no separate unresolved packet problem.
Equivalently, the packet side is closed as an independent branch of the height-selection analysis.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:routeC-height-selection-packet-no-new-local}, the only unresolved packet-side content is the global
realization/survival problem for a binary pair output on the residual strong packet.
If no residual strong packet section remains after removing the defect set, then
Theorem~\ref{thm:routeC-height-selection-packet-defect-eliminated} has already closed the packet side.
So assume the residual strong packet is nonempty.
By Corollary~\ref{cor:routeC-scalar-packet-two-service}, every remaining strong packet section still carries at least two
later blowup centers, and by Corollary~\ref{cor:routeC-height-selection-binary-unary} every remaining local tree is
binary after discarding terminal unary staircases that already give direct entry.
Choose a packet-side leaf branch in this residual strong packet.
Lemma~\ref{lem:routeC-height-selection-strong-packet-terminal-leaf} applies to its last essential binary block.
If the lemma yields alternative~\textnormal{(1)}, then
Proposition~\ref{prop:routeC-height-selection-strong-packet-leaf-old} absorbs the surviving output into the old
branching / touched-$[2]$ corridor.
If it yields alternative~\textnormal{(2)}, the same proposition shows that the final transform is already old vertical
branching geometry.
Hence no separate packet-specific residual branch remains.
\end{proof}

\begin{remark}[What remains after closing the packet side]
\label{rem:routeC-height-selection-packet-state}
Proposition~\ref{prop:routeC-packet-preparation-negative-section} and
Corollary~\ref{cor:routeC-highheight-packet-scalar} already show that, after preparation, every three-fiber
section-only packet of height at least $6$ is forced into the scalar branch.
Corollary~\ref{cor:routeC-sectiononly-five-defect-packet} shows that, after choosing a clean section, all but at most
$5$ packet members are strong.
Proposition~\ref{prop:routeC-scalar-packet-peels-to-disjoint} removes the whole common scalar core and replaces it by a
disjoint packet on $\mathbb F_0$, while Corollary~\ref{cor:routeC-scalar-packet-two-service} shows that every packet
branch still needs at least two later service centers.
Corollary~\ref{cor:routeC-scalar-packet-complementary-controlled} and
Corollary~\ref{cor:routeC-scalar-packet-rootpair-controlled} reduce every surviving smooth-fiber branch to either a
surviving scalar root or an isolated pair-collision block, and
Corollary~\ref{cor:routeC-height-selection-packet-binary} removes the remaining many-branch ambiguity on the residual
strong packet.
Proposition~\ref{prop:routeC-no-five-horizontal-hub} bounds every continuing complementary scalar packet by the same
five-defect budget.
Theorem~\ref{thm:routeC-height-selection-scalar-root-absorbed} removes the marked-scalar and surviving-scalar-root
residue, Theorem~\ref{thm:routeC-height-selection-packet-defect-eliminated} removes the defect-supported binary residue,
and Theorem~\ref{thm:routeC-height-selection-packet-side-closed} closes the remaining strong binary packet regime.
So the packet side is now closed as an independent branch of the height-selection analysis.
\end{remark}

\subsection{The special-multisection branch}

The special-multisection side now breaks into three concrete pieces:
the smooth-image separable branch,
the one-section singular-image separable branch,
and the purely inseparable ordinary singleton $[2]$-root branch.
The next results first show that any ramification or singular-image output with the expected extra contact is already
old geometry, then collapse the smooth-image side to an explicit low-degree corridor, close the one-section
singular-image side, and finally remove the purely inseparable $[2]$-root branch altogether.

\begin{lemma}[A special horizontal three-arm contributes at least one third]
\label{lem:routeC-special-horizontal-three-third}
Assume the setup of Definition~\ref{def:routeC-global-special-horizontal}.  Let $H\subset D_{\mathrm{hor}}$ be a
horizontal component such that the connected component of $D$ containing $H$ is a chain and, after $2$-tail reduction
relative to $H$, the reduced touched core is the singleton chain $[3]$.  Then $\operatorname{cf}_X(H)\ge \frac13$.
\end{lemma}

\begin{proof}
Let $(C^\sharp,H^\sharp)$ be the reduced touched pair obtained from the chain component $C$ containing $H$.
Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu} gives $\operatorname{cf}_X(H)=1-\mu(C,H)$, while
Lemma~\ref{lem:routeC-mu-monotone-tail} gives $\mu(C,H)\le \mu(C^\sharp,H^\sharp)$.  Since
$(C^\sharp,H^\sharp)=([3],\text{its unique component})$, one has $\mu(C^\sharp,H^\sharp)=\frac23$, hence
$\operatorname{cf}_X(H)\ge 1-\frac23=\frac13$.
\end{proof}

\begin{proposition}[A non-touched-two special horizontal arm with enough extra contact is already old]
\label{prop:routeC-special-horizontal-extra-contact-oldcorridor}
Let $L\subset \widetilde X$ be a $(-1)$-curve with $L\not\subset D$.  Assume that for some horizontal component
$H\subset D_{\mathrm{hor}}$ one has $m:=L\cdot H\in\{1,2\}$ and $L\cdot(D-H)\ge 3-m$.  Assume moreover that $H$ is
special in the sense of Definition~\ref{def:routeC-global-special-horizontal}, and that $H$ is not of touched-$[2]$
type.
Then one already falls into one of the previously isolated old Route~C side types:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item a branching side; or
    \item a chain side whose reduced touched core is a touched tip of weight $2$.
\end{enumerate}
\end{proposition}

\begin{proof}
If the connected component of $D$ containing $H$ is not a chain, then one is already on the branching side.
So assume it is a chain.  Because $H$ is special but not of touched-$[2]$ type,
Definition~\ref{def:routeC-global-special-horizontal} says that after $2$-tail reduction relative to $H$ the reduced
touched core is $[3]$.  Hence Lemma~\ref{lem:routeC-special-horizontal-three-third} gives
$\operatorname{cf}_X(H)\ge \frac13$.

Apply Proposition~\ref{prop:routeC-contact-defect-sieve} to $L$.  One gets
$m\,\operatorname{cf}_X(H)+\sum_{E\subset D-H}(L\cdot E)\,\operatorname{cf}_X(E)<1$, so
$\sum_{E\subset D-H}(L\cdot E)\,\operatorname{cf}_X(E)<1-\frac m3$.
If every contacted component $E\subset D-H$ satisfied $\operatorname{cf}_X(E)\ge \frac13$, then the left-hand side
would be at least $\frac{L\cdot(D-H)}{3}\ge \frac{3-m}{3}=1-\frac m3$, a contradiction.
Therefore some contacted component $E\subset D-H$ has $\operatorname{cf}_X(E)<\frac13$.
If the connected component of $D$ containing $E$ is not a chain, then one is on the branching side.
If it is a chain, Lemma~\ref{lem:routeC-chain-cf-third-forces-2tip} shows that the reduced touched core relative to
$E$ is a touched tip of weight $2$.
\end{proof}

\begin{corollary}[The ramification-fork frontier for separable special multisections is already old]
\label{cor:routeC-special-multisection-ramification-frontier-old}
Keep the setup of Corollary~\ref{cor:routeC-final-lemma-ramification-finite-bad}, and assume that the horizontal
multisection $H\subset D_{\mathrm{hor}}$ is special.
Then every ramification point $q\in S\setminus B_\nu$ of local degree at least $2$ produces a surviving primitive
ramification fork that is already absorbed by the old Route~C geometry.
Equivalently, the smooth-image ramification side of the separable special-multisection branch contributes no new local
horizontal corridor.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-final-lemma-ramification-finite-bad}, the primitive ramification fork over $q$ survives on
$\widetilde X$.  Its last exceptional curve $L$ meets $H$ once and has two units of additional vertical contact, so
$L\cdot H=1$ and $L\cdot(D-H)=2$.
If $H$ is of touched-$[2]$ type, or if the connected component of $D$ containing $H$ is not a chain, then the
horizontal side is already old by definition.  Otherwise
Proposition~\ref{prop:routeC-special-horizontal-extra-contact-oldcorridor} applies with $m=1$ and gives the same
conclusion.
\end{proof}

\begin{corollary}[The separable special-multisection branch adds no new isolated local fork corridor]
\label{cor:routeC-special-multisection-no-new-local-gap}
Let $H\subset D_{\mathrm{hor}}$ be a special horizontal multisection, and let
$L\subset \widetilde X\setminus D$ be a surviving nonexceptional $(-1)$-curve arising from an isolated local block over
the ruled-model image of $H$.  Assume that $L\cdot H\in\{1,2\}$.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $L\cdot(D-H)\ge 3-L\cdot H$, then the configuration is already absorbed by the old branching /
    touched-$[2]$ geometry;
    \item the only isolated local output not immediately reabsorbed is the pure same-component contact-$2$ case
    $L\cdot H=2$ and $L\cdot(D-H)=0$.
    In that remaining case one has $\operatorname{cf}_X(H)<\frac12$, so $H$ cannot be ordinary.
    Hence, unless the connected chain component containing $H$ has reduced touched core $[3]$, the horizontal side is
    already old.
\end{enumerate}
Equivalently, the separable special-multisection branch contributes no new isolated local fork corridor.
The only still-distinctive isolated local output is the pre-existing same-component contact-$2$ case on a mild special
multisection arm.
\end{corollary}

\begin{proof}
If $L\cdot H=1$, then \textnormal{(i)} follows from
Proposition~\ref{prop:routeC-special-horizontal-extra-contact-oldcorridor} when $H$ is not already of touched-$[2]$ or
branching type, and is tautological otherwise.
The same argument applies when $L\cdot H=2$ and $L\cdot(D-H)\ge 1$.  So only the pure same-component contact-$2$ case
can remain.

In that case Proposition~\ref{prop:routeC-contact-defect-sieve} gives $2\,\operatorname{cf}_X(H)<1$, hence
$\operatorname{cf}_X(H)<\frac12$.  By Lemma~\ref{lem:routeC-global-ordinary-horizontal-half}, an ordinary horizontal
component satisfies $\operatorname{cf}_X(H)\ge \frac12$, so $H$ cannot be ordinary.
Thus $H$ is special.  If its connected component is not a chain, or if it is a chain whose reduced touched core is a
touched tip of weight $2$, then the horizontal side is already old.  The only nonold possibility is therefore the chain
case whose reduced touched core is $[3]$.
\end{proof}

\begin{proposition}[The mild same-component residue on a special multisection is already old]
\label{prop:routeC-special-multisection-mild-contact2-old}
Keep the setup of Corollary~\ref{cor:routeC-special-multisection-no-new-local-gap}.  Assume that
$L\cdot H=2$, $L\cdot(D-H)=0$, and that the connected component of $D$ containing $H$ is a chain whose $2$-tail
reduction relative to $H$ is the singleton chain $[3]$.
Then this same-component contact-$2$ configuration is already locally sandwiched of $\delta=1$, hence root-contractible,
and therefore it gives a descendant with elliptic boundary by Theorem~\ref{thm:routeC-samecomponent-allties}.
In particular, it cannot occur on a residual counterexample.
\end{proposition}

\begin{proof}
Because the reduced touched core relative to $H$ is $[3]$, the full connected chain component containing $H$ has the
form $A=[(2)^u,3,(2)^v]$ for some integers $u,v\ge 0$, with $H$ corresponding to the unique $(-3)$-component.
By Corollary~\ref{cor:routeC-special-multisection-no-new-local-gap}, one already knows that
$\operatorname{cf}_X(H)<\frac12$.
On the other hand, Lemma~\ref{lem:routeC-chain-discrepancy-equals-mu} gives
$\mu_X(H)=\frac{d([2^u])+d([2^v])}{d(A)}$.
Since $d([2^u])=u+1$, $d([2^v])=v+1$, and $d(A)=uv+2u+2v+3$, one obtains
$\operatorname{cf}_X(H)=1-\mu_X(H)=\frac{(u+1)(v+1)}{uv+2u+2v+3}$.
The inequality $\operatorname{cf}_X(H)<\frac12$ therefore forces $uv<1$, so $uv=0$.
Up to reversing the chain, one may write $A=[3,(2)^m]$ for some $m\ge 0$.

Let $q$ be the singular point of the ruled-model image of $H$ whose isolated local block produces the surviving curve
$L$.  Because this block comes from resolving one curve singularity on a smooth ruled surface, there is a birational
morphism from a neighborhood of the divisor $E:=A+L$ on $\widetilde X$ to a smooth surface germ, contracting every
component of $E-H$ and mapping $H$ birationally onto the corresponding local branch of the ruled-model image.
By Proposition~\ref{prop:routeC-local-package}\textnormal{(1)}, one has $p_a(E)=1$.
Hence Lemma~\ref{lem:routeC-local-tie-equiv} shows that $E$ is locally sandwiched of $\delta=1$, equivalently that the
rooted graph on $A$ with double contact on $H$ is root-contractible.

Now Proposition~\ref{prop:routeC-a2-tail}\textnormal{(3)} says that on a chain $[3,(2)^m]$ a double-contact pattern is
root-contractible only in the one-vertex case $[3]^{|1,1|}$, that is, only when $m=0$.
So in fact $A=[3]$.  This rooted graph is root-contractible, again by
Proposition~\ref{prop:routeC-a2-tail}\textnormal{(3)}, and Theorem~\ref{thm:routeC-samecomponent-allties} therefore
yields a descendant with elliptic boundary.
\end{proof}

\begin{corollary}[The separable special-multisection branch contributes no new isolated local residue]
\label{cor:routeC-special-multisection-fully-old}
Keep the setup of Corollary~\ref{cor:routeC-special-multisection-no-new-local-gap}.
Then every isolated local output on the separable special-multisection side is already absorbed by the old Route~C
geometry or by the earlier same-component tie program.
Equivalently, the separable special-multisection branch contributes no new isolated local corridor at all.
\end{corollary}

\begin{proof}
Corollary~\ref{cor:routeC-special-multisection-no-new-local-gap} already shows that every isolated local output is old
unless one is in the mild pure same-component contact-$2$ case.  That remaining case is excluded by
Proposition~\ref{prop:routeC-special-multisection-mild-contact2-old}.
\end{proof}

\begin{proposition}[Extraction criteria on the separable special-multisection side]
\label{prop:routeC-special-multisection-global-extraction}
Let $H\subset D_{\mathrm{hor}}$ be a special horizontal multisection, and let
$\nu\colon \widetilde X\to Y$ be a birational morphism onto a smooth relatively minimal ruled surface, chosen so that
$\nu$ is an isomorphism at the generic point of $H$.  Put $S:=\nu(H)$.
Assume that one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item There exists a ramification point $q\in S\setminus B_\nu$ of local degree at least $2$, where $B_\nu$ is the
    finite bad set from Corollary~\ref{cor:routeC-final-lemma-ramification-finite-bad}.
    \item There exists a surviving nonexceptional $(-1)$-curve $L\subset \widetilde X\setminus D$ arising from an
    isolated local block over a singular point of $S$ and satisfying $L\cdot H\in\{1,2\}$.
\end{enumerate}
Then the separable special-multisection branch already lands in previously treated geometry: in case
\textnormal{(i)} in the old branching / touched-$[2]$ corridor, and in case \textnormal{(ii)} in the old branching /
touched-$[2]$ corridor or in the already-closed same-component tie program.
In particular, the unresolved content of the separable special-multisection branch is only the global extraction problem
of producing one of the outputs \textnormal{(i)} or \textnormal{(ii)}.
\end{proposition}

\begin{proof}
In case \textnormal{(i)}, Corollary~\ref{cor:routeC-special-multisection-ramification-frontier-old} applies directly and
shows that the surviving primitive ramification fork is already old Route~C geometry.
In case \textnormal{(ii)}, Corollary~\ref{cor:routeC-special-multisection-fully-old} applies and shows that every such
isolated local output is already absorbed by the old branching / touched-$[2]$ geometry or by the same-component tie
program.
\end{proof}

\begin{proposition}[Smooth-image special multisections leave only a ramification extraction problem]
\label{prop:routeC-special-multisection-smooth-image-extraction}
Keep the setup of Proposition~\ref{prop:routeC-special-multisection-global-extraction}, and assume in addition that
$S=\nu(H)$ is smooth on the ruled model.
Then the only unresolved content of the separable special-multisection branch is the existence of a ramification point
$q\in S\setminus B_\nu$ of local degree at least $2$.
Any such point already lands in old Route~C geometry by
Corollary~\ref{cor:routeC-special-multisection-ramification-frontier-old}.
\end{proposition}

\begin{proof}
If $S$ is smooth, then there is no singular point of the ruled-model image, so alternative \textnormal{(ii)} of
Proposition~\ref{prop:routeC-special-multisection-global-extraction} cannot occur.
Therefore the only possible unresolved output on the separable special-multisection side is the ramification
alternative \textnormal{(i)}.
\end{proof}

\begin{proposition}[The all-bad smooth-image separable special-multisection side is an at-most-two-point threaded \texorpdfstring{$[3]$}{[3]}-root corridor]
\label{prop:routeC-special-multisection-smooth-image-allbad-twopoint}
Keep the setup of Proposition~\ref{prop:routeC-special-multisection-global-extraction}, and assume in addition that
$S=\nu(H)$ is smooth and that the map $S\to \PP^1$ is separable of degree $d:=F\cdot H\ge 2$.
Assume further that the smooth-image branch is not already absorbed by the old branching / touched-$[2]$ corridor, and
that all isolated local outputs on the separable special-multisection side have already been discarded by
Corollary~\ref{cor:routeC-special-multisection-fully-old}.

If every ramification point of $S\to \PP^1$ lies in $B_\nu$, then the remaining smooth-image branch is forced into a
clean threaded $[3]$-root model of degree $d\le 5$ supported on at most two ramification points.
More precisely, if $q_1,\dots,q_r\in S$ are the ramification points and $a_i\ge 2$ is the local degree at $q_i$, then
$r\le 2$.
Hence the residual all-bad smooth-image branch consists of one root chain containing $H$ whose $2$-tail reduction is
$[3]$, together with at most two detached threaded ramification chains; in particular it is supported on at most two
fibers of the ruled model and has multisection degree at most $5$.
\end{proposition}

\begin{proof}
Because $S$ is smooth and $\nu|_H\colon H\to S$ is birational, it is an isomorphism. Thus $H\cong S\cong \PP^1$.
Since we have excluded the already old branching / touched-$[2]$ corridor, the connected component of $D$ containing
$H$ is a chain whose $2$-tail reduction relative to $H$ is the singleton chain $[3]$. In particular, $H^2=-3$.
Lemma~\ref{lem:routeC-special-horizontal-three-third} therefore gives $\operatorname{cf}_X(H)\ge \frac13$.
Applying Lemma~\ref{lem:routeC-fiber-anticanonical-cf} to a general fiber and discarding all horizontal contributions
except that of $H$ gives $2>d\,\operatorname{cf}_X(H)\ge \frac d3$, so $d<6$ and hence $d\le 5$.

Now regard $S$ as a smooth rational multisection of degree $d$ on the relatively minimal ruled surface $Y$.
The adjunction computation from the proof of Proposition~\ref{prop:routeC-root-two-clean-threaded} gives $S^2=2d$.
Each blowup centered on the current strict transform of $S$ lowers its self-intersection by $1$.
Therefore, passing from $S^2=2d$ to $H^2=-3$ forces exactly $2d+3$ blowup centers on successive strict transforms of
$S$:
\begin{equation}
\label{eq:routeC-special-multisection-smooth-image-budget}
\#\{\textnormal{centers on successive transforms of }S\}=2d+3.
\end{equation}

Let $q_1,\dots,q_r$ be the ramification points of the separable map $S\to \PP^1$, and let $a_i\ge 2$ be the local
degree at $q_i$.
For each $i$, the first $a_i$ centers of $\nu^{-1}$ over $q_i$ are the forced order-$a_i$ tangency-resolution centers
from Proposition~\ref{prop:routeC-ramification-fork-survival}.
After that forced stage there is a unique current nonexceptional $(-1)$-curve at the $H$-end of the local ramification
branch.

Because every ramification point is assumed to lie in $B_\nu$, there is at least one later center on that current
$(-1)$-curve over every $q_i$.
We claim that, on the still-unresolved branch, every such later center must be the thread center, namely the unique
point where that current $(-1)$-curve meets the current transform of $H$.
Indeed, if some later center on the current $H$-end $(-1)$-curve were chosen away from $H$, then the corresponding
descendant over $q_i$ would split off as an isolated unary local block over the smooth point $q_i\in S$.
But Corollary~\ref{cor:routeC-special-multisection-fully-old} already absorbs every isolated local output on the
separable special-multisection side.
Hence such a nonthreaded continuation is impossible on the residual all-bad branch.
Repeating the same argument after each thread blowup shows that every bad ramification branch is clean threaded at the
$H$-end.

Consequently, for each ramification point $q_i$, one has at least $a_i+1$ centers on successive strict transforms of
$S$: the $a_i$ forced tangency-resolution centers, plus at least one thread center because $q_i\in B_\nu$.
Summing over all ramification points and using \eqref{eq:routeC-special-multisection-smooth-image-budget}, we obtain
\begin{equation}
\label{eq:routeC-special-multisection-smooth-image-allbad-upper}
\sum_{i=1}^r (a_i+1)\le 2d+3.
\end{equation}

On the other hand, Riemann--Hurwitz for the separable map $S\cong \PP^1\to \PP^1$ gives
$-2=-2d+\deg R$, so $\deg R=2d-2$.
If $\delta_i$ denotes the different exponent at $q_i$, then $\delta_i\ge a_i-1$, whence
\begin{equation}
\label{eq:routeC-special-multisection-smooth-image-RH}
\sum_{i=1}^r (a_i-1)\le 2d-2.
\end{equation}
Subtracting \eqref{eq:routeC-special-multisection-smooth-image-RH} from
\eqref{eq:routeC-special-multisection-smooth-image-allbad-upper} yields
$2r\le (2d+3)-(2d-2)=5$, hence $r\le 2$.

Finally, the last paragraph of the proof of Proposition~\ref{prop:routeC-root-two-clean-threaded} is a purely local
statement about a thread blowup: it lengthens the detached ramification chain over that point by one terminal
$(-2)$-curve and does not create a new connected component of $D$.
Thus each ramification point contributes one detached threaded ramification chain, while all nonramification centers on
successive transforms of $S$ lie on the unique root chain containing $H$.
Therefore the remaining all-bad smooth-image branch is exactly an at-most-two-point threaded $[3]$-root corridor,
supported on at most two fibers of the ruled model.
\end{proof}

\begin{corollary}[Smooth-image separable special multisections reduce to good ramification or to an explicit at-most-two-point corridor]
\label{cor:routeC-special-multisection-smooth-image-good-or-twopoint}
Keep the setup of Proposition~\ref{prop:routeC-special-multisection-global-extraction}, and assume that $S=\nu(H)$ is
smooth.
After discarding the already old branching / touched-$[2]$ corridor and the already old isolated local outputs from
Corollary~\ref{cor:routeC-special-multisection-fully-old}, exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item There exists a ramification point $q\in S\setminus B_\nu$ of local degree at least $2$.
    In this case the branch is already old by
    Corollary~\ref{cor:routeC-special-multisection-ramification-frontier-old}.
    \item Every ramification point of $S\to \PP^1$ lies in $B_\nu$, and the remaining smooth-image branch is the
    explicit low-degree at-most-two-point threaded $[3]$-root corridor of
    Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint}.
\end{enumerate}
In particular, the smooth-image separable special-multisection side is no longer a general extraction problem: outside
old geometry, its only residual content is this explicit low-degree at-most-two-point threaded $[3]$-root corridor.
\end{corollary}

\begin{proof}
If the first alternative holds, apply Corollary~\ref{cor:routeC-special-multisection-ramification-frontier-old}.
Otherwise every ramification point lies in $B_\nu$, and
Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint} applies.
\end{proof}

\begin{lemma}[A unique special multisection leaves at most three units of ordinary horizontal degree]
\label{lem:routeC-special-multisection-ordinary-degree-le3}
Assume that a witnessing fibration does not lie in the several-special-horizontal corridor, so that
$H\subset D_{\mathrm{hor}}$ is the unique special horizontal multisection.
Let $d:=F\cdot H\ge 2$, and write
$M:=\sum_{T\subset D_{\mathrm{hor}}-H}(F\cdot T)=\operatorname{ht}(X)-d$.
Then
$d\,\operatorname{cf}_X(H)+\frac12 M<2$.
In particular, $M\le 3$ and hence $d\ge \operatorname{ht}(X)-3$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:routeC-fiber-anticanonical-cf}, one has
$\sum_{T\subset D_{\mathrm{hor}}}\operatorname{cf}_X(T)\,(F\cdot T)<2$.
Split this sum into the unique special component $H$ and the remaining ordinary horizontal components.
For every ordinary component $T\subset D_{\mathrm{hor}}-H$,
Lemma~\ref{lem:routeC-global-ordinary-horizontal-half} gives
$\operatorname{cf}_X(T)\ge \frac12$.
Therefore
$2>d\,\operatorname{cf}_X(H)+\sum_{T\subset D_{\mathrm{hor}}-H}\operatorname{cf}_X(T)\,(F\cdot T)
\ge d\,\operatorname{cf}_X(H)+\frac12 M$.
Since $d\,\operatorname{cf}_X(H)\ge 0$, it follows that $\frac12 M<2$, so $M<4$.
As $M$ is an integer, one gets $M\le 3$.
The formula $d=\operatorname{ht}(X)-M$ then gives $d\ge \operatorname{ht}(X)-3$.
\end{proof}

\begin{proposition}[High-height unique special multisections are horizontal \texorpdfstring{$(-2)$}{(-2)}-curves]
\label{prop:routeC-special-multisection-highheight-minus2}
Keep the setup of Lemma~\ref{lem:routeC-special-multisection-ordinary-degree-le3}, and assume that
$\operatorname{ht}(X)\ge 6$.
Then $H^2=-2$.
\end{proposition}

\begin{proof}
Assume for contradiction that $H^2=-\beta$ with $\beta\ge 3$.
By Lemma~\ref{lem:routeC-cf-selfintersection-lower},
$\operatorname{cf}_X(H)\ge 1-\frac{2}{\beta}\ge \frac13$.
Now Lemma~\ref{lem:routeC-special-multisection-ordinary-degree-le3} gives
$2>d\,\operatorname{cf}_X(H)+\frac12 M\ge \frac d3+\frac M2$.
Using $d=\operatorname{ht}(X)-M$, we obtain
$2>\frac{\operatorname{ht}(X)-M}{3}+\frac M2=\frac{2\operatorname{ht}(X)+M}{6}$.
But $\operatorname{ht}(X)\ge 6$ and $M\ge 0$, so the right-hand side is at least $2$, a contradiction.
Therefore $\beta=2$, that is, $H^2=-2$.
\end{proof}

\begin{corollary}[High-height special-multisection witnesses are already on an old side]
\label{cor:routeC-special-multisection-highheight-oldside}
Keep the assumptions of Proposition~\ref{prop:routeC-special-multisection-highheight-minus2}.
Then exactly one of the following holds.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the connected component of $D$ containing $H$ is not a chain, so one is already on the branching side;
    \item the connected component of $D$ containing $H$ is a chain, and its $2$-tail core relative to $H$ is a touched
    tip of weight $2$.
\end{enumerate}
In particular, outside the several-special-horizontal corridor, the special-multisection branch is no longer an
independent high-height corridor.
\end{corollary}

\begin{proof}
If the connected component of $D$ containing $H$ is not a chain, then by definition one is on the branching side.
Assume it is a chain. Because $H$ is special in the sense of
Definition~\ref{def:routeC-global-special-horizontal}, after $2$-tail reduction the reduced touched core is either a
touched tip of weight $2$ or the singleton chain $[3]$.
But Proposition~\ref{prop:routeC-special-multisection-highheight-minus2} gives $H^2=-2$.
Removing terminal $2$-tails away from the touched component does not change the self-intersection of the touched
component, so the reduced touched core cannot be $[3]$.
Hence the chain case is necessarily a touched tip of weight $2$.
\end{proof}

\begin{corollary}[Only a finite low-height list can support a non-\texorpdfstring{$(-2)$}{(-2)} unique special multisection]
\label{cor:routeC-special-multisection-finite-lowheight-list}
Keep the setup of Lemma~\ref{lem:routeC-special-multisection-ordinary-degree-le3}, and assume that $H^2\ne -2$.
Then necessarily $\operatorname{ht}(X)\le 5$.
More precisely:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $\operatorname{ht}(X)=5$, then $M\le 1$ and $d\in\{4,5\}$, so the only possible horizontal degree
    partitions are $5$ or $4+1$;
    \item if $\operatorname{ht}(X)=4$, then $d\in\{2,3,4\}$, so the only possible horizontal degree partitions are
    $4$, $3+1$, $2+2$, or $2+1+1$.
\end{enumerate}
Equivalently, outside the several-special-horizontal corridor the only possible non-$(-2)$ special-multisection
partitions form the finite list $5$, $4+1$, $4$, $3+1$, $2+2$, $2+1+1$.
\end{corollary}

\begin{proof}
If $H^2\ne -2$, then writing $H^2=-\beta$ one has $\beta\ge 3$.
So Lemma~\ref{lem:routeC-cf-selfintersection-lower} gives $\operatorname{cf}_X(H)\ge \frac13$.
Applying Lemma~\ref{lem:routeC-special-multisection-ordinary-degree-le3} again, we get
$2>d\,\operatorname{cf}_X(H)+\frac12 M \ge \frac d3+\frac M2 = \frac{2\operatorname{ht}(X)+M}{6}$.
Therefore $2\operatorname{ht}(X)+M<12$.
In particular, $\operatorname{ht}(X)\le 5$.
If $\operatorname{ht}(X)=5$, then the last inequality becomes $10+M<12$, so $M\le 1$.
Hence $d=5-M\in\{4,5\}$, and the only possible horizontal degree partitions are $5$ or $4+1$.
If $\operatorname{ht}(X)=4$, then $d=4-M$ with $0\le M\le 2$, so $d\in\{2,3,4\}$.
When $d=4$, one gets the partition $4$; when $d=3$, one gets $3+1$; and when $d=2$, the ordinary remainder has total
degree $2$, so the partition is either $2+2$ or $2+1+1$.
This proves the claim.
\end{proof}

\begin{proposition}[The one-section separable special-multisection residue is finite and explicit]
\label{prop:routeC-special-multisection-one-section-finite-extraction}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and assume that the unique
multisection $H$ is special and separable.  Let $C:=\nu(H)\subset \mathbb F_n$.
Then the unresolved extraction problem on this one-section separable branch has the following form.
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item If there exists a ramification point $q\in C\setminus B_\nu$ of local degree at least $2$, then
    Proposition~\ref{prop:routeC-special-multisection-global-extraction} already closes the branch.
    \item Otherwise one may restrict attention to the singular points of $C$.  Every such singular point is of one of
    the following two explicit types:
    \begin{enumerate}[label=\textnormal{(\alph*)}]
        \item a multibranch collision point, and any isolated local output there is governed by the primitive
        two-branch collision model of Proposition~\ref{prop:routeC-primitive-isolated-pair-collision};
        \item a unibranch point, and any surviving terminal output there is one of the three low-order forks listed in
        Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor}.
    \end{enumerate}
    Moreover every surviving isolated local output over such a singular point is already old Route~C geometry or
    belongs to the already-closed same-component tie program by
    Corollary~\ref{cor:routeC-special-multisection-fully-old}.
\end{enumerate}
Consequently, the one-section separable special-multisection branch is no longer a general multisection extraction
problem: after discarding the smooth-image ramification alternative, its unresolved content is only the survival of one
of finitely many explicit low-degree singular-image local blocks.
\end{proposition}

\begin{proof}
Corollary~\ref{cor:routeC-one-section-height4-partitions-singular} shows that $C$ is a singular rational curve of
degree $2$ or $3$ on $\mathbb F_n$ with $n\ge 2$.
If a ramification point $q\in C\setminus B_\nu$ of local degree at least $2$ exists, then
Proposition~\ref{prop:routeC-special-multisection-global-extraction} closes the branch immediately.
So assume not.

Now every singular point of $C$ is, by Lemma~\ref{lem:routeC-singular-horizontal-branch-or-ramification}, either a
multibranch collision point or a unibranch point of local degree at least $2$.
At an isolated multibranch collision point, Proposition~\ref{prop:routeC-primitive-isolated-pair-collision} gives the
primitive local collision model.
At a unibranch point, Corollary~\ref{cor:routeC-one-section-unibranch-finite-loworder} reduces the terminal endgame to
corner blowups and order-$2/3$ tangency blocks, and
Corollary~\ref{cor:routeC-one-section-unibranch-finite-fork-corridor} identifies any surviving terminal output with one
of exactly three explicit low-order fork types.

Finally, Corollary~\ref{cor:routeC-special-multisection-fully-old} shows that every surviving isolated local output on
the separable special-multisection side is already absorbed by old Route~C geometry or by the same-component tie
program.
\end{proof}

\begin{corollary}[Many-branch singular-image points on the separable special-multisection side add no new endpoint]
\label{cor:routeC-height-selection-special-multisection-binary}
Keep the setup of Proposition~\ref{prop:routeC-special-multisection-global-extraction}, and let $q\in S$ be a point
through which the total horizontal image on the ruled model has at least three local branches.
Then the local factorization over $q$ decomposes into binary descendants and unary descendants in the sense of
Corollary~\ref{cor:routeC-height-selection-binary-unary}.
Consequently:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the many-branch ambiguity itself disappears;
    \item the remaining unary descendants belong to the ramification or unibranch side already isolated in
    Proposition~\ref{prop:routeC-special-multisection-global-extraction};
    \item the remaining binary descendants are pair-collision trees, and in the one-section height-$4$ partitions they
    are among the explicit low-degree local blocks recorded in
    Proposition~\ref{prop:routeC-special-multisection-one-section-finite-extraction}.
\end{enumerate}
In particular, the separable special-multisection branch has no separate many-branch local endpoint.
\end{corollary}

\begin{proof}
Apply Corollary~\ref{cor:routeC-height-selection-binary-unary} to the family of all local horizontal branches through
$q$.
This removes the many-branch ambiguity itself: every descendant block is unary or binary.
A unary descendant is supported on a single horizontal branch, so it lies on the already isolated ramification or
unibranch side from Proposition~\ref{prop:routeC-special-multisection-global-extraction}.
A binary descendant is a pair-collision tree.
In the one-section height-$4$ partitions, the possible surviving local outputs were already reduced in
Proposition~\ref{prop:routeC-special-multisection-one-section-finite-extraction} to an explicit finite low-degree list.
So no separate many-branch endpoint remains.
\end{proof}

\paragraph*{Closing the degree-two one-section branch.}
The direct-cubic closure draft shows that the degree-two one-section branch can be pushed further than the finite local
fork package alone suggests.  The key point is that the singular-point elementary-transform sequence already lifts from
$\widetilde X$, and after that lift the residual conic branch also feeds into the same singular plane-cubic output.

\begin{proposition}[The singular-point transforms of the ruled-model bisection lift from \texorpdfstring{$\widetilde X$}{Xtilde}]
\label{prop:routeC-bisection-singular-eltrans-lifts}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and suppose that $d=2$.
Let $\nu_0:=\nu\colon \widetilde X\to \mathbb F_n$ and $C_0:=\nu_0(H)$.
Suppose that $q_0\in C_0$ is a singular point, and let
$\sigma_0\colon \mathbb F_n\dashrightarrow \mathbb F_{n-1}$ be the elementary transform centered at $q_0$.
Then there exists a birational morphism $\mu_1\colon \widetilde X\to \mathbb F_{n-1}$ such that
$\mu_1=\sigma_0\circ \nu_0$ as rational maps.

More generally, any successive sequence of elementary transforms centered at singular points of the current image of
$H$ lifts inductively to birational morphisms $\mu_r\colon \widetilde X\to \mathbb F_{n-r}$.
\end{proposition}

\begin{proof}
Factor the birational map $\nu_0^{-1}\colon \mathbb F_n\dashrightarrow \widetilde X$ into a sequence of point blowups
away from the negative section $S_0:=\nu_0(H_0)$.
Because $H\subset \widetilde X$ is smooth while its image $C_0$ is singular at $q_0$, at least one blowup center of
this factorization lies over $q_0$.  The first such center must be the point $q_0$ itself.  Hence $\nu_0$ factors
through the blowup $\varepsilon_0\colon \operatorname{Bl}_{q_0}(\mathbb F_n)\to \mathbb F_n$.
Write $\widehat\nu_0\colon \widetilde X\to \operatorname{Bl}_{q_0}(\mathbb F_n)$ for the induced birational morphism.
Let $\beta_0\colon \operatorname{Bl}_{q_0}(\mathbb F_n)\to \mathbb F_{n-1}$ be the contraction of the strict transform
of the fiber through $q_0$.
Then $\mu_1:=\beta_0\circ \widehat\nu_0$ is the required birational morphism.  The inductive statement follows by repeating the same argument at each later
singular point of the current image of $H$.
\end{proof}

\begin{corollary}[The conic-line endpoint of the degree-two branch already gives an elliptic descendant]
\label{cor:routeC-one-section-bisection-conic-line-descendant}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, suppose that $d=2$ and $s=0$,
and perform a lifted singular-point transform sequence of
Proposition~\ref{prop:routeC-bisection-singular-eltrans-lifts} until the image of $H$ is the smooth conic
$Q\equiv 2S_0+2f$ on $\mathbb F_1$. Assume that the final image of the second horizontal section $H_1$ has class
$L\equiv S_0+f$.
Then $X$ has a descendant with elliptic boundary.
\end{corollary}

\begin{proof}
Let $\phi\colon \widetilde X\to \mathbb F_1$ be the lifted morphism.  By construction, $\phi(H)=Q$ and
$\phi(H_1)=L$,
and the image of $H_0$ is the negative section of $\mathbb F_1$.
Let $\tau\colon \mathbb F_1\to \PP^2$ be the contraction of that negative section.
Since $Q\equiv 2S_0+2f$ and $L\equiv S_0+f$, the images $\tau(Q)$ and $\tau(L)$ are respectively a smooth plane conic
and a line.  Moreover, $Q\cdot L=(2S_0+2f)\cdot(S_0+f)=2$, so the reduced divisor
$T:=\tau(Q)+\tau(L)$ is connected and has arithmetic genus $p_a(T)=0+0+2-1=1$.
The only other boundary components on $\widetilde X$ are $H_0$ and vertical components, and under $\tau\circ\phi$ these
map to points.  Hence $T$ is a connected component of the reduced image of the boundary.
By \cite[Definition~2.10, Definition~2.11, Lemma~2.12]{PP12}, this gives a descendant with elliptic boundary.
\end{proof}

\begin{corollary}[Residual exclusion of the conic-line endpoint]
\label{cor:routeC-one-section-bisection-conic-line-not-residual}
In the residual sector $\operatorname{ht}(X)\ge 4$ and $X$ has no descendant with elliptic boundary,
the configuration of Corollary~\ref{cor:routeC-one-section-bisection-conic-line-descendant} cannot occur.
\end{corollary}

\begin{proof}
This is immediate from Corollary~\ref{cor:routeC-one-section-bisection-conic-line-descendant}.
\end{proof}

\begin{proposition}[The residual conic branch birationally reduces to a lifted singular plane-cubic model]
\label{prop:routeC-one-section-conic-reduces-to-cubic-model}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, suppose that $d=2$ and $s=0$,
and perform a lifted singular-point transform sequence of
Proposition~\ref{prop:routeC-bisection-singular-eltrans-lifts} until the image of $H$ is the smooth conic
$Q\equiv 2S_0+2f$ on $\mathbb F_1$.
Let $\phi\colon \widetilde X\to \mathbb F_1$ be the resulting morphism, and write
$B:=\phi(H_1)\equiv S_0+bf$ with $b\ge 2$.
Then there exists a birational morphism $\eta\colon \widetilde X\to \PP^2$ such that
$\eta(H)\subset \PP^2$ is an irreducible singular plane cubic.
\end{proposition}

\begin{proof}
Since $Q\cdot B=(2S_0+2f)\cdot(S_0+bf)=2b\ge 4$,
while distinct irreducible components of the reduced divisor $D$ meet at most once on $\widetilde X$, there exists a
point $q\in Q\cap B$ over which $\phi^{-1}$ is not a local isomorphism.
Because $Q\cdot S_0=0$, one has $q\notin S_0$. Blow up $q$
and contract the strict transform of the fiber through $q$.
Exactly as in the standard elementary-transform class computation
(Lemma~\ref{lem:routeC-elementary-transform-class}), this produces a birational morphism
$\psi\colon \widetilde X\to \mathbb F_0$ such that, if
$Q':=\psi(H)$, $B':=\psi(H_1)$, and $S:=\psi(H_0)$, then
$Q'\equiv 2S+f$ and $B'\equiv S+(b-1)f$.

Now choose a point $q_0\in S\setminus(Q'\cup B')$
and perform one further elementary transform centered at $q_0$.
Again by Lemma~\ref{lem:routeC-elementary-transform-class}, one obtains a birational morphism
$
\chi\colon \widetilde X\to \mathbb F_1
$
whose image $Q_{\mathrm{cub}}:=\chi(H)$ satisfies $Q_{\mathrm{cub}}\equiv 2S_0+3f$.
Contracting the negative section of this final $\mathbb F_1$ gives a birational morphism
$\eta\colon \widetilde X\to \PP^2$ whose image of $H$ is a plane cubic.
Because $p_a(Q_{\mathrm{cub}})=1$
and the normalization of $H$ is rational, that plane cubic is singular.
\end{proof}

\begin{proposition}[A birational singular plane-cubic model already yields an elliptic-boundary descendant]
\label{prop:routeC-one-section-plane-cubic-descendant}
Let $H\subset \widetilde X$ be a smooth rational curve.
Assume that there exists a birational map $\eta\colon \widetilde X \dashrightarrow \PP^2$ such that the
induced rational map $\eta|_H\colon H\dashrightarrow C$ is birational onto an irreducible singular plane cubic
$C\subset \PP^2$.
Then $X$ has a descendant with elliptic boundary.
\end{proposition}

\begin{proof}
Choose a sequence of point blowups $\sigma\colon Y\to \widetilde X$ resolving the indeterminacy of $\eta$ and any
remaining singularities of the total transform of $C$, minimal with the property that the induced map
$f:=\eta\circ \sigma\colon Y\to \PP^2$ is a morphism and the reduced total transform
$E:=\bigl(f^{-1}(C)\bigr)_{\mathrm{red}}$ is a simple normal crossings divisor.
Let $R\subset E$ be the strict transform of $H$.
Then $E$ is connected, every irreducible component of $E$ is a smooth rational curve, the restriction
$f|_R\colon R\to C$ is birational, and every component of $E-R$ is $f$-exceptional.

Because $C$ is an irreducible singular plane cubic, one has $p_a(C)=1$ and $C$ is rational.
The fibers of the induced map $E\to C$ are connected trees of rational curves, so
$f_*\mathcal O_E=\mathcal O_C$ and $R^1f_*\mathcal O_E=0$.
Hence $p_a(E)=p_a(C)=1$.
Therefore $E$ is a connected reduced simple normal crossings divisor with rational components and arithmetic genus one.
Lemma~\ref{lem:routeC-local-tie-equiv} applied with root $R$ shows that $E$ contains an elliptic tie.
By \cite[Definition~2.10, Definition~2.11, Lemma~2.12]{PP12}, this gives a descendant with elliptic boundary.
\end{proof}

\begin{theorem}[The degree-two one-section branch is closed]
\label{thm:routeC-one-section-bisection-closed}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and suppose that $d=2$.
Then the partition $2+1+1$
does not occur in the residual characteristic-$3$ sector.
\end{theorem}

\begin{proof}
Choose the singular-point elementary-transform sequence from Proposition~\ref{prop:routeC-bisection-plane-reduction}.
By Proposition~\ref{prop:routeC-bisection-singular-eltrans-lifts}, that sequence lifts to a birational morphism from
$\widetilde X$ to the final ruled surface.

If $s=1$,
then Proposition~\ref{prop:routeC-bisection-plane-reduction} shows that the final image of $H$ on $\mathbb F_1$
contracts to an irreducible singular plane cubic.
So Proposition~\ref{prop:routeC-one-section-plane-cubic-descendant} gives a descendant with elliptic boundary,
contradicting residuality.

If $s=0$,
write the final image of $H_1$ on $\mathbb F_1$ as
$S_0+bf$ with $b\ge 1$.
If $b=1$,
then Corollary~\ref{cor:routeC-one-section-bisection-conic-line-not-residual} excludes the branch.
If $b\ge 2$,
then Proposition~\ref{prop:routeC-one-section-conic-reduces-to-cubic-model} produces a lifted birational morphism to
$\PP^2$ whose image of $H$ is an irreducible singular plane cubic, so
Proposition~\ref{prop:routeC-one-section-plane-cubic-descendant} again gives a descendant with elliptic boundary.

All cases contradict residuality.
\end{proof}

\paragraph*{Sharpening the degree-three one-section branch.}
The new cubic-triple reduction removes one more ambiguity from the residual degree-three branch: after at most one
double singular-point transform, everything feeds into a pure-triple cubic branch.

\begin{proposition}[Singular-point transforms in the one-section cubic branch lift from \texorpdfstring{$\widetilde X$}{Xtilde}]
\label{prop:routeC-one-section-cubic-singular-eltrans-lifts}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and suppose that $d=3$.
Let $\nu_0:=\nu\colon \widetilde X\to \mathbb F_n$ and $C_0:=\nu_0(H)$.
Then any successive sequence of elementary transforms centered at singular points of the current image of $H$ lifts
inductively to birational morphisms $\mu_r\colon \widetilde X\to \mathbb F_{n-r}$.
At each stage:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item every chosen singular point lies off the current negative section and has multiplicity $m\in\{2,3\}$;
    \item if the current image has class $3S_0+(3\ell+s)f$ on $\mathbb F_\ell$, then after the elementary transform at
    such a point of multiplicity $m$ the new image has class
    $3S_0'+(3(\ell-1)+s+3-m)f' = 3S_0'+(3\ell+s-m)f'$ on $\mathbb F_{\ell-1}$;
    \item the arithmetic genus drops by $2m-3$, and the normalization remains $\PP^1$.
\end{enumerate}
\end{proposition}

\begin{proof}
The proof is the same local class computation as in the degree-two branch, together with the same lifting argument.
Indeed, if $q$ is a singular point of the current image $C\equiv 3S_0+(3\ell+s)f$, then $q\notin S_0$ because
$C\cdot S_0=s\le 1$, and a singular point on $S_0$ would contribute local intersection multiplicity at least $2$.
Since $C\cdot f=3$, the multiplicity at $q$ is at most $3$, hence belongs to $\{2,3\}$.

Blowing up $q$ and contracting the strict transform of the fiber through $q$ produces $\mathbb F_{\ell-1}$.
If $m$ is the multiplicity of $C$ at $q$, the same elementary-transform class calculation as in the proof of
Proposition~\ref{prop:routeC-bisection-singular-eltrans-lifts} gives the new class
$3S_0'+(3\ell+s-m)f'$, and adjunction shows that the arithmetic genus drops by $2m-3$.
Because the transform is birational, the normalization remains $\PP^1$.

For the lifting statement, factor the inverse map from the current ruled model to $\widetilde X$ into point blowups.
Since $H\subset \widetilde X$ is smooth while its current ruled-model image is singular at the chosen center, the first
blowup center over that singular point is the point itself, so the elementary transform factors through
$\widetilde X$.
Iterating gives the required birational morphisms $\mu_r$.
\end{proof}

\begin{corollary}[Exact double/triple count in the one-section cubic branch]
\label{cor:routeC-one-section-cubic-reduction-count}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and suppose that $d=3$.
Let $s:=H\cdot H_0\in\{0,1\}$.
Consider any full lifted singular-point transform sequence of
Proposition~\ref{prop:routeC-one-section-cubic-singular-eltrans-lifts}, continued until the image of $H$ becomes
smooth.
Let $N_2$ and $N_3$ denote the numbers of centers where the current image has multiplicity $2$ and $3$, respectively.
Then $N_2=1-s$ and $N_3=n+s-1$.
Equivalently:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $s=1$, then every step is triple and there are exactly $n$ triple steps;
    \item if $s=0$, then there is exactly one double step and the remaining $n-1$ steps are triple.
\end{enumerate}
Moreover, the final smooth model is always $\mathbb F_0$ with smooth image $Q_{\mathrm{sm}}\equiv 3S+f$.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:routeC-one-section-cubic-singular-eltrans-lifts}, every singular-point transform lowers the
Hirzebruch index by one and lowers the arithmetic genus by $1$ or $3$ according as the chosen center has multiplicity
$2$ or $3$.
Let the final smooth model lie on $\mathbb F_m$ and have class $Q_{\mathrm{sm}}\equiv 3S+(3m+r)f$.
Since the normalization is rational and the final curve is smooth, adjunction on $\mathbb F_m$ gives
$0=p_a(Q_{\mathrm{sm}})=3m+2r-2$.
Hence necessarily $m=0$ and $r=1$, so the final smooth model is $\mathbb F_0$ with
$Q_{\mathrm{sm}}\equiv 3S+f$.
Therefore the total number of singular-point transforms is exactly $N_2+N_3=n$.
On the other hand, the total drop in arithmetic genus equals the initial genus $p_a(C)=3n+2s-2$, so
$N_2+3N_3=3n+2s-2$.
Solving the two linear equations gives the displayed formulas.
\end{proof}

\begin{corollary}[The \texorpdfstring{$3+1$}{3+1} one-section branch reduces to the pure-triple cubic branch]
\label{cor:routeC-one-section-cubic-pure-triple}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}, and suppose that $d=3$.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item if $s=1$, every singular point of every intermediate lifted ruled-model image of $H$ has multiplicity $3$;
    \item if $s=0$, there is a unique double step, and immediately after that step the current image of $H$ has class
    $3S_0+(3m+1)f$ on some $\mathbb F_m$, while every remaining singular-point transform is triple.
\end{enumerate}
In particular, the whole one-section cubic branch reduces to the pure-triple branch with parameter $1$: singular
rational curves of class $3S_0+(3m+1)f$ all of whose later singular-point transforms are triple.
Moreover, in that pure-triple branch every singular point is locally of one of the following two types:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item a collision of exactly three smooth horizontal branches over one fiber;
    \item a unibranch ramification point of local degree exactly $3$.
\end{enumerate}
\end{corollary}

\begin{proof}
The first two claims are exactly Corollary~\ref{cor:routeC-one-section-cubic-reduction-count}.
If $s=0$, let the unique double step occur after $a$ triple steps.
After those $a$ triple steps, Proposition~\ref{prop:routeC-one-section-cubic-singular-eltrans-lifts} keeps the
parameter equal to $0$, so the current image has class $3S_0+3(n-a)f$ on $\mathbb F_{n-a}$.
Applying the unique double transform then gives a curve of class
$3S_0+(3(n-a)-2)f = 3S_0+(3(n-a-1)+1)f$ on $\mathbb F_{n-a-1}$.
Thus, after the unique double step, the branch is of the same form as the initial $s=1$ branch.

For the local classification, let $q$ be a singular point in the pure-triple branch.
By Proposition~\ref{prop:routeC-one-section-cubic-singular-eltrans-lifts}, one has $\operatorname{mult}_q=3$.
Lemma~\ref{lem:routeC-singular-horizontal-branch-or-ramification} shows that $q$ is either multibranch or unibranch
ramification.
In the multibranch case, each local branch is smooth and contributes multiplicity $1$, so multiplicity $3$ forces
exactly three branches.
In the unibranch case, if the local degree over the base were $2$, then a parametrization
$t=u^2$, $z=\varphi(u)$ would give multiplicity at most $2$, contrary to $\operatorname{mult}_q=3$.
Hence the local degree is exactly $3$.
\end{proof}

\begin{proposition}[The smooth \texorpdfstring{$3S+f$}{3S+f} endpoint birationally reduces to a singular plane cubic]
\label{prop:routeC-one-section-cubic-smooth-endpoint-plane-cubic}
Assume the setup of Corollary~\ref{cor:routeC-one-section-height4-partitions-singular}.
Assume that there exists a birational morphism $\phi\colon \widetilde X\to \mathbb F_0$ such that
$Q:=\phi(H)\subset \mathbb F_0$ is smooth and satisfies $Q\equiv 3S+f$.
Then there exists a birational map $\eta\colon \widetilde X \dashrightarrow \PP^2$ whose restriction to $H$ is
birational onto an irreducible singular plane cubic.
\end{proposition}

\begin{proof}
Choose a smooth point $q\in Q$.
Let $T\in |S|$ and $F\in |f|$ be the two ruling curves on $\mathbb F_0$ passing through $q$.
Then $Q\cdot T=1$, $Q\cdot F=3$, and $I_q(Q,T)=1$.

Let $\sigma\colon Z:=\operatorname{Bl}_q(\mathbb F_0)\to \mathbb F_0$ be the blowup of $q$, with exceptional curve
$E$.
Write $\widetilde T$, $\widetilde F$, and $\widetilde Q$ for the strict transforms of $T$, $F$, and $Q$.
Since $T$ and $F$ meet only at $q$, the curves $\widetilde T$ and $\widetilde F$ are disjoint.
Moreover, $\widetilde T^2=\widetilde F^2=-1$.
Hence one may contract them successively to obtain a birational morphism $\tau\colon Z\to W$ onto a smooth rational
surface $W$ with Picard rank $1$.
Therefore $W\cong \PP^2$.
Set $\rho:=\tau\circ \sigma^{-1}\colon \mathbb F_0 \dashrightarrow \PP^2$.

Now
\[
\widetilde Q\cdot \widetilde T
=
Q\cdot T-I_q(Q,T)
=
1-1
=0,
\]
while $\widetilde Q\cdot \widetilde F\ge 0$.
So $\widetilde Q$ is not contracted by $\tau$, and its image $\Gamma:=\tau(\widetilde Q)=\rho(Q)$ is irreducible.
Since $H$ is rational and $\phi|_H$ is birational onto the smooth curve $Q$, one has $Q\cong \PP^1$.
Therefore, because $\rho$ is birational, the rational map $\rho|_Q\colon Q\dashrightarrow \Gamma$ is birational, so
$\Gamma$ is a rational plane curve.

To compute the degree of $\Gamma$, note that on $Z$ the divisor $L:=\sigma^*S+\sigma^*f-E$ has self-intersection $1$
and intersects both $\widetilde T$ and $\widetilde F$ trivially, so $L$ is the pullback of a line on
$W\cong \PP^2$.
Because $q$ is a smooth point of $Q$, one has $\widetilde Q\equiv \sigma^*Q-E=3\sigma^*S+\sigma^*f-E$.
Therefore
\[
\deg \Gamma
=
L\cdot \widetilde Q
=
(\sigma^*S+\sigma^*f-E)\cdot(3\sigma^*S+\sigma^*f-E)
=3.
\]
Thus $\Gamma\subset \PP^2$ is an irreducible plane cubic.
Because $\Gamma$ is rational while every irreducible plane cubic has arithmetic genus $1$, the curve $\Gamma$ is
necessarily singular.

Finally set $\eta:=\rho\circ \phi\colon \widetilde X \dashrightarrow \PP^2$.
Since $\phi|_H\colon H\to Q$ is birational and $\rho|_Q$ is birational, the rational map
$\eta|_H\colon H\dashrightarrow \Gamma$ is birational as well.
\end{proof}

\begin{theorem}[The pure-triple cubic branch is closed]
\label{thm:routeC-one-section-cubic-pure-triple-closed}
Assume the setup of Corollary~\ref{cor:routeC-one-section-cubic-pure-triple}.
Suppose that after the possible unique double step of Corollary~\ref{cor:routeC-one-section-cubic-pure-triple} one is
in its pure-triple branch, so the current image of $H$ has class $3S_0+(3m+1)f$ on some $\mathbb F_m$ and every later
singular-point elementary transform is centered at a triple point.
Then $X$ has a descendant with elliptic boundary.
In particular, the pure-triple cubic branch cannot occur in the residual characteristic-$3$ sector.
\end{theorem}

\begin{proof}
Continue the lifted singular-point transform sequence until the image of $H$ becomes smooth.
By Proposition~\ref{prop:routeC-one-section-cubic-singular-eltrans-lifts}, every step lifts from $\widetilde X$.
Because every remaining step is triple, the parameter $1$ is preserved throughout the sequence, and after exactly $m$
steps one arrives at a birational morphism $\phi\colon \widetilde X\to \mathbb F_0$ whose image $Q:=\phi(H)$ is smooth
of class $Q\equiv 3S+f$; this is also the final smooth model identified in
Corollary~\ref{cor:routeC-one-section-cubic-reduction-count}.

Proposition~\ref{prop:routeC-one-section-cubic-smooth-endpoint-plane-cubic} now gives a birational map
$\eta\colon \widetilde X \dashrightarrow \PP^2$ whose restriction to $H$ is birational onto an irreducible singular
plane cubic.
Applying Proposition~\ref{prop:routeC-one-section-plane-cubic-descendant} yields a descendant with elliptic boundary.
This excludes the pure-triple branch in the residual sector.
\end{proof}

\begin{corollary}[The one-section singular-image side of the separable special-multisection branch is closed]
\label{cor:routeC-special-multisection-one-section-singular-closed}
Assume the setup of Proposition~\ref{prop:routeC-special-multisection-one-section-finite-extraction}, and assume that
$H$ is special and separable.
Then the one-section singular-image side contributes no residual branch.
More precisely:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item the degree-two partition $2+1+1$ is excluded by Theorem~\ref{thm:routeC-one-section-bisection-closed};
    \item the degree-three branch $3+1$ is excluded by Corollary~\ref{cor:routeC-one-section-cubic-pure-triple}
    together with Theorem~\ref{thm:routeC-one-section-cubic-pure-triple-closed}.
\end{enumerate}
Consequently, on the separable special-multisection side only the smooth-image ramification / two-point corridor of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint} remains.
\end{corollary}

\begin{proof}
The degree-two branch is closed by Theorem~\ref{thm:routeC-one-section-bisection-closed}.
For the degree-three branch, Corollary~\ref{cor:routeC-one-section-cubic-pure-triple} reduces the whole one-section
cubic side to the pure-triple branch, and Theorem~\ref{thm:routeC-one-section-cubic-pure-triple-closed} excludes that
branch.
Hence no one-section singular-image residue remains.
The last sentence is exactly Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint}.
\end{proof}

\paragraph*{Closing the remaining smooth-image corridor.}
We now show that outside the several-special-horizontal corridor the explicit low-degree at-most-two-point threaded
$[3]$-root model cannot occur either.

\begin{lemma}[In the residual low-degree smooth-image corridor the \texorpdfstring{$[3]$}{[3]}-root is one-sided]
\label{lem:routeC-special-multisection-smooth-image-root-onesided}
Keep the setup of Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint}, and assume in
addition that we are outside the several-special-horizontal corridor, that we are in the residual sector
$\operatorname{ht}(X)\ge 4$ with no descendant with elliptic boundary, and that
alternative~\textnormal{(ii)} of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint} holds.

Let $A=[(2)^u,3,(2)^v]$ be the connected component of $D$ containing $H$, with $H$ the unique $(-3)$-component.
Then $uv=0$.
\end{lemma}

\begin{proof}
By Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint}, one has $H^2=-3$ and
$d:=F\cdot H\le 5$.
Since we are outside the several-special-horizontal corridor, $H$ is the unique special horizontal component.
Moreover, Corollary~\ref{cor:routeC-special-multisection-highheight-oldside} removes the high-height part, while
Corollary~\ref{cor:routeC-special-multisection-finite-lowheight-list} leaves only the finite low-height list
$5$, $4+1$, $4$, $3+1$, $2+2$, $2+1+1$.
Among those, the one-section partitions $3+1$ and $2+1+1$ belong to the one-section singular-image branch and are
excluded by Corollary~\ref{cor:routeC-special-multisection-one-section-singular-closed}.
Hence the only residual partitions still left here are $5$, $4+1$, $4$, and $2+2$, so in particular
$d\in\{2,4,5\}$.

Assume first that $d\in\{4,5\}$.
If $u,v\ge 1$, then exactly as in the proof of
Proposition~\ref{prop:routeC-special-multisection-mild-contact2-old} one has
    \[
    \operatorname{cf}_X(H)=\frac{(u+1)(v+1)}{uv+2u+2v+3}\ge \frac12,
    \]
because $2(u+1)(v+1)-(uv+2u+2v+3)=uv-1\ge 0$.
Lemma~\ref{lem:routeC-special-multisection-ordinary-degree-le3} therefore gives
$2>d\,\operatorname{cf}_X(H)+\frac12 M\ge \frac d2\ge 2$, a contradiction.
Hence $uv=0$ in the cases $d=4,5$.

It remains to treat $d=2$.
Then $S\to \PP^1$ is a separable double cover of $\PP^1$, so by Riemann--Hurwitz it has exactly two ramification
points, both of local degree $2$.
By Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint}, the total number of blowup
centers on successive transforms of $S$ equals $2d+3=7$.
Over each of the two ramification points, the all-bad hypothesis forces the two tangency-resolution centers together
with at least one thread center, so already $6$ of those $7$ centers are prescribed.
By the last paragraph of Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint}, every
ramification point contributes a detached threaded ramification chain, while every nonramification center on a
successive transform of $S$ lies on the root chain containing $H$.
A nonempty $2$-tail on one side of $H$ requires at least one such nonramification center on that side.
Thus a two-sided chain $[(2)^u,3,(2)^v]$ with $u,v\ge 1$ would require at least two nonramification centers on
successive transforms of $S$, but only one center is left after the mandatory $6$ ramification-side centers.
This is impossible, so again $uv=0$.
\end{proof}

\begin{lemma}[A one-sided threaded \texorpdfstring{$[3]$}{[3]}-root ramification branch gives low height or the closed $2+1+1$ package]
\label{lem:routeC-special-multisection-one-sided-threaded-height}
Let $Z$ be a smooth surface with reduced snc divisor $D_Z$.
Assume that $D_Z$ contains a chain $A=[3,(2)^s]=H-U_1-\cdots-U_s$ with $s\ge 0$, and another chain
$B=[a,(2)^m]=E_1-E_2-\cdots-E_{m+1}$ with $a\ge 2$ and $m\ge 2$, where
$H^2=-3$, $U_i^2=-2$, $E_1^2=-a$, and $E_j^2=-2$ for $j\ge 2$.
Assume moreover that there is a $(-1)$-curve $L\not\subset D_Z$ meeting $H$ and $E_2$ once each and otherwise
disjoint from $D_Z$, and that every component of $D_Z-(A+B)$ is disjoint from $A\cup B\cup L$.

Let $\sigma\colon Z\to Y$ be the contraction of $L,E_2,E_3$, and put $F:=\sigma_*H$.
Then:
\begin{enumerate}[label=\textnormal{(\roman*)}]
    \item $F$ is a smooth rational curve with $F^2=0$, hence $|F|$ defines a $\PP^1$-fibration on $Y$;
    \item $\sigma_*E_1$ is a bisection of that ruling;
    \item if $s\ge 1$, then $\sigma_*U_1$ is a section;
    \item if $m\ge 3$, then $\sigma_*E_4$ is a section;
    \item every other component of $\sigma_*D_Z$ is vertical for $|F|$.
\end{enumerate}
Consequently a general fiber $G$ of $|F|$ satisfies
$G\cdot \sigma_*D_Z\le 2+\mathbf{1}_{\{s\ge 1\}}+\mathbf{1}_{\{m\ge 3\}}$.
In particular:
\begin{enumerate}[label=\textnormal{(\alph*)}]
    \item if $m=2$, then $G\cdot \sigma_*D_Z\le 3$;
    \item if $m\ge 3$ and $s=0$, then $G\cdot \sigma_*D_Z\le 3$;
    \item if $m\ge 3$ and $s\ge 1$, then the horizontal degree partition of the ruling $|F|$ is exactly $2+1+1$.
\end{enumerate}
\end{lemma}

\begin{proof}
Contract $L$ first.
Then the image of $H$ has self-intersection $-2$, and the image of $E_2$ is a $(-1)$-curve.
Contracting $E_2$ raises the self-intersection of the image of $H$ to $-1$ and makes the image of $E_3$ a $(-1)$-curve.
Contracting $E_3$ raises the self-intersection of the image of $H$ once more, so the final curve
$F=\sigma_*H$ satisfies $F^2=0$.
Since $H\cong \PP^1$ is smooth, so is $F$, and Lemma~\ref{lem:routeC-zero-curve-pencil} gives the ruling $|F|$.
This proves \textnormal{(i)}.

After contracting $L$ and then $E_2$, the image of $H$ meets the image of $E_1$ once and the image of $E_3$ once.
Contracting $E_3$ adds one further intersection between the images of $H$ and $E_1$, so
$F\cdot \sigma_*E_1=2$.
Thus $\sigma_*E_1$ is a bisection, proving \textnormal{(ii)}.

If $s\ge 1$, then $U_1$ is disjoint from the contracted curves except for its original transverse intersection with
$H$, so $F\cdot \sigma_*U_1=1$.
Hence $\sigma_*U_1$ is a section, proving \textnormal{(iii)}.
Likewise, if $m\ge 3$, then after the contraction of $E_3$ the image of $E_4$ meets $F$ once, so
$\sigma_*E_4$ is a section, proving \textnormal{(iv)}.

Every remaining component of $A$ or $B$ is disjoint from $F$, hence vertical for $|F|$.
By assumption every component of $D_Z-(A+B)$ is disjoint from the whole local cluster and therefore also disjoint from
$F$, so those components are vertical as well.
This proves \textnormal{(v)} and the displayed bound for a general fiber.
The three final cases are then immediate.
\end{proof}

\begin{lemma}[The chosen clean-threaded ramification cluster is isolated]
\label{lem:routeC-special-multisection-threaded-cluster-isolated}
Keep the setup of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint}, and assume that
alternative~\textnormal{(ii)} holds.
Let $A$ be the full root chain containing $H$.
Choose a bad ramification point $q$ of $S\to \PP^1$, and let
$B_q=[a,(2)^m]$ be the full detached threaded ramification chain over $q$ furnished by
Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint} together with the threading statement
from the proof of Proposition~\ref{prop:routeC-root-two-clean-threaded}.
Let $L_q\not\subset D$ be the unique $(-1)$-curve at the $H$-end of that local cluster meeting $H$ and the second
component of $B_q$ once each.
Then every component of $D-(A+B_q)$ is disjoint from $A\cup B_q\cup L_q$.
\end{lemma}

\begin{proof}
By Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint}, every component of $D$ lies either
in the unique root chain $A$ containing $H$ or in one of at most two detached threaded ramification chains.
Because $B_q$ is taken to be the full detached threaded chain over $q$, any component of $D-(A+B_q)$ lies in a second
detached threaded ramification chain corresponding to a different bad ramification point.
Such a chain is disjoint from $A$ and from $B_q$ because detached connected components of $D$ are pairwise disjoint.
Moreover $L_q$ is created entirely inside the local branch over $q$ and meets only $H$ and the second component of
$B_q$.
Hence every component of $D-(A+B_q)$ is also disjoint from $L_q$.
\end{proof}

\begin{proposition}[The explicit low-degree smooth-image corridor is impossible outside the several-special-horizontal corridor]
\label{prop:routeC-special-multisection-smooth-image-lowdegree-corridor-impossible}
Keep the setup of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint}, and assume in addition that we are
outside the several-special-horizontal corridor, that we are in the residual sector $\operatorname{ht}(X)\ge 4$ with
no descendant with elliptic boundary, and that alternative~\textnormal{(ii)} of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint} holds.
Then this is impossible.
\end{proposition}

\begin{proof}
Assume for contradiction that alternative~\textnormal{(ii)} holds.
By Lemma~\ref{lem:routeC-special-multisection-smooth-image-root-onesided}, the connected component of $D$ containing
$H$ is one-sided; after possibly reversing it we may write $A=[3,(2)^s]$ with $s\ge 0$ and with $H$ the initial
$(-3)$-curve.

Choose a bad ramification point $q$ of $S\to \PP^1$.
By Proposition~\ref{prop:routeC-ramification-fork-survival}, the forced tangency-resolution stage over $q$ produces a
local bridge whose detached side starts with a chain $[a,(2)^{a-1}]$, where $a$ is the local degree at $q$.
By Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint} together with the threading statement
from the proof of Proposition~\ref{prop:routeC-root-two-clean-threaded}, the all-bad branch over $q$ is clean threaded
at the $H$-end.
So on $\widetilde X$ itself there is a detached chain $B=[a,(2)^m]$ with $a\ge 2$ and $m\ge 2$, where $a$ is the
local degree at $q$, together with a $(-1)$-curve $L\not\subset D$ meeting $H$ and the second component of $B$ once
each.
By Lemma~\ref{lem:routeC-special-multisection-threaded-cluster-isolated}, every component of $D-(A+B)$ is disjoint
from $A\cup B\cup L$.
Hence Lemma~\ref{lem:routeC-special-multisection-one-sided-threaded-height} applies with $Z=\widetilde X$.

We split according to $d=F\cdot H$.
If $d=2$, then $S\to \PP^1$ is a separable double cover, so every ramification point has local degree $a=2$.
If $m=2$, Lemma~\ref{lem:routeC-special-multisection-one-sided-threaded-height}\textnormal{(a)} gives a witnessing
ruling of height at most $3$, contradicting $\operatorname{ht}(X)\ge 4$.
If $m\ge 3$ and $s=0$, Lemma~\ref{lem:routeC-special-multisection-one-sided-threaded-height}\textnormal{(b)} again
gives height at most $3$, the same contradiction.
If $m\ge 3$ and $s\ge 1$, then
Lemma~\ref{lem:routeC-special-multisection-one-sided-threaded-height}\textnormal{(c)} gives a witnessing ruling of
height $4$ whose horizontal degree partition is exactly $2+1+1$, contradicting
Theorem~\ref{thm:routeC-one-section-bisection-closed}.
Thus $d=2$ is impossible.

Assume next that $d\in\{4,5\}$.
If every ramification point had local degree $2$, then every ramification point would be tame with different exponent
$1$.
Since Proposition~\ref{prop:routeC-special-multisection-smooth-image-allbad-twopoint} gives at most two ramification
points, this would yield $\deg R\le 2$, contrary to Riemann--Hurwitz $\deg R=2d-2\ge 6$.
So there exists a ramification point $q$ with local degree $a\ge 3$.
Then its detached chain satisfies $m\ge 3$.

If $s=0$, Lemma~\ref{lem:routeC-special-multisection-one-sided-threaded-height}\textnormal{(b)} gives a witnessing
ruling of height at most $3$, contradicting $\operatorname{ht}(X)\ge 4$.
If $s\ge 1$, then
Lemma~\ref{lem:routeC-special-multisection-one-sided-threaded-height}\textnormal{(c)} gives a witnessing ruling of
height $4$ whose horizontal degree partition is exactly $2+1+1$.
This is again excluded by Theorem~\ref{thm:routeC-one-section-bisection-closed}.

All cases contradict residuality.
\end{proof}

\begin{corollary}[Outside the several-special-horizontal corridor, the smooth-image separable special-multisection side is already old]
\label{cor:routeC-special-multisection-smooth-image-outside-several-old}
Keep the setup of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint}, and assume that we are outside the
several-special-horizontal corridor.
Then alternative~\textnormal{(ii)} of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint} cannot occur.
Consequently there exists a ramification point $q\in S\setminus B_\nu$ of local degree at least $2$, and the
smooth-image separable special-multisection branch is already old by
Corollary~\ref{cor:routeC-special-multisection-ramification-frontier-old}.
\end{corollary}

\begin{proof}
The high-height part is already old by Corollary~\ref{cor:routeC-special-multisection-highheight-oldside}, so only the
residual low-degree corridor of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint}\textnormal{(ii)} remained.
That corridor is impossible by
Proposition~\ref{prop:routeC-special-multisection-smooth-image-lowdegree-corridor-impossible}.
Therefore alternative~\textnormal{(i)} of
Corollary~\ref{cor:routeC-special-multisection-smooth-image-good-or-twopoint} must hold, and the last assertion is
exactly Corollary~\ref{cor:routeC-special-multisection-ramification-frontier-old}.
\end{proof}

\subsection{The purely inseparable ordinary singleton \texorpdfstring{$[2]$}{[2]}-root branch}

The two-fiber support bound from Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers} can in fact be sharpened to
a complete closure of the branch.
The key point is that on the cubic purely inseparable backbone, one actual service center already produces a lifted
singular plane-cubic model.

\begin{lemma}[The cubic purely inseparable backbone is actually on \texorpdfstring{$\mathbb F_0$}{F0}]
\label{lem:routeC-root-two-backbone-F0}
Keep the setup of Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers}, and assume that we are on the purely
inseparable ordinary singleton $[2]$-root branch.
Then the horizontal map $S\to \PP^1$ has degree $3$.
Writing the relatively minimal ruled surface as $Y=\mathbb F_n$ and $S\equiv 3S_0+bf$, one has $n=0$ and $b=1$.
In particular, $Y\cong \mathbb F_0$ and $S\equiv 3\Sigma+f$ for every section $\Sigma\subset \mathbb F_0$.
\end{lemma}

\begin{proof}
By Corollary~\ref{cor:routeC-final-lemma-pureinseparable-fork}, some point of $S$ produces a surviving primitive
ramification fork.
Since the branch is ordinary, Corollary~\ref{cor:routeC-ordinary-ramification-cubic} forces the local ramification
degree at that point to be $3$.
For a purely inseparable map $S\cong \PP^1\to \PP^1$, every point has the same local degree, so the global degree is
also $3$.

Now write $Y=\mathbb F_n$ and $S\equiv 3S_0+bf$.
The first paragraph of Proposition~\ref{prop:routeC-root-two-clean-threaded} with $a=3$ gives $S^2=6$.
Hence $-9n+6b=6$, i.e. $-3n+2b=2$.
Because $S$ does not contain the negative section, one has $0\le S\cdot S_0=b-3n$.
If $n\ge 1$, then $b\ge 3n$, so $2=-3n+2b\ge -3n+6n=3n\ge 3$, a contradiction.
Therefore $n=0$, and then $2b=2$, so $b=1$.
Thus $Y\cong \mathbb F_0$ and $S\equiv 3\Sigma+f$ for every section $\Sigma$ of the chosen ruling on $\mathbb F_0$.
\end{proof}

\begin{proposition}[A service fiber on the cubic backbone yields a lifted singular plane cubic]
\label{prop:routeC-root-two-lifted-plane-cubic}
Keep the setup of Lemma~\ref{lem:routeC-root-two-backbone-F0}.
Let $\mathcal T$ be the set of fibers from Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers}, and choose any
$f_0\in \mathcal T$.
Let $q\in S\cap f_0$ be the first blowup center of $\nu^{-1}$ lying over $f_0$.
Then there exists a birational morphism $\eta\colon \widetilde X\to \PP^2$ such that
$\Gamma:=\eta(H)$ is an irreducible singular plane cubic and $\eta|_H\colon H\to \Gamma$ is birational.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:routeC-root-two-backbone-F0}, we may write $Y=\mathbb F_0$ and $S\equiv 3\Sigma+f$ for a section
$\Sigma$ passing through $q$.
After commuting disjoint initial blowups if necessary, factor the inverse birational map $\nu^{-1}$ so that its first
step is the blowup of $q$:
\[
Y \xleftarrow{\ \sigma\ } Z:=\operatorname{Bl}_q Y \xleftarrow{\ \rho\ } \widetilde X.
\]
Let $F$ be the fiber of $Y$ through $q$, and let $F'\subset Z$ be its strict transform.
Since $q\in \Sigma$ and $\Sigma^2=0$, contracting $F'$ gives the elementary transform
$\tau\colon Z\to \mathbb F_1$.
Set $\mu:=\tau\circ \rho\colon \widetilde X\to \mathbb F_1$ and $C:=\mu(H)$.
Because $\rho|_H$ and the elementary transform are birational on the image of $H$, the map
$\mu|_H\colon H\to C$ is birational.

We now compute the class of $C$.
Apply Lemma~\ref{lem:routeC-elementary-transform-class-on-negative} on $\mathbb F_0$ with $a=3$, $b=1$, and
$m=\operatorname{mult}_q(S)=1$.
This gives $C\equiv 3S_0+3f$ on $\mathbb F_1$, where $S_0^2=-1$ is the negative section.
Hence $C\cdot S_0=(3S_0+3f)\cdot S_0=0$, so $C$ is disjoint from $S_0$.
Contracting the negative section, $\kappa\colon \mathbb F_1\to \PP^2$, therefore yields an irreducible plane cubic
$\Gamma:=\kappa(C)\subset \PP^2$, and the composite $\eta:=\kappa\circ \mu\colon \widetilde X\to \PP^2$ is a birational
morphism whose restriction to $H$ is birational onto $\Gamma$.

It remains to show that $\Gamma$ is singular.
Since $H\cong \PP^1$ and $\mu|_H$ is birational, the normalization of $C$ is rational.
But adjunction on $\mathbb F_1$ gives $p_a(C)=p_a(3S_0+3f)=1$.
Thus $C$ is singular, and because $\kappa$ is an isomorphism near $C$, the plane cubic $\Gamma$ is singular as well.
\end{proof}

\begin{theorem}[The purely inseparable ordinary singleton \texorpdfstring{$[2]$}{[2]}-root branch is not residual]
\label{thm:routeC-root-two-closed}
Keep the setup of Corollary~\ref{cor:routeC-root-two-allpatterns-twofibers}, and assume that we are on the purely
inseparable ordinary singleton $[2]$-root branch.
Then $X$ has a descendant with elliptic boundary.
In particular, this branch cannot occur on a residual characteristic-$3$ counterexample.
\end{theorem}

\begin{proof}
By the first paragraph of Proposition~\ref{prop:routeC-root-two-clean-threaded} with $a=3$, exactly $8$ blowup centers
lie on successive strict transforms of $S$.
So $\mathcal T\neq \varnothing$.
Choose $f_0\in \mathcal T$.
Proposition~\ref{prop:routeC-root-two-lifted-plane-cubic} then produces a birational morphism
$\eta\colon \widetilde X\to \PP^2$ such that $\eta(H)$ is an irreducible singular plane cubic and $\eta|_H$ is
birational.
Applying Proposition~\ref{prop:routeC-one-section-plane-cubic-descendant} gives a descendant with elliptic boundary.
\end{proof}

\begin{corollary}[The height-selection problem loses the purely inseparable ordinary singleton \texorpdfstring{$[2]$}{[2]}-root branch]
\label{cor:routeC-height-selection-no-root-two-branch}
In Proposition~\ref{prop:routeC-height-selection-final-split}, case~\textnormal{(iii)} does not occur.
Equivalently, after the closure of the maximal-width corridor, the height-selection analysis at this stage is reduced to the
several-special-horizontal branch and the separable special-multisection branch.
\end{corollary}

\begin{proof}
Immediate from Theorem~\ref{thm:routeC-root-two-closed}.
\end{proof}

\begin{theorem}[Height selection in characteristic $3$]
\label{thm:routeC-height-selection-complete}
Every characteristic-$3$ klt del Pezzo surface $X$ of Picard rank $1$ with $\operatorname{ht}(X)\ge 4$ and no
descendant with elliptic boundary admits a witnessing $\PP^1$-fibration of height $4$ and maximal width.
\end{theorem}

\begin{proof}
Assume for contradiction that such a surface $X$ does not admit a witnessing $\PP^1$-fibration of height $4$ and
maximal width.
By Proposition~\ref{prop:routeC-height-selection-final-split}, it is enough to treat three residual cases.

\smallskip
\noindent\emph{Case \textnormal{(ii)}: separable special multisection.}
The one-section singular-image side is closed by
Corollary~\ref{cor:routeC-special-multisection-one-section-singular-closed}.
It remains only to consider the smooth-image side.
Since case~\textnormal{(i)} is the several-special-horizontal branch and we are here in case~\textnormal{(ii)}, we are
outside that corridor.
Corollary~\ref{cor:routeC-special-multisection-smooth-image-outside-several-old} then shows that the smooth-image
separable special-multisection branch is already old as well.
Thus case~\textnormal{(ii)} is closed.

\smallskip
\noindent\emph{Case \textnormal{(iii)}: purely inseparable ordinary singleton $[2]$-root.}
This branch is excluded by Theorem~\ref{thm:routeC-root-two-closed}.

\smallskip
\noindent\emph{Case \textnormal{(i)}: several special horizontals.}
By Lemma~\ref{lem:corr-final-split-case-i-is-packet}, after removing cases~\textnormal{(ii)} and
\textnormal{(iii)} this is exactly the several-special-horizontal packet branch.
Theorem~\ref{thm:routeC-height-selection-packet-side-closed} closes that branch.

All residual cases of Proposition~\ref{prop:routeC-height-selection-final-split} are therefore closed, contrary to the
assumption that $X$ fails to admit a witnessing $\PP^1$-fibration of height $4$ and maximal width.
\end{proof}

\begin{corollary}[The characteristic-$3$ seven-point theorem]
\label{cor:routeC-seven-point-char3-final}
Let $X$ be a characteristic-$3$ klt del Pezzo surface of Picard rank $1$.
Then $\#\Sing(X)\le 7$.
\end{corollary}

\begin{proof}
By Corollary~\ref{cor:routeC-height4-maxwidth-selection-only}, it is enough to prove the boxed height-selection
statement there.
That statement is exactly Theorem~\ref{thm:routeC-height-selection-complete}.
\end{proof}

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\end{document}
