\documentclass[11pt]{article}
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\usepackage{booktabs}
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\title{The eight surviving $\gamma$-partitions at $k = 9$:\\
       a clean structural description}
\author{}
\date{}

\newtheorem*{obs}{Observation}
\newtheorem*{prop}{Proposition}
\newtheorem*{thm}{Theorem}

\begin{document}
\maketitle

\section*{Setup}

For $T_2 = (m_2 = 9, k_2 = 9, \mathrm{chords} = \{(0,3), (3,6)\},
\mathrm{SP})$, $\gamma$ has length $9$ and $T_2$'s outerplanar
$O^{(2)}$ has three $3$-edge faces $F_A = \{0,1,2\}$,
$F_B = \{3,4,5\}$, $F_C = \{6,7,8\}$ (indices into
$B_{\mathrm{in}}^{(2)}$).  Empirically (\texttt{notes/induced\_partition\_findings.tex}):
of the $280$ triple-partitions of $\{0, \dots, 8\}$, exactly $8$ have
$\mathcal{L} \subseteq \pi_U(T_2)$.  We give a clean structural
description.

\section*{Classifying $\gamma$-edges by $T_2$'s face structure}

In the balanced annular triangulation of $T_2$, $D$-positions on
$T'_{\mathrm{ann}}$ are $\{0, 2, 4, 6, 8, 10, 12, 14, 16\}$.  $U$-position
$2i + 1$ corresponds to $\gamma$-edge $i$.  Each $\gamma$-edge is
``between'' two $D$-positions on $T'_{\mathrm{ann}}$, and is
classified by which $O^{(2)}$-faces those two $D$-positions belong to.

\begin{center}
\small
\begin{tabular}{cccc}
\toprule
$\gamma$-edge $i$ & $U$-position & adjacent $D$'s (faces) & classification \\
\midrule
$0$ & $1$  & $D{=}0, D{=}2$ (both $F_A$) & internal to $F_A$ \\
$1$ & $3$  & $D{=}2, D{=}4$ (both $F_A$) & internal to $F_A$ \\
$2$ & $5$  & $D{=}4 (F_A), D{=}6 (F_B)$ & boundary $F_A$--$F_B$ \\
$3$ & $7$  & $D{=}6, D{=}8$ (both $F_B$) & internal to $F_B$ \\
$4$ & $9$  & $D{=}8, D{=}10$ (both $F_B$) & internal to $F_B$ \\
$5$ & $11$ & $D{=}10 (F_B), D{=}12 (F_C)$ & boundary $F_B$--$F_C$ \\
$6$ & $13$ & $D{=}12, D{=}14$ (both $F_C$) & internal to $F_C$ \\
$7$ & $15$ & $D{=}14, D{=}16$ (both $F_C$) & internal to $F_C$ \\
$8$ & $17$ & $D{=}16 (F_C), D{=}0 (F_A)$ (cyclic) & boundary $F_C$--$F_A$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent
So $\gamma$ partitions into $6$ internal $\gamma$-edges ($2$ per face)
and $3$ boundary $\gamma$-edges ($1$ per pair of adjacent faces):
\[
   \text{Internal}_{F_A} = \{0, 1\},\quad
   \text{Internal}_{F_B} = \{3, 4\},\quad
   \text{Internal}_{F_C} = \{6, 7\};
\]
\[
   \delta_{AB} = 2,\quad \delta_{BC} = 5,\quad \delta_{CA} = 8.
\]

\section*{The structural description}

\begin{prop}[Face-pair connection partitions]
\label{prop:fp-connection}
A triple-partition $\widetilde{\mathcal{F}_2}$ of $\gamma$ has
$\mathcal{L}(\gamma, \widetilde{\mathcal{F}_2}) \subseteq \pi_U(T_2)$
iff it has the following structure: each block consists of
\begin{itemize}
  \item one boundary $\gamma$-edge $\delta_{ij}$ between an adjacent
        $O^{(2)}$-face pair $(F_i, F_j)$,
  \item one internal $\gamma$-edge from $F_i$ (i.e.\ one of
        $\text{Internal}_{F_i}$'s two elements),
  \item one internal $\gamma$-edge from $F_j$.
\end{itemize}
Equivalently: blocks are in bijection with adjacent $O^{(2)}$-face
pairs (here, $\{AB, BC, CA\}$), and for each face $F_i$ the two
internal $\gamma$-edges are distributed between the two blocks
``involving'' $F_i$ (one per block).
\end{prop}

\begin{proof}[Proof of count]
For $r = 3$ faces: $3$ boundary edges (one per adjacent pair), so $3$
blocks.  Each face $F_i$ has $2$ internal $\gamma$-edges, and each
internal must go to one of the two blocks involving $F_i$.  Choices:
$2$ per face, $r$ faces $\Rightarrow$ $2^r = 8$ partitions matching
the empirical $8$ survivors.
\end{proof}

\subsection*{All 8 survivors enumerated with this structure}

Writing each partition as
$\{(\text{internal}_A, \delta_{AB}, \text{internal}_B),\;
(\text{internal}_B', \delta_{BC}, \text{internal}_C),\;
(\text{internal}_C', \delta_{CA}, \text{internal}_A')\}$
where internal$_A$ and internal$_A'$ split $\{0, 1\}$, etc.:

\begin{center}
\small
\begin{tabular}{ll}
\toprule
$(F_A, F_B, F_C)$ split & resulting partition\\
\midrule
$(0|1, 3|4, 6|7)$ & $\{(0, 2, 3), (4, 5, 6), (7, 8, 1)\}$ \\
$(0|1, 3|4, 7|6)$ & $\{(0, 2, 3), (4, 5, 7), (6, 8, 1)\}$ \\
$(0|1, 4|3, 6|7)$ & $\{(0, 2, 4), (3, 5, 6), (7, 8, 1)\}$ \\
$(0|1, 4|3, 7|6)$ & $\{(0, 2, 4), (3, 5, 7), (6, 8, 1)\}$ \\
$(1|0, 3|4, 6|7)$ & $\{(1, 2, 3), (4, 5, 6), (7, 8, 0)\}$ \\
$(1|0, 3|4, 7|6)$ & $\{(1, 2, 3), (4, 5, 7), (6, 8, 0)\}$ \\
$(1|0, 4|3, 6|7)$ & $\{(1, 2, 4), (3, 5, 6), (7, 8, 0)\}$ \\
$(1|0, 4|3, 7|6)$ & $\{(1, 2, 4), (3, 5, 7), (6, 8, 0)\}$ \\
\bottomrule
\end{tabular}
\end{center}

These match the empirical survivors (up to relabeling block order).

\section*{Why the naive candidates fail}

Both \texttt{induced\_partition.py}'s candidate $1$ (next-$D$)
$\{(0,1,8),(2,3,4),(5,6,7)\}$ and candidate $2$ (prev-$D$)
$\{(0,1,2),(3,4,5),(6,7,8)\}$ \emph{group both internal $\gamma$-edges
of one face into one block}.  E.g.\ candidate $2$'s first block
$\{0,1,2\}$ contains both Internal$_{F_A}$ edges ($0$ and $1$) plus
the boundary $\delta_{AB} = 2$ --- no internal from $F_B$.

This violates the structural rule of Prop.\ \ref{prop:fp-connection}
(which requires one internal from \emph{each} of the two adjacent
faces).  Empirically these partitions' Latin sets are not contained
in $\pi_U(T_2)$.

\section*{Generalisation to $r$ faces}

For an SP tire whose $O^{(2)}$ has $r$ all-$3$ faces arranged cyclically
$F_0, F_1, \dots, F_{r-1}$ on $B_{\mathrm{in}}^{(2)}$, the $\gamma$-edge
classification gives:
\begin{itemize}
  \item $2r$ internal $\gamma$-edges (two per face),
  \item $r$ boundary $\gamma$-edges (one per cyclically-adjacent pair
        $(F_i, F_{i+1})$),
\end{itemize}
for a total of $3r$ $\gamma$-edges, which is exactly $|\gamma| = k$
when $k = 3r$ (i.e.\ $k_2 = k$, the symmetric case).

\begin{prop}[General structural description]
\label{prop:general}
For the symmetric case $k = k_2$ with $r$ all-$3$ $O^{(2)}$-faces, the
triple-partitions $\widetilde{\mathcal{F}_2}$ satisfying
$\mathcal{L}(\gamma, \widetilde{\mathcal{F}_2}) \subseteq \pi_U(T_2)$
are exactly those of the form:
\begin{itemize}
  \item Block $b_{i, i+1}$ for each adjacent pair, containing
        boundary $\delta_{i, i+1}$, one internal of $F_i$, one
        internal of $F_{i+1}$,
  \item subject to: for each face $F_i$, its two internals are
        distributed between the two blocks $b_{i-1, i}$ and
        $b_{i, i+1}$ (one per block).
\end{itemize}
Count: $2^r$ partitions.
\end{prop}

\textbf{Status.}  Proved (cleanly) at $k = 9$ by exhaustive verification
matching the $8 = 2^3$ count.  At $k = 6$ ($r = 2$), the proposition
predicts $2^2 = 4$ ``structural'' partitions, which is a strict
subset of the $10$ triple-partitions whose Latin sets fit in
$\pi_U(T_2)$; the extra $6$ are absorbed by the relatively large
$|\pi_U(T_2)| = 90$ at $k = 6$.  At $k \geq 9$ we expect (and
observe at $k = 9$) that the structural partitions are the only
ones working.

\subsection*{Open: prove the proposition for all $r$}

Prop.\ \ref{prop:general} is currently empirical-only.  A clean proof
would presumably show:

\begin{enumerate}
  \item \emph{Necessity}: a Latin partition not of the face-pair-
        connection form contains a $\sigma$ that violates a
        proper-edge-coloring constraint inside $T'_{f'}$.
  \item \emph{Sufficiency}: each of the $2^r$ structural partitions'
        Latin sets is realisable as $\pi_U$-projections, by an
        explicit construction lifting a structural assignment of
        colors at internal and boundary $\gamma$-edges into a proper
        coloring of $T'_{\mathrm{ann}}$ + spokes.
\end{enumerate}

\section*{Implications for the K\"onig lift}

The worst-case note's conjecture (\emph{t2-induces-partition}) was
that the induced $\gamma$-partition is unique (the next-$D$ or
prev-$D$ candidate).  The reality is that there are $2^r$ structurally
valid candidates; the candidates from the worst-case note are
\emph{not} among them (they violate the ``one internal per face per
block'' rule).

So the K\"onig-lift approach can be \emph{rescued} by replacing the
naive candidate $\widetilde{\mathcal{F}_2}$ with any of the $2^r$
face-pair-connection partitions, and applying the K\"onig argument
on the bipartite face-incidence graph of $\mathcal{F}_1$ versus this
new $\widetilde{\mathcal{F}_2}$.  This is the natural next step in
the program.

\end{document}
