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\title{Empirical findings on the K\"onig-lift conjecture\\
       (Conj.\ \emph{t2-induces-partition} from
       \texttt{worst\_case\_proof\_sketch.tex})}
\author{}
\date{}

\newtheorem*{obs}{Observation}

\begin{document}
\maketitle

\section*{What was tested}

The K\"onig-lift approach to chain pigeonhole
(\texttt{worst\_case\_proof\_sketch.tex}) conjectures that for an SP
tire $T_2$ with $B_{\mathrm{in}}^{(2)}$-chord structure such that
every $O^{(2)}$-face has exactly $3$ $B_{\mathrm{in}}^{(2)}$-edges,
there is an \emph{induced} face partition $\widetilde{\mathcal{F}_2}$
of $\gamma$ into triples with
\[
   \pi_U(\mathcal{C}(T_2)) \supseteq
   \mathcal{L}(\gamma, \widetilde{\mathcal{F}_2}).
\]
The candidate construction (worst-case note, ``one concrete
attempt'') groups $\gamma$-edges by the $O^{(2)}$-face whose
$D$-triangle is cyclically next (or previous) on $T'_{\mathrm{ann}}$.

Script: \texttt{experiments/induced\_partition.py}.

\section*{Findings}

\begin{center}
\small
\begin{tabular}{lll|cc|cc}
\toprule
$k = |\gamma|$ & $k_2 = |B_{\mathrm{in}}^{(2)}|$ & chords on $B_{\mathrm{in}}^{(2)}$
  & $|\pi_U|$ & $3^k$
  & candidate 1 OK? & candidate 2 OK? \\
\midrule
$6$ & $6$ & $(0,3)$  (faces $3{+}3$)
  & $90$ & $729$ & \textbf{Yes} & \textbf{Yes}\\
$6$ & $6$ & $(0,2)(3,5)$ (faces $2{+}2{+}2$)
  & $456$ & $729$ & N/A (not triples) & N/A\\
$6$ & $3$ & none (face $3$, single)
  & $84$ & $729$ & N/A (not triples) & N/A\\
$9$ & $9$ & $(0,3)(3,6)$ (faces $3{+}3{+}3$)
  & $978$ & $19683$ & \textbf{No} & \textbf{No}\\
\bottomrule
\end{tabular}
\end{center}

\subsection*{(a) The candidate works at $k = 6$}

For $k = k_2 = 6$ antipodal chord (the symmetric all-$3$-faces case),
both candidate $1$ (next-$D$) and candidate $2$ (prev-$D$) produce
the same kind of $\gamma$-partition (a two-block partition of size
$3$ each), and the Latin subset $\mathcal{L}(\gamma, \widetilde{
\mathcal{F}_2})$ of size $36$ is verified $\subseteq \pi_U$.

In fact \emph{every} two-block triple-partition of
$\{0, \ldots, 5\}$ (there are $10$ such) has Latin subset
$\subseteq \pi_U$ here, because $|\pi_U| = 90$ is much larger than
$36$ and absorbs all of them.

\subsection*{(b) The candidate \emph{fails} at $k = 9$}

This is the surprise.  For $k = k_2 = 9$ with chords $(0,3),(3,6)$
producing three $O^{(2)}$-faces of size $3$ each:
\begin{itemize}
  \item Candidate $1$ (next-$D$):
        $\widetilde{\mathcal{F}_2} = \{\{0,1,8\},\;\{2,3,4\},\;\{5,6,7\}\}$.
        $|\mathcal{L}| = 216$, but $\mathcal{L} \not\subseteq \pi_U$
        (some Latin elements are missing).
  \item Candidate $2$ (prev-$D$):
        $\widetilde{\mathcal{F}_2} = \{\{0,1,2\},\;\{3,4,5\},\;\{6,7,8\}\}$.
        $|\mathcal{L}| = 216$, but $\mathcal{L} \not\subseteq \pi_U$.
  \item Of all $280$ possible triple-partitions of $\{0, \ldots, 8\}$,
        only $8$ have $\mathcal{L} \subseteq \pi_U$.
\end{itemize}

The eight surviving partitions are not contiguous blocks.  Examples:
$\{\{0,2,3\},\{1,6,8\},\{4,5,7\}\}$,
$\{\{0,2,4\},\{1,6,8\},\{3,5,7\}\}$, etc.  They do not have an
obvious geometric interpretation in terms of $T_2$'s annular
triangulation.

\subsection*{(c) The asymmetric case ($k \ne k_2$) is outside scope}

For $k = 6, k_2 = 3$ (the configuration with $T_2 = (3, -, \mathrm{SP})$),
the candidate construction collapses to a single block of $6$
$\gamma$-edges (since there is only one $O^{(2)}$-face), so it is not a
triple-partition.  Moreover, \emph{no} triple-partition of
$\{0, \ldots, 5\}$ has Latin subset $\subseteq \pi_U$ here.

So Conj.\ \emph{t2-induces-partition} as currently stated does not
cover $k \ne k_2$, and the empirical data shows there is no
``rescue'' partition of any kind.

\section*{Implications}

\subsection*{The K\"onig lift's natural construction breaks past $k = 6$}

The candidate $\widetilde{\mathcal{F}_2}$ from the worst-case note is
the geometrically natural one (group $\gamma$-edges by their nearest
$O^{(2)}$-face $D$-triangle), and it succeeds at $k = 6$ partly by
coincidence: $|\pi_U|$ is so large that every triple-partition fits.
At $k = 9$ the gap between $|\pi_U|$ and $3^k$ widens, and the
candidate's specific partition is no longer in the small set of
``correct'' partitions.

The fact that only $8 / 280$ partitions work at $k = 9$ suggests
that whatever the right $\widetilde{\mathcal{F}_2}$ is, it is
\emph{not} just a function of $T_2$'s outerplanar face structure ---
it must encode finer information about the annular triangulation.

\subsection*{Asymmetric pairs not covered at all}

The empirical worst-case overlap $|S_1 \cap S_2| = 6$ in step-$2$
data comes from \emph{asymmetric} pairs (e.g.\
$T_1 = (6, (0,3), \mathrm{SP})$ vs $T_2 = (3, -, \mathrm{SR})$)
where $k \ne k_2$.  Even if the K\"onig lift were proved for the
symmetric case, it would not handle the asymmetric pairs that
witness the worst case.

\subsection*{Step 3 (proof) is not the right next move}

Plan-step 3 from \texttt{two\_approaches\_comparison.tex} was
``prove inclusion via transfer matrix / fibre lifting,'' assuming
the candidate partition was empirically correct.  The candidate is
\emph{not} empirically correct beyond $k = 6$, so trying to prove
the wrong statement is futile.  Instead the right next move is to
\textbf{study the $8$ surviving triple-partitions at $k = 9$} and
look for a common structural description (e.g.\ via the $T_2$ annular
triangulation, $T'_{\mathrm{ann}}$ cyclic distance, or modular
arithmetic on $D$- vs $U$-positions).  If no such description exists,
the ``$\widetilde{\mathcal{F}_2}$ is a partition'' framing should be
abandoned and a different structure on $\gamma$ sought.  This is the
new step done next; see \texttt{notes/k9\_surviving\_partitions.tex}.

\section*{Reassessment of Approach 2}

Approach 2 (K\"onig lift) was preferred in
\texttt{two\_approaches\_comparison.tex} on the grounds that ``the
hard step is already proven, only the induced-partition piece is
conjectural.''  These findings show:

\begin{itemize}
  \item The induced-partition piece is \emph{not} just conjectural ---
        the specific construction in the worst-case note is
        \emph{wrong} for $k > 6$.
  \item The K\"onig-overlap proposition (when both tires give direct
        $\gamma$-face partitions) is still cleanly proved; it just
        applies to fewer cases than was hoped.
\end{itemize}

\subsection*{Updated ranking}

Both approaches now have known structural obstacles:

\begin{itemize}
  \item \textbf{Approach 1 (2-SAT, \texttt{rainbow\_proof.tex})}:
        single open conjecture (2-SAT solvability), empirically true
        for all tested $\sigma \in \mathcal{P}_m$ at $m \in \{4, 6\}$.
        Limited to $m \in \{4, 6\}$ (SP feasibility) but at least
        empirically holds throughout that range.
  \item \textbf{Approach 2 (K\"onig lift,
        \texttt{worst\_case\_proof\_sketch.tex})}: K\"onig-overlap
        prop proved, but the natural induced-partition construction
        is empirically wrong at $k = 9$.  Asymmetric pairs (where
        the worst case actually lives) are not covered at all.
\end{itemize}

Both approaches give partial structural results.  Neither closes
the chain-pigeonhole step in its full generality.  The honest
status: chain pigeonhole has no full proof yet, and both attempted
attacks have specific empirical limits.

\end{document}
