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\title{Chain pigeonhole on cut tires (pendant-redefined version)}
\author{}
\date{}

\newtheorem*{prop}{Proposition}
\newtheorem*{lem}{Lemma}
\newtheorem*{conj}{Conjecture}
\newtheorem*{obs}{Observation}

\begin{document}
\maketitle

\section*{Setup recap}

Take $G'$ the cubic planar dual of a maximal planar graph $G$.
Suppose $G'$ is a minimum counterexample to 4CT.  Pick a $6$-edge
cut $C \subseteq E(G')$ (a matching cut), form $G'_0, G'_1$ with
pendant edges as in \texttt{cut\_depth\_label.tex}, BFS-label edges
by depth from the pendants.

The crucial change under the \emph{redefined} cut tire definition:
each cut tire $T_d^{(i, f)}$ is structurally isomorphic to a partial
tire dual $D(T)$ in \texttt{paper.tex}.  Its face boundary is the
$T'_{\mathrm{ann}}$-analogue, and its labelled pendants (out spokes
at degree-$2$ vertices incident to a depth-$(d-1)$ edge; in spokes
for depth-$(d+1)$) are the analogue of $D(T)$'s leaves.

\section*{The chain pigeonhole argument}

\subsection*{Reduction by minimality}

By minimality of $G'$, each $G'_i$ (after pendant addition) is
strictly smaller than $G'$ and admits a proper $3$-edge-colouring
$\chi_i : E(G'_i) \to \{1, 2, 3\}$.  Restricting $\chi_i$ to the $6$
depth-$0$ pendants gives a boundary configuration $\sigma_i \in
\{1, 2, 3\}^6$.

A proper $3$-edge-colouring of $G'$ exists iff some $\sigma$ is
achievable as both $\sigma_0$ and $\sigma_1$, i.e.\
$\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$ where
$\mathcal{R}_i := \{\sigma_i \mid \chi_i \text{ proper}\}$.  $G'$ a
counterexample $\Rightarrow$ $\mathcal{R}_0 \cap \mathcal{R}_1 = \emptyset$.

\subsection*{Layered decomposition via cut tires}

The depth-$0$ pendants of $G'_i$ are precisely the \emph{out spokes}
of the cut tires at depth $1$ (one out spoke per depth-$0$ edge).
So $\sigma_i = \pi_{\mathrm{out}}(\{T_1^{(i, f)}\})$, where
$\pi_{\mathrm{out}}$ projects the global colouring onto all out
spokes of depth-$1$ cut tires combined.

More generally, the cut-tire chain at depths $d = 1, 2, \ldots,
d_{\max}$ partitions the edges of $G'_i$ by depth.  Each cut tire
$T_d^{(i, f)}$ has $\chi_i$-restricted colouring that:
\begin{itemize}
  \item Properly colours the face boundary (a closed walk in $H_d$),
  \item Colours out spokes (representing depth-$(d-1)$ edges in
        $G'_i$),
  \item Colours in spokes (representing depth-$(d+1)$ edges).
\end{itemize}

\subsection*{Compatibility between adjacent tire layers}

An out spoke of $T_d$ at boundary vertex $v$ represents a specific
depth-$(d-1)$ edge $e^* = vw$ in $G'_i$.  This edge $e^*$ is a
\emph{face boundary edge} of some cut tire at depth $d - 1$ (since
$e^*$ has depth $d - 1$ and so lies in $H_{d-1}$).

Therefore: the colour of an out spoke at depth $d$ equals the colour
of the corresponding face boundary edge at depth $d - 1$.

This is the chain compatibility constraint: $T_d$'s out-spoke
projection $\pi_{\mathrm{out}}(T_d) \subseteq \{1, 2, 3\}^{|\text{out
spokes}|}$ corresponds, via the bijection $\{\text{out spokes of }
T_d\} \to \{\text{specific edges on face boundaries of } T_{d-1}
\text{'s}\}$, to a projection of $T_{d-1}$'s face-boundary colouring.

Symmetrically: in spokes of $T_d$ $\leftrightarrow$ specific face-boundary edges of
some $T_{d+1}$.

\subsection*{Result inheritance via the partial-tire-dual identification}

Because each cut tire \emph{is} a partial tire dual (up to graph
isomorphism), every counting/structural result for $D(T)$ in
\texttt{paper.tex} applies directly:

\begin{itemize}
  \item \textbf{Proposition 1.13 (edge--vertex coloring bijection):}
        For spoke-only cut tires (face boundary a simple cycle of
        length $n$, all $n$ spokes pendant), the number of proper
        $3$-edge-colourings is $2^n + 2(-1)^n$.
  \item \textbf{Definitions 1.15--1.17:} The face boundary plays the
        role of $T'_{\mathrm{ann}}$; the spokes play the role of the
        tire annular face connector $T'_{f'}$ and its inner/outer
        spokes.
  \item \textbf{Rainbow conjecture (\texttt{rainbow\_proof.tex}):}
        For cut tires whose face boundary has the antipodal-chord SP
        structure, the spoke projection support equals the
        perms-per-face set $\mathcal{P}_m$, conditional on
        Conjecture~1.5.
  \item \textbf{Face-pair-connection structural description
        (\texttt{k9\_surviving\_partitions.tex}):} Cut tires with
        $r$ all-$3$ face structure admit exactly $2^r$ valid
        ``induced partitions'' for König-lift purposes.
\end{itemize}

These results provide quantitative bounds on $\pi_{\mathrm{in}}$ and
$\pi_{\mathrm{out}}$ at each layer.

\subsection*{The chain pigeonhole step}

For the chain $T_1 \to T_2 \to \cdots \to T_{d_{\max}}$ on side $i$:
\begin{itemize}
  \item At the deep interior (depth $d_{\max}$), $T_{d_{\max}}$'s in
        spokes either don't exist or terminate at the deepest
        vertex.  In the example $G'_1$ of Holton-McKay \#0,
        $T_6$ has $2$ in spokes still going somewhere, but the chain
        does eventually terminate.
  \item Each $T_d$ constrains $T_{d-1}$ and $T_{d+1}$ via the
        in/out-spoke correspondence with face boundary edges.
  \item $\mathcal{R}_i = \pi_{\mathrm{out}}(\{T_1^{(i, f)}\})$
        as restricted by the chain of compatibility conditions
        running from depth $1$ inward to $d_{\max}$.
\end{itemize}

Chain pigeonhole says: if at each layer the realisable projection
contains enough structure (e.g.\ a full $S_3$-orbit), the chain
condition propagates through and yields a non-empty $\mathcal{R}_i$
of substantial size.  If both $\mathcal{R}_0$ and $\mathcal{R}_1$
contain a common $S_3$-orbit, they intersect, contradicting that
$G'$ is a counterexample.

\section*{What's actually new under the redefinition}

\subsection*{1. Direct result transfer}

Previously (with the original cut tire definition involving incident
edges in $G'_i$): we had to re-derive each result for cut tires,
because cut tires weren't formally $D(T)$.

Now: each cut tire \emph{is} $D(T)$ for some virtual tire $T$, so
Prop 1.13, the rainbow proof, the König-lift framework, all apply
without re-derivation.

\subsection*{2. Cubicity restored at face-boundary vertices}

Each face-boundary vertex of a cut tire has degree $2$ (in $H_d$) +
$1$ pendant (in spoke or out spoke) $= 3$ in the cut tire.  So
\emph{cut tires are cubic} (at face-boundary vertices), restoring
the cubic structure that the previous definition broke (where $H_d$
was not cubic).

This means classical cubic-graph results (König, Tait, chromatic
polynomial of $L(C_n)$, etc.) apply directly.

\subsection*{3. The chain-defined structure is more rigid}

Under the previous definition, ``the cut tire'' included variable
numbers of incident edges depending on $G'_i$'s local structure.
Under the new definition, the cut tire is determined by:
\begin{itemize}
  \item The face $f$ (in $H_d$).
  \item Which of $f$'s boundary vertices are degree-$2$ in $H_d$.
  \item For each such vertex, whether its non-$H_d$ neighbour is at
        depth $d - 1$ or $d + 1$.
\end{itemize}
This is finite, combinatorial, and exactly the data needed for
chain compatibility.

\section*{What stays open}

The fundamental obstacles \emph{don't} change:

\subsection*{(a) Chain pigeonhole at each layer}

The same open conjectures (rainbow, König-lift, 2-SAT solvability)
gate the chain argument.  They now apply to cut tires directly
rather than via translation, but their statements are unchanged.

\subsection*{(b) Chain well-definedness}

\begin{itemize}
  \item Each $H_d$ must have $\ge 1$ face.  Empirically this holds in
        the Holton-McKay example at all depths $1$--$7$, but no
        general guarantee.
  \item Some $H_d$ faces have length $2$ (just a single edge of
        $H_d$ with two adjacent faces sharing it).  These ``trivial''
        cut tires have empty face boundary cycle interior and
        contribute very little structure.
  \item Some boundary vertices have degree $> 2$ in $H_d$
        (cut-vertices or branch points of $H_d$); these don't get
        pendants under the strict reading.  The cut tire then has
        fewer than $|f|$ spokes.
\end{itemize}

\subsection*{(c) Chain length and depth-by-depth variability}

In the example, the chain has variable face counts ($1$--$3$) and
face lengths ($2$--$14$) across depths.  A uniform bound on
$|\pi_{\mathrm{out}}|$ across depths still seems unlikely; the chain
would have to handle depth-by-depth structure individually.

\section*{Net assessment}

The redefinition (cut tire $=$ face boundary $+$ labelled pendants
at degree-$2$ vertices) is a \textbf{strict improvement} over the
previous formulation, because:

\begin{enumerate}
  \item It identifies each cut tire with a partial tire dual
        $D(T)$, so all results from \texttt{paper.tex} and downstream
        notes transfer directly.
  \item It restores cubicity at face-boundary vertices, which the
        previous definition broke.
  \item The data needed for chain compatibility (face $f$, degree-$2$
        boundary vertices, in/out classification) is finite and
        combinatorial.
\end{enumerate}

But the fundamental hard step is unchanged: chain pigeonhole at
each layer reduces to the same open conjectures (rainbow / König-
lift) that gate the partial-tire-dual chain pigeonhole.  The
redefinition doesn't make those easier; it makes the transfer free
of charge.

\paragraph{Concrete next step.}  Empirically verify on the $6$
Holton-McKay graphs: for each, pick a matching $6$-cut, build cut
tire chains on both sides, compute $\mathcal{R}_0$ and
$\mathcal{R}_1$, check whether $\mathcal{R}_0 \cap \mathcal{R}_1
\neq \emptyset$.  By 4CT (which we're \emph{not} assuming for the
proof) we know it does; the question is whether the cut-tire chain
machinery successfully identifies the overlap structurally, or
whether (after composition) the chain pigeonhole inequality fails
empirically.  This is the analogue of \texttt{tire\_fiber\_step2.tex}
for the cut-tire setting.

\end{document}
