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\title{Closed-chain non-emptiness: a partial proof and the remaining gap}
\author{}
\date{}

\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[lemma]{Corollary}
\newtheorem*{conj}{Conjecture}

\begin{document}
\maketitle

\section*{What this note records}

In \texttt{absorption\_proof.tex} we proved (via the $K_3$-walk parity
invariant) that the forward-propagated state at $L_n$ of a closed
SR + PDS chain ending at the outer triangle is contained in the $6$
permutations of $\{1, 2, 3\}$.  This note attempts to prove the
remaining ``non-emptiness'' half: that the state is non-empty (and
hence, by $S_3$ invariance, equals all $6$).

\textbf{Result so far:} the proof closes \emph{conditionally} on a
specific structural property of $T_n$ that holds empirically but I
have not proven in general.

\section*{What's easy}

\begin{lemma}[$S_3$-closure preserved by chain propagation]
\label{lem:s3-closure}
For any SR chain $T_1 | \dots | T_n$ starting from a degenerate-inner
$T_1$, the forward-propagated state at every $L_i$ is closed under
the $S_3$ action on colours.
\end{lemma}

\begin{proof}
Each tire's joint support $\Pi_{T_i}$ is determined by the proper
edge $3$-colourings of $T_i$'s annular dual cycle.  The $S_3$ action
on the three colour labels acts uniformly on $C_{n_i}$-colourings,
so $\Pi_{T_i}$ is closed under the diagonal $S_3$ action.  Forward
propagation $\text{state}_{i+1} = \{ \sigma_U : \exists \sigma_D \in
\text{state}_i, (\sigma_U, \sigma_D) \in \Pi_{T_{i+1}}\}$ commutes
with $S_3$: if $\sigma$ is in $\text{state}_{i+1}$ via some
$\sigma_D \in \text{state}_i$, then $g \cdot \sigma$ is in
$\text{state}_{i+1}$ via $g \cdot \sigma_D \in \text{state}_i$
(using $S_3$-closure of $\text{state}_i$ inductively).
\end{proof}

\begin{theorem}[Conditional non-emptiness $\Rightarrow$ exactly 6]
\label{thm:six-or-zero}
The forward-propagated state at $L_n$ is either empty, or equals all
$6$ permutations of $\{1, 2, 3\}$.
\end{theorem}

\begin{proof}
By the parity invariant (proved in
\texttt{absorption\_proof.tex}), $\text{state}(L_n) \subseteq$
permutations of $\{1,2,3\}$.  The $6$ permutations form a single
$S_3$-orbit (the $S_3$ action on length-$3$ tuples with $3$ distinct
colours is transitive).  By Lem.~\ref{lem:s3-closure},
$\text{state}(L_n)$ is $S_3$-closed.  An $S_3$-closed subset of a
single $S_3$-orbit is either empty or the whole orbit.
\end{proof}

So non-emptiness $\Leftrightarrow$ state $= 6$ permutations.

\section*{Non-emptiness for intermediate steps}

\begin{lemma}[Saturation preserves non-emptiness in the outward
direction]
\label{lem:saturation-nonempty}
If $T_{i+1}$ has $m_{i+1} \geq k_{i+1}$ (the outward-PDS condition),
then $\text{state}(L_{i+1})$ is non-empty whenever $\text{state}(L_i)$
is non-empty.
\end{lemma}

\begin{proof}
By the spread-projection saturation theorem from step 1, when $m_{i+1}
\geq k_{i+1}$ the $\sigma_D$-projection of $\Pi_{T_{i+1}}$ equals all of
$\{1, 2, 3\}^{k_{i+1}}$.  Hence
$\text{state}(L_i) \subseteq \{1,2,3\}^{k_{i+1}} = \sigma_D
\text{-projection of } T_{i+1}$, so every $\sigma_D \in
\text{state}(L_i)$ is in some pair of $\Pi_{T_{i+1}}$.  Thus
$\text{state}(L_{i+1}) \ni \sigma_U$ for at least one $\sigma_U$
paired with some $\sigma_D$ in the state.  Since
$\text{state}(L_i) \neq \emptyset$, $\text{state}(L_{i+1}) \neq
\emptyset$.
\end{proof}

\begin{corollary}[Open-chain non-emptiness]
\label{cor:open-nonempty}
For any prefix $T_1 | T_2 | \dots | T_j$ of an outward-PDS chain with
$T_1$ degenerate-inner and each subsequent $T_{i+1}$ satisfying
$m_{i+1} \geq k_{i+1}$, the state at $L_j$ is non-empty.
\end{corollary}

\begin{proof}
$\text{state}(L_1)$ equals the σ-set from proper edge $3$-colourings
of $C_{m_1}$, which has $2^{m_1} + 2(-1)^{m_1} > 0$ elements for
$m_1 \geq 3$.  Iterate Lem.~\ref{lem:saturation-nonempty}.
\end{proof}

\section*{The remaining piece: the final step at $T_n$}

The final step $T_n$ has $m_n = 3$ (outer triangle).  Since $|L_{n-1}|
= k_n \geq 3$ in any non-trivial PDS, typically $k_n \geq 3 = m_n$
and the inequality is reversed.

When $m_n < k_n$ the saturation theorem fails: $T_n$'s
σ$_D$-projection is a proper subset of $\{1,2,3\}^{k_n}$.  Concretely:

\begin{center}
\small
\begin{tabular}{c r r r r}
\toprule
$k$ & $|\Pi_{T_n}|$ & $|\sigma_D\text{-proj}|$ & $|\text{parity set}|$ & $|\sigma_D \cap \text{parity}|$ \\
\midrule
3  & 63   & 27   & 6  & 6 \\
5  & 255  & 171  & 60 & 42 \\
6  & 510  & 384  & 183 & 90 \\
9  & 4095 & 3681 & 4920 & 840 \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Key observation.}
Computationally, the set of $\sigma_D$'s in $T_n$'s
$\sigma_D$-projection that are \emph{also} parity-matching equals
exactly the set of $\sigma_D$'s that pair with a permutation $\sigma_U$
under $\Pi_{T_n}$.  Call this set the ``perm-paired'' subset.

For non-emptiness at $L_n$ we need: \textbf{$\text{state}(L_{n-1})$
intersects the perm-paired subset of $T_n$}.

\section*{Restatement of the remaining gap}

\begin{conj}[Perm-paired reachability]
\label{conj:perm-reach}
For any SR + outward-PDS chain $T_1 | \dots | T_{n-1}$ with $T_1$
degenerate-inner, the forward-propagated state at $L_{n-1}$ contains
at least one $\sigma_D$ that is in the perm-paired subset of $T_n =
(3, k_n)$ (i.e., a $\sigma_D$ such that $(\sigma_U, \sigma_D) \in
\Pi_{T_n}$ for some permutation $\sigma_U$).
\end{conj}

This is the only remaining gap.  Empirically Conj.~\ref{conj:perm-reach}
holds in every tested chain; theoretically I do not yet have a proof.

\subsection*{What I know about Conj.~\ref{conj:perm-reach}}

\begin{enumerate}
\item By the parity invariant, $\text{state}(L_{n-1}) \subseteq$
      parity-matching set on $L_{n-1}$ (size 60 at $k = 5$).
\item The perm-paired subset has size $\leq |\text{parity-matching}|$
      and is generally strictly smaller (at $k = 5$, $42 < 60$).
\item Both are $S_3$-closed; both are unions of $S_3$-orbits of size $6$
      (no constant orbits, since constants violate parity).
\item Empirically, at sufficiently late stages, $\text{state}(L_{n-1})
      = $ full parity-matching set.  Since the perm-paired subset is
      strictly contained in the parity-matching set, the intersection
      is the perm-paired subset itself (non-empty).
\item A clean proof of Conj.~\ref{conj:perm-reach} would seem to
      require either (a) showing chain state always equals the full
      parity-matching set at $L_{n-1}$, or (b) an explicit
      construction of a perm-paired $\sigma_D$ reachable through any
      outward-PDS chain.
\end{enumerate}

\subsection*{Why it's not immediate from saturation}

Lem.~\ref{lem:saturation-nonempty} preserves \emph{some} non-empty
state, but doesn't characterise \emph{which} σ values are in the state.
A more refined statement is needed: state at each $L_i$ equals the
full parity-matching set (or at least a strictly-larger-than-empty
subset of perm-paired$_{T_n}$).

\subsection*{Why a Tait+4CT reduction is circular}

In general, ``state at $L_n$ non-empty'' is equivalent to the chain's
underlying cubic planar graph $G'$ admitting a proper edge
$3$-colouring, which by Tait's theorem is equivalent to $G$ being
$4$-colourable.  So Conj.~\ref{conj:perm-reach} for arbitrary
outward-PDS chains under SR is essentially the 4CT itself (or rather,
4CT restricted to graphs admitting SR + outward-PDS decompositions).

This means Conj.~\ref{conj:perm-reach} cannot be proven by invoking
4CT --- but a \emph{structural} proof of it, independent of 4CT,
\emph{would constitute} a new proof of 4CT (under the SR + PDS
modelling assumption).

\section*{Summary}

\begin{itemize}
  \item \textbf{Proven:} state at $L_n$ is either empty or equals
        all 6 permutations of $\{1,2,3\}$ (Thm.~\ref{thm:six-or-zero}).
  \item \textbf{Proven:} non-emptiness propagates through all
        intermediate tires under outward PDS
        (Cor.~\ref{cor:open-nonempty}).
  \item \textbf{Conjectured (Conj.~\ref{conj:perm-reach}):}
        non-emptiness propagates through the final tire $T_n$.
        Empirically true; structural proof would imply 4CT.
\end{itemize}

The clean conclusion: \textbf{the chain-pigeonhole story under SR + PDS
reduces to one specific reachability conjecture about chain states
hitting the perm-paired subset of the final tire}.  This is the
sharpest version of the 4CT obstruction in our framework.

\end{document}
