coloring_nested_tire_graphs: partial proof of closed-chain non-emptiness identifies the 4CT-equivalent gap
Attempts to prove item 1 (non-emptiness of state at L_n in closed
SR+PDS chains ending at outer triangle). Results:
PROVEN:
- S_3-closure preserved by chain propagation.
- State at L_n is either empty OR equals all 6 permutations of {1,2,3}
(the only non-empty S_3-closed subset of permutations).
- Non-emptiness propagates through intermediate tires under outward
PDS via step-1 saturation.
REMAINING GAP (conjecture, empirically true): state at L_{n-1}
intersects the "perm-paired" subset of T_n's σ_D-projection (the
σ_D values that pair with permutation σ_U). At the final step T_n
has m_n=3 < k_n, so saturation fails — chain state at L_{n-1} could
in principle lie entirely in the (non-perm-paired) parity-matching
σ_D's, but empirically doesn't.
KEY STRUCTURAL FINDING: for T=(3, k), the σ_D's paired with a
permutation σ_U equal exactly the (parity-matching σ_D's) ∩ (T's
σ_D-projection). Verified for k=3..10.
HONEST OBSERVATION: a structural proof of the remaining conjecture
(without invoking 4CT) would constitute a new proof of 4CT under
the SR+PDS modelling assumption. The chain-pigeonhole framework
reduces to this single reachability question.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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l.127 \bottomrule
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! LaTeX Error: Unicode character σ (U+03C3)
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See the LaTeX manual or LaTeX Companion for explanation.
|
||||||
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Type H <return> for immediate help.
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...
|
||||||
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|
||||||
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l.177 ... but doesn't characterise \emph{which} σ
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values are in the state.
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|
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\DeclareUnicodeCharacter
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[2]
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Overfull \hbox (4.61479pt too wide) in paragraph at lines 201--204
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[]\OT1/cmr/bx/n/10.95 Proven: \OT1/cmr/m/n/10.95 non-emptiness prop-a-gates thr
|
||||||
|
ough all in-ter-me-di-ate tires un-der out-ward PDS (Cor. 3[]).
|
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[]
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[3] (./nonemptiness_partial_proof.aux) )
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\documentclass[11pt]{article}
|
||||||
|
\usepackage{amsmath,amssymb,amsthm}
|
||||||
|
\usepackage{graphicx}
|
||||||
|
\usepackage{geometry}
|
||||||
|
\geometry{margin=1in}
|
||||||
|
|
||||||
|
\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}
|
||||||
Reference in New Issue
Block a user