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Hence $[](\ +OML/cmm/m/it/10.95 L[]\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^R f\OT1/cmr/m/ +n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; \OT1/c +mr/m/n/10.95 3\OMS/cmsy/m/n/10.95 g[] \OT1/cmr/m/n/10.95 = \OML/cmm/m/it/10.95 +^^[[][]T[]$\OT1/cmr/m/n/10.95 , + [] + +[1 + +{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] + +! LaTeX Error: Unicode character σ (U+03C3) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.103 $\text{state}(L_1)$ equals the σ + -set from proper edge $3$-colourings +You may provide a definition with +\DeclareUnicodeCharacter + + +! LaTeX Error: Unicode character σ (U+03C3) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.115 σ + $_D$-projection is a proper subset of $\{1,2,3\}^{k_n}$. Concretely: +You may provide a definition with +\DeclareUnicodeCharacter + +! Undefined control sequence. +l.120 \toprule + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. + \midrule + +l.122 \midrule + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. + \bottomrule + +l.127 \bottomrule + +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Unicode character σ (U+03C3) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.177 ... but doesn't characterise \emph{which} σ + values are in the state. +You may provide a definition with +\DeclareUnicodeCharacter + +[2] +Overfull \hbox (4.61479pt too wide) in paragraph at lines 201--204 +[]\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[]). + [] + +[3] (./nonemptiness_partial_proof.aux) ) +Here is how much of TeX's memory you used: + 3210 strings out of 478268 + 47563 string characters out of 5846347 + 351987 words of memory out of 5000000 + 21398 multiletter control sequences out of 15000+600000 + 482094 words of font info for 77 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 55i,10n,62p,243b,218s stack positions out of 10000i,1000n,20000p,200000b,200000s +{/usr/local/texlive/2022/texmf-dist/fon +ts/enc/dvips/cm-super/cm-super-ts1.enc} +Output written on nonemptiness_partial_proof.pdf (3 pages, 201148 bytes). +PDF statistics: + 100 PDF objects out of 1000 (max. 8388607) + 60 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.pdf b/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.pdf new file mode 100644 index 0000000..a2c253b Binary files /dev/null and b/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.tex b/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.tex new file mode 100644 index 0000000..dbbd4ef --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.tex @@ -0,0 +1,214 @@ +\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}