From 1513922decc32d6d9693fa2dca460b913a49243c Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 12:58:20 -0400 Subject: [PATCH] coloring_nested_tire_graphs: partial proof of closed-chain non-emptiness identifies the 4CT-equivalent gap MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../notes/nonemptiness_partial_proof.aux | 8 + .../notes/nonemptiness_partial_proof.log | 376 ++++++++++++++++++ .../notes/nonemptiness_partial_proof.pdf | Bin 0 -> 201148 bytes .../notes/nonemptiness_partial_proof.tex | 214 ++++++++++ 4 files changed, 598 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.log create mode 100644 papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.tex diff --git a/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.aux b/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.aux new file mode 100644 index 0000000..4112b3f --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/nonemptiness_partial_proof.aux @@ -0,0 +1,8 @@ +\relax 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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}