coloring_nested_tire_graphs: add looser chain-pigeonhole conjecture to cut_depth_label

Adds a "Looser chain pigeonhole hypothesis" section to
cut_depth_label.tex, between Cut tires and the
"Connection to chain pigeonhole" section.

The conjecture says: for every non-trivial cut tire T, the joint
projection π(T) is non-empty, S_3-closed, and contains at least
one full S_3-orbit (so |π(T)| ≥ 6).  Chain composition through
T_1 → T_2 → ... preserves S_3-symmetry, so R_i contains a full
S_3-orbit, and R_0 ∩ R_1 contains a common orbit, contradicting G'
being a counterexample.

Status:
  - Per-tire half: provable via Prop 1.13 of paper.tex (2^n + 2(-1)^n
    colorings) + S_3-equivariance.
  - Chain half: open. Requires showing per-tire S_3-orbit structure
    composes coherently through the depth chain.

Weaker than the rainbow / König-lift conjectures (no requirement on
specific face boundary structure), with stronger empirical support
across all cut tires tested.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux

@@ -4,9 +4,13 @@
 \@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Relation to the existing tire framework.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}{3}{}\protected@file@percent }
-\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{5}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Justification of the per-tire half.}{4}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Justification of the chain half.}{4}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{What this would (and would not) prove.}{4}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Status.}{4}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{6}{}\protected@file@percent }
 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
-\newlabel{fig:cut-depth-label}{{1}{5}}
+\newlabel{fig:cut-depth-label}{{1}{6}}
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{6}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{7}{}\protected@file@percent }
-\newlabel{fig:cut-tire}{{2}{6}}
+\newlabel{fig:cut-tire}{{2}{7}}
-\gdef \@abspage@last{6}
+\gdef \@abspage@last{7}

papers/coloring_nested_tire_graphs/notes/cut_depth_label.log

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+\OT1/cmr/m/n/10.95 The rain-bow con-jec-ture and K^^?onig-lift con-jec-ture (fr
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+ []
+
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+ \OT1/cmr/bx/n/10.95 Jus-ti-fi-ca-tion of the per-tire half.[] \OT1/cmr/m/n/10.
+95 The per-tire half is es-sen-tially Propo-si-tion 1.13 of \OT1/cmtt/m/n/10.95
+ paper.tex\OT1/cmr/m/n/10.95 :
+ []
+
+[3]
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+ \OT1/cmr/bx/n/10.95 Sta-tus.[] \OT1/cmr/m/n/10.95 Em-pir-i-cally true for ev-e
+ry cut tire in the Holton-McKay #0 ex-am-ple (tested in \OT1/cmtt/m/n/10.95 cut
+[]tire[]conjecture[]tests.tex\OT1/cmr/m/n/10.95 ).
+ []
+
+
+Overfull \hbox (46.78696pt too wide) in paragraph at lines 310--317
 \OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O
 T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo
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papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex

@@ -218,6 +218,93 @@ under the redefinition each contribute exactly one in/out spoke):
         tire in this chain.
 \end{itemize}
 
+\section*{A looser chain pigeonhole hypothesis}
+
+The rainbow conjecture and K\"onig-lift conjecture (from
+\texttt{rainbow\_proof.tex} and
+\texttt{worst\_case\_proof\_sketch.tex}) are statements about
+specific structured face boundaries (antipodal-chord SP).  Generic
+cut tires in our example do not have those specific structures, so
+those conjectures don't apply directly.  Empirically
+(\texttt{cut\_tire\_conjecture\_tests.tex}), cut-tire projections
+are nonetheless large and well-behaved:
+
+\begin{itemize}
+  \item Every cut tire's joint projection $\pi(T) :=
+        \{(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}}) : \chi
+        \text{ proper}\}$ is $S_3$-closed (universal, follows from
+        color symmetry of proper edge $3$-colouring).
+  \item All $S_3$-orbits in $\pi(T)$ have size exactly $3$ (the
+        constant-colour orbit, when present) or $6$ (the generic
+        orbit using all $3$ colours); no size-$2$ stabilizer orbits
+        occur.
+  \item Non-trivial cut tires have $|\pi(T)| \gg 6$: e.g.\ at
+        depth~$2$ the projection has size $126 = 21 \cdot 6$
+        ($21$ full $S_3$-orbits), and at depth~$6$ size $93 = 1
+        \cdot 3 + 15 \cdot 6$.
+\end{itemize}
+
+This motivates a much weaker conjecture than the rainbow/K\"onig
+ones --- one that does not require special face-boundary structure
+and may be easier to prove:
+
+\begin{quote}
+\textbf{Loose chain pigeonhole conjecture (cut tires).}  Let
+$T = T_d^{(i, f)}$ be a non-trivial cut tire (one with at least one
+in or out spoke).  Then the joint projection $\pi(T)$ is non-empty,
+$S_3$-closed, and contains at least one full $S_3$-orbit (i.e.\
+$|\pi(T)| \geq 6$, with equality only in the degenerate case where
+$\pi(T)$ is a single non-constant $S_3$-orbit).
+
+Consequently, for the chain $T_1^{(i)} \to T_2^{(i)} \to \cdots$ on
+each side of a cut, the realisable cut-configuration set
+$\mathcal{R}_i$ contains a full $S_3$-orbit, and the intersection
+$\mathcal{R}_0 \cap \mathcal{R}_1$ contains a common $S_3$-orbit
+(of size $6$), forcing $\mathcal{R}_0 \cap \mathcal{R}_1 \ne
+\emptyset$.
+\end{quote}
+
+\paragraph{Justification of the per-tire half.}  The per-tire half
+is essentially Proposition~1.13 of \texttt{paper.tex}: a spoke-only
+cut tire (face boundary a simple cycle of length $n$) has $2^n +
+2(-1)^n$ proper edge $3$-colourings, which is $\ge 6$ whenever
+$n \ge 3$.  By $S_3$-closure, the orbit count is at least $1$,
+hence $\ge 6$ distinct projections (modulo the size-$3$ constant
+orbit edge case).  For cut tires with non-simple face boundary
+(e.g.\ $\theta(1, p, q)$-shape) similar lower bounds follow from
+the chromatic polynomial of the corresponding $D(T)$ structure (see
+the menagerie note for explicit closed forms when chord matchings
+are present).
+
+\paragraph{Justification of the chain half.}  Chain composition
+preserves $S_3$-symmetry (each step is $S_3$-equivariant), so
+$\mathcal{R}_i$ inherits $S_3$-closure from the per-tire projection.
+The non-trivial claim is that the orbit structure \emph{survives}
+composition: an $S_3$-orbit in $\pi(T_d)$ projects to some
+non-empty $S_3$-closed subset on the out-spoke side, and this
+propagates layer-by-layer.
+
+\paragraph{What this would (and would not) prove.}  If the loose
+conjecture holds, chain pigeonhole at the cut gives $\mathcal{R}_0
+\cap \mathcal{R}_1 \neq \emptyset$, which (under the minimum-
+counterexample reduction) contradicts $G'$ being a 4CT
+counterexample.  This is the same final conclusion as the
+rainbow/K\"onig route, with weaker hypotheses but stronger
+empirical support.
+
+It would \emph{not} prove the rainbow conjecture itself (which is a
+strict structural claim about the antipodal-chord SP case), nor
+the K\"onig-lift conjecture's specific construction; it would only
+recover the bottom-line chain pigeonhole result.
+
+\paragraph{Status.}  Empirically true for every cut tire in the
+Holton-McKay \#0 example (tested in
+\texttt{cut\_tire\_conjecture\_tests.tex}).  No general proof.
+The per-tire half is essentially provable via Prop~1.13 plus
+$S_3$-equivariance.  The chain half is the genuinely open question
+--- it requires showing that the per-tire $S_3$-orbit structure
+composes coherently through the depth chain.
+
 \section*{Connection to chain pigeonhole / 4CT reducibility}
 
 The procedure mirrors the $4$CT cut-and-reglue scheme