Run Heawood pigeonhole between nested connected tire clusters

Add the two-sided cluster decomposition proposition: a vertex's full
Heawood face-sum splits as exactly one child-cluster contribution plus
one parent-cluster contribution (the at-most-two-clusters bound makes the
pairing binary and complete). Explain why this fails per-tire -- a vertex
on many same-depth tires has only a fragment of its face-star in any one
tire -- and recast the chain-pigeonhole and 4CT conjectures to nested
clusters with a cluster restriction relation.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex

@@ -303,34 +303,130 @@ $\{+1,-1\}$ face-labelling of $G$ satisfying
 proper $4$-vertex-colouring of $G$.
 \end{remark}
 
+\subsection*{Why the programme runs between nested clusters}
+
+The vanishing condition \eqref{eq:heawood-face-sum-dual} at a vertex $v$
+is a constraint on the \emph{full} face-star of $v$.  To run a
+pigeonhole between two objects --- a child and a parent --- we need that
+full sum to split as exactly two one-sided contributions, so that each
+vertex label is the combination of a single child value and a single
+parent value.  This is true at the level of connected tire clusters, and
+\emph{false} at the level of individual tires.  Extend a Heawood
+face-labelling to a connected tire cluster $\mathsf{K}$ by labelling
+every annular face of every tire of $\mathsf{K}$, and for $v \in
+V(\mathsf{K})$ write
+\[
+   \lambda^{\!*}_{\mathsf{K}}(v) \;:=\;
+   \sum_{f} \lambda(f) \;\bmod 3,
+\]
+the sum over the annular faces of $\mathsf{K}$ incident to $v$.
+
+\begin{proposition}[Two-sided cluster decomposition at a vertex]
+\label{prop:two-sided-decomposition}
+Let $v \in V(G)$ have level $\ell = \ell_G(v)$, and let
+$\mathsf{K}_{\ell}$ and $\mathsf{K}_{\ell-1}$ be the at most two
+connected tire clusters containing $v$, of depths $\ell$ and $\ell-1$
+respectively (Proposition~\ref{prop:two-clusters-per-vertex}).  Then the
+bounded faces of $G$ incident to $v$ partition into the annular faces of
+$\mathsf{K}_{\ell}$ at $v$ and the annular faces of $\mathsf{K}_{\ell-1}$
+at $v$, and
+\[
+   \sum_{f \ni v} \lambda(f)
+   \;\equiv\;
+   \lambda^{\!*}_{\mathsf{K}_{\ell}}(v) +
+   \lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v)
+   \pmod 3 .
+\]
+Each one-sided value $\lambda^{\!*}_{\mathsf{K}_d}(v)$ is the
+\emph{complete} sum over all depth-$d$ faces at $v$, so the Heawood
+condition \eqref{eq:heawood-face-sum-dual} at $v$ reads
+\[
+   \lambda^{\!*}_{\mathsf{K}_{\ell}}(v) +
+   \lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v) \;\equiv\; 0 \pmod 3 ,
+\]
+a pairing between the single child cluster $\mathsf{K}_{\ell}$ and the
+single parent cluster $\mathsf{K}_{\ell-1}$.  (When $\ell = 0$, or when
+$v$ bounds no depth-$\ell$ face, only one term is present.)
+\end{proposition}
+
+\begin{proof}
+By Proposition~\ref{prop:two-clusters-per-vertex} (Step~1) every bounded
+face incident to $v$ has depth $\ell-1$ or $\ell$, partitioning the
+incident faces by depth; by Step~2 all depth-$\ell$ faces at $v$ lie in
+the single cluster $\mathsf{K}_{\ell}$ and all depth-$(\ell-1)$ faces at
+$v$ in $\mathsf{K}_{\ell-1}$.  Hence the depth-$\ell$ part is exactly the
+annular faces of $\mathsf{K}_{\ell}$ at $v$, the depth-$(\ell-1)$ part
+those of $\mathsf{K}_{\ell-1}$, and summing $\lambda$ over the two parts
+gives the identity; \eqref{eq:heawood-face-sum-dual} is its vanishing.
+\end{proof}
+
+\begin{remark}[Failure at the tire level]
+\label{rem:why-clusters}
+Proposition~\ref{prop:two-sided-decomposition} is what makes the binary
+parent/child pairing possible, and it requires the cluster.  A vertex
+$v$ may lie on many depth-$\ell$ tires --- the unbounded case of
+Section~\ref{sec:tire-clusters} --- and the per-tire value
+$\lambda^{\!*}(v)$ of Definition~\ref{def:heawood-labelling} then records
+only the faces of \emph{one} tire at $v$, a fragment of $v$'s face-star.
+No single child tire carries the complete depth-$\ell$ sum, so the label
+$\sum_{f \ni v}\lambda(f)$ cannot be written as one child value plus one
+parent value, and per-tire compatibility
+(Definition~\ref{def:heawood-compatible}) fails to assemble to
+\eqref{eq:heawood-face-sum-dual}.  Clustering repairs this:
+Proposition~\ref{prop:two-clusters-per-vertex} guarantees exactly one
+cluster meets $v$ on each side, so $\lambda^{\!*}_{\mathsf{K}_{\ell}}(v)$
+is the complete child contribution and
+$\lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v)$ the complete parent
+contribution.  Every vertex label is then realised as the combination of
+a single child-cluster value with a single parent-cluster value, and the
+pigeonhole programme below chains \emph{nested connected tire clusters}
+rather than individual tires.
+\end{remark}
+
+We write $R_{\mathsf{K}}$ for the \emph{cluster Heawood restriction
+relation}: the set of (outer, inner) boundary Heawood sequence pairs
+realisable by a face-labelling of $\mathsf{K}$, defined as in
+Definition~\ref{def:boundary-sequences} but with the outer and inner
+boundaries of the cluster and the complete one-sided values
+$\lambda^{\!*}_{\mathsf{K}}$ in place of a single tire's, read up to
+rotation and global sign-flip.  By
+Proposition~\ref{prop:two-sided-decomposition} two nested clusters are
+compatible along their shared interface exactly when the inner sequence
+of the parent is the pointwise negation mod $3$ of the outer sequence of
+the child (after the orientation reversal of
+Definition~\ref{def:heawood-compatible}).
+
 \begin{conjecture}[Heawood chain-pigeonhole principle]
 \label{conj:heawood-chain-pigeonhole}
 There is a function $N(k)$ such that the following holds.  Let
 \[
-   T_0 \supset T_1 \supset \cdots \supset T_{N(k)}
+   \mathsf{K}_0 \supset \mathsf{K}_1 \supset \cdots \supset
+   \mathsf{K}_{N(k)}
 \]
-be a nested chain of tires in $\mathcal{T}(G, S)$ whose shared interface
-cycles have length at most $k$.  Then two adjacent Heawood restriction
-relations $R_{T_i}, R_{T_{i+1}}$ in the chain admit compatible
-face-labellings along their shared interface
-(Definition~\ref{def:heawood-compatible}), after rotation and global
-sign-flip.  Equivalently, the chain contains a local gluing step that
-cannot be obstructed by disjoint Heawood boundary restrictions.
+be a nested chain of connected tire clusters in $\mathcal{T}(G, S)$ whose
+shared interfaces have length at most $k$.  Then two adjacent cluster
+restriction relations $R_{\mathsf{K}_i}, R_{\mathsf{K}_{i+1}}$ in the
+chain admit compatible face-labellings along their shared interface,
+after rotation and global sign-flip.  Equivalently, the chain contains a
+local gluing step that cannot be obstructed by disjoint Heawood boundary
+restrictions.
 \end{conjecture}
 
-\begin{conjecture}[Heawood tire route to the Four Colour Theorem]
+\begin{conjecture}[Heawood cluster route to the Four Colour Theorem]
 \label{conj:heawood-route-fct}
 For every plane triangulation $G$ and every level source $S$, the
-Heawood restriction relations $\{R_T : T \in \mathcal{T}(G, S)\}$ admit
-a selection of face-labellings that is compatible along every interface
-of the tire tree.  By Remark~\ref{rem:compat-is-heawood} this yields a
-$\{+1,-1\}$ face-labelling of $G$ satisfying
-\eqref{eq:heawood-face-sum-dual}, hence $G$ is properly
-$4$-vertex-colourable.
+cluster Heawood restriction relations
+$\{R_{\mathsf{K}} : \mathsf{K} \text{ a connected tire cluster}\}$ admit
+a selection of face-labellings that is compatible along every cluster
+interface.  By Proposition~\ref{prop:two-sided-decomposition} and
+Remark~\ref{rem:compat-is-heawood} this yields a $\{+1,-1\}$
+face-labelling of $G$ satisfying \eqref{eq:heawood-face-sum-dual}, hence
+$G$ is properly $4$-vertex-colourable.
 \end{conjecture}
 
-%% TODO: realisability of $R_T$ per tire; counting / pigeonhole bound
-%% giving $N(k)$; orientation/reversal bookkeeping on $\gamma$.
+%% TODO: realisability of $R_{\mathsf{K}}$ per cluster; counting /
+%% pigeonhole bound giving $N(k)$; orientation/reversal bookkeeping on
+%% the shared interface.
 
 \begin{thebibliography}{9}