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\begin{document}

\title{Heawood Restrictions on Nested Tire Graph Duals}

%    author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}

\subjclass[2010]{Primary }

\keywords{plane graph, triangulation, plane depth, level edge, dual graph,
tire graph, Heawood number}

\date{}

\dedicatory{}

\begin{abstract}
%% TODO: abstract. Following \cite{bauerfeld-nested-tires}, which establishes
%% the basic vocabulary of tire graphs and dual depth, we study the Heawood
%% (mod-3 / face-sum) restrictions imposed on the duals of nested tire graphs.
\end{abstract}

\maketitle

\section{Introduction}

A classical theorem of Tait recasts the Four Colour Theorem in dual,
edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable
if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a
minimal counterexample to the Four Colour Theorem -- a smallest triangulation
admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
admitting no proper $3$-edge-colouring.

This paper continues the series studying that structure through the
lens of \emph{nested level duals}.  The foundational vocabulary ---
level sources, levels, the inner planar dual $G'$ and its dual depth,
and tire graphs --- is developed in the companion paper
\cite{bauerfeld-nested-tires}; we refer to that paper for those
definitions and rely on them throughout.  In particular we use,
without restating, the notions of:
\begin{itemize}
  \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$;
  \item the inner planar dual $G'$
        (\cite[Definition~1.3]{bauerfeld-nested-tires});
  \item \emph{dual depth} $\delta_G(d_f)$
        (\cite[Definition~1.4]{bauerfeld-nested-tires});
  \item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$
        with outer/inner boundaries and annular edges
        (\cite[Definition~1.5]{bauerfeld-nested-tires});
  \item the \emph{tire-component lemma}
        (\cite[Lemma~1.8]{bauerfeld-nested-tires}); and
  \item the \emph{tire-tread partition theorem}
        (\cite[Theorem~1.9]{bauerfeld-nested-tires}).
\end{itemize}

Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
and $G$ has $2n - 4$ triangular faces.

The classical input is Heawood's face-sum identity \cite{Heawood1898}:
for any proper $3$-edge-colouring of a cubic plane graph $H$, assigning
each face of $H$ a number in $\{+1, -1\}$ can be done so that the labels
around every vertex of $H$ sum to $0 \pmod 3$.  In the triangulation
$G$ dual to $H$ this becomes a $\{+1, -1\}$ labelling of the
\emph{faces} of $G$ whose incident-face sum at every vertex of $G$
vanishes mod $3$.  Our aim is to record what this restriction forces
along the boundary cycles of a nested tire graph, and to formulate a
chain-pigeonhole programme in this Heawood labelling parallel to the
medial programme of \cite{bauerfeld-medial-tires}.

\section{Connected tire clusters}
\label{sec:tire-clusters}

The tire treads at a fixed depth partition the depth-$d$ faces of $G$
\cite{bauerfeld-nested-tires}, but distinct depth-$d$ tires need not be
vertex-disjoint: a single vertex of $G$ may lie on the source-side
boundary of several depth-$d$ tires at once (this occurs exactly when
the depth-$d$ faces around that vertex are split into more than one arc
by depth-$(d{-}1)$ faces).  We organise the depth-$d$ tires by this
sharing.

\begin{lemma}[Same-depth tires meet only in vertices]
\label{lem:same-depth-vertex-meet}
Let $T \neq T'$ be two distinct tire treads at the same depth $d$ in
$\mathcal{T}(G, S)$, arising from connected components $C', C''$ of the
depth-$d$ dual subgraph $G'_d$.  Then $T$ and $T'$ share no edge of $G$;
any intersection $V(T) \cap V(T')$ consists of isolated vertices.
\end{lemma}

\begin{proof}
An edge $e$ of $G$ shared by two depth-$d$ annular faces $f_1, f_2$ is,
by definition of the inner dual \cite[Definition~1.3]{bauerfeld-nested-tires},
a dual edge of $G'$ joining $d_{f_1}$ and $d_{f_2}$; since $\delta(d_{f_1})
= \delta(d_{f_2}) = d$, this edge lies in $G'_d$, so $d_{f_1}$ and
$d_{f_2}$ belong to the same component of $G'_d$.  Hence no edge of $G$
is shared by annular faces of two \emph{different} components, and
distinct depth-$d$ tires share no edge.  Their intersection is therefore
a set of isolated vertices.
\end{proof}

\begin{definition}[Connected tire cluster]
\label{def:connected-tire-cluster}
Fix a nested tire decomposition $\mathcal{T}(G, S)$ and a depth $d$.  On
the set of depth-$d$ tire treads define the relation
\[
   T \sim T' \quad\Longleftrightarrow\quad V(T) \cap V(T') \neq \varnothing .
\]
A \emph{connected tire cluster} at depth $d$ is the subgraph of $G$
\[
   \mathsf{K} \;=\; \bigcup_{i} T_i \;\subseteq\; G
\]
obtained as the union (of underlying plane graphs) of the tires in a
single connected component $\{T_i\}$ of the transitive closure of
$\sim$.  A cluster consisting of a single tire is \emph{trivial}; the
connected tire clusters at depth $d$ partition the depth-$d$ tires.
\end{definition}

\begin{remark}
\label{rem:cluster-cut-vertices}
By Lemma~\ref{lem:same-depth-vertex-meet} the constituent tires of a
connected tire cluster are joined only at shared vertices, each of which
is a cut vertex of $\mathsf{K}$; a connected tire cluster is thus a
``cactus of tires'' and is in general \emph{not} itself a tire graph,
since the annulus structure of \cite[Definition~1.5]{bauerfeld-nested-tires}
fails at each such pinch.  The shared (cut) vertices are precisely the
vertices that belong to more than one depth-$d$ tire.
\end{remark}

A single vertex may belong to several tires at one depth --- the
high-degree case where its depth-$d$ faces split into many arcs --- so
the number of \emph{tires} through a vertex is unbounded.  Clustering
collapses exactly this multiplicity: all tires through a vertex at a
fixed depth share that vertex, hence lie in one cluster.  The cluster
count is therefore controlled.

\begin{proposition}[A vertex meets at most two clusters]
\label{prop:two-clusters-per-vertex}
Every vertex $v \in V(G)$ belongs to at most two connected tire
clusters, namely at most one at each of the two consecutive depths
$\ell_G(v) - 1$ and $\ell_G(v)$.  In particular a source vertex
($\ell_G(v) = 0$) belongs to a single cluster.
\end{proposition}

\begin{proof}
Write $\ell = \ell_G(v)$.

\emph{Step 1: every bounded face incident to $v$ has dual depth
$\ell - 1$ or $\ell$.}  Let $f$ be a bounded triangular face with
$v \in V(f)$.  Then
$\delta_G(d_f) = \min_{u \in V(f)} \ell_G(u) \le \ell_G(v) = \ell$.
The other two vertices of $f$ are adjacent to $v$ in $G$, and the level
function $\ell_G(\cdot) = \mathrm{dist}_G(\cdot, S)$ is $1$-Lipschitz
along edges, so each has level at least $\ell - 1$; hence
$\delta_G(d_f) \ge \ell - 1$.  Thus $\delta_G(d_f) \in \{\ell-1, \ell\}$
(only $\delta_G(d_f) = 0$ when $\ell = 0$), so $v$ bounds faces of, and
therefore belongs to tires of, no depth other than $\ell - 1$ or
$\ell$.

\emph{Step 2: at each depth, all tires through $v$ lie in one cluster.}
Fix $d \in \{\ell-1, \ell\}$ and let $T, T'$ be depth-$d$ tires with
$v \in V(T) \cap V(T')$.  Then $V(T) \cap V(T') \ne \varnothing$, so
$T \sim T'$ in the sense of
Definition~\ref{def:connected-tire-cluster}, and all depth-$d$ tires
containing $v$ lie in a single connected component of $\sim$ --- one
connected tire cluster $\mathsf{K}_d$.

Combining the two steps, $v$ belongs to at most the clusters
$\mathsf{K}_{\ell-1}$ and $\mathsf{K}_{\ell}$, i.e.\ to at most two
connected tire clusters; when $\ell = 0$ only $\mathsf{K}_0$ occurs.
\end{proof}

\section{Heawood restrictions on the tire dual}
\label{sec:heawood-restrictions}

We work inside a fixed nested tire decomposition $\mathcal{T}(G, S)$ of
$G$ from a single-vertex level source $S$ \cite{bauerfeld-nested-tires},
and use the tire data $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ with
annular faces $F_{\mathrm{ann}}$, outer boundary $B_{\mathrm{out}}$, and
inner boundary $B_{\mathrm{in}}$
(\cite[Definition~1.5]{bauerfeld-nested-tires}).  Since $O$ is
outerplanar, every vertex of a tire lies on $B_{\mathrm{out}}$ or on the
inner-boundary walk $B_{\mathrm{in}}$; a tire has no interior vertices.

\begin{definition}[Heawood face-labelling of a tire]
\label{def:heawood-labelling}
A \emph{Heawood face-labelling} of a tire graph $T$ is a map
\[
   \lambda : F_{\mathrm{ann}} \longrightarrow \{+1, -1\}
\]
assigning a sign to each annular face of $T$.  For a vertex
$v \in V(T)$, write $F_{\mathrm{ann}}(v) \subseteq F_{\mathrm{ann}}$ for
the set of annular faces of $T$ incident to $v$, and define the
\emph{induced vertex value}
\[
   \lambda^{\!*}(v) \;:=\; \sum_{f \in F_{\mathrm{ann}}(v)} \lambda(f)
   \;\;\bmod 3 \;\in\; \{0, 1, -1\}.
\]
The value $\lambda^{\!*}(v)$ is the \emph{partial} face-sum at $v$ taken
over the annular faces of $T$ alone, not over all faces of $G$ incident
to $v$.
\end{definition}

\begin{remark}
\label{rem:no-interior-constraint}
Because a tire has no interior vertices, every annular face of $T$ is
incident to $B_{\mathrm{out}} \cup B_{\mathrm{in}}$, and a Heawood
face-labelling is subject to \emph{no} internal constraint: all
$2^{|F_{\mathrm{ann}}|}$ sign assignments are admissible.  The Heawood
restriction is felt only on the two boundary cycles, through the induced
vertex values $\lambda^{\!*}$.
\end{remark}

\begin{definition}[Induced boundary sequences]
\label{def:boundary-sequences}
Let $\lambda$ be a Heawood face-labelling of $T$.  Reading the vertices
of $B_{\mathrm{out}}$ in clockwise order $v_0, v_1, \dots, v_{p-1}$, the
\emph{outer Heawood sequence} of $(T, \lambda)$ is
\[
   \sigma_{\mathrm{out}}(T, \lambda)
   \;:=\; \bigl(\lambda^{\!*}(v_0), \dots, \lambda^{\!*}(v_{p-1})\bigr)
   \;\in\; \{0, 1, -1\}^{p}.
\]
Reading the inner-boundary walk $B_{\mathrm{in}}$ in clockwise order
$w_0, \dots, w_{q-1}$ gives the \emph{inner Heawood sequence}
$\sigma_{\mathrm{in}}(T, \lambda) \in \{0, 1, -1\}^{q}$.  The
\emph{Heawood restriction relation} of $T$ is the set
\[
   R_T \;:=\; \bigl\{\,
     \bigl(\sigma_{\mathrm{out}}(T, \lambda),\,
            \sigma_{\mathrm{in}}(T, \lambda)\bigr)
     \;:\; \lambda : F_{\mathrm{ann}} \to \{+1, -1\}
   \,\bigr\}
\]
of all (outer, inner) sequence pairs realisable by a single
face-labelling, read up to rotation and the global sign-flip
$\lambda \mapsto -\lambda$ (equivalently
$\sigma \mapsto -\sigma$).
\end{definition}

\begin{definition}[Heawood compatibility across an interface]
\label{def:heawood-compatible}
Let $T$ be a tire and $T' \in \mathcal{T}(G, S)$ a child of $T$, so the
outer boundary cycle $B_{\mathrm{out}}^{(T')}$ coincides with a bounded
face of $O^{(T)}$; let $\gamma$ be this shared cycle, of length $L$, and
let $v$ range over its vertices.  Heawood face-labellings $\lambda$ of
$T$ and $\lambda'$ of $T'$ are \emph{compatible along $\gamma$} if at
every shared vertex $v$,
\[
   \lambda^{\!*}(v) + (\lambda')^{\!*}(v) \;\equiv\; 0 \pmod 3,
\]
i.e.\ $0$ is paired with $0$ and $+1$ with $-1$.  Equivalently, the
inner Heawood sequence of $T$ on $\gamma$ is the pointwise negation
mod $3$ of the outer Heawood sequence of $T'$ on $\gamma$, after
reversing one of the two clockwise readings to account for the opposite
rotational senses in which $T$ and $T'$ traverse $\gamma$.
\end{definition}

\begin{remark}
\label{rem:compat-is-heawood}
Call $v$ \emph{interior} if it is not incident to the outer face of
$\Pi_G$.  For an interior vertex every incident face is bounded, and
compatibility along $\gamma$ at $v$ is exactly the statement that the
incident-face sum at $v$ --- over the parent's annular faces together
with the child's --- vanishes mod $3$:
\begin{equation}
\label{eq:heawood-face-sum-dual}
   \sum_{f \ni v} \lambda(f) \;\equiv\; 0 \pmod 3
   \qquad\text{for every interior vertex } v \in V(G),
\end{equation}
the sum ranging over the bounded faces incident to $v$.  The interfaces
of $\mathcal{T}(G, S)$ are interior level cycles, so cluster
compatibility only ever constrains interior vertices and is untouched by
the outer face.

To pass from \eqref{eq:heawood-face-sum-dual} to a colouring one must
account for the outer face: an outer-boundary vertex is incident to the
unbounded face $f_\infty$, whose label is omitted from the bounded sum.
Extend $\lambda$ by a single label $\lambda(f_\infty) \in \{+1, -1\}$ on
$f_\infty$.  Then a family of Heawood face-labellings that is pairwise
compatible along every interface of $\mathcal{T}(G, S)$ assembles into a
$\{+1,-1\}$ labelling of \emph{all} faces of $G$ for which
$\sum_{f \ni v} \lambda(f) \equiv 0 \pmod 3$ holds at every vertex ---
the outer-boundary vertices now carrying $\lambda(f_\infty)$ in their
sum.  This is Heawood's face-sum identity \cite{Heawood1898} for a
proper $3$-edge-colouring of the full cubic dual of $G$, hence (by Tait)
a 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
\[
   \mathsf{K}_0 \supset \mathsf{K}_1 \supset \cdots \supset
   \mathsf{K}_{N(k)}
\]
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 cluster route to the Four Colour Theorem]
\label{conj:heawood-route-fct}
For every plane triangulation $G$ and every level source $S$, the
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 $R_{\mathsf{K}}$ per cluster; counting /
%% pigeonhole bound giving $N(k)$; orientation/reversal bookkeeping on
%% the shared interface.

\section{The constraint floor}
\label{sec:constraint-floor}

A nested substructure constrains its outer interface through the set of
Heawood boundary sequences it can realise.  By the self-similarity of the
tire decomposition (\cite{bauerfeld-nested-tires}), the region $G_T$
enclosed by a tire's outer cycle, away from the source, is itself a
triangulated disk; we record how tightly any such disk can constrain its
boundary.  The bound below depends only on the disk triangulation, not on
a tire-tree labelling.

\begin{definition}[Achievable boundary set of a disk]
\label{def:achievable-boundary-set}
Let $D$ be a triangulated disk whose boundary is a simple $n$-cycle
$C = (v_0, \dots, v_{n-1})$.  Call a Heawood face-labelling
$\lambda : F(D) \to \{+1,-1\}$ \emph{interior-valid} if
$\sum_{f \ni w} \lambda(f) \equiv 0 \pmod 3$ at every interior vertex $w$
of $D$ (no condition on $C$).  The \emph{achievable boundary set} of $D$
is
\[
   \Phi(D) \;:=\; \bigl\{\,
     (\lambda^{*}(v_0), \dots, \lambda^{*}(v_{n-1}))
     \;:\; \lambda \text{ interior-valid} \,\bigr\}
   \;\subseteq\; \{0,1,-1\}^{n} .
\]
\end{definition}

\begin{proposition}[Constraint floor]
\label{prop:constraint-floor}
For every triangulated disk $D$ with boundary an $n$-cycle,
\[
   |\Phi(D)| \;\ge\; 2^{\,n-2},
\]
and the bound is attained --- already by the triangulation of the
$n$-gon with no interior vertices.  Consequently no nested structure
constrains the outer cycle below $2^{\,n-2}$ achievable Heawood
sequences; the trivial tire is already maximally constraining.
\end{proposition}

\begin{proof}[Proof of attainment]
Triangulate the $n$-gon as a fan from $v_0$, with faces
$\{v_0, v_i, v_{i+1}\}$ for $1 \le i \le n-2$ and labels
$\lambda_i := \lambda(\{v_0, v_i, v_{i+1}\})$; there are no interior
vertices, so every labelling is interior-valid.  The induced boundary
values are
\[
   \lambda^{*}(v_1) = \lambda_1, \quad
   \lambda^{*}(v_i) = \lambda_{i-1} + \lambda_i \ \ (1 < i < n-1), \quad
   \lambda^{*}(v_{n-1}) = \lambda_{n-2}, \quad
   \lambda^{*}(v_0) = \textstyle\sum_{j} \lambda_j .
\]
From $\lambda^{*}(v_1)$ and the relations
$\lambda_i = \lambda^{*}(v_i) - \lambda_{i-1}$ the tuple
$(\lambda_1, \dots, \lambda_{n-2}) \in \{+1,-1\}^{n-2}$ is recovered from
the boundary sequence, so the map $\lambda \mapsto \lambda^{*}|_C$ is
injective and $|\Phi(D)| = 2^{\,n-2}$.
\end{proof}

\begin{remark}[Depth is freedom-positive]
\label{rem:freedom-positive}
The lower bound is plausible from a counting balance.  A triangulated
disk with $k$ interior vertices has $2k + n - 2$ faces (Euler) and
imposes exactly $k$ interior Heawood constraints, one per interior
vertex.  So each interior vertex contributes \emph{two} faces --- two new
$\{+1,-1\}$ degrees of freedom --- against only \emph{one} constraint,
and the free dimension $(2k + n - 2) - k = k + n - 2$ \emph{grows} with
depth.  Going deeper is freedom-positive on balance: the boundary
projection $\Phi(D)$ can only retain or enlarge its options, never drop
below the interior-free value $2^{\,n-2}$.  (Empirically $|\Phi(D)|$ does
grow with $k$; e.g.\ on the $4$-cycle the central-apex wheel realises $5$
sequences against the fan's $4$.)  The constraints relate only
interior-incident faces and cannot collapse the $n-2$ degrees of freedom
carried by the boundary-incident faces --- which is the content the lower
bound must make precise.
\end{remark}

\begin{remark}
\label{rem:floor-consequences}
Two consequences.  First, $\Phi(D)$ is a $\mathbb{Z}/3$ zonotope --- a
projected cube, sign-closed but not a $\mathrm{GF}(3)$ subspace --- and at
the floor it has size $2^{\,n-2}$ with affine hull of dimension $n-2$.
Second, since the floor is exponential in the interface length $n$, a
maximally-constraining child still offers $2^{\,n-2}$ outer options, so
the gluing of Conjecture~\ref{conj:heawood-chain-pigeonhole} has the least
slack at \emph{short} interfaces (e.g.\ $n = 4$ leaves $4$ options) and is
easy at long ones; the difficulty of the programme is concentrated at
short level cycles.
\end{remark}

\begin{thebibliography}{9}

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\bibitem{bauerfeld-depth}
E.~Bauerfeld,
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manuscript (math-research repository), 2026.

\bibitem{bauerfeld-nested-tires}
E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
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\bibitem{bauerfeld-medial-tires}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.

\bibitem{bauerfeld-nested-tire-duals}
E.~Bauerfeld,
\emph{Coloring Nested Tire Dual Graphs},
manuscript (math-research repository), 2026.

\end{thebibliography}

\end{document}