Scaffold the 2^(n-2) constraint-floor proposition

Add section 4: define the achievable boundary set Phi(D) of a triangulated
disk and state the constraint-floor proposition |Phi(D)| >= 2^(n-2), with
the attainment direction proved (fan injectivity) and the lower bound left
as a marked gap with strategy. Remark records the zonotope structure and
the short-interface concentration of difficulty.

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

papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex

@@ -441,6 +441,84 @@ $G$ is properly $4$-vertex-colourable.
 %% 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}
+
+%% TODO (lower bound): show |Phi(D)| >= 2^{n-2} for EVERY triangulated
+%% disk D. Strategy: the n boundary-incident faces (one per boundary edge)
+%% carry n-2 independent binary degrees of freedom after the interior
+%% Heawood constraints are imposed; those constraints relate only
+%% interior-incident faces and cannot collapse the boundary freedom below
+%% 2^{n-2}. (See notes/boundary_restriction_structure.tex.)
+
+\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}
 
 \bibitem{Heawood1898}