Add the freedom-positive counting balance to the constraint floor

Remark: a disk with k interior vertices has 2k+n-2 faces (Euler) but only
k interior constraints, so each interior vertex adds two degrees of
freedom against one constraint -- depth is freedom-positive and Phi can
only retain or enlarge below the interior-free floor 2^(n-2). Motivates
the lower bound and replaces the prior TODO sketch.

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

papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex

@@ -499,12 +499,23 @@ 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}[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}