\subsection{Statistical graphs for either small weak wall}
\label{sec:small-section-common-statistics}

The common return has all the properties used to construct
statistical entrance graphs: a decreasing output radius, a
square-root tangent cone and a bound retaining the initial
weak-wall product. We apply the same explicit graph construction
to the enlarged label set. The uniform phase lower bound then
converts the statistical loss back to the original shear
coordinate with a single constant for both orderings.

Fix $h,q,D,P,a,\varepsilon$ under the thresholds of
Theorem~\ref{thm:common-shear-return}. Write
\begin{equation}
 \begin{split}
 I&=[a,2a],\qquad t_0=\frac q2\log(1/a),\\
 \Lambda_c&=\{\nu=(n_2,n_3):0<n_2,n_3\leq2\varepsilon a,
                         \ \min(n_2,n_3)\leq a^P\},\\
 G_\nu(\rho)&=G_{t_0}(n_2,n_3,\rho),\qquad
 \xi_\nu(\rho)=2/g_\nu(\rho),\qquad
 B_\nu=\frac{800n_2n_3}{a^4}.
 \end{split}
 \label{eq:common-statistical-family}
\end{equation}
In particular $\Lambda_c$ is a Borel subset of $\R^2$.

\begin{proposition}
\label{prop:common-statistical-preparation}
Given any $B_*>0$, a sufficiently small common scale threshold
gives, for every $\nu\in\Lambda_c$, an admissible graph
\[
 \Gamma_\nu(r)=(r,\widetilde u_\nu(r),\widetilde w_\nu(r))
\]
at amplitude $B_\nu$, with $0<B_\nu\leq B_*$.
Its wall functions are positive and $C^1$ on $\R$, and
\begin{gather}
 \widetilde u_\nu+\widetilde w_\nu\leq B_\nu,\qquad
 |\widetilde u_\nu'|\leq\sqrt{\widetilde u_\nu},\qquad
 |\widetilde w_\nu'|\leq\sqrt{\widetilde w_\nu},
 \label{eq:common-statistical-graph-bounds}\\
 \Gamma_\nu(g_\nu(\rho))=G_\nu(\rho),\qquad
 \widetilde u_\nu'(g_\nu(\rho))
      =\frac{\widehat u_\nu'(\rho)}{g_\nu'(\rho)},\qquad
 \widetilde w_\nu'(g_\nu(\rho))
      =\frac{\widehat w_\nu'(\rho)}{g_\nu'(\rho)}
 \quad(\rho\in I).
 \label{eq:common-statistical-graph-match}
\end{gather}
The state $G_\nu(\rho)$ is jointly Borel on $\Lambda_c\times I$,
and the four extended wall and wall-derivative functions are
jointly Borel on $\Lambda_c\times\R$. Moreover,
\begin{equation}
 0<\xi_\nu(\rho)<1,\qquad
 \xi_\nu'(\rho)\geq\frac1{4\sqrt3},\qquad
 \Gamma_\nu(2/\xi_\nu(\rho))=G_\nu(\rho).
 \label{eq:common-statistical-phase}
\end{equation}
\end{proposition}

\begin{proof}
For each label, $g_\nu'<0$ on $I$. Construct its inverse
on $[g_\nu(2a),g_\nu(a)]$ and the two radius graphs as in
\eqref{eq:shear-radius-graphs}. The $C^2$ return
gives the continuous one-sided endpoint derivatives in
\eqref{eq:shear-radius-graph-slopes}.
Dividing~\eqref{eq:common-shear-cone} by $|g_\nu'|$
gives the unit square-root slope bounds for these graphs.
Their sum is at most $512n_2n_3/a^4$.

Use the explicit extension of
Lemma~\ref{lem:positive-amplitude-extension} with $\theta=1$.
It preserves values and derivatives on the closed interval,
and its amplitude cost is $25/16$. Thus
\[
 \widetilde u_\nu+\widetilde w_\nu
   \leq\frac{25}{16}\frac{512n_2n_3}{a^4}=B_\nu.
\]
This proves the graph bounds and matching statements on
the same extended graphs used in
Section~\ref{sec:small-section-statistical-families}.

The measurability argument of
Proposition~\ref{prop:shear-extension-measurability} uses
only a Borel label domain, a jointly Borel return, a strictly
decreasing radius and positive $C^1$ wall graphs. All these
conditions hold here. In detail, joint ordinary regularity
makes the restricted map $G$ continuous, and
$(\nu,\rho)\mapsto(\nu,g_\nu(\rho))$ is an injective
Borel map between standard Borel spaces. Its inverse is
Borel on the varying radius intervals. The interior samples
and adjacent secants in~\eqref{eq:shear-interior-sampling-limits}
recover measurable endpoint values and slopes. Applying the
same explicit extension formulas proves joint measurability
of the four extended functions.

For the amplitude threshold, $2\varepsilon\leq1$ gives
$n_2,n_3\leq a$. At least one of them is at most
$a^P\leq a^4$ when $a\leq1$, since $P>6D+2>4$.
In either case,
\begin{equation*}
 n_2n_3\leq a^5,\qquad 0<B_\nu\leq800a.
\end{equation*}
Require $a\leq B_*/800$ and $a\leq1/384$.
The radius bound $g_\nu\geq1/(96a)$ then gives
$g_\nu\geq4$, hence $0<\xi_\nu<1$.
The lower slope follows from
Theorem~\ref{thm:common-shear-return}. Finally
$2/\xi_\nu=g_\nu$, so the value matching proves
the last identity in~\eqref{eq:common-statistical-phase}.
\end{proof}

Fix admissible $\eta,J,\sigma$ and the constants
$B_*,B_{\mathrm{cap}}$ of
Theorem~\ref{thm:era-graph-statistics}.
Use the same Borel statistical set
$\mathcal S_{\mathrm{stat}}$ throughout the enlarged family.
For $\nu\in\Lambda_c$ define
\begin{equation}
 \mathcal F_\nu
   =\{\rho\in I:(B_\nu,G_\nu(\rho))
                         \notin\mathcal S_{\mathrm{stat}}\}.
 \label{eq:common-statistical-failure}
\end{equation}

\begin{theorem}[Statistical entrances for both wall orderings]
\label{thm:common-shear-statistical-loss}
For sufficiently small $h>0$, every $q>3$, $D>4$,
$P>6D+2$, and all sufficiently small $a>0$, the following
holds uniformly for $0\leq\varepsilon\leq1/128$.
One has $0<B_\nu\leq B_*\leq B_{\mathrm{cap}}$ for
all $\nu\in\Lambda_c$, and the family
\[
 \{(\nu,\rho)\in\Lambda_c\times I:
                                      \rho\in\mathcal F_\nu\}
\]
is Borel. For every Borel $A\subset I$ and nonnegative
Borel weight $\omega\leq M$ on $A$, $M\geq0$,
\begin{equation}
 \int_{A\cap\mathcal F_\nu}\omega(\rho)\dd\rho
   \leq\min\left\{\int_A\omega(\rho)\dd\rho,
                    320\sqrt3\,M B_\nu^\sigma\right\}.
 \label{eq:common-statistical-weighted-loss}
\end{equation}
Outside $\mathcal F_\nu$, the selected era digits and
phases from the returned state $G_\nu(\rho)$
satisfy all six statistical limits of Theorem~\ref{thm:main}.
\end{theorem}

\begin{proof}
Apply Proposition~\ref{prop:common-statistical-preparation}
with the chosen $B_*$. The graph $\Gamma_\nu$ is admissible
at $B_\nu$, and~\eqref{eq:common-statistical-phase}
verifies the phase and representation hypotheses of
Corollary~\ref{cor:radial-statistics} with
$c=1/(4\sqrt3)$. Its weighted bound is
\[
 \min\left\{\int_A\omega(\rho)\dd\rho,
                         \frac{80M}{c}B_\nu^\sigma\right\},
\]
which is~\eqref{eq:common-statistical-weighted-loss}.
The same corollary gives all six limits for the actual
state whenever $\rho\notin\mathcal F_\nu$.

The map $\nu\mapsto B_\nu$ is continuous, and $G$ is
jointly Borel. Hence the family of excluded sets is the
inverse image of $\mathcal S_{\mathrm{stat}}^c$ under
$(\nu,\rho)\mapsto(B_\nu,G_\nu(\rho))$. It is Borel,
as is each of its sections $\mathcal F_\nu$.
\end{proof}

Both orderings now contribute to the same statistical family.
The product $n_2n_3$ remains in the loss bound, and the
minimum with the original mass is retained for integration
over the weak-wall labels.