\subsection{Statistical entrance families from the small section}
\label{sec:small-section-statistical-families}

The returned shear curves have a phase with a uniform
positive derivative and two small transverse walls.
To apply the statistical estimates of
Section~\ref{sec:era-radial-families}, we express each
curve as a graph over its output radius. The square-root
cone bounds allow this graph to be extended to all radii
with a fixed amplitude cost. We then verify that the
construction is measurable in the two weak-wall labels.
This makes the excluded set a single Borel
set on the original family of initial conditions.

Fix $h,q,\ell,a,\varepsilon$ as in
Section~\ref{sec:small-section-return-phase}, and put
\begin{equation}
 \begin{split}
 I&=[a,2a],\\
 \Lambda&=\{\nu=(n_2,n_3):
       0<n_2\leq2\varepsilon a,\quad
       0<n_3\leq2\varepsilon a,\quad n_3\leq a^\ell\}.
 \end{split}
 \label{eq:shear-label-domain}
\end{equation}
We display the label by writing
\[
 G_\nu(\rho)=(g_\nu(\rho),\widehat u_\nu(\rho),
                         \widehat w_\nu(\rho)),\qquad
 \xi_\nu(\rho)=2/g_\nu(\rho).
\]
All scale thresholds below are uniform over
$\Lambda\times I$. The time
$T=(q/2)\log(1/a)$ remains fixed when differentiating
in the labels or in $\rho$.

\begin{proposition}
\label{prop:shear-phase-intervals}
Under the thresholds of
Proposition~\ref{prop:positive-shear-return-phase},
for every $\nu\in\Lambda$ and
$a\leq b\leq c\leq2a$, the phase $\xi_\nu$ is
strictly increasing on $[b,c]$ and
\begin{equation}
 \xi_\nu([b,c])=[\xi_\nu(b),\xi_\nu(c)],\qquad
 \frac{c-b}{8\sqrt3}\leq\xi_\nu(c)-\xi_\nu(b)
                            \leq C_h(c-b).
 \label{eq:shear-phase-intervals}
\end{equation}
For every Borel set $A\subset[b,c]$ and every set
$E\subset\R$ with $\xi_\nu(A)\subset E$,
\begin{equation}
 \lambda(A)\leq
       \min\{c-b,\,8\sqrt3\,\lambda(E)\}.
 \label{eq:shear-phase-deleted-length}
\end{equation}
Here $\lambda$ is Lebesgue measure, interpreted as
outer measure for a set $E$ that is not measurable.
\end{proposition}

\begin{proof}
The mean value theorem applied on each subinterval
gives
\[
 \frac{t-s}{8\sqrt3}\leq\xi_\nu(t)-\xi_\nu(s)
                         \leq C_h(t-s)
       \quad(b\leq s\leq t\leq c).
\]
It proves strict monotonicity; continuity and the
intermediate value theorem give the exact interval
image. For the measure estimate, use the
one-dimensional change of variables argument in
\eqref{eq:radial-deleted-length}:
\[
 \frac1{8\sqrt3}\lambda(A)
       \leq\int_A|\xi_\nu'(\rho)|\dd\rho
       =\lambda(\xi_\nu(A))\leq\lambda(E).
\]
Combining this with $A\subset[b,c]$ proves
\eqref{eq:shear-phase-deleted-length}.
\end{proof}

Since $g_\nu'<0$, its image of $I$ is the interval
\begin{equation}
 J_\nu=[l_\nu,r_\nu],\qquad
 l_\nu=g_\nu(2a)<g_\nu(a)=r_\nu.
 \label{eq:shear-output-radius-interval}
\end{equation}
Let $j_\nu:J_\nu\to I$ be its inverse, and define
\begin{equation}
 u_\nu(r)=\widehat u_\nu(j_\nu(r)),\qquad
 w_\nu(r)=\widehat w_\nu(j_\nu(r)).
 \label{eq:shear-radius-graphs}
\end{equation}
Their derivatives on the closed interval, including
one-sided derivatives at its endpoints, are
\begin{equation}
 u_\nu'(r)=
       \frac{\widehat u_\nu'(j_\nu(r))}{g_\nu'(j_\nu(r))},
 \qquad
 w_\nu'(r)=
       \frac{\widehat w_\nu'(j_\nu(r))}{g_\nu'(j_\nu(r))}.
 \label{eq:shear-radius-graph-slopes}
\end{equation}

The radius extends to a $C^1$ function with negative derivative
on a neighborhood of $I$. The inverse function theorem gives
$j_\nu'=1/g_\nu'(j_\nu(r))$ up to both endpoints of $J_\nu$.
Thus the expressions in~\eqref{eq:shear-radius-graph-slopes}
are continuous on the closed interval.

Define $\widetilde u_\nu,\widetilde w_\nu$ by the
explicit square-root extension of
Lemma~\ref{lem:positive-amplitude-extension}, applied
on $J_\nu$ to \eqref{eq:shear-radius-graphs} and
\eqref{eq:shear-radius-graph-slopes}. Put
\begin{equation}
 B_\nu=\frac{800n_2n_3}{a^4},\qquad
 \Gamma_\nu(r)=(r,\widetilde u_\nu(r),
                         \widetilde w_\nu(r)).
 \label{eq:shear-extended-graph}
\end{equation}

\begin{proposition}
\label{prop:shear-radius-extension}
After a common decrease of the preceding scale
thresholds, both extensions are positive $C^1$
functions on $\R$, and
\begin{equation}
 \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:shear-extended-graph-bounds}
\end{equation}
For every $\rho\in I$ they agree with the actual
output in both value and radial derivative:
\begin{gather}
 \Gamma_\nu(g_\nu(\rho))=G_\nu(\rho),\notag\\
 \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)}.
 \label{eq:shear-extended-graph-match}
\end{gather}
In particular $\Gamma_\nu$ is admissible at amplitude
$B_\nu$.
\end{proposition}

\begin{proof}
Divide the two cone inequalities in
\eqref{eq:positive-shear-all-active-cone} by
$|g_\nu'|$. Equations
\eqref{eq:shear-radius-graphs}--\eqref{eq:shear-radius-graph-slopes}
give
\[
 |u_\nu'|\leq\sqrt{u_\nu},\qquad
 |w_\nu'|\leq\sqrt{w_\nu}
       \quad\hbox{on }J_\nu.
\]
The functions are positive, with continuous
one-sided derivatives, and their sum is at most
$512n_2n_3/a^4$ by
\eqref{eq:positive-shear-return-amplitude}.
Lemma~\ref{lem:positive-amplitude-extension}, with
$\theta=1$, gives global positivity and the two
slope bounds. Its amplitude cost is exactly
\[
 \frac{25}{16}\frac{512n_2n_3}{a^4}
                       =\frac{800n_2n_3}{a^4}=B_\nu.
\]
The same lemma preserves values and derivatives on
the whole closed interval. Using
$j_\nu(g_\nu(\rho))=\rho$ proves
\eqref{eq:shear-extended-graph-match}.
These global estimates imply the required
admissibility conditions on $r>2$.
\end{proof}

The intervals $J_\nu$ depend on the labels. We verify
measurability for the explicit extensions just
constructed, so the same graphs can be used for all
labels in the statistical estimate.

\begin{proposition}
\label{prop:shear-extension-measurability}
The map $(\nu,\rho)\mapsto G_\nu(\rho)$ is Borel
on $\Lambda\times I$. The four functions
\begin{equation}
 (\nu,r)\longmapsto
 \widetilde u_\nu(r),\quad\widetilde w_\nu(r),\quad
 \widetilde u_\nu'(r),\quad\widetilde w_\nu'(r)
 \label{eq:shear-extension-measurability}
\end{equation}
are Borel on $\Lambda\times\R$. Extending any of
them by zero outside $\Lambda\times\R$ gives a
Borel function on $\R^2\times\R$.
\end{proposition}

\begin{proof}
The set $\Lambda$ is Borel. The joint $C^2$
regularity from
Corollary~\ref{cor:positive-shear-common-connection}
implies continuity of the restricted map
$(\nu,\rho)\mapsto G_\nu(\rho)$, including at
parameters where $V=1$. In particular it is Borel,
as are the two endpoint functions $l_\nu,r_\nu$.

Consider the Borel map
\[
 H:\Lambda\times I\longrightarrow\Lambda\times\R,
 \qquad H(\nu,\rho)=(\nu,g_\nu(\rho)).
\]
It is injective: equality of its values first fixes
the label, and strict decrease of $g_\nu$ then fixes
$\rho$. Both spaces are standard Borel spaces.
The injective Borel mapping theorem therefore gives
a Borel inverse on its image
\[
 \mathcal J=\{(\nu,r):r\in J_\nu\}.
\]
Transporting $\widehat u_\nu,\widehat w_\nu$ by
this inverse proves that the radius graphs are
jointly Borel on $\mathcal J$. In particular their
values restricted to the interval interiors,
extended by zero elsewhere, are jointly Borel.

We next recover the endpoint values and slopes from
interior samples. This also gives a direct measurable
description of the derivative used in the extension.
Let $v_\nu$ be either radius graph, or its positive
square root, and let $v_\nu'$ be its continuous
one-sided derivative. For $t\in J_\nu$ and
$n\geq0$ set
\begin{equation}
 h_n(\nu,t)=
 \begin{cases}
 (r_\nu-t)/(3(n+1)),&t<r_\nu,\\
 (l_\nu-t)/(3(n+1)),&t=r_\nu.
 \end{cases}
 \label{eq:shear-interior-sampling-step}
\end{equation}
This nonzero step tends to zero, and both $t+h_n$
and $t+2h_n$ lie in $(l_\nu,r_\nu)$. Continuity
and the one-sided derivative imply
\begin{equation}
 \begin{split}
 v_\nu(t)&=\lim_{n\to\infty}v_\nu(t+h_n),\\
 v_\nu'(t)&=\lim_{n\to\infty}
       \frac{v_\nu(t+2h_n)-v_\nu(t+h_n)}{h_n}.
 \end{split}
 \label{eq:shear-interior-sampling-limits}
\end{equation}
For the second identity, write the quotient as
twice the secant from $t$ to $t+2h_n$, minus the
secant from $t$ to $t+h_n$. Each secant tends to
$v_\nu'(t)$.

If $t$ is any Borel selector of $J_\nu$, all the
samples and quotients in
\eqref{eq:shear-interior-sampling-limits} are Borel.
Their pointwise limits are therefore Borel. Apply
this first to $t=l_\nu,r_\nu$. For an arbitrary
$(\nu,r)$ apply it also, on the product label space,
to the selector
\[
 t=\max\{l_\nu,\min\{r,r_\nu\}\}.
\]
Thus the endpoint data and the interior value and
slope are measurable wherever the piecewise extension
uses them. The formulas
\eqref{eq:root-extension-formula}--\eqref{eq:root-curve-extension}
now show that the extended square roots and their
derivatives are jointly Borel. Squaring gives
the amplitudes, and the derivative is twice the
extended root times its derivative. This proves
\eqref{eq:shear-extension-measurability}.

Finally $\Lambda\times\R$ is Borel in the ambient
space. A Borel function on this Borel subspace,
set equal to zero on its complement, is Borel.
\end{proof}

Fix $\eta,J,\sigma$ and the common constants
$B_*,B_{\mathrm{cap}}$ of
Theorem~\ref{thm:era-graph-statistics}, as in
Section~\ref{sec:era-radial-families}. The amplitude
in \eqref{eq:shear-extended-graph} can be placed
below any prescribed positive bound by decreasing
$a$ uniformly in the labels. Indeed, for $a\leq1$,
\begin{equation}
 n_2\leq a,\qquad n_3\leq a^\ell\leq a^4,
 \qquad0<B_\nu\leq800a.
 \label{eq:shear-budget-smallness}
\end{equation}
The first bound uses $2\varepsilon\leq1$.
Choose also $a\leq1/256$. The lower radius bound
then gives $g_\nu\geq4$, so
\begin{equation}
 0<\xi_\nu<1,\qquad
 \xi_\nu'\geq\frac1{8\sqrt3},\qquad
 \Gamma_\nu(2/\xi_\nu(\rho))=G_\nu(\rho).
 \label{eq:shear-statistical-entrance}
\end{equation}
The last identity is
\eqref{eq:shear-extended-graph-match}, since
$2/\xi_\nu=g_\nu$.

\begin{theorem}[Statistical entrances for the first wall ordering]
\label{thm:positive-shear-statistical-loss}
There is $h_*>0$ such that, for every
$0<h\leq h_*$, $q>3$, $\ell>4$, and all
sufficiently small $a>0$, the following holds for
each $0\leq\varepsilon\leq1/64$.
For $\nu\in\Lambda$ define
\begin{equation}
 \mathcal F_\nu=\{\rho\in I:
       (B_\nu,G_\nu(\rho))\notin\mathcal S_{\mathrm{stat}}\}.
 \label{eq:shear-statistical-failure}
\end{equation}
Then $0<B_\nu\leq B_*\leq B_{\mathrm{cap}}$,
each $\mathcal F_\nu$ is Borel, and the family
$\{(\nu,\rho)\in\Lambda\times I:
\rho\in\mathcal F_\nu\}$ is Borel.
For every Borel $A\subset I$ and nonnegative
Borel weight $\omega\leq M$ on $A$, with $M\geq0$,
\begin{equation}
 \int_{A\cap\mathcal F_\nu}\omega(\rho)\dd\rho
 \leq\min\left\{\int_A\omega(\rho)\dd\rho,
                   640\sqrt3\,M B_\nu^\sigma\right\}.
 \label{eq:shear-statistical-weighted-loss}
\end{equation}
For $\rho\in I\setminus\mathcal F_\nu$, the
selected era digits and phases starting at the
returned state $G_\nu(\rho)$ satisfy all
six statistical limits of Theorem~\ref{thm:main}.
\end{theorem}

\begin{proof}
Take the common small-$h$ threshold of the return,
cone and phase estimates. For each $h,q,\ell$ take
their common small-$a$ threshold and require further
\[
 a\leq\min\{1/256,B_*/800\}.
\]
Equation~\eqref{eq:shear-budget-smallness} gives
$0<B_\nu\leq B_*$. The graph is admissible at
this amplitude by
Proposition~\ref{prop:shear-radius-extension}, and
\eqref{eq:shear-statistical-entrance} verifies the
phase and representation hypotheses of
Corollary~\ref{cor:radial-statistics}.
That corollary gives the Borel excluded set and
\eqref{eq:shear-statistical-weighted-loss}, because
\[
 \frac{80M}{1/(8\sqrt3)}B_\nu^\sigma
                         =640\sqrt3\,M B_\nu^\sigma.
\]
It also gives all six limits outside the excluded
set for the same entrance state.

For the joint assertion, $\nu\mapsto B_\nu$ is
Borel by its explicit formula, and
$(\nu,\rho)\mapsto G_\nu(\rho)$ is Borel by
Proposition~\ref{prop:shear-extension-measurability}.
The family in the statement is the inverse image
of the complement of the common Borel set
$\mathcal S_{\mathrm{stat}}$ under
$(\nu,\rho)\mapsto(B_\nu,G_\nu(\rho))$.
\end{proof}

This completes the entrance construction for positive shear under
the wall bounds~\eqref{eq:shear-label-domain}. The factor
$B_\nu^\sigma$ retains both initial weak walls in the measure
estimate, and joint measurability permits integration over their
labels. The next section extends the construction to the other
wall ordering by allowing an additional interval of slow decay.