\subsection{Flux through the small section}
\label{sec:small-section-flux}

The statistical estimate on a shear interval must now be integrated
over the two weak walls. To compare repeated crossings we use
a weighted flux of the normalized flow through the section. In logarithmic wall
coordinates its density stays uniformly comparable to Lebesgue
measure near the Taub point. This leaves two logarithmic factors
to be paid for by the product in the statistical loss estimate.

For $x=(p,s,N_1,N_2,N_3)$, write the constraint as
\[
 \mathcal C(x)=p^2+s^2+Q(N_1,N_2,N_3).
\]
For positive walls put
\begin{equation}
 H(x)=(N_1N_2N_3)^{2/3},\qquad
 \mathcal W(x)=N_1N_2N_3(1+4H(x)).
 \label{eq:section-volume-weight}
\end{equation}
Let $\mathcal V$ be the standard volume form on $\R^5$.
On $\mathcal C=1$ define the four-form
\begin{equation}
 \Omega_x(v_1,v_2,v_3,v_4)
   =\frac{\mathcal V(x/2,v_1,v_2,v_3,v_4)}{\mathcal W(x)},
 \qquad v_i\in\ker d\mathcal C_x.
 \label{eq:constraint-volume-form}
\end{equation}
The quadratic constraint gives $d\mathcal C_x(x/2)=1$.
Thus $x/2$ can be replaced by any vector $X$ satisfying
$d\mathcal C_x(X)=1$: the difference is tangent, and five
vectors in the four-dimensional tangent space have zero
determinant. Contracting $\Omega$ with the normalized vector
field $F$ defines the section flux
$\omega_x(v_1,v_2,v_3)=\Omega_x(F(x),v_1,v_2,v_3)$.

Use the logarithmic chart $\mathcal X_h$ from
\eqref{eq:signed-logarithmic-chart}, on its open domain
\[
 \mathcal D_h=\{(s,\beta_2,\beta_3):
       1-s^2-Q(h,e^{-\beta_2},e^{-\beta_3})>0\}.
\]
Here $p=-\sqrt{1-s^2-Q(h,e^{-\beta_2},e^{-\beta_3})}<0$.
Define
\begin{equation}
 w_h(s,\beta_2,\beta_3)
   =\frac{p^2+s^2-2p}
          {2|p|\,[1+4(he^{-\beta_2-\beta_3})^{2/3}]},
 \qquad
 d\mu_h=\ind_{\mathcal D_h}w_h\dd s\dd\beta_2\dd\beta_3.
 \label{eq:physical-section-measure}
\end{equation}
This is a positive continuous density on $\mathcal D_h$.

\begin{proposition}
\label{prop:physical-section-density}
For $h>0$, the pullback of the signed flux to this chart is
\begin{equation}
 \mathcal X_h^*\omega
        =-w_h\,ds\wedge d\beta_2\wedge d\beta_3.
 \label{eq:physical-section-signed-flux}
\end{equation}
Consequently $\mu_h$ is its absolute flux measure and has
the same null sets as Lebesgue measure restricted to
$\mathcal D_h$.
\end{proposition}

\begin{proof}
Take $X=(2p)^{-1}\partial_p$ as the normalized transverse
vector in~\eqref{eq:constraint-volume-form}. If
$n_i=e^{-\beta_i}$, the three chart tangent vectors have
the form
\[
 (*,1,0,0,0),\qquad (*,0,0,-n_2,0),\qquad
 (*,0,0,0,-n_3).
\]
The determinant with columns $X,F$ and these vectors is
$-F_{N_1}n_2n_3/(2p)$. By~\eqref{eq:normalized-flow},
$F_{N_1}=(-p^2-s^2+2p)h$. Dividing by
$\mathcal W=hn_2n_3(1+4H)$ gives $-w_h$, proving
\eqref{eq:physical-section-signed-flux}. The numerator
$p^2+s^2-2p$ and both denominator factors are strictly
positive on $\mathcal D_h$. Integrating this positive
density proves equivalence of the null sets.
\end{proof}

For use on a radial slice, abbreviate
\begin{equation}
 w_h(\rho;n_2,n_3)=w_h(\rho,-\log n_2,-\log n_3),
 \qquad
 \mu_{h,n_2,n_3}(A)=\int_A w_h(\rho;n_2,n_3)\dd\rho.
 \label{eq:radial-section-flux}
\end{equation}

\begin{lemma}
\label{lem:small-section-flux-bounds}
Suppose $0<h\leq1/8$, $0<a\leq h/8$,
$0\leq\varepsilon\leq1/64$, and
$0<n_2,n_3\leq2\varepsilon a$. Then for
$\rho\in[a,2a]$ the chart radicand is positive and
\begin{equation}
 \frac12\leq w_h(\rho;n_2,n_3)\leq3.
 \label{eq:small-section-flux-bounds}
\end{equation}
In particular $\mu_{h,n_2,n_3}(A)\leq3a$ for every
measurable $A\subset[a,2a]$.
\end{lemma}

\begin{proof}
We have $n_i\leq h/8$, $|\rho|\leq1/8$ and
$h+n_2+n_3\leq5h/4$. Since
$|Q(h,n_2,n_3)|\leq(h+n_2+n_3)^2$, the radicand
is at least $983/1024$. On its negative square-root branch,
\[
 p\leq-15/16,\qquad p^2+\rho^2\leq2.
\]
The uncorrected density is therefore
\[
 \frac{p^2+\rho^2-2p}{2|p|}
       =1+\frac{p^2+\rho^2}{2|p|}\in[1,3].
\]
Also $0\leq(hn_2n_3)^{2/3}\leq h^2\leq1/64$.
Dividing by $1+4(hn_2n_3)^{2/3}$ proves
\eqref{eq:small-section-flux-bounds}. The interval has
length $a$, which gives the last assertion.
\end{proof}

The reflection $\mathsf s(s,\beta_2,\beta_3)
=(-s,\beta_3,\beta_2)$ preserves $\mathcal D_h$ and
$w_h$. Its determinant is $1$: reversing the shear and
exchanging the two wall coordinates each contributes a
minus sign. The change of variables formula consequently gives
\begin{equation}
 \mu_h(\mathsf s E)=\mu_h(E),\qquad
 \mu_h(E\cup\mathsf s E)\leq2\mu_h(E)
 \label{eq:section-reflection-measure}
\end{equation}
for every Borel set $E$. The domain restriction in
\eqref{eq:physical-section-measure} is preserved as well.

The scale in the weak walls disappears from the logarithmic
Jacobian. For $a>0$ define
\begin{equation}
 L_a(\rho,x,y)=(\rho,-\log(ax),-\log(ay)),
 \qquad x,y>0.
 \label{eq:scaled-logarithmic-chart}
\end{equation}
This map is smooth and injective, with derivative
$\operatorname{diag}(1,-1/x,-1/y)$ and determinant
$1/(xy)$.

Let $\Lambda\subset(0,a)^2$ be Borel and let
$\mathcal F_\nu\subset[a,2a]$ have a jointly Borel
membership relation on $\Lambda\times[a,2a]$.
The corresponding set on the logarithmic section is
\begin{equation}
 E=\{(\rho,-\log n_2,-\log n_3):
                \nu=(n_2,n_3)\in\Lambda,
                \ \rho\in\mathcal F_\nu\}.
 \label{eq:logarithmic-radial-family}
\end{equation}
Assume its chart radicand is positive. Then $E$ is Borel
and
\begin{equation}
 \mu_h(E)
   =\int_0^1\!\int_0^1
      \ind_{\Lambda}(ax,ay)
      \frac{\mu_{h,ax,ay}(\mathcal F_{(ax,ay)})}{xy}
           \dd y\dd x.
 \label{eq:logarithmic-radial-flux}
\end{equation}
The integrand is taken to be zero outside the selected label set.

Indeed, the set
\[
 A=\{(\rho,x,y):0<x,y<1,\ (ax,ay)\in\Lambda,
                                  \rho\in\mathcal F_{(ax,ay)}\}
\]
is Borel, and $E=L_a(A)$. A continuous injective map on
this standard Borel domain has Borel image. Applying the
Jacobian formula to $L_a$ gives the integral over $A$ with
density $w_h(\rho;ax,ay)/(xy)$. Tonelli's theorem, with
the radial coordinate integrated first, gives
\eqref{eq:logarithmic-radial-flux}. All integrands are
nonnegative, so this identity requires no prior finiteness
assumption. Finiteness will follow from retaining the
minimum in the radial statistical estimate.