\subsection{Radial families through zero shear}
\label{sec:angular-reference-phase}

Vary the shear and the larger weak wall along the affine rays
\begin{equation}
 Z_{\theta,\zeta}(\rho)=(\theta\rho,\rho,\zeta),
 \qquad |\theta|\leq L,\quad a\leq\rho\leq2a,
 \qquad 0\leq\zeta\leq a/9,
 \label{eq:angular-initial-family}
\end{equation}
where $L\geq0$. The labels $\theta,\zeta$ stay fixed
when differentiating in $\rho$.
Taking $\theta=0$ gives the zero-shear family.
The initial physical state is $\Xi_h(Z_{\theta,\zeta}(\rho))$.

Write $A=A(h)$, $C=C(h)$ and $K=K(h)$ for the
transport coefficients of Section~\ref{sec:small-section-transport}.
The reference phase on~\eqref{eq:angular-initial-family} is
\begin{equation}
 F_{\theta,\zeta}(\rho)
   =\Phi_h(Z_{\theta,\zeta}(\rho))
   =\frac1{\sqrt3}\left|
        \rho(A\theta+K,C)-\zeta(K,C)\right|_2.
 \label{eq:angular-reference-phase}
\end{equation}
Here $|\cdot|_2$ denotes the Euclidean norm in two
dimensions. The following elementary estimate explains why
no division by the shear is needed.

\begin{lemma}
\label{lem:affine-radial-slope}
Let $v,w\in\R^2$, $|v|_2>0$, $\rho>0$ and $\zeta\geq0$.
If $3\zeta|w|_2\leq\rho|v|_2$, then
$r(\rho)=|\rho v-\zeta w|_2$ is differentiable near
$\rho$, and
\begin{equation}
 \frac{|v|_2}{2}\leq r'(\rho)\leq|v|_2.
 \label{eq:affine-radial-slope}
\end{equation}
\end{lemma}

\begin{proof}
Put $P=|v|_2$ and $Q=|w|_2$. Cauchy's inequality,
applied after expanding the squared norm, gives
\[
 0<\rho P-\zeta Q\leq r\leq\rho P+\zeta Q.
\]
Thus the square root is smooth near the point and
\[
 r'=\frac{\rho P^2-\zeta\langle v,w\rangle}{r}.
\]
The numerator is at least $P(\rho P-\zeta Q)$.
Since $\rho P\geq3\zeta Q$, this is at least
$P(\rho P+\zeta Q)/2\geq Pr/2$.
The upper bound follows from
$\langle v,\rho v-\zeta w\rangle\leq Pr$.
\end{proof}

For $0\leq h\leq1/16$, the previously proved coefficient
bounds imply
\[
 1\leq A,C\leq2,\qquad 0\leq K\leq1.
\]
For $v=(A\theta+K,C)$ and $w=(K,C)$, it follows that
\[
 1\leq|v|_2\leq2L+3,\qquad |w|_2\leq3.
\]
The condition $\zeta\leq a/9\leq\rho/9$ verifies the
hypothesis of Lemma~\ref{lem:affine-radial-slope}.
Consequently
\begin{equation}
 \frac1{2\sqrt3}\leq F_{\theta,\zeta}'(\rho)
                   \leq\frac{2L+3}{\sqrt3}.
 \label{eq:angular-reference-slope}
\end{equation}
This includes the direction $A\theta+K=0$, in which
the first component of the reference ray vanishes.
The wall component $C$ supplies its positive length.

On the families~\eqref{eq:angular-initial-family},
\begin{equation}
 \|Z_{\theta,\zeta}(\rho)\|\leq2(L+1)a,
 \qquad \varrho(Z_{\theta,\zeta}(\rho))\geq a/2,
 \qquad \|Z_{\theta,\zeta}'\|\leq L+1.
 \label{eq:angular-initial-annulus}
\end{equation}
The middle inequality follows from
$\varrho(Z)\geq\rho-\zeta\geq8a/9$; the stated
weaker bound will be convenient.
For $q>2$, choose the common time
\[
 T=\frac q2\log(1/a),\qquad
 f(\rho)=\frac{\varrho(H_{h,T}(Z_{\theta,\zeta}(\rho)))}{\sqrt3}.
\]

\begin{proposition}
\label{prop:angular-fast-phase}
There is $h_*>0$ such that, for each $0<h\leq h_*$
and $L\geq0$, one can choose $K_0>0$ before choosing
$q>2$. For all sufficiently small $a>0$, uniformly over
\eqref{eq:angular-initial-family}, the actual orbit exists
at $T$, $f$ has an ordinary derivative on $[a,2a]$, and
\begin{equation}
 \begin{split}
 |f-F_{\theta,\zeta}|&\leq K_0a^2,\qquad
 |f'-F_{\theta,\zeta}'|\leq K_0a,\\
 \frac1{4\sqrt3}\leq f'&\leq\frac{2L+3}{\sqrt3}+1.
 \end{split}
 \label{eq:angular-fast-phase}
\end{equation}
For $a\leq x\leq y\leq2a$,
\begin{equation}
 \frac{y-x}{4\sqrt3}\leq f(y)-f(x)
         \leq\left(\frac{2L+3}{\sqrt3}+1\right)(y-x).
 \label{eq:angular-fast-secants}
\end{equation}
\end{proposition}

\begin{proof}
Apply Theorem~\ref{thm:small-section-radial-approximation}
with $c_0=1/2$ and $C_0=2(L+1)$, using
\eqref{eq:angular-initial-annulus}. Its error constant
is chosen before $q$. Restricting the ordinary initial
derivative to $Z_{\theta,\zeta}'=(\theta,1,0)$ multiplies
its bound by at most $L+1$. Enlarge the constant by this
factor to obtain the first line of~\eqref{eq:angular-fast-phase}.

Reduce the scale threshold so that
$K_0a\leq1/(4\sqrt3)\leq1$.
Comparison with~\eqref{eq:angular-reference-slope}
gives the derivative bounds. The mean value theorem on
each subinterval gives~\eqref{eq:angular-fast-secants}.
Ordinary differentiability at both endpoints follows from
the ordinary derivative in the radial approximation theorem.
\end{proof}

The phase is therefore monotone throughout these families,
including zero shear and the zero weak-wall boundary.
We next express the same states in core coordinates.
The positive active wall keeps the projection denominator
away from zero after rescaling.