\subsection{Uniform Taub avoidance and bounded era lengths}
\label{sec:compact-phase-exclusion}

For sufficiently small transverse walls, uniform avoidance of the Taub
points separates core radii from the ends of an era. We use this separation
to continue its finite histories and bound their digits. The local
coding estimates then give a null set of such core entrances.

Write $\mathcal I(r,u,w)$ for the physical state whose projective
coordinates are $\iota_+(r,u,w)$. We say that a physical state has
\emph{uniform Taub avoidance at distance $\epsilon$} if
\begin{equation}
 \dist(\phi_tq,T_i)\geq\epsilon
       \qquad(t\geq0,\ i=1,2,3),
 \label{eq:uniform-taub-avoidance}
\end{equation}
where distance is Euclidean in the normalized variables. This
property is preserved by forward time shifts and by axis permutations.

\begin{lemma}
\label{lem:core-even-separation}
There are $\delta>0$ and $C\geq1$ such that, for each $\epsilon>0$,
one can choose $0<\rho\leq1/16$ with the following property.
If $Y=(r,u,w)$ satisfies
\[
 u,w>0,\qquad u+w\leq\frac{\delta}{1+r},\qquad
 C(u+w)\leq\rho/4,
\]
and $\mathcal I(Y)$ satisfies~\eqref{eq:uniform-taub-avoidance}, then
\begin{equation}
 |r-2n|>\rho/2\qquad(n\geq1).
 \label{eq:core-even-separation}
\end{equation}
\end{lemma}

\begin{proof}
First consider the single-wall orbit starting at radius two. On its
descending part its projective state is
\[
 (1-\sqrt{4-v^2},0,v,0),\qquad v\downarrow0.
\]
The inverse formula~\eqref{eq:projective-inverse} shows that these
states tend to $T_2$. Every positive $v$ on this part is reached in
finite positive original time. Choose one such time at which the
distance is less than $\epsilon$. Continuous dependence of the
physical flow at this fixed time gives $0<\rho\leq1/16$ such that
\begin{equation}
 |r-2|\leq\rho,\quad |u|,|w|\leq\rho
 \quad\Longrightarrow\quad
 \dist(\phi_T\mathcal I(r,u,w),T_2)<\epsilon.
 \label{eq:near-two-finite-visit}
\end{equation}
This comparison uses a neighborhood of the initial state at the
same fixed time $T$.

We next record the amplitude estimate that permits a finite sequence
of reflections to reach this neighborhood. After decreasing $\delta$,
Lemma~\ref{lem:core-reflection-step} applies whenever
$r\geq63/16$ and $b=u+w\leq\delta/(1+r)$. Its core exit radius obeys
$|R-r|\leq1/16$, so $R\geq31/8$. If
$G(Y)=(r',u',w')$ and $b'=u'+w'$, its conclusions give
\[
 3/2\leq r-r'\leq5/2,\qquad
 0<u'\leq u,\quad0<w'\leq w,\quad
 b'\leq e^{-c/r}b.
\]
Since $1-e^{-c/r}\geq(c/r)/(1+c/r)$ and $r\geq2$,
\[
 \frac b r\leq\frac{1+c/2}{c}(b-b').
\]
Thus the same lemma, with a larger $C\geq1$, yields
\begin{equation}
 |r'-(r-2)|\leq C(b-b').
 \label{eq:reflection-paid-phase}
\end{equation}
Each step is an actual positive-time passage, followed by an axis
permutation, so it preserves~\eqref{eq:uniform-taub-avoidance}.

Suppose now that $|r-2n|\leq\rho/2$. Starting from $Y_0=Y$, construct
$Y_{j+1}=G(Y_j)$ for $j<n-1$, and put $b_j=u_j+w_j$.
Induction using~\eqref{eq:reflection-paid-phase} gives
\begin{equation}
 b_j\leq b_0,\quad r_j\leq r_0,\quad
 |r_j-(r_0-2j)|\leq C(b_0-b_j)\leq\rho/4.
 \label{eq:near-even-prefix}
\end{equation}
For each step that must be taken, $j\leq n-2$, and hence
$r_j\geq4-3\rho/4\geq63/16$. Also
$b_j\leq\delta/(1+r_0)\leq\delta/(1+r_j)$. These inequalities
justify the induction.
At $j=n-1$,~\eqref{eq:near-even-prefix} gives
$|r_j-2|\leq3\rho/4$, while $u_j,w_j\leq b_0\leq\rho$.
Equation~\eqref{eq:near-two-finite-visit} contradicts the avoidance
property at this actual point of the orbit. The argument also covers
$n=1$, when no reflection is taken.
\end{proof}

For $N\in\N$ and $0<\gamma<1/4$, define the compact range of radii
\begin{equation}
 \mathcal K_{N,\gamma}
 =\{(r,u,w):2+2\gamma\leq r\leq2N,\quad
             \gamma\leq r/2-\lfloor r/2\rfloor\leq1-\gamma\}.
 \label{eq:compact-era-phase}
\end{equation}
The transverse amplitudes are restricted separately.

\begin{proposition}
\label{prop:avoidance-compact-phase}
For each $\epsilon>0$, there are $N$, $0<\gamma<1/4$ and $B_0>0$
such that every $Y=(r,u,w)$ with
\[
 r\geq2-\gamma/2,\qquad u,w>0,\qquad u+w\leq B_0,
\]
whose physical state satisfies~\eqref{eq:uniform-taub-avoidance},
belongs to $\mathcal K_{N,\gamma}$.
\end{proposition}

\begin{proof}
For $r\geq1$ and $0\leq u,w\leq1/100$, put
$m=1+\sqrt{r^2-1}$ and $D_0=4+m^2+u^2+w^2$. The denominator in
\eqref{eq:projective-inverse} is
$D=D_0-2(u+uw+w)\geq D_0/2>0$, and $D_0\geq r^2$.
Direct substitution gives
\begin{equation*}
 \dist(\mathcal I(Y),T_1)^2
       =\frac{12D_0}{D^2}\leq\frac{48}{r^2}.
\end{equation*}
Thus avoidance implies $r<K:=8/\epsilon$.

Take $\rho,C,\delta$ from Lemma~\ref{lem:core-even-separation}, and set
\[
 \gamma=\rho/4,\qquad N=\lceil K/2\rceil,\qquad
 B_0=\min\{1/100,\delta/(1+K),\rho/(4C)\}.
\]
The assumed lower bound on $r$ is greater than one, so the preceding
estimate applies. The lemma excludes every positive even integer by
a distance greater than $\rho/2$. At $2$ the alternative below
$2-\rho/2$ is incompatible with $r\geq2-\rho/8$; hence
$r\geq2+2\gamma$. Dividing the separation inequalities by two
separates $r/2$ from every positive integer by at least $\gamma$.
Zero and the negative integers are already farther away. Applying
these bounds to $\lfloor r/2\rfloor$ and its successor proves
\eqref{eq:compact-era-phase}.
\end{proof}

\begin{lemma}
\label{lem:compact-histories-survive}
Fix the parameters of the selected era return, $N$, and
$0<\gamma<1/4$. For sufficiently small $B_*>0$, suppose
$0<B\leq B_*$, $u,w>0$, and $u+w\leq B$.
If the endpoint of every existing finite selected history from
$(B,Y)$ belongs to $\mathcal K_{N,\gamma}$, then $(B,Y)$ survives
indefinitely and all its selected digits are at most $N$.
\end{lemma}

\begin{proof}
Choose $B_*\leq B_{\mathrm{cap}}$, $B_*<1$, below the era estimate
threshold and such that
\[
 b^\sigma<\min\{\gamma/4,1/(N+2)\}
                      \qquad(0<b\leq B_*).
\]
At an endpoint $X=(r,u,w)\in\mathcal K_{N,\gamma}$ with amplitude
bound $b$, put $a=\lfloor r/2\rfloor$ and $\beta=b^\sigma$.
Then $1\leq a\leq N$, $a-1<\lfloor1/\beta\rfloor$, and the
entrance inequalities~\eqref{eq:era-selected-entrance} hold.
The actual output phase $y$ satisfies
\[
 |y-(r/2-a)|<\beta/4.
\]
Since $r/2-a\in[\gamma,1-\gamma]$, both
$y-2\beta$ and $1-2\beta-y$ exceed $7\gamma/16$.
Thus the return is selected and its positive output walls have sum
at most $b^{4/3}$.

Starting with the empty history, this extends a history of length
$n$ to one of length $n+1$. Its amplitude is
$B_n=B^{(4/3)^n}\leq B\leq B_*$, and the assumed compactness applies
at its endpoint. Histories therefore exist at every finite length.
The definition~\eqref{eq:era-state-survivors} gives survival, and
uniqueness of selected histories identifies these digits with the
survivor digits. Their bound follows from $a\leq N$ at every step.
\end{proof}

\begin{proposition}
\label{prop:taub-avoiding-core-null}
For each $\epsilon>0$, there is $B_*>0$ such that, for
$0<B\leq B_*$, the set
\begin{equation}
 \mathcal A_{\epsilon,B}
 =\{(x,u,w):0<x<1,\ u,w>0,\ u+w\leq B,
       \ \mathcal I(2/x,u,w)\text{ satisfies }
                  \eqref{eq:uniform-taub-avoidance}\}
 \label{eq:taub-avoiding-core-set}
\end{equation}
has three-dimensional outer Lebesgue measure zero.
\end{proposition}

\begin{proof}
Use Proposition~\ref{prop:avoidance-compact-phase} to choose $N,\gamma$.
Choose $B_*$ below its amplitude threshold and those for the actual
era return and Lemma~\ref{lem:compact-histories-survive}.
Along every existing finite selected history the amplitude bound
decreases. The actual orbit identity preserves avoidance, and the
next radius is greater than two. Induction therefore places each
endpoint in $\mathcal K_{N,\gamma}$. The lemma gives a survivor
with bounded digits.

For any admissible graph, almost every survivor has a Gauss-typical
code, by the coding measure estimate used in
Theorem~\ref{thm:era-graph-statistics}. Such a code cannot have bounded
digits: for every fixed $N$, the limiting frequency of a digit at
least $N+1$ is $\log(1+1/(N+1))/\log2>0$. Consequently the bounded
digit survivors have zero phase length on every admissible graph.

The set of bounded digit survivors in entrance state space is Borel.
Indeed, on the Borel survivor set its defining condition is
$\bigcup_N\bigcap_n\{a_n\leq N\}$, and the digits are Borel by
Proposition~\ref{prop:era-state-coding}. Fixing positive $u,w$ with
$u+w\leq B$ gives the admissible constant graph $r\mapsto(r,u,w)$.
Its phase slice of this Borel set is null. All other slices are
empty after imposing the amplitude conditions. Tonelli's theorem
now gives a null set in $(x,u,w)$ coordinates containing
$\mathcal A_{\epsilon,B}$, as required.
\end{proof}

The nullity follows from the local statistics on continued survivors.
The next two steps use product decay to find such entrances and
transfer this nullity to initial sections.