==== TABLE OF CONTENTS ====
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//main.tex
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/conclusion.tex
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/energy.tex
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/geometry.tex
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/introduction.tex
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/rigidity.tex
Weak-mixing-of-triangular-billiards-with-an-irrational-angle//references.bib
===========================
==== BEGIN Weak-mixing-of-triangular-billiards-with-an-irrational-angle//main.tex ====
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\hypersetup{pdfauthor={OpenAI},pdftitle={Weak mixing of triangular billiards with an irrational angle}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
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\title{Weak mixing of triangular billiards\\with an irrational angle}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that the unit-speed billiard flow in every nondegenerate Euclidean
triangle with at least one angle irrational relative to \(\pi\) is weakly
mixing for normalized Liouville measure. Equivalently, the product of the
flow with itself is ergodic.
\end{abstract}
\input{sections/introduction}
\input{sections/geometry}
\input{sections/energy}
\input{sections/rigidity}
\input{sections/conclusion}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Weak-mixing-of-triangular-billiards-with-an-irrational-angle//main.tex ====
==== BEGIN Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/conclusion.tex ====
\section{Side reflections and weak mixing}\label{sec:conclusion}
The analytic argument is complete. We now apply Proposition~\ref{prop:rigidity}
to the lift of a billiard eigenfunction. Its local form will make the angle
condition decisive.
\begin{proposition}\label{prop:eigenfunctions}
Let \(Q\) be a nondegenerate Euclidean triangle with an angle irrational
relative to \(\pi\). If \(F\in L^2(Q\times S^1,\mu)\) satisfies
\[
F\circ\Phi_t=e^{i\lambda t}F\quad\text{in }L^2
\qquad(t\in\R)
\]
for some \(\lambda\in\R\), then \(F\) is constant almost everywhere.
If this constant is nonzero, then \(\lambda=0\).
\end{proposition}
\begin{proof}
First suppose \(F\) is bounded. Put the origin of the triangle coordinates
at a vertex whose angle \(\alpha\) satisfies \(\alpha/\pi\notin\mathbb Q\).
Let \(f\) be the lift of \(F\) to the double, with its orientation chosen
so that these coordinates are oriented on the first sheet.
Proposition~\ref{prop:rigidity} and the eigenfunction equation give
\[
Xf=i\lambda f,\qquad Yf=0,
\qquad\text{hence}\qquad \nabla f=i\lambda v f
\]
in every oriented flat chart, where \(v=(\cos\theta,\sin\theta)\).
Thus \(e^{-i\lambda x\cdot v}f(x,v)\) has zero spatial distributional
gradient. On a connected chart it is independent of position for almost
every direction. One can see this by taking its angular Fourier coefficients:
each has zero spatial distributional gradient and is therefore constant
on the connected base chart. In the first triangle interior this gives
\begin{equation}\label{eq:plane-wave}
F(x,v)=c(v)e^{i\lambda x\cdot v}
\end{equation}
for a measurable bounded function \(c\) on \(S^1\).
Consider an open side. Reflection in its supporting line is the affine map
\(p\mapsto Rp+d\), where \(R\) is the linear orthogonal reflection and
\(d\in\R^2\). A crossing chart agrees with the first-sheet coordinates
and assigns coordinates \(x=Rp+d\) to the point \(p\) of the second sheet.
The chart direction \(v\) folds to the billiard direction \(Rv\) there.
Since \(R^T=R=R^{-1}\), Equation~\eqref{eq:plane-wave} gives
\[
f(x,v)=F(p,Rv)=c(Rv)e^{i\lambda (x-d)\cdot v}
\]
on the second portion of the chart. The phase-removed function is independent
of position throughout the connected crossing chart, so its expressions on
the two open portions agree:
\begin{equation}\label{eq:side-relation}
c(v)=c(Rv)e^{-i\lambda d\cdot v}
\quad\text{for almost every }v\in S^1.
\end{equation}
This comparison uses open subsets of a regular chart and requires no
boundary trace for \(F\).
For the two sides through the origin, \(d=0\). Their reflections generate
a rotation through \(2\alpha\), under which \(c\) is invariant.
If \(\widehat c(k)\) is its \(k\)-th circle Fourier coefficient, this gives
\(\widehat c(k)=e^{2ik\alpha}\widehat c(k)\). Irrationality of
\(\alpha/\pi\) forces every coefficient except \(k=0\) to vanish.
Thus \(c\) is constant almost everywhere.
The third side has \(d\ne0\), since its supporting line does not pass
through the chosen vertex. If the constant \(c\) is nonzero,
\eqref{eq:side-relation} says \(e^{-i\lambda d\cdot v}=1\) almost
everywhere on the circle, and hence everywhere by continuity. The values
of \(d\cdot v\) fill the interval \([-|d|,|d|]\), so this is possible
only when \(\lambda=0\). Equation~\eqref{eq:plane-wave} now makes \(F\)
constant, proving the bounded case.
For an arbitrary \(L^2\) eigenfunction, use
\(\widetilde F=F/(1+|F|)\). The map \(z\mapsto z/(1+|z|)\) is bounded,
injective, and commutes with multiplication by unit complex scalars.
Consequently \(\widetilde F\) is a bounded eigenfunction with the same
frequency. The bounded case and injectivity give the assertion for \(F\).
\end{proof}
We recall the standard compact-operator proof of the spectral implication
needed to finish the argument; see, for example,
\cite[Theorem~6.1.4]{Peterson2015}. It applies
to the full continuous-time flow.
\begin{lemma}\label{lem:spectral}
Let \((\Omega,\mu)\) be a standard probability space and \(\Phi_t\) a
measure-preserving flow whose composition operators form a strongly
continuous unitary group on complex \(L^2(\mu)\). If the only flow
eigenfunctions are constants, then the product flow
\(\Phi_t\times\Phi_t\) is ergodic for \(\mu\otimes\mu\).
\end{lemma}
\begin{proof}
Write \(U_t h=h\circ\Phi_t\). The frequency-zero hypothesis first implies
ergodicity of \(\Phi_t\). If the product flow were not ergodic, there
would be a nonzero invariant kernel \(K\in L^2(\mu\otimes\mu)\) of mean
zero. Each one-variable marginal of \(K\) is invariant under the single
flow and thus constant. The zero total mean makes both marginals zero.
Define the complex-linear Hilbert--Schmidt operator
\[
(Th)(x)=\int_\Omega K(x,y)h(y)\,d\mu(y).
\]
The zero marginals imply \(T1=0\) and that its range is contained in
\(L^2_0(\mu)=\{h:\int h\,d\mu=0\}\). Its restriction to this space
is nonzero, since a zero integral operator has zero \(L^2\) kernel.
Product invariance gives \(U_tT=TU_t\) by a change of variables in the
integral. Hence \(T^*T\), acting on \(L^2_0(\mu)\), is a nonzero
positive compact operator commuting with every \(U_t\).
A positive eigenvalue of \(T^*T\) has a nonzero finite-dimensional
eigenspace invariant under every \(U_t\). Commuting unitary operators
on that space have a common eigenvector \(h\ne0\). The associated
scalar character is continuous by strong continuity of \(U_t\), so
it has the form \(e^{i\lambda t}\) for some \(\lambda\in\R\).
This gives a flow eigenfunction in \(L^2_0(\mu)\), a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
The billiard composition group is strongly continuous by the construction
in Section~\ref{sec:geometry}. Proposition~\ref{prop:eigenfunctions}
eliminates every nonconstant eigenfunction, so Lemma~\ref{lem:spectral}
proves ergodicity of the product flow.
\end{proof}
==== END Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/conclusion.tex ====
==== BEGIN Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/energy.tex ====
\section{Localized energy and individual coefficient bounds}\label{sec:energy}
The first estimate uses the full spatial gradient only where the
eigenfunction equation fails. After smoothing and cutting off near
the vertices, this will confine the resulting error to small cone
annuli. The zero-frequency version of this argument appears in
\cite[Sections~3--4]{Companion}; the spectral correction below is
needed when \(\lambda\ne0\). The equal complex-derivative norms and
coefficient recurrences also occur in Forni and Moll's Sobolev
arguments \cite[Lemma~3.3, Corollary~4.2, and Lemma~5.1]{FM}.
Twisted complex-derivative energy identities provide a related
spectral precedent in Forni's work \cite[Section~3]{Forni2022}.
\begin{lemma}[Localized coefficient estimate]\label{lem:energy}
Let \(w\in L^2(SM)\) have compact spatial support in \(M\) and be
smooth in position with values in angular \(L^2\) in every flat chart.
Suppose \(Aw\) vanishes over \(M\setminus E\), where \(E\subset M\)
is measurable. Define
\(D_j=\norm{\pa w_j}_2^2\). Then, for every \(m\in\Z\),
\begin{equation}\label{eq:localized-energy}
\tfrac12D_m+D_{m+1}
\le \norm{Aw}_2\norm{\ind_E\nabla w}_2
+\tfrac{\lambda^2}{2}\norm{w_{m+1}}_2^2,
\qquad
D_j=\tfrac14\norm{\nabla w_j}_2^2.
\end{equation}
\end{lemma}
\begin{proof}
Compact support in the regular surface permits integration by parts
between the coefficient bundles. The local formal adjoints are
\(\pa^*=-\pb\) and \(\pb^*=-\pa\); the derivatives commute in
flat charts. Hence
\begin{equation}\label{eq:complex-energies}
\norm{\pa w_j}_2^2=\norm{\pb w_j}_2^2
=\tfrac14\norm{\nabla w_j}_2^2.
\end{equation}
These identities are intrinsic, since all transition factors are
constant and unitary. In particular, no integration takes place across
a cone point.
Put
\(C_j=\operatorname{Re}\ip{i\lambda w_{j+1}}{\pa w_j}\).
Equation~\eqref{eq:modes} gives
\begin{equation}\label{eq:spectral-telescope}
D_j-D_{j+2}
=\operatorname{Re}\ip{(Aw)_{j+1}}{\pa w_j-\pb w_{j+2}}
+C_j-C_{j+1}.
\end{equation}
For the spectral term, the sign follows from
\[
\operatorname{Re}\ip{i\lambda w_{j+1}}{\pb w_{j+2}}
=\operatorname{Re}\ip{-i\lambda\pa w_{j+1}}{w_{j+2}}
=C_{j+1}.
\]
The correction \(C_j-C_{j+1}\) couples consecutive indices, so a sum
over one parity class would not telescope. Sum
\eqref{eq:spectral-telescope} over \emph{every} integer
\(j=m,\ldots,N\). The left side is
\(D_m+D_{m+1}-D_{N+1}-D_{N+2}\), and the last two terms on the
right telescope to \(C_m-C_{N+1}\). All high endpoints tend to zero:
the coefficients of \(w\) and their spatial derivatives are square
summable, by the hypotheses on \(w\). The remaining sum of pairings
converges absolutely by Cauchy--Schwarz. Since \(Aw\) vanishes off
the part over \(E\), its absolute value is at most
\[
\norm{Aw}_2
\left(\int_E\sum_{j\ge m}
|\pa w_j-\pb w_{j+2}|^2\,d\bar A\right)^{1/2}
\le \norm{Aw}_2\norm{\ind_E\nabla w}_2.
\]
The last inequality uses the pointwise bound
\[
\sum_{j\ge m}|\pa w_j-\pb w_{j+2}|^2
\le 2\sum_{j\in\Z}(|\pa w_j|^2+|\pb w_j|^2)
=\sum_{j\in\Z}|\nabla w_j|^2.
\]
Finally,
\[
|C_m|\le |\lambda|\norm{w_{m+1}}_2\sqrt{D_m}
\le \tfrac12D_m+\tfrac{\lambda^2}{2}\norm{w_{m+1}}_2^2,
\]
which proves \eqref{eq:localized-energy}.
\end{proof}
We now construct regularizations to which Lemma~\ref{lem:energy}
applies. Write \(r(x)=d(x,V)\) in \(\overline M\), also using \(r\)
for its pullback to \(SM\). Near each cone point, an embedded flat
disk about \(x\) has radius at least \(c r(x)\), for a fixed
\(c>0\): this follows by choosing uniformly sized disks on the compact
circle at radius one in the model cone and scaling. Compactness away
from the tips therefore gives a constant \(K>2\) such that, for all
sufficiently small \(\eps>0\), every point with
\(r>(K-2)\eps\) has an embedded flat disk of radius \(4\eps\).
For each fixed \(L>0\), the cone area formula also gives
\begin{equation}\label{eq:cone-area}
\bar A\{r\le L\eps\}=O_L(\eps^2).
\end{equation}
Choose a nonnegative radial function
\(\rho\in C_c^\infty(\R^2)\), supported in the unit disk and with
Euclidean integral one, and set
\(\rho_\eps(y)=\eps^{-2}\rho(y/\eps)\). For \(r(x)>K\eps\),
define spatial convolution at fixed parallel direction by
\begin{equation}\label{eq:smoothing-definition}
(S_\eps u)(x,\theta)
=\int_{\R^2}\rho_\eps(y)u(x-y,\theta)\,dy
\end{equation}
in the embedded flat disk. Radiality makes this definition independent
of its oriented chart. Unless a derivative is being taken, extend
\(S_\eps u\) by zero to \(r\le K\eps\).
\begin{lemma}[Spatial regularization]\label{lem:smoothing}
For every integer \(\ell\ge0\), the preceding operators satisfy
\begin{equation}\label{eq:smoothing-bounds}
\eps^\ell\norm{\nabla^\ell S_\eps u}_{L^2(r>K\eps)}
\le C_\ell\norm{u}_2.
\end{equation}
Their derivatives here are taken before zero extension. The output
is smooth in position with values in angular \(L^2\), and
\(S_\eps u\to u\) in \(L^2(SM)\) for every fixed \(u\in L^2(SM)\).
For bounded input,
\[
\norm{S_\eps u}_\infty\le\norm{u}_\infty,
\qquad
\eps^\ell\norm{\nabla^\ell S_\eps u}_{L^\infty(r>K\eps)}
\le C_\ell\norm{u}_\infty.
\]
Convolution commutes locally with \(X,Y\), and \(A\), in
distributions. If \(L>K+4\) and \(g=0\) on \(r(K-1)\eps\), and all centers
which can contribute to it lie in a disk of area \(O(\eps^2)\).
Integrating the bound over centers proves
\eqref{eq:smoothing-bounds}. The same kernel estimates give the
\(L^\infty\) bounds; nonnegativity and mass one give the contraction
bound at order zero. Constant-coefficient differentiation in each
chart proves local commutation and spatial smoothness.
For a smooth compactly supported input on \(SM\), the usual
approximate-identity argument gives \(L^2\) convergence, including
the zero extension for sufficiently small \(\eps\). Such inputs
are dense in \(L^2(SM)\), so the uniform bound at order zero proves
convergence for every fixed input.
The function \(r\) is 1-Lipschitz. If \(r(x)<(L-1)\eps\), every
point sampled by its convolution disk has \(r0,\ \beta>0,\ \alpha+\beta<\pi\}:
\]
the excluded pairs with both \(\alpha/\pi\) and \(\beta/\pi\) rational
form a countable set. The irrationality assumption is sharp for the full
phase space. In a rational triangle, side reflections generate a finite
group on directions. A nonconstant function on the circle invariant under
this finite group is then an invariant function for the billiard flow, so
the full flow is not ergodic.
\paragraph{Historical context.}
Polygonal billiards connect elementary reflection geometry with dynamics
on flat surfaces. The unfolding construction of Zemlyakov and Katok
\cite{ZK} replaces reflections by straight continuation on a surface.
For rational polygons, it produces a compact translation surface and
separates the full phase space into invariant directional surfaces.
Kerckhoff, Masur, and Smillie proved unique ergodicity in almost every
direction \cite{KMS}. They also obtained ergodicity of the full billiard
flow for a dense \(G_\delta\) family of polygons. Vorobets gave a
quantitative rational-approximation criterion and explicit ergodic
examples, including irrational right triangles \cite{Vorobets}.
These results establish ergodicity under genericity or sufficiently rapid
approximation assumptions rather than for each irrational triangle.
Chaika and Forni proved that the full billiard flow is weakly mixing for a
dense \(G_\delta\) set of polygons with any fixed number of sides at least
three \cite{CF}. Their argument uses the absence of common nontrivial
eigenvalues for directional flows in almost every pair of directions on a
translation surface. The Baire-category conclusion does not assert
Lebesgue-almost-everywhere weak mixing in the space of triangle angles.
For rational billiards, recent work of Arana-Herrera, Chaika, and Forni
characterizes weak mixing in almost every direction \cite{ACF}; this is
a statement on directional invariant surfaces, rather than on the full
Liouville phase space considered here.
Numerical investigations have long suggested weak mixing in irrational
right triangles and mixing in triangles with all angles irrational
\cite{ACG,CP}. Other computations have suggested nonergodicity in some
irrational right or symmetric triangles \cite{WCP,ZSBJ}. These conflicting
finite-time observations underscore the difficulty of identifying
measurable invariants from trajectories with slow statistical convergence.
Theorem~\ref{thm:main} treats every irrational triangle without an
approximation or symmetry restriction.
\paragraph{Analytic contribution.}
The regularity of an eigenfunction is the central issue. Glue two oppositely
oriented copies of \(Q\) along their sides. The resulting flat double is
regular across the open sides and has cone points at the vertices. On its
unit tangent bundle, let \(X\) denote differentiation along a geodesic and
\(Y\) differentiation in the perpendicular parallel direction. Locally
these vector fields commute. Forni and Moll developed angular Fourier and
Cauchy--Riemann methods in this setting \cite[Sections~3.2--3.5]{FM}.
Their rigidity results distinguish invariance under both horizontal
directions from geodesic invariance under an additional horizontal Sobolev
hypothesis \cite[Theorem~4.1 and Corollary~4.2]{FM}.
A measurable flow eigenfunction supplies only \(Xf=i\lambda f\) at the
outset; the missing transverse derivative cannot be assumed.
The companion manuscript \cite[Sections~2--6]{Companion} removes this
regularity obstacle for invariant functions, proving ergodicity of
irrational triangular billiards. We extend its coefficient-energy and
long-average method to every real spectral parameter \(\lambda\).
Summing the coefficient-energy identities over consecutive angular indices
makes the terms containing \(\lambda\) telescope to a single boundary term.
A separate pairing with the two boundary modes produced by differentiating
an angular Fourier tail gives the sharp bound
\[
\norm{Xf}_2^2+\norm{Yf}_2^2\le\lambda^2\norm{f}_2^2.
\]
Since \(Xf=i\lambda f\), it follows that \(Yf=0\).
The use of a spectral parameter in Cauchy--Riemann energy estimates has a
related precedent in the twisted cohomological equations studied by Forni
\cite[Section~3]{Forni2022}. Here the argument starts with bounded measurable
eigenfunctions and proves the needed derivative bounds. We give a
self-contained proof of this analytic extension below.
\paragraph{Proof strategy.}
Expand a bounded eigenfunction on the double in angular Fourier modes.
Spatial smoothing away from the cone points and cutoffs near the tips
first give a uniform bound on the gradient of each coefficient. These
individual bounds do not
yet give a square-summable family of gradients. To obtain the missing
summability, take the sum of all modes above a fixed index. Its derivative
under \(X-i\lambda\) consists of two endpoint terms.
We test this equation using long time averages supported on orbit segments
that stay clear of the cone points. There are two approximation parameters.
At a fixed bound on the discarded measure, the smoothing limit gives a
bounded solution of the exact local eigenfunction equation and passes the
endpoint pairing to it.
The individual-coefficient estimate then supplies a new derivative bound
independent of the discarded measure. A second limit passes the pairing
to the original eigenfunction. Summing the resulting endpoint inequalities
proves \(Yf=0\). This order of limits is what permits the initial
eigenfunction to be merely measurable.
The remaining argument is geometric. On the triangle interior,
\(Xf=i\lambda f\) and \(Yf=0\) give a plane wave with a measurable
direction-dependent amplitude. Reflection in the two sides through an
irrational vertex makes this amplitude constant. Reflection in the third
side excludes every nonzero frequency. The standard spectral criterion
then gives weak mixing.
Section~\ref{sec:geometry} sets up the double and its Fourier coefficients.
Section~\ref{sec:energy} proves the coefficient and smoothing estimates.
Section~\ref{sec:rigidity} establishes transverse rigidity by the two
successive limits. Section~\ref{sec:conclusion} applies side reflections
and proves ergodicity of the product flow.
==== END Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/introduction.tex ====
==== BEGIN Weak-mixing-of-triangular-billiards-with-an-irrational-angle//sections/rigidity.tex ====
\section{Transverse rigidity of bounded eigenfunctions}\label{sec:rigidity}
The coefficient estimates of Section~\ref{sec:energy} do not yet give
square summability of the horizontal derivatives. We obtain that summability
from an identity involving the two modes at the lower end of an angular
Fourier tail.
The long averages used below adapt the construction for invariant functions
in \cite[Sections~4--5]{Companion}; the phase in the average and the endpoint
terms retain the spectral parameter throughout the argument.
\begin{proof}[Proof of Proposition~\ref{prop:rigidity}]
Write $H=\norm{f}_\infty$ and $A=X-i\lambda$. The eigenfunction equation
implies $Af=0$ locally in distributions. Lemma~\ref{lem:bounded-modes}
therefore gives
\begin{equation}\label{eq:rigidity-mode-bound}
\norm{\nabla f_j}_2\le C_\lambda H
\qquad(j\in\Z).
\end{equation}
Here and below constants may depend on the fixed triangle, but not on an
angular index or on the approximation parameters.
\paragraph{The two boundary modes of a Fourier tail.}
Fix $m\in\Z$. Define the $L^2$ field and the two coefficient derivatives
\[
U=\sum_{j\ge m}P_jf,\qquad
b=\pb f_m,\qquad a=\pa f_{m-1},
\]
and put
\[
B=b(x)e^{i(m-1)\theta}-a(x)e^{im\theta}.
\]
The factors in this expression respect the coefficient transition rules,
so $B$ is a global $L^2(SM)$ field. The mode equations give
\begin{equation}\label{eq:tail-equation}
AU=B\qquad\text{locally in distributions on }SM.
\end{equation}
For completeness, for $N\ge m+2$ the finite tail satisfies
\[
A\sum_{j=m}^N P_jf
=B-(\pb f_{N+1})e^{iN\theta}
+(\pa f_N)e^{i(N+1)\theta}.
\]
The last two terms are bounded in $L^2$ by
\eqref{eq:rigidity-mode-bound} and converge weakly to zero: their angular
frequencies tend to infinity. Since the finite tails converge strongly
to $U$ in $L^2$, this proves \eqref{eq:tail-equation}.
We will prove the endpoint identity
\begin{equation}\label{eq:endpoint-identity}
\ip{B}{(P_{m-1}+P_m)Yf}=0.
\end{equation}
At this stage $Yf$ denotes a distribution; its two displayed modes are
in $L^2$ by \eqref{eq:modes} and \eqref{eq:rigidity-mode-bound}.
The difficulty is that $Yf$ itself is not yet known to be an admissible
test in \eqref{eq:tail-equation}. We construct compactly supported
approximations whose error under $A$ becomes negligible in that pairing.
\paragraph{Long averages away from the vertices.}
Let $K$ be the constant in Lemma~\ref{lem:smoothing} and fix $L>K+4$.
For $00$, set $T=s/\eps$.
Let $G_\eps$ consist of the regular states $z$ whose entire trajectory
segment $\{\Psi_tz:0\le t\le T\}$ lies over $\{r>L\eps\}$.
Then
\begin{equation}\label{eq:bad-segments}
\nu(SM\setminus G_\eps)\le C(s+\eps^2).
\end{equation}
Indeed, $r$ is $1$-Lipschitz along a unit-speed geodesic. If a segment
meets $\{r\le L\eps\}$, a time mesh of spacing at most $\eps$ detects
a visit to $\{r\le(L+1)\eps\}$. There are at most $T/\eps+2$ mesh
points. Measure preservation and the cone-area bound
$\nu\{r\le(L+1)\eps\}\le C\eps^2$ prove
\eqref{eq:bad-segments}. The discarded trajectories that hit a vertex
form a null set.
Define
\begin{equation}\label{eq:segment-average}
q_\eps=f\ind_{G_\eps},\qquad
g_\eps=\frac1T\int_0^T e^{i\lambda t}
q_\eps\circ\Psi_{-t}\,dt.
\end{equation}
The integral may be taken in $L^2$; bounded measurable representatives
and Fubini give the same field. The phase is chosen so that replacing
$q_\eps$ by $f$ in the integrand gives $f$. Consequently,
\begin{equation}\label{eq:average-bounds}
\norm{g_\eps}_\infty\le H,\qquad
\norm{g_\eps-f}_2\le CH\sqrt{s+\eps^2},\qquad
\norm{Ag_\eps}_2\le\frac{2H}{T}.
\end{equation}
The first bound follows directly from \eqref{eq:segment-average}; the
second follows by integrating the $L^2$ distance
$\norm{q_\eps-f}_2\le H\nu(G_\eps^c)^{1/2}$.
For the third we use the distributional identity
\begin{equation}\label{eq:average-endpoints}
Ag_\eps=\frac1T
\bigl(q_\eps-e^{i\lambda T}q_\eps\circ\Psi_{-T}\bigr).
\end{equation}
Here no derivative of $\ind_{G_\eps}$ is taken. To verify the identity,
test against $\varphi\in C_c^\infty(SM)$ and change variables by the
flow. Along every regular orbit,
\[
\frac{d}{dt}\left(e^{i\lambda t}
\overline{\varphi(\Psi_tz)}\right)
=e^{i\lambda t}\overline{(A\varphi)(\Psi_tz)}.
\]
Integrating this formula in $t$, and using the formal skew adjointness
of $A$, proves \eqref{eq:average-endpoints}. The test and its horizontal
derivative are bounded, which also justifies Fubini in this calculation.
Both $g_\eps$ and $Ag_\eps$ vanish over $\{rL\eps\}$; this also applies to both endpoint
terms in \eqref{eq:average-endpoints}.
Apply the spatial smoothing of Lemma~\ref{lem:smoothing} and set
\[
w_\eps=S_\eps g_\eps.
\]
Before its prescribed zero extension this field vanishes on
$\{K\eps