Autor: A. Jaster | Datum: 2026/05/24 | DOI: 0.5281/zenodo.20362994
The causal-Machian theory of emergent time of [Jaster2026a] employs a sharp top-hat averaging kernel over the causal horizon domain, identified there as an idealisation requiring a smooth replacement. This paper addresses that open task by replacing the sharp characteristic function of the causal domain boundary with a C∞ causal bump function of width ℓP = √ℏG/c3, constructed from a locally smooth boundary-defining function χH. A concrete form for χH encoding the full four-dimensional retarded causal structure is given for the flat FLRW case. The smeared kernel is a genuine volume-averaging kernel that recovers the original top-hat pointwise in the regular sector as ℓP → 0. The interpolation function S(r) of the parent theory is simultaneously replaced by a fully C∞ saturating function S∞(r), ensuring that second and higher derivatives of the scalar time field ΘℓP are finite and continuous in the regular sector Mreg. All structural properties of the parent theory are preserved on Mreg; inherited open tasks, in particular the explicit construction of the constraint stress tensor TCμν, are identified and listed. The construction introduces a natural Planck-energy scale and suppresses ultraviolet contributions from causal-boundary gradients in the weak-field limit; full UV-finiteness of a quantised theory requires a separate analysis. No new dimensionful parameter is introduced; dimensionless functional choices remain part of the model definition.
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% =============================================================================
\begin{document}
% =============================================================================
\begin{center}
{\LARGE\bfseries Planck-Scale Smearing of the Causal Time Field:\\[4pt]
Boundary Regularisation of the Emergent Time Theory}\\[1.2em]
{\large A.~Jaster}\\[0.4em]
{\normalsize \today}
\end{center}
\vspace{0.8em}
% ── Abstract ──────────────────────────────────────────────────────────────────
\begin{abstract}
The causal-Machian theory of emergent time of \citet{Jaster2026a}
employs a sharp top-hat averaging kernel over the causal horizon domain,
identified there as an idealisation requiring a smooth replacement.
This paper addresses that open task by replacing the sharp characteristic
function of the causal domain boundary with a $C^\infty$ causal bump
function of width $\lP=\sqrt{\hbar G/c^3}$, constructed from a locally
smooth boundary-defining function $\chi_H$. A concrete form for $\chi_H$
encoding the full four-dimensional retarded causal structure is given
for the flat FLRW case. The smeared kernel is a genuine volume-averaging
kernel that recovers the original top-hat pointwise in the regular sector
as $\lP\to0$. The interpolation function $S(r)$ of the parent theory is
simultaneously replaced by a fully $C^\infty$ saturating function
$S_\infty(r)$, ensuring that second and higher derivatives of the scalar
time field $\Theta_{\lP}$ are finite and continuous in the regular
sector $\Mreg$. All structural properties of the parent theory are
preserved on $\Mreg$; inherited open tasks, in particular the explicit
construction of the constraint stress tensor $T^{\rm C}_{\mu\nu}$,
are identified and listed. The construction introduces a natural
Planck-energy scale and suppresses ultraviolet contributions from
causal-boundary gradients in the weak-field limit; full UV-finiteness
of a quantised theory requires a separate analysis. No new
dimensionful parameter is introduced; dimensionless functional choices
remain part of the model definition.
\end{abstract}
\vspace{1em}
% =============================================================================
\section{Motivation}
\label{sec:motivation}
% =============================================================================
\citet{Jaster2026a} proposed a causal-Machian extension of General
Relativity~(GR) in which physical proper time is defined by matter and
energy in the causally relevant past of each event. The retarded
total energy-momentum of causal-past matter defines a preferred rest
frame $U_H^\mu(x)$; a bounded scalar field $\Theta(x)\in[0,1]$,
constructed from the normalised causal matter connection
$\Omega_H/\Omega_{\rm ref}$ through a smooth interpolation $S(r)$,
controls the strength of temporal structure. The averaging kernel is
a uniform top-hat over the causal horizon domain $\Dm(x)$, with a
step-function boundary explicitly identified as an idealisation
requiring regularisation \citep[Open Problem~3]{Jaster2026a}.
Two mathematical consequences of the sharp boundary motivate this
work. First, the characteristic function $\mathbf{1}_{\Dm}$ is
discontinuous, so $\nabla\Theta$ and $\nabla\nabla\Theta$ contain
distributional contributions at the causal boundary and the geodesic
equation in $\hat{g}_{\mu\nu}$ is singular there. Second, the
original interpolation $S(r)=3r^2-2r^3$ is only $C^1$ at its
saturation point $r=1$, which is insufficient for the field equations
that involve second derivatives of $\Theta$.
Both issues are resolved simultaneously: the kernel boundary is
smeared over the Planck length $\lP$, and the interpolation is
replaced by a $C^\infty$ function. The Planck length is the natural
smearing scale because quantum fluctuations of the metric render the
concept of a sharp causal boundary physically untenable at that scale.
This paper regularises the kinematic and geometric sector of the
emergent time theory. It does not close the gravitational dynamics;
the construction of the constraint action $S_{\rm constr}$ and the
associated stress tensor $T^{\rm C}_{\mu\nu}$ remain the central open
tasks of the parent theory \citep[Open Problems 1 and 5]{Jaster2026a}.
% =============================================================================
\section{Regular Sector and Structural Assumptions}
\label{sec:sector}
% =============================================================================
Several quantities in the theory are well-defined only under specific
conditions; we collect these as explicit assumptions.
\paragraph{Assumptions.}
\begin{enumerate}[label=(A\arabic*)]
\item \label{A1} Finite causal-horizon volume:
$00$.
\item \label{A5} Smooth matter sources on scales $\geq\lP$;
ideal point particles are not elements of the regular sector.
\end{enumerate}
\begin{definition}[Regular sector]
The regular sector is the open set
\begin{equation}
\Mreg = \bigl\{x\in\mathcal{M}
\;\big|\;
\textup{\ref{A1}--\ref{A5} hold at }x
\bigr\}.
\label{eq:Mreg}
\end{equation}
All regularity and field-theoretic statements are understood to hold
on $\Mreg$.
\end{definition}
The case $\Theta_{\lP}=0$ (completely matter-free causal domain)
is a boundary of the clock sector, not a regular solution of the
field equations. Events with null or spacelike $\Pi_{H,\lP}^\mu$
lie outside $\Mreg$.
% =============================================================================
\section{Boundary-Defining Function and Causal Bump}
\label{sec:bump}
% =============================================================================
\subsection{Boundary-Defining Function}
\label{sec:chi}
Geodesic distance in Lorentzian geometry is not positive-definite, is
ill-defined for null boundaries, and may fail to be smooth at caustics
or cut loci. We instead characterise the causal domain $\Dm(x)$
through a \emph{boundary-defining function}
$\chi_H(x,\cdot):\mathcal{M}\to\mathbb{R}$ satisfying
\begin{equation}
\chi_H(x,x') \begin{cases}
>0, & x'\in\Dm(x),\\
=0, & x'\in\partial\Dm(x),\\
<0, & x'\notin\Dm(x).
\end{cases}
\label{eq:chi}
\end{equation}
Under assumption~\ref{A2}, $\chi_H$ is locally $C^\infty$ near
$\partial\Dm(x)$. At caustics, cut loci, or topological changes of
$\Dm(x)$, assumption~\ref{A2} may fail and the event lies
outside $\Mreg$.
\paragraph{Standard case.}
In spatially flat FLRW with cosmic time $T_q$ (of the reference
geometry $q_{\mu\nu}$) and comoving coordinates $\vec{x}$, the causal
past of event $x=(T_q(x),\vec{x})$ consists of all events
$x'=(T_q(x'),\vec{x}')$ satisfying $T_q(x')0,\\
0, & y\leq0.
\end{cases}
\label{eq:varphi}
\end{equation}
All derivatives of $\beta$ vanish at $y=0$, so
$\varphi\in C^\infty(\mathbb{R})$ with all derivatives zero at $y=0$
and $y=1$.
\subsection{Causal Bump Function}
\label{sec:B}
The Planck-scale causal bump function is
\begin{equation}
\boxed{
\mathcal{B}_{\lP}(x,x')
= \varphi\!\left(\frac{\chi_H(x,x')}{\lP}\right).
}
\label{eq:B}
\end{equation}
Under assumption~\ref{A2}, $\mathcal{B}_{\lP}$ is $C^\infty$
wherever $\chi_H$ is $C^\infty$. Its properties are:
\begin{itemize}
\item $\mathcal{B}_{\lP}=1$ for $\chi_H>\lP$ (interior of $\Dm(x)$);
\item $\mathcal{B}_{\lP}=0$ for $\chi_H\leq0$ (outside $\Dm(x)$),
ensuring strict causality with respect to the reference geometry
$q_{\mu\nu}$;
\item $\mathcal{B}_{\lP}$ transitions $C^\infty$-smoothly from 0 to 1
in the layer $0<\chi_H<\lP$ inside the boundary.
\end{itemize}
The smearing reduces the effective averaging domain by a boundary
layer of width $\lP$, reflecting the physical Planck-scale uncertainty
of the causal boundary.
\subsection{Smeared Averaging Kernel}
\label{sec:kernel}
The top-hat kernel $W_H=V_H^{-1}\mathbf{1}_{\Dm}$ of
\citet{Jaster2026a} is replaced by
\begin{equation}
\boxed{
W_{H,\lP}(x,x')
= \frac{\mathcal{B}_{\lP}(x,x')}{\mathcal{N}(x)},
\qquad
\mathcal{N}(x)
= \int\mathcal{B}_{\lP}(x,x'')\sqrt{-q(x'')}\,\dint^4x''.
}
\label{eq:kernel}
\end{equation}
Under assumption~\ref{A1}, $\mathcal{N}(x)$ is finite and positive;
the kernel satisfies $\int W_{H,\lP}\sqrt{-q}\,\dint^4x'=1$.
\begin{proposition}[Limit $\lP\to0$]
As $\lP\to0$, $\mathcal{B}_{\lP}(x,x')\to\mathbf{1}_{\Dm(x)}(x')$
pointwise and $\mathcal{N}(x)\to V_H(x)$; hence, pointwise in
$\Mreg$ and for smooth fixed domains,
\begin{equation}
W_{H,\lP}(x,x')
\;\xrightarrow{\lP\to0}\;
\frac{\mathbf{1}_{\Dm(x)}(x')}{V_H(x)}
= W_H(x,x').
\label{eq:limit}
\end{equation}
The smeared kernel is a volume-averaging kernel throughout; the
limit produces no light-cone distribution.
\end{proposition}
% =============================================================================
\section{$C^\infty$ Interpolation Function}
\label{sec:interp}
% =============================================================================
The function $S(r)=3r^2-2r^3$ of \citet{Jaster2026a} is only $C^1$
at its saturation point $r=1$: the second derivative is discontinuous
there. Since the field equations for $\hat{g}_{\mu\nu}$ involve
second derivatives of $\Theta$, $C^1$ regularity is insufficient.
We replace $S$ by the $C^\infty$ saturating function
\begin{equation}
S_\infty(r) = \begin{cases}
0, & r\leq0,\\[4pt]
\dfrac{\beta(r)}{\beta(r)+\beta(1-r)}, & 00$ and
$S_\infty(r)\leq1$.
\item $\Theta_{\lP}=1$ in homogeneous FLRW: by
equation~\eqref{eq:Oref}, $r_{\lP}=1$ identically.
\item $\Theta_{\lP}=0$ iff matter-free: $\mathcal{E}_H=0$ implies
$r_{\lP}=0$ implies $\Theta_{\lP}=0$; excluded from $\Mreg$
by~\textup{\ref{A4}}.
\item Strict causality with respect to $q_{\mu\nu}$:
$\mathcal{B}_{\lP}=0$ for $\chi_H\leq0$, i.e.\ outside
$J_q^-(x)$.
\item GR recovery (conditional): $r_{\lP}\geq1$ implies
$\Theta_{\lP}=1$ implies $\hat{g}_{\mu\nu}=q_{\mu\nu}$; equation
\eqref{eq:field} reduces to GR when $T^{\rm C}_{\mu\nu}\to0$.
\item Overdense saturation: $S_\infty(r)=1$ for $r\geq1$; clocks
do not run faster than the GR rate.
\item $\lP\to0$ limit: $W_{H,\lP}\to W_H$ pointwise on $\Mreg$
for smooth fixed domains (Proposition~1).
\item $C^\infty$ regularity: on $\Mreg$ the functions
$\Omega_{H,\lP}$, $r_{\lP}$, $\Theta_{\lP}$ and their first and
second derivatives are finite and continuous; in particular
$\nabla\nabla\Theta_{\lP}$ is regular and the geodesic equation
in $\hat{g}_{\mu\nu}$ is well-posed.
\end{enumerate}
\end{proposition}
\begin{table}[h]
\centering
\caption{Comparison of the sharp and smeared theories on $\Mreg$.}
\label{tab:comparison}
\renewcommand{\arraystretch}{1.45}
\begin{tabular}{>{\bfseries}p{5.1cm} p{3.4cm} p{3.9cm}}
\toprule
Property
& Sharp \citep{Jaster2026a}
& Smeared (this work) \\
\midrule
Kernel type & Volume (top-hat) & Volume ($C^\infty$ bump) \\
$\lP\to0$ limit & --- & Top-hat (pointwise, $\Mreg$) \\
Boundary regularity & Step function & $C^\infty$ transition \\
$\nabla\Theta$, $\nabla\nabla\Theta$ & Distributional & Finite, continuous \\
Interpolation $S$ & $C^1$ at $r=1$ & $C^\infty$ everywhere \\
Geodesic eq.\ at boundary & Singular & Regular \\
Strict causality (wrt $q_{\mu\nu}$) & Yes & Yes \\
$\Theta\in[0,1]$ & Yes & Yes \\
FLRW: $\Theta=1$ & Yes & Yes \\
GR recovery$^\dagger$ & Yes & Yes \\
Planck-energy scale $E_P$ & None & $E_P=\hbar c/\lP$ \\
UV suppression (weak field) & None & Superpolynomial \\
Full UV-finiteness & Not addressed & Open \\
New dimensionful parameters & None & None$^\ddagger$ \\
\midrule
\multicolumn{3}{l}{$^\dagger$Conditional on $T^{\rm C}_{\mu\nu}\to0$
in the GR sector (open task).}\\
\multicolumn{3}{l}{$^\ddagger$Dimensionless functional choices
($\chi_H$, $\varphi$, $S_\infty$) remain part of the model
definition.}\\
\bottomrule
\end{tabular}
\end{table}
% =============================================================================
\section{Planck-Scale UV Suppression}
\label{sec:UV}
% =============================================================================
The $C^\infty$ function $\mathcal{B}_{\lP}$ introduces the scale
$\lP$ into the causal averaging kernel. In the weak-field,
flat-space approximation the Fourier transform of $\mathcal{B}_{\lP}$
with respect to the boundary coordinate decays faster than any power
of $|k|$ for $|k|\gg\hbar/\lP$, a standard property of $C^\infty$
functions with compact support. The exact form of this decay depends
on $\chi_H$ and $\varphi$; it is superpolynomial but not necessarily
exponential.
This rapid decay provides effective suppression of ultraviolet
contributions from causal-boundary gradients at scales $\lesssim\lP$,
establishing the Planck energy $E_P=\hbar c/\lP$ as a natural scale
for boundary-related effects. Two limitations apply. First, the
Fourier argument is valid only in flat space; in curved spacetime a
global momentum space does not exist and the statement is locally
meaningful only. Second, classical UV suppression of boundary
contributions does not imply UV-finiteness of a quantised theory; the
latter requires an action principle, a Hilbert space, and an explicit
quantisation scheme.
% =============================================================================
\section{Structural Remarks}
\label{sec:structural}
% =============================================================================
\paragraph{Machian construction with auxiliary reference geometry.}
The clock field $\Theta_{\lP}$ is determined by the matter
distribution in the causal past; this is the Machian content of
the theory. The construction employs an FLRW-normalised reference
geometry $q_{\mu\nu}$ for causal averaging, foliation, and
normalisation. The reference geometry is an auxiliary structure,
not a second physical metric; it coincides with $\hat{g}_{\mu\nu}$
in the GR sector. Its dynamical status is an open task of the parent
theory \citep[Open Problem~2]{Jaster2026a}.
\paragraph{Causal structure relative to two metrics.}
The kernel $W_{H,\lP}$ and the causality condition~\ref{A3} are
defined with respect to $q_{\mu\nu}$. Physical fields propagate on
$\hat{g}_{\mu\nu}$, which has a generally different null cone for
$\Theta_{\lP}<1$ \citep{Jaster2026a}. Since $\Theta_{\lP}\approx1$
in all tested regimes, the two light cones are virtually identical in
practice; the general relation between the two causal structures is an
open problem.
\paragraph{Problem of time.}
The Wheeler--DeWitt equation \citep{DeWitt1967} contains no explicit
time parameter. In the present framework this is a consequence rather
than a puzzle: time is emergent and defined by matter through
$\Theta_{\lP}$. Time arises from correlations between material
subsystems \citep{Page1983}, which is what $U_{H,\lP}^\mu$ and
$\Omega_{H,\lP}$ implement classically. A formal mathematical
embedding into canonical quantum gravity remains open.
\paragraph{Connection to discrete causal approaches.}
The Planck length appears as the fundamental causal grain in several
discrete approaches to quantum gravity \citep{Bombelli1987}. The
$C^\infty$ bump kernel can be interpreted as a continuum
regularisation ansatz consistent with a discrete causal structure of
characteristic spacing $\lP$. The precise derivation of the continuum
bump kernel from such a discrete framework is not established here;
the connection is motivational.
% =============================================================================
\section{Open Problems}
\label{sec:open}
% =============================================================================
\begin{enumerate}[label=(\arabic*)]
\item \textbf{Smoothness of $\chi_H$ at caustics (new).}
Assumption~\ref{A2} requires $\chi_H\in C^\infty$ near
$\partial\Dm(x)$. At caustics, cut loci, or topology changes of
$\Dm(x)$ this fails; a rigorous treatment, for example via Synge's
world function, lies outside the scope of this paper.
\item \textbf{Smoothness of $\mathcal{N}(x)$ (new).}
The normalisation factor may fail to be $C^\infty$ when the
topology of $\Dm(x)$ changes (a consequence of problem~1).
\item \textbf{UV suppression in curved spacetime (new).}
The UV argument of Section~\ref{sec:UV} uses flat-space Fourier
analysis; in curved spacetime no global momentum space exists and
the statement is only locally meaningful.
\item \textbf{Full quantisation (new).}
Classical UV suppression does not establish UV-finiteness; a
complete quantum theory requires an action, Hilbert space, and
propagating degrees of freedom.
\item \textbf{Infrared regulator for infinite causal pasts (new).}
Assumption~\ref{A1} excludes infinite-volume causal pasts;
such spacetimes require a large-scale regulator not defined here.
\item \textbf{Numerical implementation (new).}
Evaluating $\chi_H$, $\mathcal{B}_{\lP}$, $\mathcal{N}$, and
$\Omega_{H,\lP}$ in realistic curved spacetimes requires a
concrete numerical scheme not provided here.
\item \textbf{Constraint action $S_{\rm constr}$ and
$T^{\rm C}_{\mu\nu}$ (inherited from \citealt{Jaster2026a},
Open Problems~1 and~5).}
Dynamical closure of equation~\eqref{eq:field}.
\item \textbf{Status of $q_{\mu\nu}$ (inherited, Open Problem~2).}
Whether the reference geometry is prescribed or dynamically
generated.
\item \textbf{Fixed-point existence (inherited, Open Problem~4).}
Convergence of the self-consistent map
$T_{\mu\nu}\!\to\!\Pi_{H,\lP}^\mu\!\to\!
U_{H,\lP}^\mu\!\to\!\Omega_{H,\lP}\!\to\!\Theta_{\lP}
\!\to\!\hat{g}_{\mu\nu}\!\to\!T_{\mu\nu}$.
\item \textbf{Timelikeness of $\Pi_{H,\lP}^\mu$ (inherited,
Open Problem~11).}
Admissibility condition~\ref{A3} is not universally guaranteed;
edge cases with null or spacelike horizon energy-momentum require
analysis.
\item \textbf{Saturating function $S_\infty$ (inherited,
Open Problem~8).}
The functional form is a $C^\infty$ model ansatz; physical
derivation or observational constraint remains open.
\end{enumerate}
% =============================================================================
\section{Outlook}
\label{sec:outlook}
% =============================================================================
The Planck-scale structure of the kernel raises the question of
whether it induces energy-dependent modifications to the propagation
of matter fields. If the smearing generates an effective
energy-dependent metric, dimensional analysis suggests a possible
leading correction of the form
\begin{equation}
E^2 = p^2c^2 + m^2c^4
+ \alpha\,\frac{E^3}{E_P}
+ \mathcal{O}\!\left(\frac{E^4}{E_P^2}\right),
\label{eq:MDR}
\end{equation}
with $E_P=\hbar c/\lP$. Equation~\eqref{eq:MDR} is a
phenomenological parametrisation, not a derived result: the existence
of a cubic correction is not established, the coefficient $\alpha$ is
not computed here (it is a computable output of the kernel geometry,
not a free parameter of the Lagrangian), and a quantitative comparison
with astrophysical bounds \citep{Abdo2009} requires both an explicit
calculation of $\alpha$ and a careful treatment of the preferred-frame
structure introduced by $U_{H,\lP}^\mu$. Computing $\alpha$ and
establishing whether the dispersion relation~\eqref{eq:MDR} is the
appropriate effective description are identified as tasks for future
work.
% =============================================================================
\section{Summary}
\label{sec:summary}
% =============================================================================
The sharp top-hat kernel of \citet{Jaster2026a} is replaced by a
normalised $C^\infty$ causal bump function $\mathcal{B}_{\lP}$,
constructed through the boundary-defining function $\chi_H$ and the
$C^\infty$ saturating function $\varphi$. A concrete form of $\chi_H$
encoding the full four-dimensional retarded causal structure is given
for the flat FLRW case in equation~\eqref{eq:chiflat}. The smeared
kernel is a genuine volume-averaging kernel and recovers the
parent-theory top-hat pointwise in the regular sector $\Mreg$ as
$\lP\to0$. Simultaneously, the $C^1$ interpolation $S(r)$ is
replaced by $S_\infty(r)\in C^\infty(\mathbb{R})$, ensuring that
second derivatives of $\Theta_{\lP}$ are finite and continuous on
$\Mreg$.
Within $\Mreg$ (smooth $\chi_H$, finite $V_H$, timelike
$\Pi_{H,\lP}^\mu$, $\Theta_{\lP}>0$, smooth matter sources) all
structural properties of the parent theory are preserved: strict
causality with respect to the reference geometry $q_{\mu\nu}$,
$\Theta_{\lP}\in(0,1]$, exact GR recovery in homogeneous FLRW,
overdense saturation. New properties are: $C^\infty$ regularity of
$\Theta_{\lP}$ and its first and second derivatives; regular geodesic
equation at the causal boundary; a natural Planck-energy scale $E_P$;
and superpolynomial UV suppression of causal-boundary contributions in
the weak-field limit. No new dimensionful parameter is introduced;
dimensionless functional choices ($\chi_H$, $\varphi$, $S_\infty$)
remain part of the model definition. The constraint action
$S_{\rm constr}$, the quantisation of the theory, and the coefficient
$\alpha$ of the possible modified dispersion relation are identified as
tasks for future work.
\begin{table}[h]
\centering
\caption{Summary of core properties of the parent and smeared
theories.}
\label{tab:summary}
\renewcommand{\arraystretch}{1.45}
\begin{tabular}{>{\bfseries}p{5.3cm} p{3.3cm} p{4.0cm}}
\toprule
Property & Parent \citep{Jaster2026a} & Smeared (this work) \\
\midrule
Kernel & Top-hat & $C^\infty$ bump \\
$\chi_H$ (flat FLRW) & Not needed & Eq.~\eqref{eq:chiflat} \\
$\lP\to0$ limit & --- & Top-hat (pointwise, $\Mreg$) \\
Regular sector $\Mreg$ & Implicit & Explicit (Sec.~\ref{sec:sector}) \\
$\nabla\nabla\Theta$ at boundary & Distributional & Finite (on $\Mreg$) \\
Interpolation & $C^1$ at $r=1$ & $C^\infty$ everywhere \\
Planck-energy scale & None & $E_P=\hbar c/\lP$ \\
UV suppression (weak field) & None & Superpolynomial \\
UV-finiteness (quantum) & Not addressed & Open \\
MDR coefficient $\alpha$ & Not applicable & Computable; not computed \\
$T^{\rm C}_{\mu\nu}$ & Open task & Open task (inherited) \\
New dimensionful parameters & None & None$^\ddagger$ \\
\midrule
\multicolumn{3}{l}{$^\ddagger$Dimensionless functional choices remain
part of the model definition.}\\
\bottomrule
\end{tabular}
\end{table}
% =============================================================================
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\end{document}