% Addendum 249: Does w=x Self-Adjointness of D²_{B⁴} Force x*_∞ = x*_CZ?
%   Commutator [O-hat, x-hat] Analysis and w=x Galerkin Computation.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
\newcommand{\Ohat}{\hat{\mathcal{O}}}
\newcommand{\Dtwo}{D^2_{B^4}}
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\title{Does $w=x$ Self-Adjointness of $\Dtwo$\\
       Force $x^*_\infty = x^*_{\mathrm{CZ}}$?\\
       Commutator Analysis and $w=x$ Galerkin\\[0.5em]
\large Addendum 249 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~248 (A248) established that $\Dtwo = 16x^2\partial^4 + 96x\partial^3
+ 96\partial^2$ is formally self-adjoint under the unique weight $w(x)=x$
(Sturm-Liouville eigenvalue $\lambda=0$, Euler ODE $x^2w''+xw'-w=0$).
This addendum answers the question: does that self-adjointness condition
structurally force $x^*_\infty = x^*_{\mathrm{CZ}} = (\pi-1)/(48\pi)$?

\textbf{Answer: No.}
The commutator $[\Ohat, \hat{x}]$ is computed explicitly.
For the ground mode $n=0$, $[\Ohat,\hat{x}] = 64x^2\partial^3 + 288x\partial^2 + 192\partial$,
a third-order operator.
The commutator-zero equation $[\Ohat,\hat{x}]\psi = 0$ reduces to the Euler
ODE $x^2 u'' + \tfrac{9}{2}x u' + 3u = 0$ (with $u = \psi'$), whose indicial
roots are $r_{1,2} = -\tfrac{3}{2}, -2$ (both negative).
The only regular solution on $[0,1]$ is $\psi = \mathrm{const}$, which gives
$x^*(w{=}x) = \tfrac{2}{3}$ --- far from $x^*_{\mathrm{CZ}} \approx 0.01420$.
The $w=x$ Galerkin computation (Section~\ref{sec:galerkin}) confirms this
structurally: $x^*(N)$ under $w=x$ is monotone \emph{increasing} from
$0.9627$ at $N=6$ to $0.9956$ at $N=20$, extrapolating to
$x^*_\infty \approx 0.9995$, which is $69\times x^*_{\mathrm{CZ}}$.
The $w=x$ inner product emphasises large $x$, pulling the ground state toward
the far end of $[0,1]$; it is the \emph{wrong} weight for locating $x^*_{\mathrm{CZ}}$.
All 30 checks in \texttt{verify\_P249.py} pass at \texttt{mpmath} \texttt{dps}\,=\,60.
\end{abstract}

%==========================================================================
\section{Setup and Context}
\label{sec:setup}
%==========================================================================

The master operator is
\begin{equation}
  \Ohat = \Dtwo + \DeltaS + \gamma\,T + \alpha\,\rho(x),
  \qquad T = \DeltaS \circ \Dtwo,
\end{equation}
where $\Dtwo = 16x^2\partial^4 + 96x\partial^3 + 96\partial^2$ acts on
$[0,1]$; $\DeltaS$ is diagonal with mode-$n$ eigenvalue $-n(n+2)$;
$\gamma = 3/4$; $\alpha = 1/\Ov$ with $\Ov = 4\pi^3+\pi^2+\pi$; and
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$.  The commutator-zero
fixed point is
\begin{equation}
  x^*_{\mathrm{CZ}} = \frac{\pi-1}{48\pi} \approx 0.014\,202.
\end{equation}
A248 proved that the unique weight making $\Dtwo$ formally self-adjoint
is $w(x) = x$ (satisfying the Euler ODE $x^2 w'' + x w' - w = 0$ with
Sturm-Liouville eigenvalue $\lambda = 0$).  The question is whether this
geometric fact forces the ground-state expectation $x^* \to x^*_{\mathrm{CZ}}$.

%==========================================================================
\section{Commutator \texorpdfstring{$[\Ohat, \hat{x}]$}{[O,x]} Analysis}
\label{sec:commutator}
%==========================================================================

\subsection{Commutator of \texorpdfstring{$\Dtwo$}{D2\_B4} with \texorpdfstring{$\hat{x}$}{x-hat}}
\label{ssec:comm_D2}

For a differential operator $L$ and the multiplication operator $\hat{x}$,
the commutator $[L, \hat{x}]g = L(xg) - x\cdot Lg$ is computed by expanding
$(xg)^{(k)}$ via the Leibniz rule:
$(xg)' = xg' + g$, $(xg)'' = xg'' + 2g'$, $(xg)''' = xg''' + 3g''$,
$(xg)'''' = xg'''' + 4g'''$.

\begin{proposition}[Commutator of $\Dtwo$ with $\hat{x}$]\label{prop:comm}
  \begin{equation}\label{eq:comm_D2}
    [\Dtwo, \hat{x}] = 64x^2\partial^3 + 288x\partial^2 + 192\partial.
  \end{equation}
  On monomials $x^n$, this yields
  \begin{equation}\label{eq:comm_monomial}
    [\Dtwo, \hat{x}]\,x^n = 32\,n(n+1)(2n+1)\,x^{n-1}.
  \end{equation}
\end{proposition}

\begin{proof}
  Expanding $\Dtwo(xg)$ and subtracting $x\cdot\Dtwo g$:
  \begin{align*}
    \Dtwo(xg) &= 16x^2(xg'''' + 4g''') + 96x(xg''' + 3g'') + 96(xg'' + 2g')\\
               &= x\cdot\Dtwo g + 64x^2 g''' + 288x g'' + 192 g'.
  \end{align*}
  For $g = x^n$: inserting $g''' = n(n-1)(n-2)x^{n-3}$,
  $g'' = n(n-1)x^{n-2}$, $g' = nx^{n-1}$ gives
  $64n(n-1)(n-2) + 288n(n-1) + 192n = n[64n^2 + 96n + 32] = 32n(2n+1)(n+1)$.
\end{proof}

\noindent
Numerically verified: $[\Dtwo, \hat{x}]\,x^n$ for $n = 1, 2, 3, 4$ gives
$192, 960, 2688, 5760$, matching $32n(n+1)(2n+1)$ exactly (checks C01--C04).

\subsection{Full commutator \texorpdfstring{$[\Ohat, \hat{x}]$}{[O-hat, x-hat]}}
\label{ssec:comm_full}

\begin{proposition}[Full commutator of $\Ohat$ with $\hat{x}$]\label{prop:full_comm}
  For a fixed mode $n$ (where $\DeltaS$ acts as the scalar $\lambda_n = -n(n+2)$):
  \begin{equation}\label{eq:full_comm}
    [\Ohat, \hat{x}] = (1 + \gamma\lambda_n)\,[\Dtwo, \hat{x}].
  \end{equation}
  In particular, for the ground mode $n = 0$ ($\lambda_0 = 0$):
  \begin{equation}
    [\Ohat, \hat{x}]\big|_{n=0} = 64x^2\partial^3 + 288x\partial^2 + 192\partial.
  \end{equation}
\end{proposition}

\begin{proof}
  Three terms contribute to $[\Ohat, \hat{x}]$:
  \begin{enumerate}[label=(\roman*)]
    \item $[\rho(x)\cdot, \hat{x}] = 0$: multiplication operators commute.
    \item $[\DeltaS, \hat{x}] = [\lambda_n\cdot\mathrm{Id}, \hat{x}] = 0$: in
      mode $n$, $\DeltaS$ is the scalar $\lambda_n$, which commutes with everything.
    \item $[T, \hat{x}] = [\DeltaS\circ\Dtwo, \hat{x}] = \DeltaS\circ[\Dtwo,\hat{x}]$
      (since $\DeltaS = \lambda_n\cdot\mathrm{Id}$ in mode $n$)
      $= \lambda_n[\Dtwo,\hat{x}]$.
  \end{enumerate}
  Summing: $[\Ohat,\hat{x}] = [\Dtwo,\hat{x}] + \gamma\lambda_n[\Dtwo,\hat{x}] = (1+\gamma\lambda_n)[\Dtwo,\hat{x}]$.
\end{proof}

\begin{remark}
  For mode $n=1$: $\lambda_1 = -3$, factor $= 1 - 9/4 = -5/4$ (check C08).
  For mode $n=2$: $\lambda_2 = -8$, factor $= 1 - 6 = -5$ (check C26).
  The commutator grows more negative for higher modes; the ground mode $n=0$
  has the simplest form.
\end{remark}

%==========================================================================
\section{Commutator-Zero Subspace}
\label{sec:cz_subspace}
%==========================================================================

In mode $n=0$, the commutator-zero condition $[\Ohat,\hat{x}]\psi = 0$ is
\begin{equation}\label{eq:cz_eq}
  64x^2\psi''' + 288x\psi'' + 192\psi' = 0.
\end{equation}
Setting $u = \psi'$ yields the Euler ordinary differential equation
\begin{equation}\label{eq:euler_ode}
  x^2 u'' + \tfrac{9}{2}\,x u' + 3u = 0.
\end{equation}

\begin{theorem}[Commutator-zero subspace is trivial]\label{thm:cz_trivial}
  The only function $\psi \in C^3([0,1])$ that is regular at $x=0$ and
  satisfies~\eqref{eq:cz_eq} is $\psi = \mathrm{const}$.
\end{theorem}

\begin{proof}
  The Euler ODE~\eqref{eq:euler_ode} has the ansatz $u = x^r$, giving the
  indicial equation
  \begin{equation}
    r(r-1) + \tfrac{9}{2}r + 3 = r^2 + \tfrac{7}{2}r + 3 = 0.
  \end{equation}
  The discriminant is $49/4 - 12 = 1/4$, yielding
  \begin{equation}
    r_{1,2} = \frac{-7/2 \pm 1/2}{2} = -\frac{3}{2},\; -2.
  \end{equation}
  Both roots are negative; the general solution is
  $u(x) = Ax^{-3/2} + Bx^{-2}$, which is singular at $x = 0$ unless $A = B = 0$.
  Hence $u = \psi' = 0$, i.e.\ $\psi = \mathrm{const}$.
\end{proof}

\begin{corollary}[Commutator-zero $x^*$ under $w=x$]\label{cor:cz_xstar}
  The commutator-zero regular ground state $\psi = 1$ has
  \begin{equation}
    x^*_{w=x}(\mathrm{const}) = \frac{\int_0^1 x\cdot 1^2\cdot x\,\d x}{\int_0^1 1^2\cdot x\,\d x}
    = \frac{1/3}{1/2} = \frac{2}{3} \approx 0.667.
  \end{equation}
  This is $47$ times larger than $x^*_{\mathrm{CZ}} \approx 0.01420$.
  The $w=x$ self-adjointness condition does \emph{not} force
  $x^*_\infty = x^*_{\mathrm{CZ}}$.
\end{corollary}

\begin{remark}[Why $x^*_{\mathrm{CZ}}$ and commutator-zero are different conditions]
  $x^*_{\mathrm{CZ}}$ is the Cauchy-zero of the $\Ohat$ operator density
  (A243): the $x$-value where a certain spectral density function passes
  through zero.  Theorem~\ref{thm:cz_trivial} concerns a completely
  different condition: the subspace of functions whose position expectation
  is \emph{sharp} (i.e.\ $\hat{x}$ commutes with $\Ohat$ on that function).
  The two conditions coincide only if the ground state is simultaneously
  an eigenstate of $\hat{x}$, which requires $\hat{x}$ to commute with
  $\Ohat$ --- and the only regular such state is a constant, giving
  $x^* = 2/3$, not $x^*_{\mathrm{CZ}}$.
\end{remark}

%==========================================================================
\section{$w=x$ Inner Product Matrix}
\label{sec:inner_product}
%==========================================================================

\subsection{Monomial matrix elements}
\label{ssec:mono}

Under the $w=x$ inner product $\langle f, g\rangle_x = \int_0^1 f(x)g(x)x\,\d x$,
the matrix elements of $\Dtwo$ in the monomial basis $\{x^n\}$ are

\begin{equation}\label{eq:M_exact}
  M[m,n] \;=\; \int_0^1 x^m\,[\Dtwo(x^n)]\,x\,\d x
  \;=\; \frac{16n^2(n^2-1)}{m+n}, \quad n \geq 2;\quad M[m,n] = 0 \;\text{for}\; n < 2.
\end{equation}

The Gram matrix is
\begin{equation}\label{eq:gram}
  G[m,n] = \int_0^1 x^{m+n+1}\,\d x = \frac{1}{m+n+2}.
\end{equation}
Both formulas are verified for $m,n = 0, 1, 2, 3$ at $10^{-10}$ precision (checks C12--C13).

\subsection{Jacobi orthonormal basis under $w=x$}
\label{ssec:jacobi}

The Jacobi polynomials $P^{(0,1)}_n$ on $[-1,1]$ (weight $(1+t)$) shifted to
$[0,1]$ via $t = 2x-1$ are orthogonal under $x\,\d x$.  Their norms satisfy
\begin{equation}
  \int_0^1 [P^{(0,1)}_n(2x-1)]^2\,x\,\d x = \frac{1}{2(n+1)},
\end{equation}
so the orthonormal basis is $\varphi_n(x) = \sqrt{2(n+1)}\,P^{(0,1)}_n(2x-1)$.
Orthonormality $\langle\varphi_m,\varphi_n\rangle_x = \delta_{mn}$ is verified for
$m,n = 0,\ldots,4$ (25 pairs, check C14).

%==========================================================================
\section{$w=x$ Galerkin Computation}
\label{sec:galerkin}
%==========================================================================

\subsection{Setup}
\label{ssec:galerkin_setup}

The Galerkin matrix is built using the same Gegenbauer basis as A246/A248 but
with the $w=x$ inner product for all matrix entries.  For a basis function
$U_m$:
\begin{align}
  h_m &= \int_0^1 U_m^2\,x\,\d x,\\
  [D_2^{(wx)}]_{mn} &= \frac{1}{h_m}\int_0^1 U_m(x)\,[\Dtwo U_n](x)\,x\,\d x,\\
  [D_{\Delta}]_{mn} &= -n(n+2)\,\delta_{mn}\quad (\text{mode eigenvalue}),\\
  H^{(wx)} &= D_2^{(wx)} + D_\Delta + \gamma\,D_\Delta D_2^{(wx)} + \alpha\,\rho^{(wx)}.
\end{align}

The ground-state expectation value is
\begin{equation}\label{eq:xstar_wx}
  x^*(N;\,w{=}x) = \frac{\int_0^1 x^2|\psi_0|^2\,\d x}{\int_0^1 x|\psi_0|^2\,\d x}
  = \frac{\langle x\rangle_{w=x}^{\mathrm{num}}}{\langle 1\rangle_{w=x}^{\mathrm{den}}}.
\end{equation}
This is the $w=x$ expectation of $\hat{x}$, equal to
$\langle x^2\rangle_{\mathrm{flat}}/\langle x\rangle_{\mathrm{flat}}$.
All integrals use $N_q = 4000$-point Gauss-Legendre quadrature.

\subsection{Results}
\label{ssec:galerkin_results}

\begin{table}[ht]
\centering
\caption{$x^*(N;\,w{=}x)$ computed by $w=x$ Galerkin diagonalisation.}
\label{tab:xstar_wx}
\begin{tabular}{rrrr}
\toprule
$N$ & $x^*(N;\,w{=}x)$ & $|x^*-x^*_{\mathrm{CZ}}|$ & drift \\
\midrule
6  & 0.96273116 & 0.94852929 & --- \\
8  & 0.97801486 & 0.96381298 & $+0.01528$ \\
10 & 0.98518057 & 0.97097869 & $+0.00717$ \\
12 & 0.98923229 & 0.97503041 & $+0.00405$ \\
14 & 0.99177729 & 0.97757542 & $+0.00255$ \\
16 & 0.99349177 & 0.97928989 & $+0.00171$ \\
18 & 0.99470685 & 0.98050497 & $+0.00122$ \\
20 & 0.99560202 & 0.98140014 & $+0.00090$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent
All seven consecutive steps are \emph{monotone increasing}.  The sequence
moves away from $x^*_{\mathrm{CZ}}$ as $N$ grows.  A power-law fit
$x^*(N) = x^*_\infty + c/N^\alpha$ gives $x^*_\infty \approx 0.9995$
(RMSE $5.6\times10^{-5}$, $\alpha \approx 1.85$), which is
$70\times x^*_{\mathrm{CZ}}$.

\begin{table}[ht]
\centering
\caption{$x^*(N=12)$ under three inner product weights.}
\label{tab:compare_weights}
\begin{tabular}{lrl}
\toprule
Weight & $x^*(N=12)$ & Note \\
\midrule
$w = x$ & $0.9892$ & This addendum; converges to $\approx 1$ \\
$w = (1-x)^{3/2}$ (Haar exp.) & $0.0955$ & A248, monotone decreasing \\
$x^*_{\mathrm{CZ}}$ & $0.0142$ & Target fixed point \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Interpretation}
\label{ssec:galerkin_interp}

Under $w=x$ the inner product assigns weight proportional to $x$, so the
operator matrix heavily favours the region $x \approx 1$.  The ground state
of $H^{(wx)}$ concentrates near $x = 1$, giving $x^* \to 1$ as $N \to \infty$.
This is geometrically opposite to $x^*_{\mathrm{CZ}} \approx 0.014$, which
sits near $x = 0$.

By contrast, the $w = (1-x)^{3/2}$ weight (A248) assigns most weight near
$x = 0$, and the corresponding $x^*(N)$ decreases monotonically toward
$x^*_{\mathrm{CZ}}$ --- though still far from it at $N = 20$.

The Galerkin result thus confirms the analytic conclusion of
Section~\ref{sec:cz_subspace}: $w = x$ is the \emph{wrong} weight for
locating $x^*_{\mathrm{CZ}}$.  Its role in A248 was purely structural
(SL self-adjointness of $\Dtwo$), not physical (locating the fixed point).

\subsection{$\Dtwo$ alone under $w=x$}
\label{ssec:d2_alone}

For completeness, $\Dtwo$ alone (no $\DeltaS$, $T$, or $\rho$ terms) under
$w=x$ at $N=20$ gives $x^*(D^2; w{=}x; N{=}20) = 0.98885$, consistent
with the full-$\Ohat$ result.  The $\rho(x)$ potential term contributes
only a small correction ($\Delta x^* \approx 0.007$ at $N=20$), confirming
that the dominant effect is the $w=x$ weighting geometry, not the potential.

%==========================================================================
\section{Structural Explanation: Why $x^*_{\mathrm{CZ}}$ Is Not Forced}
\label{sec:structure}
%==========================================================================

The following diagram summarises the logical chain:

\begin{enumerate}[label=\arabic*.]
  \item \textbf{A248 result:} $w = x$ is the unique weight making $\Dtwo$
    self-adjoint (SL eigenvalue $\lambda = 0$, Euler ODE).
  \item \textbf{Commutator-zero condition} ($[\Ohat,\hat{x}]\psi = 0$): forces
    $\psi = \mathrm{const}$ by Theorem~\ref{thm:cz_trivial}.
  \item \textbf{$x^*$ for constant $\psi$ under $w=x$:} equals $2/3$
    (Corollary~\ref{cor:cz_xstar}).
  \item \textbf{$x^*_{\mathrm{CZ}} = 2/3$?} No; $x^*_{\mathrm{CZ}} \approx 0.014 \neq 2/3$.
  \item \textbf{$w=x$ Galerkin:} $x^*(N) \to 1$, moving away from $x^*_{\mathrm{CZ}}$.
\end{enumerate}

The chain breaks at step~4.  $x^*_{\mathrm{CZ}}$ is not the expectation of
$\hat{x}$ for any commutator-zero state; it enters from a different
structural source (the Cauchy-zero of the spectral density).  The $w=x$
self-adjointness of $\Dtwo$ is a real fact, but it locates neither
$x^*_{\mathrm{CZ}}$ nor the physical ground state.

%==========================================================================
\section{Summary}
\label{sec:summary}
%==========================================================================

\begin{table}[ht]
\centering
\caption{A249 key results.}
\label{tab:summary}
\begin{tabular}{ll}
\toprule
Quantity & Value \\
\midrule
$[\Dtwo, \hat{x}](x^n)$              & $32n(n+1)(2n+1)x^{n-1}$ (exact) \\
$[\Dtwo, \hat{x}]$ as operator       & $64x^2\partial^3 + 288x\partial^2 + 192\partial$ \\
$[\Ohat, \hat{x}]$ in mode $n$       & $(1+\tfrac{3}{4}\lambda_n)[\Dtwo,\hat{x}]$ \\
Comm.-zero ODE indicial roots        & $r_1 = -3/2$,\; $r_2 = -2$ \\
Regular null space dimension         & 1 (only $\psi = \mathrm{const}$) \\
$x^*(\psi{=}\mathrm{const};\,w{=}x)$ & $2/3 \approx 0.667$ \\
$x^*_{\mathrm{CZ}}$                  & $(\pi-1)/(48\pi) \approx 0.01420$ \\
$w=x$ forces $x^*_\infty = x^*_{\mathrm{CZ}}$? & \textbf{No} \\
$x^*(\Dtwo;\,w{=}x;\,N{=}20)$       & $0.98885$ \\
$x^*(\Ohat;\,w{=}x;\,N{=}12)$       & $0.98923$ \\
$x^*(\Ohat;\,w{=}x;\,N{=}20)$       & $0.99560$ \\
$x^*_\infty(w{=}x)$ extrapolated    & $\approx 0.9995$ \\
Convergence direction under $w=x$    & Increasing (away from $x^*_{\mathrm{CZ}}$) \\
Comparison weight at $N=12$          & $x^*(w{=}(1-x)^{3/2}) = 0.0955$ (A248) \\
Verifier checks                      & $30/30$ PASS \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Conclusions.}
(1) $[\Ohat,\hat{x}]$ in the ground mode is $64x^2\partial^3 + 288x\partial^2 + 192\partial$.
(2) Its null space on regular functions is one-dimensional, spanned by $\psi = \mathrm{const}$.
(3) The constant function gives $x^*(w{=}x) = 2/3$, not $x^*_{\mathrm{CZ}}$.
(4) The $w=x$ Galerkin confirms this: $x^*(N)$ increases monotonically to
$\approx 1$ under $w=x$, the geometric opposite of $x^*_{\mathrm{CZ}} \approx 0.014$.
(5) The $w=x$ self-adjointness of $\Dtwo$ is a valid SL fact but does not
link to $x^*_{\mathrm{CZ}}$, which arises from the Cauchy-zero structure of
the spectral density (A243), not from commutator algebra.

%==========================================================================
\section{Verifier}
\label{sec:verifier}
%==========================================================================

All claims are certified by
\texttt{Lumen/corpus/addenda/verify/verify\_P249.py}
(30 independent checks at \texttt{mpmath} \texttt{dps}\,=\,60):
commutator algebra C01--C08,
commutator-zero subspace C09--C11,
inner product matrices C12--C14,
$\Dtwo$-alone ground state C15--C16,
full Galerkin C17--C19,
structural conclusion C20--C22,
and sanity checks C23--C30. All 30 pass.

%==========================================================================
\begin{thebibliography}{9}
\bibitem{A242} L.\,F.\,Vlegels, \textit{Addendum 242: $\hat{\mathcal{O}}$ Eigentrajectory and $x^*_{\mathrm{CZ}}$ Fixed Point}, TOE Corpus, 2026.
\bibitem{A243} L.\,F.\,Vlegels, \textit{Addendum 243: Gegenbauer Basis Galerkin Approach to $x^*$}, TOE Corpus, 2026.
\bibitem{A246} L.\,F.\,Vlegels, \textit{Addendum 246: Unbiased Stable Basis for $\hat{\mathcal{O}}$ Eigentrajectory}, TOE Corpus, 2026.
\bibitem{A248} L.\,F.\,Vlegels, \textit{Addendum 248: Sturm-Liouville Weight Derivation and $x^*(N)\to x^*_{\mathrm{CZ}}$ Convergence}, TOE Corpus, 2026.
\end{thebibliography}

\end{document}
