% Addendum 250: Can x*_CZ Be Identified as a Fixed Point of the Spectral Density of Ô?
%   Local density rho(x), spectral flow J(x), eigenvalue-weighted position,
%   inward-falling attractor interpretation.
% 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}}
\newcommand{\DeltaS}{\Delta_{S^3}}
\newcommand{\Ov}{\Omega}
\newcommand{\al}{\alpha}
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\title{Can $x^*_{\mathrm{CZ}}$ Be Identified as a Fixed Point\\
       of the Spectral Density of $\Ohat$?\\[0.5em]
\large Addendum 250 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~249 showed that the $w=x$ self-adjointness condition for $\Dtwo$ does
not force $x^*_\infty = x^*_{\mathrm{CZ}}$ --- the commutator-zero subspace
forces $x^* = 2/3$, not the Cauchy-zero. Here we pursue the complementary
spectral approach: if $x^*_{\mathrm{CZ}} \approx 0.01420$ is the ``inward
falling from every direction'' attractor, the spectral density
$\rho(x) = \sum_n |\psi_n(x)|^2$ should be maximised there. We compute
$\rho(x)$, the spectral flow $J(x) = \sum_n |\psi_n(x)|^2\langle\psi_n|[\Ohat,\hat{x}]|\psi_n\rangle$,
the eigenvalue-weighted centroid $x^*_\lambda$, and exponentially-weighted
positions $x^*_{\exp}(\beta)$ in the $N=12$ Gegenbauer-Galerkin truncation.
The finding is a clean negative: every spectral statistic concentrates near
$x = 1$ rather than $x^*_{\mathrm{CZ}}$, and the spectral centroid
$\langle x\rangle_\rho(N)$ diverges from $x^*_{\mathrm{CZ}}$ as $N$ grows.
The sole partial alignment is that $|J(x^*_{\mathrm{CZ}})| < |J(x^*_{\mathrm{spec}})|$,
meaning the spectral flow at the CZ point is smaller than at the density
maximum --- but the absolute value ($\approx 2\times 10^9$) is far from zero.
We conclude that the ``inward falling'' interpretation, while physically
motivated, does not survive contact with the Galerkin truncation of $\Ohat$
in the Gegenbauer basis.  The open question of which basis or weight correctly
exposes $x^*_{\mathrm{CZ}}$ as an attractor is sharpened.
\end{abstract}

\tableofcontents

\section{Context and Motivation}

The Cauchy-zero $x^*_{\mathrm{CZ}} = \frac{\pi-1}{48\pi} \approx 0.014202$ has
appeared throughout the addenda series as the empirical ground-state position
in the flat-weight Gegenbauer-Galerkin truncation of the master operator $\Ohat$
(Addenda 243--248). Addendum~249 asked whether the self-adjointness condition
$w(x)=x$ forces this value; the answer was no.

The present addendum asks the complementary question from spectral theory: if
$x^*_{\mathrm{CZ}}$ is genuinely the attractor of the system, the \emph{spectral
density} of $\Ohat$, defined as the diagonal of the spectral projector
\begin{equation}
  \rho(x) = \sum_{n=0}^{N-1} |\psi_n(x)|^2,
  \label{eq:rho_def}
\end{equation}
should be concentrated near $x^*_{\mathrm{CZ}}$. In a dissipative or
convergent system, all trajectories relax toward the ground state; summing
$|\psi_n|^2$ over all modes weights the ground-state region most heavily as
higher modes oscillate away. This is the ``inward falling from every direction''
intuition.

We also examine the \emph{spectral flow}
\begin{equation}
  J(x) = \sum_n |\psi_n(x)|^2 \langle\psi_n|[\Ohat,\hat{x}]|\psi_n\rangle,
  \label{eq:J_def}
\end{equation}
which measures the local drift of $x$ under the operator flow. A fixed point
of the flow satisfies $J(x^*) = 0$.

\section{Setup: Operator and Galerkin Basis}

\subsection{The master operator}

Following Papers 18 and 30 and Addenda 243--249, the truncated operator is
\begin{equation}
  \Ohat = \Dtwo + \DeltaS + \gamma\,\DeltaS\,\Dtwo + \al\,\rho_{\mathrm{pot}}(x),
\end{equation}
with constants $\Omega = 4\pi^3+\pi^2+\pi$, $\al = 1/\Omega$, $\gamma = 3/4$,
and CZ-attractor potential
\begin{equation}
  \rho_{\mathrm{pot}}(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x.
\end{equation}
The Gegenbauer basis on $[0,1]$ is $\{U_n\}_{n=0}^{N-1}$ with
$U_0=1$, $U_1=4x-2$, and three-term recurrence $U_n=(4x-2)U_{n-1}-U_{n-2}$.
Derivatives are computed by the differentiated recurrence (numerically stable).
The Galerkin inner product uses flat weight $w(x)=1$.

\subsection{Eigenstates at $N=12$}

The $N=12$ Galerkin matrix has $10$ real eigenvalues after filtering negligible
imaginary parts. The first six eigenvalues are
\begin{equation*}
  \lambda_0 \approx -5.11\times10^9,\quad
  \lambda_1 \approx -3.58\times10^7,\quad
  \lambda_2 \approx -0.509,\quad
  \lambda_3 \approx -0.509,\quad
  \lambda_4 \approx 3.27\times10^3,\quad
  \lambda_5 \approx 5.28\times10^4.
\end{equation*}
The extreme spread (nine orders of magnitude between $\lambda_0$ and $\lambda_4$)
is characteristic of this operator: the $\Dtwo$ component generates very large
negative eigenvalues for the lowest modes, while the $\rho_{\mathrm{pot}}$
term contributes a large positive shift at higher basis indices.

Each eigenfunction $\psi_n(x) = \sum_k c_k^{(n)} U_k(x)$ is $L^2([0,1])$-normalised
using Gauss--Legendre quadrature with $N_Q = 3000$ points.

\section{Local Spectral Density $\rho(x)$}

The local spectral density \eqref{eq:rho_def} is evaluated on a 500-point
uniform grid over $(0.001,\,0.999)$. Key values at $N=12$:

\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Formula \\
\midrule
$\int_0^1 \rho(x)\,\d x$ & $9.694$ & $\approx N_{\mathrm{real}} = 10$ \\
$x^*_{\mathrm{spec}}$ (argmax $\rho$) & $0.99900$ & \\
$\langle x\rangle_\rho$ (centroid) & $0.76931$ & $\int x\rho / \int\rho$ \\
$x_{\min}(\rho)$ & $0.303$ & argmin $\rho$ \\
$x^*_{\mathrm{CZ}}$ & $0.014202$ & $({\pi-1})/(48\pi)$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent The completeness check $\int\rho = 9.694 \approx 10$ confirms the
normalisation is consistent with the number of real eigenstates ($<\!5\%$ error).
However, the maximum of $\rho(x)$ is at $x \approx 0.999$ --- far from
$x^*_{\mathrm{CZ}} \approx 0.014$.

\paragraph{Ground-state density.}
The ground-state contribution $\rho_0(x) = |\psi_0(x)|^2$ has its maximum also
at $x \approx 0.999$, with centroid $\langle x\rangle_{\rho_0} = 0.977$. The
ground state concentrates near $x=1$, not near $x^*_{\mathrm{CZ}}$.

\paragraph{Eigenvalue-weighted density.}
The eigenvalue-weighted density $\sigma(x) = \sum_n \lambda_n|\psi_n(x)|^2$
has its $|\sigma|$ centroid at $0.975$, also near $x=1$.

\begin{remark}
The concentration of all eigenstates near $x=1$ reflects the structure of
$\Dtwo = 16x^2\partial^4 + 96x\partial^3 + 96\partial^2$ on $[0,1]$: the
fourth-order operator with boundary condition at $x=0$ (ball boundary $\partial B^4$)
and $x=1$ (equator) forces large-amplitude modes near $x=1$ in the Gegenbauer
basis, which lacks an explicit zero at $x=1$.
\end{remark}

\section{Spectral Flow $J(x)$}

\subsection{Commutator matrix}

The commutator $[\Ohat, \hat{x}]$ is computed in the Galerkin representation as
\begin{equation}
  [H, X]_{mn} = (H X - X H)_{mn},
  \quad X_{mn} = \frac{\int_0^1 U_m(x)\,x\,U_n(x)\,\d x}{\int_0^1 U_m^2\,\d x}.
\end{equation}
The diagonal elements $\langle\psi_n|[\Ohat,\hat{x}]|\psi_n\rangle$ are
\begin{equation*}
  +4.14\times10^8,\quad -3.70\times10^8,\quad -3.32\times10^6,\quad -3.32\times10^6,\;\ldots
\end{equation*}
The spectral flow $J(x)$ ranges over $[-1.72\times10^{10},\, 1.27\times10^{10}]$,
with a single sign change (zero) at
\begin{equation*}
  x^*_J = 0.99519.
\end{equation*}

\subsection{Comparison with $x^*_{\mathrm{CZ}}$}

\begin{center}
\begin{tabular}{lll}
\toprule
Point & $|J(x)|$ & Notes \\
\midrule
$x^*_{\mathrm{CZ}} = 0.01420$ & $2.03\times10^9$ & candidate fixed point \\
$x^*_{\mathrm{spec}} = 0.99900$ & $1.27\times10^{10}$ & density maximum \\
$x^*_J = 0.99519$ & $\approx 0$ & true J-zero \\
\bottomrule
\end{tabular}
\end{center}

\noindent Although $|J(x^*_{\mathrm{CZ}})| < |J(x^*_{\mathrm{spec}})|$ by a
factor of $\approx 6$, the absolute value at $x^*_{\mathrm{CZ}}$ remains large
($\approx 2\times10^9$). The true zero of $J$ is the sign change at $x^*_J = 0.995$,
which is near the density maximum but not near $x^*_{\mathrm{CZ}}$.

\begin{remark}
The partial ordering $|J(x^*_{\mathrm{CZ}})| < |J(x^*_{\mathrm{spec}})|$
is consistent with $x^*_{\mathrm{CZ}}$ being in a ``quieter'' region of the
spectral flow --- the density is low there, so the flow magnitude is suppressed.
This is a consequence of $\rho(x^*_{\mathrm{CZ}}) \ll \rho(x^*_{\mathrm{spec}})$,
not of $x^*_{\mathrm{CZ}}$ being a genuine fixed point.
\end{remark}

\section{$N$-Scaling of the Spectral Centroid}

\begin{center}
\begin{tabular}{rrr}
\toprule
$N$ & $\langle x\rangle_\rho(N)$ & $|\langle x\rangle_\rho - x^*_{\mathrm{CZ}}|$ \\
\midrule
6  & 0.60907 & 0.59487 \\
8  & 0.67139 & 0.65719 \\
10 & 0.72424 & 0.71004 \\
12 & 0.76931 & 0.75511 \\
14 & 0.79075 & 0.77655 \\
16 & 0.80324 & 0.78904 \\
\bottomrule
\end{tabular}
\end{center}

The spectral centroid \emph{diverges} from $x^*_{\mathrm{CZ}}$ as $N$ grows,
increasing monotonically from $0.609$ at $N=6$ to $0.803$ at $N=16$. The
centroid is converging toward $x \approx 1$, not toward
$x^*_{\mathrm{CZ}} \approx 0.014$. The ground-state $N$-scaling series
(Addendum~248) showed slow drift toward $x^*_{\mathrm{CZ}}$ in the ground-state
position; the spectral centroid moves in the opposite direction.

\begin{remark}
The divergence of the spectral centroid from $x^*_{\mathrm{CZ}}$ does not
contradict Addendum~248. The spectral centroid $\langle x\rangle_\rho$ averages
over \emph{all} eigenstates with equal weight. As $N$ grows, more high-energy
modes (concentrated near $x=1$) enter the sum and drag the centroid rightward.
The ground-state centroid $\langle x\rangle_{\rho_0}$ is the relevant quantity
for the $x^*_{\mathrm{CZ}}$ hypothesis, and it also concentrates near $x=1$ in
this basis at $N=12$.
\end{remark}

\section{Eigenvalue-Weighted and Exponentially-Weighted Positions}

\subsection{Eigenvalue-weighted centroid}

The eigenvalue-weighted centroid
\begin{equation}
  x^*_\lambda = \frac{\sum_n \lambda_n\langle\psi_n|x|\psi_n\rangle}{\sum_n \lambda_n}
  \label{eq:xstar_lambda}
\end{equation}
evaluates to $x^*_\lambda = 0.97630$ at $N=12$. The large negative ground-state
eigenvalue ($\lambda_0 \approx -5.1\times10^9$) dominates, and since the
ground state concentrates near $x=1$, the weighted centroid is also near $x=1$.

\subsection{Exponential weighting}

The exponentially-weighted centroid $x^*_{\exp}(\beta)$ is defined by
\begin{equation}
  x^*_{\exp}(\beta) = \frac{\sum_n e^{-\beta\lambda_n}\langle\psi_n|x|\psi_n\rangle}{\sum_n e^{-\beta\lambda_n}}.
\end{equation}

\begin{center}
\begin{tabular}{lll}
\toprule
$\beta$ & $x^*_{\exp}(\beta)$ & $|x^*_{\exp} - x^*_{\mathrm{CZ}}|$ \\
\midrule
0.001 & 0.97856 & 0.96436 \\
0.01  & 0.97856 & 0.96436 \\
0.1   & 0.97856 & 0.96436 \\
1.0   & 0.97856 & 0.96436 \\
\bottomrule
\end{tabular}
\end{center}

The $x^*_{\exp}(\beta)$ is constant across all tested $\beta$ values because
the dominant exponential weight always belongs to $\lambda_0 \approx -5.1\times10^9$
(shifting eigenvalues by $-\lambda_{\min}$ before exponentiation, the ground state
weight $e^{0}=1$ while all others are $e^{-\beta(\lambda_n - \lambda_0)} \approx 0$
for any $\beta > 0$). The function $x^*_{\exp}(\beta)$ does \emph{not} cross
$x^*_{\mathrm{CZ}}$ for any positive $\beta$.

\section{Relation to Paper 18 and the Operator Foundation}

Paper~18 defines $\Ohat$ and states that the eigenvalue equation
$\Ohat\psi_n = E_n\psi_n$ encodes fundamental physics through the ground-state
normalisation ($\alpha^{-1}$), the spectral gap ($M_\text{Pl}$), and the
discrete spectrum (fermion masses). The paper defines $\rho(x)$ as the
geometric density $16\pi^3x^3+3\pi^2x^2+2\pi x$ and establishes the potential
coupling. The value $x^*_{\mathrm{CZ}} = (\pi-1)/(48\pi)$ is not defined in
Paper~18 as a fixed point; it emerged from the Galerkin series (Addendum~243
onward) as the empirical ground-state position in the flat-weight truncation.

The flow-zero interpretation --- that $x^*_{\mathrm{CZ}}$ is the point where
``inward falling terminates'' --- is consistent with Paper~18's framing of
$\rho_{\mathrm{pot}}(x)$ as a self-lensing attractor, but the spectral density
analysis shows this interpretation is not borne out quantitatively in the $N=12$
Gegenbauer truncation.

\section{Conclusions and Open Questions}

\begin{theorem}[Spectral density does not concentrate at $x^*_{\mathrm{CZ}}$]
In the $N=12$ Gegenbauer-Galerkin truncation of $\Ohat$, with flat weight
$w(x)=1$:
\begin{enumerate}[label=(\roman*)]
  \item The local spectral density $\rho(x) = \sum_n|\psi_n(x)|^2$ is maximised
        at $x^*_{\mathrm{spec}} \approx 0.999$, not at $x^*_{\mathrm{CZ}}$.
  \item The spectral centroid $\langle x\rangle_\rho \approx 0.769$ and diverges
        from $x^*_{\mathrm{CZ}}$ as $N$ grows.
  \item The spectral flow $J(x)$ has its zero at $x^*_J \approx 0.995$, not
        at $x^*_{\mathrm{CZ}}$.
  \item The eigenvalue-weighted position $x^*_\lambda \approx 0.976$ and
        $x^*_{\exp}(\beta) \approx 0.979$ for all tested $\beta$.
  \item None of the above quantities crosses or converges to
        $x^*_{\mathrm{CZ}} \approx 0.014$.
\end{enumerate}
\end{theorem}

\paragraph{Partial alignment.} The spectral flow at the CZ point satisfies
$|J(x^*_{\mathrm{CZ}})| \approx 2.0\times10^9 < |J(x^*_{\mathrm{spec}})| \approx 1.3\times10^{10}$,
a factor-6 suppression. This is consistent with $x^*_{\mathrm{CZ}}$ being in a
low-density region of the flow, but does not constitute proximity to a fixed point.

\begin{openproblem}
Identify the inner product (weight $w(x)$), basis, or regularisation under which
the spectral density of $\Ohat$ is maximised at $x^*_{\mathrm{CZ}}$, or prove
that no such choice exists within the family of polynomial Galerkin truncations.
\end{openproblem}

\begin{openproblem}
The ground-state position $\langle x\rangle_{\rho_0}(N)$ drifts toward
$x^*_{\mathrm{CZ}}$ under the Jacobi weight (Addendum~248) but the spectral
centroid $\langle x\rangle_\rho(N)$ diverges from it under flat weight. Determine
whether any convex combination of eigenstate contributions $\sum_n \alpha_n \langle\psi_n|x|\psi_n\rangle$
with $\alpha_n \geq 0$ and $\sum_n\alpha_n = 1$ converges to $x^*_{\mathrm{CZ}}$
as $N\to\infty$.
\end{openproblem}

\paragraph{Summary.} The series A246--A250 constitutes a systematic elimination:
flat-weight ground state (A246--A248) converges slowly to $x^*_{\mathrm{CZ}}$
under Jacobi weight; $w=x$ self-adjointness (A249) forces a different fixed point
($x=2/3$); spectral density (A250) concentrates near $x=1$. The status of
$x^*_{\mathrm{CZ}}$ as a spectral attractor of $\Ohat$ remains an open problem
requiring either a different inner product or a different definition of ``attractor.''

\subsection*{Numerical Setup}

All computations use \texttt{numpy} for matrix algebra and \texttt{mpmath}
(dps=60) for high-precision constants. Gauss--Legendre quadrature with
$N_Q=3000$ points on $[0,1]$. The full verifier with 30 checks is at
\texttt{Lumen/corpus/addenda/verify/verify\_P250.py}.

\vspace{1em}
\noindent\textbf{Checks passed:} 30/30.

\vspace{0.5em}
\noindent\textbf{Constants:}
$x^*_{\mathrm{CZ}} = 0.01420188\ldots$\,; $\Omega = 4\pi^3+\pi^2+\pi \approx 137.036$\,;
$\alpha = 1/\Omega$\,; $\gamma = 3/4$.

\end{document}
