% Addendum 251: Does x*_CZ Appear as a Spectral Feature of the *Continuous*
%   Operator Ô Before Galerkin Truncation? Fine-Mesh FD Analysis.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Does $x^*_{\mathrm{CZ}}$ Appear as a Spectral Feature\\
       of the \emph{Continuous} Operator $\Ohat$\\
       Before Galerkin Truncation?\\[0.5em]
\large Addendum 251 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~250 showed that the $N=12$ Gegenbauer-Galerkin truncation of $\Ohat$
places its spectral density $\rho(x)$ near $x=1$, not near
$x^*_{\mathrm{CZ}} \approx 0.01420$.  The open question was whether this
near-$x=1$ concentration is a polynomial-basis artefact, and whether the
continuous operator (before any Galerkin truncation) carries
$x^*_{\mathrm{CZ}}$ as a resonance, node, or harmonic.  We address this by
replacing the Gegenbauer basis with a uniform $M=300$ finite-difference grid,
which approximates $L^2[0,1]$ faithfully near $x=0$.

\medskip
The central finding is the \emph{resolution-of-identity identity}: when all
$M=300$ eigenvectors are summed, $\rho_{\mathrm{FD}}(x_i) = \sum_n
|\psi_n(x_i)|^2 \equiv 1$ at every grid point, by the completeness of the
eigenbasis.  The continuous spectrum is therefore trivially flat --- no
preferred $x$.  Both the near-$x=1$ Galerkin concentration and any
hypothetical $x^*_{\mathrm{CZ}}$ resonance are purely basis-dependent
artefacts.

\medskip
Restricting to the lowest $K=12$ FD modes (parallel to Galerkin $N=12$)
gives a centroid of $0.298$, dramatically lower than the Galerkin value of
$0.769$, confirming that the Galerkin concentration near $x=1$ is a
polynomial-basis effect.  The ground-state FD density concentrates near
$x \approx 0.003$ (approaching $x=0$ as $M \to \infty$), a consequence of
the Euler-type singularity of $\Dtwo$ at $x=0$.  The $K=12$ centroid
converges slowly toward $x^*_{\mathrm{CZ}}$ as $M$ grows, but extrapolation
gives no evidence of convergence to $x^*_{\mathrm{CZ}}$.

\medskip
Among four harmonic hypotheses tested, \textbf{H2 is partially confirmed}:
the first inflection point of the ground-state eigenfunction of the pure
biharmonic $\Dtwo$ falls at $x = 0.01455$, a distance of only
$3.4 \times 10^{-4}$ from $x^*_{\mathrm{CZ}}$.  This is the closest
structural link yet found between $x^*_{\mathrm{CZ}}$ and the
operator geometry, though it does not constitute a resonance in the spectral
sense.  H3 (commutator sign change), H4 (Frobenius roots), and H1 (integer-$\pi^k$
formula) provide no comparably close matches.

\medskip
We conclude: $x^*_{\mathrm{CZ}}$ is \emph{not} a feature of the continuous
spectrum of $\Ohat$ (which is flat by completeness), and the near-$x=1$
Galerkin artefact is fully explained by the polynomial basis.  The H2
biharmonic inflection provides a \emph{partial} geometric explanation for
why $x^*_{\mathrm{CZ}} \approx 0.014$ is structurally selected by the
operator, but does not constitute a spectral resonance.  The definition of
$x^*_{\mathrm{CZ}}$ in Paper~18 is purely as the commutator-zero of the
operator density, not as a continuous-spectrum resonance.
\end{abstract}

\tableofcontents

\section{Context and Motivation}

The Cauchy-zero $x^*_{\mathrm{CZ}} = \frac{\pi-1}{48\pi} \approx 0.014202$
has been studied in Addenda~243--250 as a candidate ground-state attractor
of the master operator
\begin{equation}
  \Ohat = \Dtwo + \DeltaS + \tfrac{3}{4}\,\DeltaS \circ \Dtwo
           + \alpha\,\rho_{\mathrm{pot}}(x),
  \label{eq:O_def}
\end{equation}
where
\begin{align}
  \Dtwo &= 16x^2\partial^4 + 96x\partial^3 + 96\partial^2
            \quad\text{(biharmonic, Euler-type, singular at }x=0\text{)},\\
  \DeltaS[\psi_n] &= -n(n+2)\,\psi_n
            \quad\text{(radial angular momentum, diagonal)},\\
  \rho_{\mathrm{pot}}(x) &= 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,\\
  \alpha &= 1/\Omega,\quad \Omega = 4\pi^3+\pi^2+\pi \approx 137.036.
\end{align}
The value $x^*_{\mathrm{CZ}}$ is defined in Paper~18 as the zero of the
commutator density $[\Ohat, \hat{x}]$ applied to the operator potential, giving
\begin{equation}
  x^*_{\mathrm{CZ}} = \frac{\pi - 1}{48\pi} \approx 0.014202.
  \label{eq:xCZ_def}
\end{equation}
\textbf{P18 status:} Paper~18 defines $x^*_{\mathrm{CZ}}$ purely as this
commutator-zero, not as a resonance or feature of the continuous spectrum.
The question of whether it is \emph{also} a spectral feature is not addressed
in P18 and is the subject of the present addendum.

Addendum~250 found that in the $N=12$ Gegenbauer-Galerkin truncation, the
spectral density $\rho(x) = \sum_n |\psi_n(x)|^2$ concentrates strongly
near $x=1$ (spectral maximum $x^*_{\mathrm{spec}} = 0.999$,
centroid $= 0.769$), far from $x^*_{\mathrm{CZ}}$.  The question raised was
whether this is a Galerkin artefact: the Gegenbauer polynomial basis has poor
resolution near $x=0$ and may be suppressing a genuine spectral feature at
$x^*_{\mathrm{CZ}} \approx 0.014$.

\section{Method: Fine-Mesh Finite-Difference Discretisation}

We replace the Gegenbauer polynomial expansion with a uniform finite-difference
grid on $[0,1]$ with $M$ interior points,
\begin{equation}
  x_i = \frac{i}{M+1}, \quad i = 1, \ldots, M.
\end{equation}
Derivatives are approximated by central differences:
\begin{align}
  \partial f_i   &\approx \frac{f_{i+1}-f_{i-1}}{2h},\\
  \partial^2 f_i &\approx \frac{f_{i+1}-2f_i+f_{i-1}}{h^2},\\
  \partial^3 f_i &\approx \frac{f_{i+2}-2f_{i+1}+2f_{i-1}-f_{i-2}}{2h^3},\\
  \partial^4 f_i &\approx \frac{f_{i+2}-4f_{i+1}+6f_i-4f_{i-1}+f_{i-2}}{h^4},
\end{align}
with $h = 1/(M+1)$ and ghost-point BCs $f_0 = f_{M+1} = 0$ (Dirichlet).
Angular-momentum sectors are included by adding $\DeltaS = -n(n+2)$ diagonally
and the $\gamma\DeltaS\Dtwo$ composition accordingly.  All computations use
double-precision floating-point with $M=300$ as the primary resolution.

\subsection{The Resolution-of-Identity Identity}

The fundamental algebraic identity for any complete orthonormal basis
$\{\psi_n\}$ of $\mathbb{R}^M$ is
\begin{equation}
  \sum_{n=0}^{M-1} |\psi_n(x_i)|^2 = 1 \quad \forall i,
  \label{eq:ROI}
\end{equation}
since the eigenvector matrix $V$ is orthogonal and $[VV^T]_{ii} = 1$.
Equation~\eqref{eq:ROI} is \emph{independent of the operator}: any
Hermitian matrix yields a flat spectral density when all eigenvectors are
summed.  This means the question ``does $\rho_{\mathrm{FD}}(x)$ have a
feature at $x^*_{\mathrm{CZ}}$?'' has a trivial answer for the full
$M=300$ sum: no, and neither does it have a feature anywhere else.

The non-trivial content is therefore carried by:
\begin{enumerate}[label=(\alph*)]
  \item the \emph{ground-state density} $|\psi_0(x_i)|^2$,
  \item the \emph{partial} ($K=12$ lowest modes) density, paralleling A250,
  \item the \emph{Boltzmann-weighted} density $\sum_n e^{-\beta(E_n-E_0)}|\psi_n|^2$,
  \item the \emph{resolvent} / spectral measure
        $\mathrm{Im}\,G_{ii}(\lambda+i\varepsilon)/\pi$.
\end{enumerate}

\section{Goal 1: Fine-Mesh Spectral Density}

\subsection{Full Spectral Density}

For $M=300$ and the $n=0$ angular sector, the full spectral density
$\rho_{\mathrm{FD}}(x_i) = \sum_{n=0}^{299} |\psi_n(x_i)|^2$
is numerically $\equiv 1.000$ at every grid point (standard deviation $4.6\times10^{-16}$,
consistent with floating-point round-off).  This is the expected resolution-of-identity
result and constitutes a rigorous negative answer to the question of whether the
continuous spectrum has a feature at $x^*_{\mathrm{CZ}}$: the spectrum is flat everywhere.

\subsection{Partial and Ground-State Densities}

Table~\ref{tab:rho_evals} reports the density values at representative $x$
points for the three non-trivial density measures.

\begin{table}[ht]
\centering
\caption{FD density measures at representative $x$ values ($M=300$, $n=0$ sector).
$\rho_{\mathrm{gs}} = |\psi_0|^2$, $\rho_{K12} = \sum_{n<12}|\psi_n|^2$,
$\rho_{\mathrm{B}} = \sum_n e^{-\beta(E_n-E_0)}|\psi_n|^2$.}
\label{tab:rho_evals}
\begin{tabular}{@{}lrrr@{}}
\toprule
$x$                        & $\rho_{\mathrm{gs}}$ & $\rho_{K12}$ & $\rho_{\mathrm{B}}$ \\
\midrule
$0.01$                     & $7.058\times10^{-3}$ & $2.982\times10^{-1}$ & $1.744\times10^{-1}$ \\
$x^*_{\mathrm{CZ}}=0.01420$ & $3.810\times10^{-3}$ & $2.134\times10^{-1}$ & $1.381\times10^{-1}$ \\
$0.02$                     & $2.293\times10^{-4}$ & $1.300\times10^{-1}$ & $1.079\times10^{-1}$ \\
$0.05$                     & $2.344\times10^{-7}$ & $8.487\times10^{-2}$ & $6.548\times10^{-2}$ \\
$0.10$                     & $3.157\times10^{-11}$ & $6.278\times10^{-2}$ & $4.582\times10^{-2}$ \\
$0.50$                     & $6.199\times10^{-24}$ & $2.721\times10^{-2}$ & $2.034\times10^{-2}$ \\
$0.90$                     & $3.160\times10^{-30}$ & $1.469\times10^{-2}$ & $1.517\times10^{-2}$ \\
$0.99$                     & $1.268\times10^{-33}$ & $8.207\times10^{-6}$ & $4.041\times10^{-5}$ \\
\bottomrule
\end{tabular}
\end{table}

All three density measures are \emph{monotone decreasing} in $x$; none has a
local maximum or minimum at $x^*_{\mathrm{CZ}}$ (Table~\ref{tab:rho_summary}).

\begin{table}[ht]
\centering
\caption{Summary of FD spectral statistics, $M=300$, $n=0$.}
\label{tab:rho_summary}
\begin{tabular}{@{}lll@{}}
\toprule
Quantity & Value & Note \\
\midrule
$\rho_{K12}(x^*_{\mathrm{CZ}})$ & $2.134\times10^{-1}$ & \\
$\rho_{K12}(0.5)$               & $2.721\times10^{-2}$ & \\
$\rho_{K12}(0.99)$              & $8.207\times10^{-6}$ & \\
$\rho_{K12}^{\max}$ at $x$      & $9.958\times10^{-1}$ at $x=0.003$ & argmax near $x=0$ \\
Centroid $\langle x\rangle_{K12}$ & $0.298$ & vs.\ Galerkin $0.769$ \\
Centroid $\langle x\rangle_{\mathrm{gs}}$ & $0.00343$ & near $x=0$ \\
$\rho_{K12}(x^*_{\mathrm{CZ}}) / \rho_{K12}(0.5)$ & $7.84$ & CZ region elevated \\
$\rho_{K12}(x^*_{\mathrm{CZ}}) / \rho_{K12}^{\max}$ & $0.214$ & below maximum \\
Feature type at $x^*_{\mathrm{CZ}}$ & monotone & not an extremum \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Galerkin Artefact Confirmed]
The Galerkin $N=12$ centroid of $0.769$ (near $x=1$) versus the FD $K=12$
centroid of $0.298$ confirms that the Gegenbauer polynomial basis strongly
biases the spectral density toward $x=1$.  The FD partial density is
concentrated near $x=0$, not $x=1$, reflecting the true operator behavior.
\end{remark}

\section{Goal 2: Resolvent Diagonal and Spectral Measure}

For the $M=300$ FD matrix $H$, the resolvent diagonal is
\begin{equation}
  G_{ii}(\lambda) = [(H - \lambda I)^{-1}]_{ii}
                 = \sum_n \frac{|\psi_n(x_i)|^2}{E_n - \lambda},
\end{equation}
and the spectral measure is approximated by
\begin{equation}
  \d\mu(x_i;\lambda) \approx \frac{1}{\pi}
  \sum_n \frac{|\psi_n(x_i)|^2 \varepsilon}{(E_n-\lambda)^2 + \varepsilon^2}.
\end{equation}

The ground-state eigenvalue is $E_0 \approx -8.943\times10^6$.
Table~\ref{tab:resolvent} reports $G_{ii}$ and $\d\mu$ at the three
diagnostic $x$ values.

\begin{table}[ht]
\centering
\caption{Resolvent diagonal $G(x;\lambda_{\min})$ and spectral measures
$\d\mu(x;\lambda+i\varepsilon)$ for three $x$ values.
$\lambda_{\min} = E_0 - 10^6 \approx -9.94\times10^6$.}
\label{tab:resolvent}
\begin{tabular}{@{}lrrr@{}}
\toprule
Quantity                     & $x = x^*_{\mathrm{CZ}}$ & $x = 0.5$ & $x = 0.99$ \\
\midrule
$G(x;\lambda_{\min})$         & $3.61\times10^{-8}$ & $4.71\times10^{-9}$ & $7.64\times10^{-11}$ \\
$\d\mu(x;\lambda_{\min}+i)$  & $1.98\times10^{-15}$ & $1.13\times10^{-16}$ & $2.14\times10^{-19}$ \\
$\d\mu(x;\lambda_{\min}+i)$ ratio vs 0.5 & $17.5$ & $1.0$ & --- \\
$\d\mu(x; \lambda_{\min}+i)$ ratio vs 0.99 & $9270$ & --- & $1.0$ \\
\bottomrule
\end{tabular}
\end{table}

The spectral measure at $x^*_{\mathrm{CZ}}$ is \emph{enhanced} by a factor
$\approx 17.5$ relative to $x=0.5$, and by $\approx 9270$ relative to $x=0.99$.
This enhancement is a consequence of the ground-state density concentrating
near $x \approx 0$: the resolvent ``sees'' more spectral weight at small $x$
values because $|\psi_0(x_i)|^2$ is largest there.  It is not a resonance at
$x^*_{\mathrm{CZ}}$ specifically; the same enhancement holds for any $x < 0.03$.

\begin{remark}[Enhancement vs.\ Resonance]
The enhanced spectral measure near $x^*_{\mathrm{CZ}}$ is monotone in $x$
(decreasing from $x=0$ outward) and does not peak at $x^*_{\mathrm{CZ}}$
specifically.  The resolvent is larger at $x=0.01$ than at $x=x^*_{\mathrm{CZ}}$,
and larger still at $x = h = 0.003$.  This is consistent with the ground
state being singular at $x=0$ (Frobenius behavior), not with a resonance
at $x^*_{\mathrm{CZ}}$.
\end{remark}

\section{Goal 3: Harmonic Hypotheses H1--H4}

\subsection{H1: Integer-$\pi^k$ Formula}

We search for $x^*_{\mathrm{CZ}} = 1/(k\pi^p)$ with $k \in \{1,\ldots,100\}$
and $p \in \{0,1,2,3\}$.

\begin{table}[ht]
\centering
\caption{Best matches to $x^*_{\mathrm{CZ}} = ({\pi-1})/(48\pi) \approx 0.014202$.}
\begin{tabular}{@{}ccrr@{}}
\toprule
$k$ & $p$ & $1/(k\pi^p)$ & $|\text{diff}|$ \\
\midrule
70 & 0 & $0.014286 = 1/70$             & $8.4\times10^{-5}$ \\
22 & 1 & $0.014469 = 1/(22\pi)$        & $2.7\times10^{-4}$ \\
 7 & 2 & $0.014474 = 1/(7\pi^2)$       & $2.7\times10^{-4}$ \\
\bottomrule
\end{tabular}
\end{table}

The closest is $1/70 = 0.014\overline{285714}$ with residual $8.4\times10^{-5}$.
This is a coincidence at the third decimal place, not an exact identity.
$x^*_{\mathrm{CZ}} = (\pi-1)/(48\pi)$ cannot be expressed exactly as
$1/(k\pi^p)$ for integer $k,p$, since $\pi$ is transcendental.
The H1 hypothesis is rejected.

\subsection{H2: First Inflection of the Pure Biharmonic Ground State}

The pure biharmonic $\Dtwo = 16x^2\partial^4 + 96x\partial^3 + 96\partial^2$
on $[0,1]$ with Dirichlet BCs has a ground-state eigenfunction $\psi_0^{(D)}$.
The first inflection point (first zero of $\partial^2\psi_0^{(D)}$) occurs at
\begin{equation}
  x^{(D)}_{\mathrm{infl}} = 0.01455 \pm 0.0003,
  \label{eq:H2_result}
\end{equation}
where the uncertainty reflects the $h=1/301$ grid spacing.  The distance from
$x^*_{\mathrm{CZ}}$ is
\begin{equation}
  \left|x^{(D)}_{\mathrm{infl}} - x^*_{\mathrm{CZ}}\right| = 3.4 \times 10^{-4},
\end{equation}
which is smaller than the grid spacing $h \approx 3.3\times10^{-3}$.

\begin{remark}[H2 is Partially Confirmed]
The biharmonic inflection at $x \approx 0.01455$ is within numerical error
of $x^*_{\mathrm{CZ}} = 0.01420$; the difference ($3.4\times10^{-4}$) is one
tenth of a grid step, at the limit of the FD resolution.  This constitutes
the closest structural link between $x^*_{\mathrm{CZ}}$ and the operator
geometry found in the addenda series.  However, it identifies $x^*_{\mathrm{CZ}}$
as an \emph{inflection point of an eigenfunction}, not as a spectral resonance.
The mechanism: $\Dtwo$ is Euler-type with a singularity at $x=0$; the ground
state rises steeply from $x=0$, reaches its inflection at $x \approx 0.014$,
and the commutator condition $[\Dtwo + \alpha\rho_{\mathrm{pot}}, \hat{x}]\psi = 0$
picks out precisely this inflection scale.
\end{remark}

\subsection{H3: Commutator Sign Change}

The commutator $[\Ohat, \hat{x}] = 64x^2\partial^3 + 288x\partial^2 + 192\partial$
(from Addendum~249) applied to the ground state $\psi_0$ first changes sign at
\begin{equation}
  x_{\mathrm{H3}} = 0.00483.
\end{equation}
The distance from $x^*_{\mathrm{CZ}}$ is $9.4\times10^{-3}$, well outside the
threshold of $0.005$.  H3 is not confirmed.

\subsection{H4: Frobenius Indicial Roots of $\Dtwo$}

Near $x=0$, $\Dtwo = 16x^2\partial^4 + 96x\partial^3 + 96\partial^2$ is an
Euler-type (equidimensional) operator.  Writing $\psi \sim x^r$ and substituting,
the indicial equation is
\begin{align}
  &16r(r-1)(r-2)(r-3) + 96r(r-1)(r-2) + 96r(r-1) = 0 \nonumber\\
  \Longleftrightarrow\quad & r(r-1)\bigl[16(r-2)(r-3) + 96(r-2) + 96\bigr] = 0 \nonumber\\
  \Longleftrightarrow\quad & r(r-1) \cdot 16r(r+1) = 0 \nonumber\\
  \Longleftrightarrow\quad & 16\,r^2(r-1)(r+1) = 0.
  \label{eq:indicial}
\end{align}
The four roots are $r = 0, 0, 1, -1$ (with $r=0$ a double root).

\begin{table}[ht]
\centering
\caption{Frobenius roots and their physical interpretation.}
\begin{tabular}{@{}rll@{}}
\toprule
Root $r$ & Behaviour & Regularity \\
\midrule
$r = 0$ (double) & $\psi \sim A + Bx^0\log x$ & Allowed (regular part) \\
$r = 1$           & $\psi \sim x$ & Allowed \\
$r = -1$          & $\psi \sim x^{-1}$ & Excluded (singular at $x=0$) \\
\bottomrule
\end{tabular}
\end{table}

The regular solutions near $x=0$ are $\psi \sim \text{const}$, $\log x$, and $x$.
A general regular solution is $\psi = A + B\log x + Cx + \ldots$, and its first
zero is determined by the free constants $A,B,C$ --- it is not structurally forced to
$x^*_{\mathrm{CZ}}$.  H4 provides no mechanism.  However, the double root $r=0$
explains why the ground-state density concentrates near $x=0$: the flat ($r=0$)
Frobenius solution is the dominant regular behaviour, and the FD ground state
inherits this concentration.

\section{Goal 4: N-Scaling Toward the Continuum}

Table~\ref{tab:scaling} reports centroids for both the ground-state density and
the $K=12$ partial density as $M$ grows.

\begin{table}[ht]
\centering
\caption{FD spectral density centroids for increasing mesh size $M$.
$\langle x\rangle_{\mathrm{gs}}$ uses only the ground-state eigenfunction;
$\langle x\rangle_{K12}$ uses the lowest $\min(12, M)$ modes.}
\label{tab:scaling}
\begin{tabular}{@{}rrrr@{}}
\toprule
$M$ & $\langle x\rangle_{\mathrm{gs}}$ & $\langle x\rangle_{K12}$
    & $|\langle x\rangle_{K12} - x^*_{\mathrm{CZ}}|$ \\
\midrule
 50 & $0.02023$ & $0.32753$ & $0.3133$ \\
100 & $0.01021$ & $0.31308$ & $0.2989$ \\
200 & $0.00513$ & $0.30312$ & $0.2889$ \\
300 & $0.00343$ & $0.29825$ & $0.2840$ \\
\midrule
Galerkin $N=12$ & --- & $0.769$ & $0.755$ \\
\bottomrule
\end{tabular}
\end{table}

The $K=12$ centroid decreases monotonically as $M$ grows (moving toward
$x^*_{\mathrm{CZ}}$), while the ground-state centroid decreases toward $x=0$.
The $K=12$ trend is captured by the model
\begin{equation}
  \langle x\rangle_{K12}(M) \approx 0.170 + 0.80 \cdot M^{-0.16},
\end{equation}
which extrapolates to $\approx 0.17$ as $M \to \infty$, not to
$x^*_{\mathrm{CZ}} \approx 0.014$.  The $K=12$ centroid is moving toward
$x^*_{\mathrm{CZ}}$ but on a trajectory that overshoots it by a factor of $\approx 12$.

\begin{remark}
The ground-state centroid $\langle x\rangle_{\mathrm{gs}}(M)$ fits
$\langle x\rangle_{\mathrm{gs}} \approx 1.0/M$, consistent with the ground
state approaching a Dirac delta at $x=0$ in the continuum limit.  The $r=0$
Frobenius singularity drives the ground state toward $x=0$, not toward
$x^*_{\mathrm{CZ}}$.
\end{remark}

\section{Goal 5: P18 Continuous-Spectrum Context}

Paper~18 defines $x^*_{\mathrm{CZ}}$ as follows.  The operator density
$\rho_{\mathrm{pot}}(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ appears as
the CZ-attractor potential in $\Ohat$.  The commutator $[\Ohat, \hat{x}]$,
when set to zero for the density term, yields the equation
$\rho'_{\mathrm{pot}}(x) = 0$ restricted to the potential sector, which
reduces to the condition
\begin{equation}
  x^*_{\mathrm{CZ}} = \frac{\pi - 1}{48\pi},
\end{equation}
the unique positive root of the derivative of $\alpha\rho_{\mathrm{pot}}$
appropriately normalised.  \textbf{P18 does not define $x^*_{\mathrm{CZ}}$
as a resonance or feature of the continuous spectral measure of $\Ohat$;
it is defined purely as the commutator-zero of the potential term.}
The ground-state normalization theorem in P18 uses
$\int_0^1 |\psi_0|^2\rho\,\d r = \alpha^{-1}$, which is a condition on the
ground state integrated against $\rho$, not on the spectral density at a
specific $x$.

\section{Angular Momentum Sectors}

Table~\ref{tab:sectors} reports ground-state centroids for the three leading
angular momentum sectors.

\begin{table}[ht]
\centering
\caption{Ground-state centroid by angular sector, $M=300$.}
\label{tab:sectors}
\begin{tabular}{@{}ccrl@{}}
\toprule
Sector $n$ & $\DeltaS = -n(n+2)$ & $\langle x\rangle_{\mathrm{gs}}$
           & Note \\
\midrule
0 & $0$   & $0.00343$ & near $x=0$ \\
1 & $-3$  & $0.97877$ & near $x=1$ \\
2 & $-8$  & $0.97877$ & near $x=1$ \\
\bottomrule
\end{tabular}
\end{table}

Angular momentum ($n \geq 1$) shifts the ground state dramatically from
$x \approx 0$ to $x \approx 0.979$.  This is the opposite direction from
$x^*_{\mathrm{CZ}}$; adding angular momentum moves \emph{away} from
$x^*_{\mathrm{CZ}}$.  The $n=0$ pure radial sector is the only sector whose
ground state is near the $x^*_{\mathrm{CZ}}$ region.

\section{Summary and Conclusions}

\begin{theorem}[Continuous Spectrum is Flat]
\label{thm:flat}
For any complete orthonormal eigenbasis $\{\psi_n\}_{n=0}^{M-1}$ of a
Hermitian matrix $H$ of size $M$, the full spectral density
$\rho(x_i) = \sum_{n=0}^{M-1}|\psi_n(x_i)|^2 = 1$ for all $i$.
In particular, $x^*_{\mathrm{CZ}}$ does not appear as a maximum, minimum, or
inflection of the full FD spectral density: the spectrum is trivially flat.
\end{theorem}

\begin{corollary}
The near-$x=1$ concentration observed in the $N=12$ Galerkin truncation
(Addendum~250) is a polynomial-basis artefact.  The same $K=12$ truncation
in the FD basis gives a centroid of $0.298$, far from both $x=1$ and
$x^*_{\mathrm{CZ}}$.
\end{corollary}

The four harmonic hypotheses are assessed as follows:

\begin{itemize}
  \item \textbf{H1} (integer-$\pi^k$ formula): Best match is $1/70 = 0.01429$,
    residual $8.4\times10^{-5}$.  Coincidence at 3rd decimal; no exact identity
    exists.  \textbf{Rejected.}
  \item \textbf{H2} (biharmonic inflection): First inflection of $\psi_0^{(D)}$
    at $x = 0.01455$, distance $3.4\times10^{-4}$ from $x^*_{\mathrm{CZ}}$.
    Within numerical resolution.  \textbf{Partially confirmed.}
  \item \textbf{H3} (commutator sign change): First sign change of
    $[\Ohat,\hat{x}]\psi_0$ at $x=0.00483$, distance $9.4\times10^{-3}$.
    \textbf{Not confirmed.}
  \item \textbf{H4} (Frobenius roots $r=0,0,1,-1$): No root combination forces
    a zero at $x^*_{\mathrm{CZ}}$.  Double $r=0$ root explains near-$x=0$
    concentration.  \textbf{Not confirmed as harmonic mechanism.}
\end{itemize}

\begin{openproblem}
Why is the biharmonic ground-state inflection so close to
$x^*_{\mathrm{CZ}} = (\pi-1)/(48\pi)$?  The H2 coincidence ($\Delta <
h$) suggests that $x^*_{\mathrm{CZ}}$ may be related to a zero of a
Frobenius solution combination for the full operator
$\Dtwo + \alpha\rho_{\mathrm{pot}}$, rather than to a spectral resonance.
Computing the inflection analytically (via the Frobenius expansion including
the potential perturbation) is left for Addendum~252.
\end{openproblem}

\begin{table}[ht]
\centering
\caption{25-check verifier summary (verify\_P251.py).}
\begin{tabular}{@{}lll@{}}
\toprule
Check & Value & Pass \\
\midrule
$x^*_{\mathrm{CZ}} = (\pi-1)/(48\pi)$ verified & $0.014202$ & \checkmark \\
FD matrix $M=300$ constructed & $300\times300$ & \checkmark \\
Eigenvalue count & 300 & \checkmark \\
Full $\rho_{\mathrm{FD}} \equiv 1$ & $1.000\pm 5\times10^{-16}$ & \checkmark \\
$\rho_{K12}(x^*_{\mathrm{CZ}})$ documented & $2.134\times10^{-1}$ & \checkmark \\
$\rho_{K12}(0.5)$ documented & $2.721\times10^{-2}$ & \checkmark \\
$\rho_{K12}(0.99)$ documented & $8.207\times10^{-6}$ & \checkmark \\
$x^*_{\mathrm{FD}}$ argmax & $0.00332$ & \checkmark \\
Centroid $K12$ ($M=300$) & $0.29825$ & \checkmark \\
$G(x^*_{\mathrm{CZ}}; \lambda_{\min})$ & $3.61\times10^{-8}$ & \checkmark \\
$\d\mu(x^*_{\mathrm{CZ}})$ at 3 $\lambda$ & enhanced & \checkmark \\
H1 nearest formula $1/70$ & $\Delta = 8.4\times10^{-5}$ & \checkmark \\
H2 biharmonic inflection & $0.01455$ & \checkmark \\
H3 commutator sign change & $0.00483$ & \checkmark \\
H4 Frobenius roots & $\{0,0,1,-1\}$ & \checkmark \\
FD centroid $M=50$--$300$ (4 values) & monotone & \checkmark \\
$K12$ centroid toward $x^*_{\mathrm{CZ}}$? & True & \checkmark \\
gs centroid toward $x^*_{\mathrm{CZ}}$? & False (toward 0) & \checkmark \\
$n=1$ sector gs centroid & $0.97877$ & \checkmark \\
$n=2$ sector gs centroid & $0.97877$ & \checkmark \\
Angular momentum toward $x^*_{\mathrm{CZ}}$? & False & \checkmark \\
$\rho_{\mathrm{CZ}}/\rho_{0.5}$ ratio & $7.84$ & \checkmark \\
$\rho_{\mathrm{CZ}}/\rho_{\max}$ ratio & $0.214$ & \checkmark \\
H2 match ($|\Delta|<0.005$) & True & \checkmark \\
Continuous spectrum supports CZ as resonance? & No (flat) & \checkmark \\
\bottomrule
\end{tabular}
\end{table}

\bigskip

\textbf{Answer to the central question:}
$x^*_{\mathrm{CZ}} \approx 0.014203$ does \emph{not} appear as a resonance,
node, or harmonic of the continuous spectral density of $\Ohat$: the
full FD spectrum is trivially flat (resolution of identity).  The H2
biharmonic inflection ($\Delta = 3.4\times10^{-4}$) provides the closest
structural link to $x^*_{\mathrm{CZ}}$ found so far, identifying it as the
scale at which the Euler-type biharmonic ground state changes concavity.
This is a geometric property of the eigenfunction, not a spectral resonance.

\end{document}
