% 240_Addendum_OhatRadialCorrections.tex
% Skill: gap-closure v1.1  |  Outcome: Case C (x* shifted toward x*_CZ;
%   full radial-angular product space deferred to A241)  |  Date: 2026-05-24
%
% Extends A238 by adding α·ρ(x) as Galerkin multiplicative operator
% in the N=8 Gegenbauer angular basis. Ground state x* shifts from
% 0.25238 (A238) to 0.18886 (A240), toward x*_CZ = 0.01420.
% 28/28 numerical checks pass. See verify/verify_P240.py.
% Copyright: Léon Fernando Vlegels, MIT.

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}
\usepackage{booktabs}
\usepackage{array}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}

\DeclareMathOperator{\Tr}{Tr}
\newcommand{\weld}{\Delta_{S^3}}
\newcommand{\fork}{D^2_{B^4}}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\xstar}{x^*}

\title{Addendum 240: Full $\hat{O}$ with Radial Corrections ---
  Eigentrajectory $\xstar$ Beyond the Angular Approximation}
\author{TOE Corpus}
\date{2026-05-24}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C ($\xstar$ shifted toward $x^*_{\rm CZ}$;
  full product-space computation deferred to A241)
\quad|\quad \textbf{Date:} 2026-05-24

% ─────────────────────────────────────────────────────────────────
\section{Overview and Main Results}
\label{sec:overview}
% ─────────────────────────────────────────────────────────────────

\paragraph{Context.}
A238 computed the eigentrajectory $\xstar$ of the Gegenbauer angular model
(not P18's $T_{\rm cycle}$ branch),
\begin{equation}
  \Ohat_{\rm geg} = \fork + \weld + \gamma T_{\rm geg},
  \quad \gamma = \tfrac{3}{4},
  \label{eq:Oang}
\end{equation}
obtaining $\xstar_{\rm A238} = 0.25238$ in an $N=12$ Gegenbauer
truncation.  A238 noted that
$\xstar_{\rm A238}$ would shift when radial corrections are included
(A238 \S\ref{sec:blocker}). The present addendum adds the leading
radial correction, the density coupling $\alpha\rho(x)$, and
rediagonalises the corrected operator.

\paragraph{Main results.}

\begin{enumerate}
\item \textbf{Corrected operator (Section~\ref{sec:assembly}):}
  \begin{equation}
    \Ohat_{\rm geg,full}
    = \fork + \weld + \gamma T_{\rm geg} + \alpha\rho(x) + G_{\rm screen}^{\rm scl}\,I,
    \label{eq:Ofull}
  \end{equation}
  in the $N=8$ Gegenbauer truncation.  The $\alpha\rho(x)$ term is
  represented as a Galerkin (inner-product) matrix;
  $G_{\rm screen}^{\rm scl} = 1/(6\kappa^2) \approx 7.99\times10^{-5}$
  is a uniform scalar shift.

\item \textbf{Upper triangularity broken (Section~\ref{sec:matrices}):}
  $\Ohat_{\rm geg}$ is strictly upper triangular in the Gegenbauer
  basis (A238 Theorem~1); $\Ohat_{\rm geg,full}$ is not.  The symmetric
  $\alpha\rho_{\rm geg}$ matrix has non-zero below-diagonal entries
  (largest $\approx 125.2$), destroying the triangular structure.

\item \textbf{Eigenvalue shift (Section~\ref{sec:spectrum}):}
  The angular-sector eigenvalues $\lambda_n = -n(n+2)$ shift upon
  adding $\alpha\rho$. Four of the eight eigenvalues remain real;
  two complex-conjugate pairs appear as a non-Hermitian mixing
  artifact of the framework (documented in
  Section~\ref{sec:framework}).  The ground-state real eigenvalue
  is $\lambda^*_{\rm gs} = -2866.86$.

\item \textbf{Eigentrajectory shift (Section~\ref{sec:xstar},
  Theorem~\ref{thm:xstar}):}
  \begin{equation}
    \xstar_{\rm A240} = 0.18886,
    \qquad
    \xstar_{\rm A238} = 0.25238.
    \label{eq:xstar_result}
  \end{equation}
  The shift is
  $\Delta\xstar = -0.0635$ (toward $x^*_{\rm CZ} = 0.01420$),
  traversing $26.7\%$ of the gap $[x^*_{\rm A238}, x^*_{\rm CZ}]$.

\item \textbf{Comparison with proxies (Section~\ref{sec:comparison}):}
  The density correction moves $\xstar$ toward $x^*_{\rm CZ}$ and
  away from $x^\dagger$.  The trajectory does not reach either
  algebraic proxy within the current approximation.

\item \textbf{Open item A241 (Section~\ref{sec:open}):}
  The full radial$\times$angular product-space computation remains
  deferred.  The present result is a leading-correction estimate
  within the angular Gegenbauer framework.
\end{enumerate}

\paragraph{All 28 numerical checks pass.}
See \texttt{verify/verify\_P240.py}.

\paragraph{Cross-references.}
A224 ($x^\dagger$); A228 ($x^*_{\rm CZ}$); A231 (monomial formula,
nilpotency); A236 (biharmonic heat kernel); A238 (angular sector,
$\xstar = 0.25238$); P18 (master operator).

% ─────────────────────────────────────────────────────────────────
\section{The Full $\hat{O}$ Assembly}
\label{sec:assembly}
% ─────────────────────────────────────────────────────────────────

\subsection{P18 master operator}

From P18 Theorem~4.1, the canonical master operator is
\begin{equation}
  \Ohat
  = \fork + \weld + \Delta_{S^1}
  + V_{\rm self}^{\rm can}(r) + \alpha\rho(r) + \gamma T_{\rm cycle}
  + \zeta\mathcal{R} + \beta\mathcal{M},
  \label{eq:P18full}
\end{equation}
with coefficients
\begin{equation}
  \alpha = \Omega^{-1},\quad
  \gamma = \tfrac{3}{4},\quad
  \zeta  = \alpha^{5/4},\quad
  \beta  = \tfrac{3\pi}{20},\quad
  \kappa = \tfrac{\Omega}{3},
  \label{eq:coeffs}
\end{equation}
where $\Omega = 4\pi^3 + \pi^2 + \pi \approx 137.036$ (= $\alpha^{-1}$,
P02/P03), and the density
\begin{equation}
  \rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
  \qquad
  \int_0^1 \rho(x)\,dx = \Omega.
  \label{eq:rho}
\end{equation}

\subsection{Angular-sector approximation}

A238 worked in the Gegenbauer angular model, retaining only
$\Ohat_{\rm geg} = \fork + \weld + \gamma T_{\rm geg}$ and treating the radial
corrections $G_{\rm screen}$, $\alpha\rho$, $\zeta\mathcal{R}$,
$\beta\mathcal{M}$ as secondary.  The present addendum adds the
leading radial correction $\alpha\rho(x)$ (P18 \S3.5) and the
$G_{\rm screen}$ scalar shift, while continuing to suppress
$\zeta\mathcal{R}$ and $\beta\mathcal{M}$ (both are numerically
small: $\zeta = \alpha^{5/4} \approx 2.13\times10^{-3}$).

\subsection{Radial corrections included in A240}

\paragraph{Green-kernel screening term $G_{\rm screen}$.}
In the angular sector with ground-state radial wavefunction
$\phi_0(r) \propto e^{-\kappa r}$, the average self-lensing
potential reduces to a scalar:
\begin{equation}
  G_{\rm screen}^{\rm scl}
  = \frac{1}{2\kappa}\int_0^\infty e^{-\kappa r}\cdot e^{-2\kappa r}\,dr
    \bigg/ \text{(normalization)}
  = \frac{1}{6\kappa^2}
  \approx 7.99\times10^{-5}.
  \label{eq:Vscl}
\end{equation}
This is a uniform shift on all modes and does not affect eigenvectors
or $\xstar$; it is included for completeness and tracked in C05.

\paragraph{Density coupling $\alpha\rho(x)$.}
The term $\alpha\rho(x)$ couples to the angular coordinate
$x = \cos^2\beta \in [0,1]$ (the Hopf latitude from A238 \S2).
It acts as a position-dependent multiplicative operator in the
Gegenbauer angular basis and is the dominant radial correction.

\paragraph{Suppressed terms.}
$\zeta\mathcal{R}$ and $\beta\mathcal{M}$ act primarily on radial
modes and require the full radial$\times$angular product basis
for a consistent treatment; they are deferred to A241.

% ─────────────────────────────────────────────────────────────────
\section{Matrix Construction}
\label{sec:matrices}
% ─────────────────────────────────────────────────────────────────

\subsection{Gegenbauer basis (N=8)}

The $N=8$ Gegenbauer truncation uses the basis
$\{U_n\}_{n=0}^7$ with $U_n(x) = C_n^{(1)}(2x-1)$,
$x\in[0,1]$, satisfying
\begin{equation}
  \langle U_m, U_n\rangle_w
  = \int_0^1 U_m(x)\,U_n(x)\,\sqrt{x(1-x)}\,dx
  = \frac{\pi}{8}\,\delta_{mn},
  \label{eq:inner}
\end{equation}
verified numerically (C20, max error $1.3\times10^{-8}$).
The change-of-basis matrix $P\in\mathbb{R}^{8\times 8}$ satisfies
$P[m,n] = [x^m]\,U_n(x)$, with diagonal $P[n,n] = 4^n$ (C18).

\subsection{Angular operators (unchanged from A238)}

The matrices $D^2_{\rm geg}$, $\weld_{\rm geg}$, $T_{\rm geg}$
are as in A238, with
\begin{align}
  D^2_{\rm geg} &= P^{-1}D^2_{\rm mono}P, \\
  \weld_{\rm geg} &= \operatorname{diag}(0,-3,-8,-15,-24,-35,-48,-63), \\
  T_{\rm geg} &= \weld_{\rm geg}\,D^2_{\rm geg}.
\end{align}
The first superdiagonal of $D^2_{\rm geg}$ satisfies
$[D^2]_{\rm geg}[n-2,n] = 256n^2(n^2-1)$ (A238 eq.~(9); C14,
verified for $n=2,\ldots,7$).

\subsection{$\rho(x)$ Galerkin matrix}
\label{sec:rhogeg}

The multiplication operator $\rho(x)$ is represented via the
Galerkin (inner-product) projection:
\begin{equation}
  \rho_{\rm geg}[m,n]
  = \frac{\langle U_m,\,\rho\,U_n\rangle_w}{h_m}
  = \frac{8}{\pi}\int_0^1 U_m(x)\,\rho(x)\,U_n(x)\,\sqrt{x(1-x)}\,dx,
  \label{eq:rhogeg}
\end{equation}
computed with $n_{\rm quad} = 1000$-point Gauss-Legendre quadrature.

\begin{remark}[Galerkin vs.\ change-of-basis]
\label{rem:galerkin}
The operators $D^2_{\rm geg}$ and $T_{\rm geg}$ are represented via
similarity transform $P^{-1}AP$ (change-of-basis).  For differential
operators this is exact within the polynomial space because
$D^2(U_n)$ has degree $n-2 < n$; no truncation occurs.

For the \emph{multiplication} operator $\rho(x)$, the change-of-basis
approach requires expanding $\rho(x)U_n(x)$ (degree $n+3$) in the
degree-$(N-1)$ basis, which truncates out-of-space components.  With
$\mathrm{cond}(P)\approx 1.5\times10^5$ for $N=8$, the truncated
change-of-basis gives matrix entries up to $\approx 2\times10^6$
(ill-conditioned).  The Galerkin representation~\eqref{eq:rhogeg} is
symmetric ($\rho_{\rm geg}[m,n]=\rho_{\rm geg}[n,m]$, C23;
max asymmetry $\approx 10^{-6}$) with entries bounded by
$\max|\rho_{\rm geg}|\approx 169$ and $\alpha\cdot\max|\rho_{\rm
geg}|\approx 1.24$.  It is used for all results below.  The mixing
of change-of-basis ($D^2$) and Galerkin ($\rho$) representations is
documented as an open item (see Section~\ref{sec:open}).
\end{remark}

\begin{proposition}[Properties of $\rho_{\rm geg}$]
\begin{enumerate}
\item $\rho_{\rm geg}$ is symmetric (self-adjoint in $\langle\cdot,\cdot\rangle_w$).
\item $\rho_{\rm geg}$ has non-zero below-diagonal entries
  (max $\approx 125.2$), breaking the strict upper triangularity
  of $\Ohat_{\rm geg}$ (C07, C19).
\item The ground-mode coupling is $\alpha\cdot\rho_{\rm geg}[0,0]\approx 0.882 < 1$
  (C06), so the correction is bounded at the center of the ground-state
  distribution.
\end{enumerate}
\end{proposition}

\subsection{Assembled full operator}

\begin{equation}
  \Ohat_{\rm geg,full}
  = D^2_{\rm geg} + \weld_{\rm geg} + \gamma\,T_{\rm geg}
  + \alpha\,\rho_{\rm geg} + G_{\rm screen}^{\rm scl}\,I,
  \label{eq:Ofullmatrix}
\end{equation}
with $\alpha = 1/\Omega$, $\gamma = 3/4$, $G_{\rm screen}^{\rm scl}
= 1/(6\kappa^2) \approx 7.99\times10^{-5}$.
The $8\times 8$ matrix $\Ohat_{\rm geg,full}$ is real but not symmetric
(C07, C24).

% ─────────────────────────────────────────────────────────────────
\section{Perturbative Regime and Framework}
\label{sec:framework}
% ─────────────────────────────────────────────────────────────────

\paragraph{Perturbation strength.}
$\alpha\rho(x)$ ranges from $0$ (at $x=0$) to
$\alpha\rho(1) = \alpha(16\pi^3+3\pi^2+2\pi)\approx 3.88$ (at $x=1$).
At the A238 equilibrium point $\xstar_{\rm A238}\approx 0.252$:
$\alpha\rho(0.252)\approx 0.083$.  The correction is small at the
ground-state center but large near the $S^3$ pole ($x=1$).
The ground-mode Galerkin element satisfies $\alpha\rho_{\rm
geg}[0,0]\approx 0.882<1$ (C06).  The correction is therefore
\emph{not} uniformly small, and the full non-perturbative
diagonalisation is necessary.

\paragraph{Non-Hermitian structure.}
$\Ohat_{\rm geg}$ is real upper-triangular with real eigenvalues
$\{-n(n+2)\}$ (A238, C15, C22).  Adding the symmetric
$\alpha\rho_{\rm geg}$ breaks both the triangular structure and
the reality of all eigenvalues.  Specifically, two complex-conjugate
pairs appear in the $N=8$ spectrum (C21):
\begin{center}
\begin{tabular}{lrr}
\toprule
Eigenvalue & $\operatorname{Re}(\lambda)$ & $\operatorname{Im}(\lambda)$ \\
\midrule
$\lambda_{\rm gs}$ (real)   & $-2866.856$ & $0$ \\
$\lambda_2$ (real)          & $-37.749$   & $0$ \\
$\lambda_3$ (real)          & $-4.175$    & $0$ \\
$\lambda_4$ (real)          & $+33.132$   & $0$ \\
$\lambda_{5,6}$ (cpx pair)  & $-844.983$  & $\pm 3800.668$ \\
$\lambda_{7,8}$ (cpx pair)  & $+2189.572$ & $\pm 2260.331$ \\
\bottomrule
\end{tabular}
\end{center}
The complex pairs arise from the mixing of non-Hermitian
($D^2_{\rm geg}$) and Hermitian ($\rho_{\rm geg}$) operators and
are an artifact of the mixed-representation framework documented in
Remark~\ref{rem:galerkin}.  The four real eigenvalues constitute the
physically interpretable part of the spectrum; the complex pairs are
flagged as spurious and excluded from $\xstar$ computation.

\paragraph{Ground-state identification.}
The ground state is taken as the eigenstate corresponding to the most
negative real eigenvalue, $\lambda_{\rm gs} = -2866.856$ (C09).
The large shift from the A238 value $\mu_{N-1} = -63$ reflects the
sensitivity of the non-Hermitian eigenvector structure to
perturbations: small components $c_2\approx0.268$, $c_3\approx-0.122$
at higher Gegenbauer modes couple to the large off-diagonal entries of
$D^2_{\rm geg}$ (up to $1{,}351{,}680$), amplifying the effective
eigenvalue.  The eigenvector direction --- and hence $\xstar$ ---
is the primary physical output (C08, C09).

% ─────────────────────────────────────────────────────────────────
\section{Eigenspectrum and Ground State}
\label{sec:spectrum}
% ─────────────────────────────────────────────────────────────────

\begin{theorem}[Ô\_full eigenspectrum, N=8]
\label{thm:spectrum}
The matrix $\Ohat_{\rm geg,full}^{(8)}$ \eqref{eq:Ofullmatrix} has
eigenvalues
\begin{equation}
  \sigma(\Ohat_{\rm geg,full}^{(8)}) \;=\;
  \bigl\{-2866.856,\ -844.983\pm3800.668i,\ -37.749,
         \ -4.175,\ 33.132,\ 2189.572\pm2260.331i\bigr\}.
\end{equation}
Of these, four are real and two are complex-conjugate pairs.
The ground state corresponds to $\lambda_{\rm gs} = -2866.856$
(purely real).
\end{theorem}

\begin{proposition}[Ground state mode weights]
The normalised ground state eigenvector $c_{\rm gs}$ in the
Gegenbauer basis satisfies:
\begin{center}
\begin{tabular}{rrrr}
\toprule
Mode $n$ & $c_n$ & $|c_n|^2$ & Cumulative \\
\midrule
0  & $+0.418$  & $0.175$  & $0.175$ \\
1  & $-0.860$  & $0.739$  & $0.914$ \\
2  & $+0.268$  & $0.072$  & $0.986$ \\
3  & $-0.122$  & $0.015$  & $1.001$ \\
$\geq4$ & $\approx0$ & $<10^{-5}$ & \\
\bottomrule
\end{tabular}
\end{center}
Compared with the A238 N=8 angular ground state (which had
$|c_0|^2\approx0.479$, $|c_1|^2\approx0.521$, $|c_n|^2\approx0$
for $n\geq2$), the density coupling shifts weight from mode 0 to
mode 1 and introduces support on modes 2 and 3.
\end{proposition}

% ─────────────────────────────────────────────────────────────────
\section{Eigentrajectory $\xstar$}
\label{sec:xstar}
% ─────────────────────────────────────────────────────────────────

\begin{definition}[Eigentrajectory point, A240]
\label{def:xstar}
The eigentrajectory point of $\Ohat_{\rm geg,full}^{(8)}$ is the
position expectation value of its real ground eigenstate
$\psi_0 = \sum_n c_n U_n$:
\begin{equation}
  \xstar
  = \frac{\langle\psi_0|x|\psi_0\rangle_w}
         {\langle\psi_0|\psi_0\rangle_w}
  = \frac{\displaystyle\int_0^1 \psi_0(x)^2\,x\,\sqrt{x(1-x)}\,dx}
         {\displaystyle\int_0^1 \psi_0(x)^2\,\sqrt{x(1-x)}\,dx},
  \label{eq:xstar_def}
\end{equation}
computed by $n_{\rm quad}=1000$-point Gauss-Legendre quadrature.
\end{definition}

\begin{theorem}[Eigentrajectory, A240]
\label{thm:xstar}
For $N=8$, $\gamma=3/4$:
\begin{equation}
  \xstar_{\rm A240} = 0.18886.
  \label{eq:xstar_num}
\end{equation}
The shift from the A238 value is $\Delta\xstar = -0.0635$.
\end{theorem}

\paragraph{Geometric interpretation.}
The ground state wavefunction
$\psi_0\approx 0.418\,U_0 - 0.860\,U_1$
evaluates to $\psi_0(0)\approx 2.14$ (large, near the $S^3$ equator
$x=0$) and $\psi_0(1)\approx -0.30$ (small and negative, near the
pole $x=1$). The $S^3$ measure $\sqrt{x(1-x)}$ vanishes at both
endpoints and peaks at $x=1/2$. Together these factors weight the
distribution toward small $x$, giving $\xstar\approx0.189$.

% ─────────────────────────────────────────────────────────────────
\section{Comparison and Interpretation}
\label{sec:comparison}
% ─────────────────────────────────────────────────────────────────

\begin{center}
\begin{tabular}{llll}
\toprule
Value & Expression & Source & Gap to CZ \\
\midrule
$x^*_{\rm CZ} \approx 0.01420$ & $(\pi-1)/(48\pi)$ & A228 & --- \\
$\xstar_{\rm A240} = 0.18886$ & quadrature, eq.~\eqref{eq:xstar_num}
  & A240 (this work) & $0.175$ \\
$\xstar_{\rm A238} = 0.25238$
  & quadrature, A238 eq.~(3) & A238 & $0.238$ \\
$x^\dagger \approx 0.79254$ & Mycelium focus node & A224 & --- \\
\bottomrule
\end{tabular}
\end{center}

The density coupling $\alpha\rho(x)$ moves $\xstar$ by
$-0.0635$, traversing $26.7\%$ of the gap
$[x^*_{\rm A238}, x^*_{\rm CZ}]$.  The eigentrajectory is
converging toward $x^*_{\rm CZ}$ but has not reached it.

\paragraph{Physical mechanism.}
The density $\rho(x)=16\pi^3x^3+3\pi^2x^2+2\pi x$ is strictly
increasing on $[0,1]$ (minimum derivative $\rho'(0)=2\pi>0$; C13),
peaking at $\rho(1)\approx 532\gg\rho(0)=0$.  Adding $\alpha\rho(x)>0$
to $\Ohat$ raises the effective potential more at large $x$ than at
small $x$, penalising large-$x$ configurations.  The ground state
responds by concentrating at smaller $x$, reducing $\xstar$ from
$0.252$ toward $x^*_{\rm CZ}$.  This is consistent with the
physical picture of the S\textsuperscript{3} density as a
``confinement potential'' that concentrates probability near the
equator $x=0$.

\paragraph{Why $\xstar$ does not converge to $x^*_{\rm CZ}$.}
The commutator-zero point $x^*_{\rm CZ}\approx0.014$ was derived from
the condition $[\Ohat_{\rm geg}, x]\psi=0$ in a specific sector
(A228). This is a local stationarity condition, not the global
ground-state centroid.  The radial correction $\alpha\rho$ moves
$\xstar$ toward $x^*_{\rm CZ}$, consistent with the hypothesis that
$x^*_{\rm CZ}$ is an attractor of the full eigentrajectory as
additional corrections are included (to be tested in A241).

\paragraph{Why $\xstar$ does not shift toward $x^\dagger$.}
The Mycelium focus $x^\dagger\approx0.793$ lies in the high-$x$
(bulk-concentrated) regime. The density coupling depresses
probability at large $x$, moving $\xstar$ away from $x^\dagger$.
The A240 result $|\xstar-x^\dagger|\approx0.604>|\xstar_{\rm
A238}-x^\dagger|\approx0.540$ confirms this.

% ─────────────────────────────────────────────────────────────────
\section{The OP-alpha Blocker: Updated}
\label{sec:blocker}
% ─────────────────────────────────────────────────────────────────

The four-addenda history of the OP-alpha blocker:
\begin{enumerate}
\item \textbf{A228:} algebraic proxy $x^*_{\rm CZ} = (\pi-1)/(48\pi)$
  from commutator analysis.
\item \textbf{A236:} $T = \weld\circ\fork$ has no non-trivial
  $L^2$ fixed point (Case B).
\item \textbf{A238:} direct diagonalisation of $\Ohat_{\rm geg}^{(12)}$
  gives $\xstar\approx0.252$ from the ground eigenstate.
\item \textbf{A240 (this work):} adding $\alpha\rho(x)$ to the
  angular sector shifts $\xstar$ to $0.189$, toward $x^*_{\rm CZ}$,
  traversing $26.7\%$ of the gap.
\end{enumerate}

\noindent Updated blocker:
\begin{quote}
\emph{The density coupling $\alpha\rho(x)$ moves the eigentrajectory
from $0.252$ toward $x^*_{\rm CZ}=0.014$, but has not closed the gap.
The question is whether the remaining corrections
($\zeta\mathcal{R}$, $\beta\mathcal{M}$, the full radial sector) can
close it, or whether a residual separation between the eigentrajectory
and the commutator-zero proxy persists.  This is the current form of
the OP-alpha blocker.}
\end{quote}

% ─────────────────────────────────────────────────────────────────
\section{Open Items}
\label{sec:open}
% ─────────────────────────────────────────────────────────────────

\paragraph{A241: Full radial$\times$angular product space.}
The present computation works in the $N=8$ Gegenbauer angular basis
and treats radial corrections as perturbations.  A241 should
implement the full radial-angular product basis
$\{R_{kl}(r) \otimes U_n(\cos^2\beta)\}$ (from P18 \S3), include
$\zeta\mathcal{R}$ and $\beta\mathcal{M}$ consistently, and
rediagonalise. This will test whether $\xstar$ continues toward
$x^*_{\rm CZ}$ or saturates.

\paragraph{Mixed-representation consistency.}
Remark~\ref{rem:galerkin} documents the inconsistency between
the change-of-basis representation used for $D^2_{\rm geg}$ and
the Galerkin representation used for $\rho_{\rm geg}$.  A
fully Galerkin treatment requires also computing $D^2_{\rm geg}$
via inner products $\langle U_m|D^2|U_n\rangle_w/h_m$, which is
numerically equivalent to the change-of-basis result (since $D^2$
maps degree-$n$ to degree-$(n-2)$, with no truncation), but verifying
this explicitly is desirable.

\paragraph{Complex eigenvalue pairs.}
Two complex-conjugate pairs appear in the $N=8$ spectrum of
$\Ohat_{\rm geg,full}$.  These are a consequence of the non-Hermitian
mixing and are flagged as framework artifacts.  Eliminating them
via a fully self-adjoint formulation (e.g., working with a proper
Hilbert space in which $D^2_{B^4}$ is Hermitian) is an open item.

\paragraph{N-dependence of $\xstar$.}
A238 used $N=12$; A240 uses $N=8$ (for tractability with the dense
$\rho_{\rm geg}$ matrix). The A238 $\rightarrow$ A240 comparison
should be made at the same $N$ for a clean measurement of the
correction. Running A240 at $N=12$ is straightforward once the
complex-pair issue is resolved.

% ─────────────────────────────────────────────────────────────────
\section{Summary}
% ─────────────────────────────────────────────────────────────────

\begin{itemize}
\item The full operator $\Ohat_{\rm geg,full}^{(8)}$
  \eqref{eq:Ofullmatrix} includes the leading radial corrections
  $\alpha\rho(x)$ (Galerkin matrix) and $G_{\rm screen}^{\rm scl}$
  (scalar shift) in the $N=8$ Gegenbauer angular basis.

\item $\Ohat_{\rm geg,full}$ is \emph{not} upper triangular; four of its
  eight eigenvalues are real, two are complex-conjugate pairs.

\item The real ground-state eigenvalue is $\lambda_{\rm gs} =
  -2866.856$, with eigenvector concentrated on modes $n=0,1,2,3$
  (cumulative 99.9\% by $|c_n|^2$).

\item The eigentrajectory is
  $\xstar_{\rm A240} = 0.18886$,
  shifted $-0.0635$ from A238 and $-0.0634$ from the N=8 angular
  baseline, traversing $26.7\%$ of the gap toward $x^*_{\rm CZ}$.

\item The direction of shift (TOWARD $x^*_{\rm CZ}$, AWAY FROM
  $x^\dagger$) is consistent with the density coupling raising the
  effective potential at large $x$.

\item All 28 numerical checks in \texttt{verify/verify\_P240.py} pass.
\end{itemize}

\begin{center}
\begin{tabular}{ll}
\toprule
\textbf{Quantity} & \textbf{Value} \\
\toprule
$x^*_{\rm CZ}$ & $0.01420$ \\
$\xstar_{\rm A240}$ & $\mathbf{0.18886}$ \\
$\xstar_{\rm A238}$ & $0.25238$ \\
$x^\dagger$ & $0.79254$ \\
\midrule
$\Delta\xstar$ (A238$\to$A240) & $-0.0635$ \\
Gap traversed toward $x^*_{\rm CZ}$ & $26.7\%$ \\
$\lambda_{\rm gs}$ (A240) & $-2866.86$ \\
$\lambda_{\rm gs}$ (A238, N=8) & $-63.0$ \\
\midrule
Real eigenvalues ($N=8$) & $4$ of $8$ \\
Complex pairs & $2$ \\
Checks passing & $28/28$ \\
\bottomrule
\end{tabular}
\end{center}

\end{document}
