% 241_Addendum_OhatProductBasis.tex
% Skill: gap-closure v1.1  |  Outcome: Case D (x* unchanged by radial corrections;
%   radial sector decoupled from angular eigentrajectory at current operator scale)
%   |  Date: 2026-05-24
%
% Builds the full Laguerre-radial × Gegenbauer-angular product basis for the
% P18 master operator Ô.  Adiabatic separation and full 48×48 product-space
% diagonalisation both confirm x*_A241 = 0.18887 ≈ x*_A240.
% The radial corrections (V_self, R_0) are negligible at the scale set by D²_B4.
% 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}_{\rm geg}}
\newcommand{\xstar}{x^*}

\title{Addendum 241: Full $\hat{O}$ in the Radial$\times$Angular Product Basis ---
  Eigentrajectory Decoupled from Radial Corrections}
\author{TOE Corpus}
\date{2026-05-24}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case D (radial sector decoupled; $\xstar$ unchanged from A240)
\quad|\quad \textbf{Date:} 2026-05-24

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

\paragraph{Context.}
A240 added the density coupling $\alpha\rho(x)$ to the angular sector,
shifting $\xstar$ from $0.25238$ (A238) to $0.18886$ and traversing
$26.7\%$ of the gap $[\xstar_{\rm A238}, x^*_{\rm CZ}]$.  The open item
in A240 was whether the full radial-angular product space
--- including the screening kernel $G_{\rm screen}(r)$ computed with
the correct 4D volume element $r^3\,dr$ and the radial kinetic operator
$R_0$ --- would close the remaining $73.3\%$.  A241 answers this question.

\paragraph{Main results.}

\begin{enumerate}
\item \textbf{Product basis (Section~\ref{sec:basis}):}
  The M$\times$N = $6\times8 = 48$-dimensional product basis is
  $\{\phi_k(r)\}_{k=0}^{5} \otimes \{U_n(x)\}_{n=0}^{7}$,
  where $\phi_k$ are Laguerre-type radial functions with scale
  $\kappa = \Omega/3 \approx 45.68$.

\item \textbf{Radial matrix elements (Section~\ref{sec:radial}):}
  In the correct 4D volume $r^3\,dr$, the ground-state self-lensing expectation is
  \begin{equation}
    \langle\phi_0|G_{\rm screen}|\phi_0\rangle_{r^3\!dr}
    = 1.70\times10^{-8},
  \end{equation}
  four orders of magnitude below the A240 scalar estimate
  $1/(6\kappa^2)\approx 7.99\times10^{-5}$.  The radial kinetic energy
  of the ground state is
  \begin{equation}
    \langle\phi_0|R_0|\phi_0\rangle_{r^3\!dr} = 1.64\times10^{-2}.
  \end{equation}
  Both are negligible relative to the ground-state eigenvalue scale
  $|\lambda_{\rm gs}| \approx 2867$.

\item \textbf{Adiabatic result (Section~\ref{sec:adiabatic}):}
  The effective angular Hamiltonian for radial ground state $\phi_0$ is
  \begin{equation}
    H^{\rm eff} = D^2_{\rm geg} + \weld_{\rm geg} + \gamma T_{\rm geg}
    + \alpha\rho_{\rm geg} + \langle G_{\rm screen}\rangle_0 \cdot I,
  \end{equation}
  with $\langle G_{\rm screen}\rangle_0 = 1.70\times10^{-8}$.  The eigentrajectory is
  \begin{equation}
    \xstar_{\rm A241}^{\rm (ad)} = 0.18887.
  \end{equation}

\item \textbf{Full product space (Section~\ref{sec:full}):}
  The $48\times48$ operator $\Ohat_{\rm full}^{(48)}$ (which includes the full
  off-diagonal $G_{\rm screen}$ radial mixing and $R_0$ in each angular channel)
  yields the same ground-state eigenvector and
  \begin{equation}
    \xstar_{\rm A241}^{(48)} = 0.18887.
  \end{equation}

\item \textbf{Eigentrajectory unchanged (Theorem~\ref{thm:xstar}):}
  \begin{equation}
    \xstar_{\rm A241} = 0.18887,
    \qquad
    \Delta\xstar = \xstar_{\rm A241} - \xstar_{\rm A240} = +4.7\times10^{-6}.
  \end{equation}
  The shift is smaller than $10^{-5}$ in absolute value and carries a
  positive sign (away from $x^*_{\rm CZ}$), indicating that the radial
  corrections do not contribute to gap closure.

\item \textbf{Cumulative gap unchanged (Section~\ref{sec:comparison}):}
  \begin{center}
  \begin{tabular}{lll}
  \toprule
  Addendum & $\xstar$ & Cumulative gap traversed \\
  \midrule
  A238 & $0.25238$ & $0.0\%$ \\
  A240 & $0.18886$ & $26.7\%$ \\
  A241 & $0.18887$ & $26.7\%$ \\
  \bottomrule
  \end{tabular}
  \end{center}
  The $73.3\%$ gap between A240 and $x^*_{\rm CZ}$ is not closed by
  the radial sector.

\item \textbf{N-convergence (Section~\ref{sec:convergence}):}
  The angular truncation $N = 8$ used in A240--A241 is not in a
  converged regime.  $\xstar$ varies non-monotonically with $N$
  (Table~\ref{tab:N}), ranging from $0.148$ ($N=10$) to $0.537$ ($N=6$).
  This erratic behavior is a consequence of the mixed-representation
  instability documented in A240 Remark~2.1.

\item \textbf{Open item A242 (Section~\ref{sec:open}):}
  Closing the gap requires either a fully Galerkin treatment of
  $\fork$ (resolving the mixed-representation instability) or
  inclusion of the suppressed operators $\zeta\mathcal{R}$ and
  $\beta\mathcal{M}$.
\end{enumerate}

\paragraph{All 20 numerical checks pass.}
See \texttt{verify/verify\_P241.py}.

\paragraph{Cross-references.}
A228 ($x^*_{\rm CZ}$); A238 (angular sector, $\xstar=0.252$);
A240 (density coupling, $\xstar=0.189$); P18 (master operator).

% ─────────────────────────────────────────────────────────────────
\section{Product Basis Construction}
\label{sec:basis}
% ─────────────────────────────────────────────────────────────────

\subsection{Laguerre-type radial functions}

The screening kernel decays as $G_{\rm screen}(r) = (1/2\kappa)e^{-\kappa r}$
with $\kappa = \Omega/3$.  The natural radial basis is:
\begin{equation}
  \phi_k(r) = \sqrt{2\kappa}\, L_k(2\kappa r)\, e^{-\kappa r},
  \quad k = 0, 1, \ldots, M-1,
  \label{eq:phidef}
\end{equation}
where $L_k$ are Laguerre polynomials.  Under the substitution $t = 2\kappa r$:
\begin{equation}
  \phi_k = \sqrt{2\kappa}\, L_k(t)\, e^{-t/2}.
  \label{eq:phi_t}
\end{equation}

\begin{proposition}[Radial overlap]
\label{prop:radnorm}
In 4D with volume element $r^3\,dr$ and variable $t=2\kappa r$,
\begin{equation}
  \langle\phi_m|\phi_n\rangle_{r^3\!dr}
  = \frac{1}{(2\kappa)^3}\int_0^\infty L_m(t)\,L_n(t)\,e^{-t}\,t^3\,dt.
  \label{eq:phioverlap}
\end{equation}
The diagonal ground-state norm $\langle\phi_0|\phi_0\rangle = 8.00\times10^{-6}$.
\end{proposition}

\subsection{Angular basis}

The angular basis $\{U_n\}_{n=0}^{N-1}$ with $U_n(x)=C_n^{(1)}(2x-1)$,
$x=\cos^2\!\beta\in[0,1]$ is unchanged from A238/A240.
For $N=8$, norms $h_n = \int_0^1 U_n^2\sqrt{x(1-x)}\,dx = \pi/8$ to
machine precision.

\subsection{Product space}

The full product space has dimension $M\times N = 6\times8 = 48$.
States are indexed $(k,n)$, $k=0\ldots5$ (radial), $n=0\ldots7$ (angular),
flattened as $k\cdot N + n$.

% ─────────────────────────────────────────────────────────────────
\section{Radial Matrix Element Computations}
\label{sec:radial}
% ─────────────────────────────────────────────────────────────────

All radial integrals are computed by 500-point Gauss-Legendre quadrature
on $t\in[0, 60]$, with $e^{-t}$ included explicitly (no overflow).

\subsection{Self-lensing potential}

\begin{proposition}[$G_{\rm screen}$ in 4D]
\label{prop:Vself}
\begin{equation}
  \langle\phi_m|G_{\rm screen}|\phi_n\rangle_{r^3\!dr}
  = \frac{1}{(2\kappa)^4}
    \int_0^\infty L_m(t)\,L_n(t)\,e^{-3t/2}\,t^3\,dt.
  \label{eq:Vself4D}
\end{equation}
The ground-state diagonal matrix element is
\begin{equation}
  \langle\phi_0|G_{\rm screen}|\phi_0\rangle_{r^3\!dr} = 1.701\times10^{-8}.
  \label{eq:V00}
\end{equation}
\end{proposition}

\begin{remark}[Comparison with A240 estimate]
A240 estimated $G_{\rm screen}^{\rm scl} = 1/(6\kappa^2)\approx 7.99\times10^{-5}$
by a 1D radial integral without the 4D weight.  The correct 4D value
\eqref{eq:V00} is smaller by a factor $\approx 2.1\times10^{-4}$.
The extra suppression arises from the $(2\kappa)^{-4}$ prefactor in \eqref{eq:Vself4D}
together with the short-range character of both $e^{-t/2}$ (from $\phi_0$) and
the additional $e^{-t/2}$ from $G_{\rm screen}$: at $\kappa\approx45.7$,
the four powers of $2\kappa$ in the denominator give $(91.4)^4 \approx 6.97\times10^7$.
\end{remark}

\subsection{Radial kinetic operator}

The $l=0$ radial kinetic operator in 4D is
\begin{equation}
  R_0 = -\frac{d^2}{dr^2} - \frac{3}{r}\frac{d}{dr}.
  \label{eq:R0def}
\end{equation}
Using $t=2\kappa r$, $\phi_n = \sqrt{2\kappa}\,L_n(t)\,e^{-t/2}$:
\begin{align}
  \frac{d\phi_n}{dr} &= (2\kappa)\sqrt{2\kappa}\bigl[L_n'(t)-\tfrac{1}{2}L_n(t)\bigr]e^{-t/2}, \\
  \frac{d^2\phi_n}{dr^2} &= (2\kappa)^2\sqrt{2\kappa}\bigl[L_n''(t)-L_n'(t)+\tfrac{1}{4}L_n(t)\bigr]e^{-t/2}.
\end{align}
Thus
\begin{equation}
  R_0\phi_n = \sqrt{2\kappa}\,e^{-t/2}\left\{
    -(2\kappa)^2\Bigl[L_n''-L_n'+\tfrac{L_n}{4}\Bigr]
    -\frac{12\kappa^2}{t}\Bigl[L_n'-\tfrac{L_n}{2}\Bigr]
  \right\},
  \label{eq:R0action}
\end{equation}
and
\begin{equation}
  \langle\phi_m|R_0|\phi_n\rangle_{r^3\!dr}
  = \frac{1}{(2\kappa)^3}\int_0^\infty L_m(t)\,\mathcal{K}_n(t)\,e^{-t}\,t^3\,dt,
  \label{eq:R0mat}
\end{equation}
where $\mathcal{K}_n(t)$ is the bracketed expression in \eqref{eq:R0action}.

\begin{proposition}[Ground-state kinetic energy]
\begin{equation}
  \langle\phi_0|R_0|\phi_0\rangle_{r^3\!dr} = 1.642\times10^{-2}.
  \label{eq:R0_00}
\end{equation}
\end{proposition}

\subsection{Scale comparison}

\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Relative to $|\lambda_{\rm gs}|=2867$ \\
\midrule
$\langle\phi_0|G_{\rm screen}|\phi_0\rangle_{r^3\!dr}$ & $1.70\times10^{-8}$ & $5.9\times10^{-12}$ \\
$\langle\phi_0|R_0|\phi_0\rangle_{r^3\!dr}$ & $1.64\times10^{-2}$ & $5.7\times10^{-6}$ \\
$\max|D^2_{\rm geg}|$ & $2.67\times10^{6}$ & $932$ \\
$\alpha\max|\rho_{\rm geg}|$ & $1.24$ & $4.3\times10^{-4}$ \\
\bottomrule
\end{tabular}
\end{center}

Both radial corrections are negligible compared to the angular operator scale.
The dominant energy scale is set by $D^2_{\rm geg}$ (entries $\sim 10^6$).

% ─────────────────────────────────────────────────────────────────
\section{Adiabatic Separation}
\label{sec:adiabatic}
% ─────────────────────────────────────────────────────────────────

In the adiabatic approximation, the effective angular Hamiltonian
for radial eigenstate $\phi_0$ is:
\begin{equation}
  H^{\rm eff}_{k=0}
  = D^2_{\rm geg} + \weld_{\rm geg} + \gamma T_{\rm geg}
  + \alpha\rho_{\rm geg}
  + \langle\phi_0|G_{\rm screen}|\phi_0\rangle_{r^3\!dr}\cdot I.
  \label{eq:Heff}
\end{equation}
The shift from A240 is the replacement of the 1D estimate
$1/(6\kappa^2)\approx7.99\times10^{-5}$ (A240 eq.~(4))
with the correct 4D value $1.70\times10^{-8}$.
The difference is $\Delta V = -7.99\times10^{-5}$, which
moves all eigenvalues by $-7.99\times10^{-5}$ (a rigid diagonal shift)
and has no effect on eigenvectors or $\xstar$.

The inclusion of $R_0$ in the adiabatic sense would add
$\langle\phi_0|R_0|\phi_0\rangle_{r^3\!dr} = 1.64\times10^{-2}$ as an additional
scalar shift.  At the scale of the eigenvalue $\lambda_{\rm gs}\approx-2867$,
this is a relative correction of $5.7\times10^{-6}$ and similarly leaves
eigenvectors unchanged.

\begin{theorem}[Adiabatic eigentrajectory, A241]
\label{thm:xstar}
For $N=8$, $M=6$, $\gamma=3/4$, with radial ground state $\phi_0$:
\begin{equation}
  \xstar_{\rm A241}^{\rm (ad)} = 0.18887.
  \label{eq:xstar_ad}
\end{equation}
The shift from A240 is $\Delta\xstar = +4.7\times10^{-6}$, which is
$1.3\%$ of $x^*_{\rm CZ}$ and constitutes no measurable gap closure.
The ground-state eigenvector mode weights are unchanged from A240:
\begin{center}
\begin{tabular}{rrrr}
\toprule
Mode $n$ & $c_n$ & $|c_n|^2$ & Cumulative \\
\midrule
0 & $-0.4179$ & $0.1747$ & $0.175$ \\
1 & $+0.8596$ & $0.7389$ & $0.914$ \\
2 & $-0.2676$ & $0.0716$ & $0.986$ \\
3 & $+0.1218$ & $0.0148$ & $1.001$ \\
$\geq4$ & $\approx0$ & $<10^{-5}$ & \\
\bottomrule
\end{tabular}
\end{center}
\end{theorem}

% ─────────────────────────────────────────────────────────────────
\section{Full 48$\times$48 Product Basis}
\label{sec:full}
% ─────────────────────────────────────────────────────────────────

The full $48\times48$ operator is:
\begin{equation}
  \Ohat^{(48)}_{\rm full}
  = I_{\rm rad}\otimes(D^2_{\rm geg}+\weld_{\rm geg}+\gamma T_{\rm geg}+\alpha\rho_{\rm geg})
  + G_{\rm screen,mat}\otimes I_{\rm ang}
  + R_{0,\rm mat}\otimes I_{\rm ang},
  \label{eq:O48}
\end{equation}
where $G_{\rm screen,mat}$ and $R_{0,\rm mat}$ are the $M\times M$ radial matrices
\eqref{eq:Vself4D} and \eqref{eq:R0mat} respectively.

Diagonalisation of $\Ohat^{(48)}_{\rm full}$ gives ground eigenvalue
$\lambda^{(48)}_{\rm gs} = -2866.87$ and ground eigenvector
$\{c_{kn}\}$.  The radial-marginal x-centroid is:
\begin{equation}
  \xstar^{(48)}
  = \frac{\displaystyle\sum_{k=0}^{M-1}
    \int_0^1\!\bigl(\textstyle\sum_n c_{kn}U_n(x)\bigr)^2\!x\,w(x)\,dx}
         {\displaystyle\sum_{k=0}^{M-1}
    \int_0^1\!\bigl(\textstyle\sum_n c_{kn}U_n(x)\bigr)^2\!w(x)\,dx}
  = 0.18887.
  \label{eq:xstar_48}
\end{equation}
The full-product-space result agrees with the adiabatic result to all
printed digits.

% ─────────────────────────────────────────────────────────────────
\section{Comparison Table and Gap Analysis}
\label{sec:comparison}
% ─────────────────────────────────────────────────────────────────

\begin{center}
\begin{tabular}{lllll}
\toprule
Source & $\xstar$ & $\Delta\xstar$ & Gap remaining & Cumulative traversed \\
\midrule
$x^*_{\rm CZ}$ & $0.01420$ & --- & $0$ & $100\%$ \\
A241 (this work) & $0.18887$ & $+4.7\times10^{-6}$ & $0.17467$ & $26.7\%$ \\
A240 & $0.18886$ & $-0.0635$ & $0.17466$ & $26.7\%$ \\
A238 & $0.25238$ & (baseline) & $0.23818$ & $0.0\%$ \\
$x^\dagger$ & $0.79254$ & --- & --- & --- \\
\bottomrule
\end{tabular}
\end{center}

The A240$\to$A241 step adds $+4.7\times10^{-6}$ to $\xstar$ (a negligible
\emph{widening} of the gap, not closure).  The cumulative traversed fraction
from A238 remains $26.7\%$.

\paragraph{Why the radial corrections cannot close the gap.}
The angular operator $D^2_{\rm geg}$ has off-diagonal entries of order
$2.7\times10^6$ in the $N=8$ basis.  These set the scale of the
ground-state eigenvalue $\lambda_{\rm gs}\approx-2867$ and the eigenvector
weights $c_0, c_1$ (via back-substitution through these large entries).
The radial corrections $G_{\rm screen}\approx10^{-8}$ and $R_0\approx10^{-2}$
are smaller by 6--14 orders of magnitude.  They cannot shift the eigenvector
direction and hence cannot shift $\xstar$.

Physically: the 4D volume element $r^3\,dr$ strongly suppresses contributions
near $r=0$, and the $\kappa^4\approx(45.7)^4\approx4.4\times10^6$ in the
denominator of the $G_{\rm screen}$ matrix element
\eqref{eq:Vself4D} ensures that the self-lensing coupling is negligible.
The radial scale $1/\kappa\approx0.022$ is so short that the radial ground
state $\phi_0$ is essentially a point mass at the origin from the angular
operator's perspective.

% ─────────────────────────────────────────────────────────────────
\section{Convergence Analysis}
\label{sec:convergence}
% ─────────────────────────────────────────────────────────────────

\begin{table}[h]
\centering
\caption{$\xstar$ as a function of angular basis size $N$
  (adiabatic, A241 radial ground state).}
\label{tab:N}
\begin{tabular}{rrl}
\toprule
$N$ & $\xstar$ & Cumulative gap \\
\midrule
4  & $0.43715$ & $-77.6\%$ \\
6  & $0.53648$ & $-119.3\%$ \\
8  & $0.18887$ & $+26.7\%$ \\
10 & $0.14810$ & $+43.8\%$ \\
12 & $0.19923$ & $+22.3\%$ \\
\bottomrule
\end{tabular}
\end{table}

The values in Table~\ref{tab:N} are erratic and non-monotonic.
This is not a numerical artifact of the quadrature but a structural
feature of the mixed-representation operator.

\begin{proposition}[Instability of mixed representation]
\label{prop:mixed}
The operator $\Ohat$ mixes two incompatible representations:
$D^2_{\rm geg}$ is assembled via change-of-basis
$P^{-1}D^2_{\rm mono}P$ (exact within polynomial space),
while $\rho_{\rm geg}$ is assembled via Galerkin projection.
As $N$ increases:
\begin{enumerate}
\item $\mathrm{cond}(P)$ grows as $4^N$ (diagonal entries $\{4^n\}$),
  so $P^{-1}$ introduces larger and larger rounding errors.
\item The Galerkin matrix $\rho_{\rm geg}$ remains well-conditioned
  (bounded entries $\leq\Omega$) but interacts with the ill-conditioned
  $D^2_{\rm geg}$ through the non-Hermitian eigenvalue problem.
\item The ground eigenstate can flip between the physical branch
  (supported on modes $n=0,1$) and a spurious high-mode branch
  as $N$ increases, producing the swings visible in Table~\ref{tab:N}.
\end{enumerate}
\end{proposition}

The $N=8$ basis is the largest at which the physical ground state
remains identifiable by the criterion that most of the probability
sits on modes $n=0,1$.

\paragraph{Extrapolation.}
Because of the instability, a rigorous $N\to\infty$ extrapolation of
$\xstar$ is not possible within the mixed-representation framework.
If we restrict to the two stable points $N=8$ and $N=10$:
$\xstar(N=8) = 0.189$, $\xstar(N=10) = 0.148$.  A linear extrapolation
in $1/N$ would give $\xstar(\infty)\approx 0.07$, which would traverse
approximately $77\%$ of the total gap from A238.  However, this
extrapolation is unreliable given the instability.

% ─────────────────────────────────────────────────────────────────
\section{Physical Interpretation}
\label{sec:phys}
% ─────────────────────────────────────────────────────────────────

\paragraph{Radial-angular decoupling.}
The P18 operator has two very different scales.
The angular sector $\fork + \weld + \gamma T_{\rm geg}$ operates at the scale of
$\fork$ ($\sim 10^6$ in Gegenbauer basis) and produces a ground eigenvalue
$\lambda_{\rm gs}\approx -2867$.  The radial sector $G_{\rm screen} + R_0$
operates at the scale of $\kappa^{-2}\sim 10^{-3}$ to $\kappa^{-4}\sim10^{-8}$.
The ratio of radial to angular scale is $\sim10^{-9}$, making the radial
corrections a perturbation that is undetectable in double precision and
immaterial for eigenvector orientation.

This scale separation is not an approximation but a structural feature
of the P18 operator.  The screening kernel $G_{\rm screen}(r)
= (1/2\kappa)e^{-\kappa r}$ is concentrated at radii $r\lesssim1/\kappa
\approx0.022$ and is suppressed by $(2\kappa)^4\approx 1.7\times10^7$
when projected onto the 4D radial ground state with volume $r^3\,dr$.

\paragraph{Implication for the OP-alpha blocker.}
The eigentrajectory $\xstar\approx0.189$ is now confirmed robust against
the radial corrections at scales $\sim\kappa^{-1}$.  The path to
$x^*_{\rm CZ}\approx0.014$ must come from:
\begin{enumerate}
\item A fully Galerkin treatment of $\fork$ (A242), which would resolve
  the mixed-representation instability and allow a convergent $N\to\infty$
  study.
\item Inclusion of $\zeta\mathcal{R}$ and $\beta\mathcal{M}$ operators,
  which act in the full $(r,x)$ space and may carry cross-coupling not
  captured by the adiabatic ansatz.
\item A physically different identification of the eigentrajectory, in
  which $x^*_{\rm CZ}$ arises from a \emph{different sector} of $\Ohat$
  than the one computed here.
\end{enumerate}

% ─────────────────────────────────────────────────────────────────
\section{Open Items for A242}
\label{sec:open}
% ─────────────────────────────────────────────────────────────────

\paragraph{A242: Fully Galerkin treatment of $\fork$.}
Replace $D^2_{\rm geg} = P^{-1}D^2_{\rm mono}P$ with the Galerkin matrix
$D^2_{\rm Gal}[m,n] = \langle U_m | D^2 | U_n\rangle_w / h_m$ computed
by inner-product quadrature.  Since $D^2(U_n)$ is a polynomial of degree
$n-2\leq n-2$, the Galerkin and change-of-basis representations should
agree for the entries that matter.  But the Galerkin matrix will be
well-conditioned at all $N$, potentially stabilising the $N$-convergence
and enabling a reliable extrapolation of $\xstar(N)\to x^*_{\rm CZ}$.

\paragraph{A242: $\zeta\mathcal{R}$ and $\beta\mathcal{M}$ operators.}
These have $\zeta = \alpha^{5/4}\approx2.1\times10^{-3}$ and
$\beta=3\pi/20\approx0.471$.  The $\beta\mathcal{M}$ operator (mass term)
acts on both radial and angular modes; at $\beta\approx0.5$ it may
contribute comparably to $\alpha\rho_{\rm geg}\approx1.24$ and shift
$\xstar$ toward $x^*_{\rm CZ}$.

\paragraph{A242: Commutator-zero connection.}
$x^*_{\rm CZ}$ was derived in A228 from $[\Ohat, x]\psi=0$ restricted to
a specific sector.  It would be valuable to identify which sector, whether it
coincides with the ground state computed here, and whether making the
representation fully Galerkin causes the ground-state centroid to converge
toward $x^*_{\rm CZ}$.

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

\begin{itemize}
\item The Laguerre$\times$Gegenbauer product basis ($M=6$, $N=8$, 48 states)
  has been constructed and all radial matrix elements
  ($G_{\rm screen}$, $R_0$) computed with the correct 4D volume $r^3\,dr$.

\item The correct 4D self-lensing expectation is $1.70\times10^{-8}$,
  smaller than the A240 estimate by a factor $2.1\times10^{-4}$.
  The radial kinetic energy of the ground state is $1.64\times10^{-2}$.
  Both are negligible at the eigenvalue scale $|\lambda_{\rm gs}|\approx2867$.

\item Adiabatic and full 48-state diagonalisations both give
  $\xstar_{\rm A241} = 0.18887$, unchanged from A240 to within $5\times10^{-6}$.

\item The cumulative gap traversed from A238 remains $26.7\%$.
  The radial sector does not contribute to gap closure.

\item N-convergence is erratic (Table~\ref{tab:N}) due to the
  mixed-representation instability.  A242 should implement a fully
  Galerkin $\fork$ matrix to stabilise convergence.

\item All 20 checks in \texttt{verify/verify\_P241.py} pass.
\end{itemize}

\begin{center}
\begin{tabular}{ll}
\toprule
\textbf{Quantity} & \textbf{Value} \\
\toprule
$x^*_{\rm CZ}$ & $0.01420$ \\
$\xstar_{\rm A241}$ & $\mathbf{0.18887}$ \\
$\xstar_{\rm A240}$ & $0.18886$ \\
$\xstar_{\rm A238}$ & $0.25238$ \\
$x^\dagger$ & $0.79254$ \\
\midrule
$\Delta\xstar$ (A240$\to$A241) & $+4.7\times10^{-6}$ \\
Cumulative gap from A238 & $26.7\%$ \\
$\lambda_{\rm gs}$ (A241) & $-2866.87$ \\
$\langle\phi_0|G_{\rm screen}|\phi_0\rangle_{r^3dr}$ & $1.70\times10^{-8}$ \\
$\langle\phi_0|R_0|\phi_0\rangle_{r^3dr}$ & $1.64\times10^{-2}$ \\
Product basis size $M\times N$ & $48$ \\
Checks passing & $20/20$ \\
\bottomrule
\end{tabular}
\end{center}

\end{document}
