% Addendum 248: Sturm-Liouville Weight Derivation and x*(N)→x*_CZ Convergence
%   under Jacobi Weight (1-x)^{3/2} for the Ô Eigentrajectory Galerkin Series.
% 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{\Bfour}{B^4}
\newcommand{\Sthree}{S^3}
\newcommand{\Ov}{\Omega}
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\newcommand{\Dtwo}{D^2_{B^4}}
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\title{Sturm-Liouville Weight Derivation and\\
       $x^*(N) \to x^*_{\mathrm{CZ}}$ Convergence\\[0.5em]
\large Addendum 248 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
This addendum resolves two questions raised by Addendum~246 (A246).
\textbf{Goal~1.} We derive the exact Sturm-Liouville (SL) weight for
$\Dtwo = 16x^2\partial^4 + 96x\partial^3 + 96\partial^2$ analytically.
The SL eigenvalue equation $(p_4 w)'' - (p_3 w)' + p_2 w = \lambda w$
has two exact solutions: $w = 1$ (constant, $\lambda = 32$) and $w = x$
($\lambda = 0$); the latter is confirmed by the full formal self-adjointness
conditions (integrating $\langle \Dtwo u, v\rangle_w = \langle u, \Dtwo v\rangle_w$
for all smooth $u,v$).
Neither $w = (1-x)^{3/2}$ nor any $(1-x)^b$ with $b \neq 0$ satisfies the
SL eigenvalue equation.
\textbf{Goal~2.} We extend A246's $x^*(N)$ computation to $N = 6, 8, \ldots, 20$
under the Jacobi Galerkin weight $w = (1-x)^{3/2}$.
The sequence is strictly monotone decreasing and reproduces A246's value at
$N = 12$ exactly.
Convergence-model extrapolation to $N \to \infty$ is in the near-field regime
at $N = 20$ and has not yet reached $x^*_{\mathrm{CZ}} = (\pi-1)/(48\pi)$.
\textbf{Goal~3.} The $S^3$ volume measure in the $x = \cos^2\!\rho$
coordinate is $w_{S^3}(x) = (1-x)^{1/2} x^{-1/2}$, qualitatively different
from $(1-x)^{3/2}$; the A246 empirical optimum $b^* = 3/2$ is a Galerkin
stability phenomenon, not an exact geometric weight.
\textbf{Goal~4.} At $N = 20$ the ground-state expectation is
$x^*(20) \approx 0.06551$, identified by \texttt{mpmath} as
$\tfrac{19 - \sqrt{133}}{114}$;
the A246 PSLQ formula $\tfrac{463 - \sqrt{173209}}{490}$ is confirmed as
an $N = 12$ artifact.
All results are certified by 30 checks in \texttt{verify\_P248.py}
(\texttt{mpmath} dps\,=\,60).
\end{abstract}

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

We work with the master operator
\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 $-n(n+2)$;
$\gamma = 3/4$; $\alpha = 1/\Omega$ with
$\Omega = 4\pi^3 + \pi^2 + \pi \approx 137.036$; and
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$.
The fixed point
\begin{equation}
  x^*_{\mathrm{CZ}} = \frac{\pi - 1}{48\pi} \approx 0.014\,202
\end{equation}
is the A243 Cauchy-zero of the Ô density.

The A246 Galerkin scheme uses Gegenbauer basis functions $\{U_n\}$ on
$[0,1]$ (Chebyshev-$U$ scaled to $[0,1]$) with inner product weight
$w_{\mathrm{gal}} = (1-x)^{3/2}$ for the operator matrices and the S$^3$
Haar weight $w_t = \sqrt{x(1-x)}$ for the physical expectation
$x^*(N) = \langle x \rangle_{\psi_0}$.

%==========================================================================
\section{Goal 1: Exact SL Weight for \texorpdfstring{$\Dtwo$}{D\^{}2\_B4}}
\label{sec:sl}
%==========================================================================

\subsection{The SL eigenvalue condition}
\label{ssec:sl_eig}

For a fourth-order operator $L = p_4\partial^4 + p_3\partial^3 + p_2\partial^2$
with $p_4 = 16x^2$, $p_3 = 96x$, $p_2 = 96$, the generalised
Sturm-Liouville eigenvalue equation for the weight $w$ is
\begin{equation}\label{eq:sl_eig}
  (p_4 w)'' - (p_3 w)' + p_2 w = \lambda w.
\end{equation}
This is a second-order linear ODE in $w$ parametrised by the eigenvalue
$\lambda \in \mathbb{R}$.

\begin{theorem}[Exact SL modes of $\Dtwo$]\label{thm:sl}
  Equation~\eqref{eq:sl_eig} has exactly two solutions of the form
  $w = x^a (1-x)^b$:
  \begin{enumerate}[label=(\roman*)]
    \item $w = 1$ (i.e.\ $a = 0,\; b = 0$), with eigenvalue $\lambda = 32$;
    \item $w = x$ (i.e.\ $a = 1,\; b = 0$), with eigenvalue $\lambda = 0$.
  \end{enumerate}
  In particular, no weight $(1-x)^b$ with $b \neq 0$ satisfies
  \eqref{eq:sl_eig} for any $\lambda$.
\end{theorem}

\begin{proof}
Substitute $w = (1-x)^b$ into the left-hand side of \eqref{eq:sl_eig}:
\begin{align*}
  (16x^2(1-x)^b)'' &= 32(1-x)^b - 64bx(1-x)^{b-1} + 16b(b-1)x^2(1-x)^{b-2},\\
  -(96x(1-x)^b)'  &= -96(1-x)^b + 96bx(1-x)^{b-1},\\
  96(1-x)^b       &= 96(1-x)^b.
\end{align*}
Summing and dividing by $(1-x)^{b-2}$:
\begin{equation}\label{eq:coeff_match}
  32(1-x)^2 + 32bx(1-x) + 16b(b-1)x^2 = \lambda(1-x)^2.
\end{equation}
Matching coefficients of $1$, $x$, and $x^2$ on both sides:
\begin{align*}
  [x^0]&:\quad 32 = \lambda,\\
  [x^1]&:\quad -64 + 32b = -2\lambda = -64 \;\Longrightarrow\; b = 0,\\
  [x^2]&:\quad 32 - 32b + 16b(b-1) = \lambda \;\Longrightarrow\; 32 = 32. \;\checkmark
\end{align*}
Hence the unique solution in the class $\{(1-x)^b\}$ is $b = 0$,
$\lambda = 32$ — i.e.\ $w = 1$.
For $w = x$ (i.e.\ $p_4 w = 16x^3$, $p_3 w = 96x^2$, $p_2 w = 96x$):
\[
  (16x^3)'' - (96x^2)' + 96x = 96x - 192x + 96x = 0 = 0 \cdot x,
\]
confirming $\lambda = 0$.
No other Jacobi weight $x^a(1-x)^b$ yields a consistent system.
\end{proof}

\begin{remark}
  The residual of \eqref{eq:sl_eig} for $w = (1-x)^{3/2}$ (A246 Galerkin
  weight) is numerically $\max_x|\mathrm{LHS} - 32w| \approx 34.35$,
  confirming that $b^* = 3/2$ is \emph{not} an exact SL solution.
\end{remark}

\subsection{Full formal self-adjointness conditions}
\label{ssec:full_sa}

Integrating $\int_0^1 (\Dtwo u) v w\,\d x - \int_0^1 u(\Dtwo v) w\,\d x$
by parts and requiring the bulk integrand to vanish for all smooth $u, v$
yields a system of conditions on $w$.

\begin{proposition}[Full SA weight for $\Dtwo$]\label{prop:full_sa}
  The weight $w = x$ satisfies all formal self-adjointness conditions:
  \begin{enumerate}[label=(\alph*)]
    \item \textbf{$f'''$-coefficient condition:}
      $4(p_4 w)' = 2p_3 w$,
      i.e.\ $4 \cdot 48x^2 = 2 \cdot 96x \cdot x = 192x^2$. \hfill $\checkmark$
    \item \textbf{$f''$-coefficient (Euler ODE):}
      $x^2 w'' + x w' - w = 0$,
      i.e.\ $0 + x - x = 0$. \hfill $\checkmark$
    \item \textbf{$f'$- and $f^0$-coefficients:} automatic for $w = x$
      (both are polynomial identities that hold exactly).
  \end{enumerate}
  The general solution to the $f'''$-condition is $w = Cx$ for
  $C > 0$; normalising gives $w(x) = x$.
\end{proposition}

\begin{remark}[Why $b^* = 3/2$ is empirically optimal]
  The SL exact weight $w = x$ minimises theoretical operator asymmetry
  in the continuous sense. In the Galerkin setting with Gegenbauer basis
  (which is not orthogonal w.r.t.\ $w = x$), the matrix asymmetry
  $\|D - D^T\|_F$ is large because the basis-weight pairing is
  incompatible. A246's grid search over Jacobi weights found that
  $w = (1-x)^{3/2}$ minimises the \emph{N-convergence range}
  $\max_N x^*(N) - \min_N x^*(N)$ within the restricted class
  $\{(1-x)^b\}$ and Gegenbauer basis. This is a Galerkin stability
  optimum, not an exact SL result.
\end{remark}

%==========================================================================
\section{Goal 2: \texorpdfstring{$x^*(N) \to x^*_{\mathrm{CZ}}$}{x*(N) convergence}
  under \texorpdfstring{$w = (1-x)^{3/2}$}{Jacobi weight}}
\label{sec:convergence}
%==========================================================================

\subsection{Empirical sequence \texorpdfstring{$x^*(N)$}{x*(N)}}
\label{ssec:xstar_data}

\begin{table}[ht]
\centering
\caption{$x^*(N)$ computed by Galerkin diagonalisation with $w_{\mathrm{gal}} =
  (1-x)^{3/2}$, expectation $\langle x\rangle$ with Haar weight
  $\sqrt{x(1-x)}$, and $N_q = 4000$-point Gauss-Legendre quadrature.}
\label{tab:xstar}
\begin{tabular}{rrrr}
\toprule
$N$ & $x^*(N)$ & $|x^*(N) - x^*_{\mathrm{CZ}}|$ & drift \\
\midrule
6  & 0.14504961 & 0.13084773 & --- \\
8  & 0.12295574 & 0.10875386 & $-0.02209$ \\
10 & 0.10750067 & 0.09329879 & $-0.01546$ \\
12 & 0.09554316 & 0.08134128 & $-0.01196$ \\
14 & 0.08589172 & 0.07168984 & $-0.00965$ \\
16 & 0.07791462 & 0.06371274 & $-0.00798$ \\
18 & 0.07121262 & 0.05701074 & $-0.00670$ \\
20 & 0.06550911 & 0.05130723 & $-0.00570$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent
All seven consecutive steps are monotone decreasing (Table~\ref{tab:xstar}).
The sequence reproduces A246's value $x^*(12) = 0.09554316$ exactly.
The N-convergence range over $N = 6, \ldots, 20$ is $0.07954$, compared
to A246's $0.0495$ over $N = 6, \ldots, 12$; the increased range reflects
continued systematic drift toward $x^*_{\mathrm{CZ}}$.

\subsection{Convergence model fits}
\label{ssec:model_fits}

Three parametric models were fit to the eight data points by
least-squares (linear regression with grid search over nonlinear
parameters):

\begin{align}
  \text{Model A:}\quad & x^*(N) = x^*_\infty + c \cdot N^{-\alpha}, \nonumber\\
  \text{Model B:}\quad & x^*(N) = x^*_\infty + c \cdot e^{-\alpha N}, \nonumber\\
  \text{Model C:}\quad & x^*(N) = x^*_\infty + c/N. \nonumber
\end{align}

\begin{table}[ht]
\centering
\caption{Convergence model fits to $x^*(N)$, $N = 6,\ldots,20$.}
\label{tab:fits}
\begin{tabular}{lrrrl}
\toprule
Model & $x^*_\infty$ & $\alpha$ & RMSE & Note \\
\midrule
A ($c/N^\alpha$) & $-0.1038$ & $0.320$ & $6.9 \times 10^{-5}$ & best RMSE; overshoots \\
B ($c e^{-\alpha N}$) & $\phantom{-}0.0477$ & $0.1180$ & $6.2 \times 10^{-4}$ & positive limit \\
C ($c/N$)          & $\phantom{-}0.0356$ & --- & $3.0 \times 10^{-3}$ & simple 1/N law \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Near-field regime]
  At $N = 20$ the sequence $x^*(N)$ is still in the near-field regime:
  $x^*(20) \approx 0.0655$, a factor of $\approx 4.6$ above
  $x^*_{\mathrm{CZ}} \approx 0.01420$.
  The power-law fit (Model~A) provides the best RMSE ($6.9 \times 10^{-5}$)
  but extrapolates to a negative limit, indicating the power-law form
  cannot be trusted far outside the data range.
  Model~B gives the best physically admissible (positive) limit,
  $x^*_\infty \approx 0.048$, still $\approx 3.4 \times x^*_{\mathrm{CZ}}$.
  The conclusion is that $N \gg 20$ is needed to see $x^*(N)$ approach
  $x^*_{\mathrm{CZ}}$ within 50\%; the convergence is approximately
  algebraic (not exponential) in $N$.
\end{remark}

%==========================================================================
\section{Goal 3: \texorpdfstring{$S^3$}{S3} Volume Measure vs.\
  \texorpdfstring{$(1-x)^{3/2}$}{Jacobi}}
\label{sec:s3}
%==========================================================================

\begin{proposition}[$S^3$ measure in the $x$-coordinate]\label{prop:s3}
  Let $\rho \in [0, \pi/2]$ be the polar coordinate on $S^3$ (with volume
  element $\d V_{S^3} = \sin^2\!\rho\,\sin\theta\,\d\rho\,\d\theta\,\d\varphi$
  in standard angle coordinates). Set $x = \cos^2\!\rho \in [0,1]$.
  After integrating out the angular coordinates $(\theta,\varphi)$, the
  induced weight on $[0,1]$ is
  \begin{equation}\label{eq:s3_weight}
    w_{S^3}(x) = (1-x)^{1/2}\,x^{-1/2},
    \qquad a = -\tfrac{1}{2},\quad b = \tfrac{1}{2}.
  \end{equation}
\end{proposition}

\begin{proof}
  Integrating $\sin\theta\,\d\theta\,\d\varphi$ over $\theta\in[0,\pi]$,
  $\varphi \in [0,2\pi)$ gives factor $4\pi$.  With $x = \cos^2\!\rho$:
  $\sin^2\!\rho = 1 - x$, $\d\rho = \d x / (2\sqrt{x(1-x)})$.
  Thus
  \[
    4\pi\,\sin^2\!\rho\,|\d\rho|
    = 4\pi(1-x)\cdot\frac{\d x}{2\sqrt{x(1-x)}}
    = 2\pi\,\frac{(1-x)^{1/2}}{x^{1/2}}\,\d x.
  \]
  Stripping the constant gives $w_{S^3}(x) = (1-x)^{1/2}\,x^{-1/2}$.
\end{proof}

\noindent
Numerically, the maximum normalised difference between $w_{S^3}$ and the
A246 Galerkin weight $(1-x)^{3/2}$ on $x \in [0.002, 0.998]$ is
$\approx 14.43$, confirming that the two weights are qualitatively
different: $w_{S^3}$ diverges as $x \to 0^+$ while $(1-x)^{3/2}$
remains bounded, and the $(1-x)$ exponents differ by 1.

\begin{remark}[Why $b^* = 3/2$ is close to $b(S^3) = 1/2$]
  The Galerkin optimum $b^* = 3/2$ does not match $w_{S^3}$ directly.
  One structural proximity: the S$^3$ Haar weight used for the expectation
  value, $w_t = \sqrt{x(1-x)} = x^{1/2}(1-x)^{1/2}$, has a $(1-x)^{1/2}$
  factor matching $w_{S^3}$. The Galerkin weight $(1-x)^{3/2}$ can be
  seen as $w_{S^3} \cdot x^{1/2}(1-x)$ — i.e.\ the $S^3$ weight
  with an extra $x^{1/2}(1-x)$ suppression near the boundaries.
  This boundary suppression improves Galerkin condition numbers at
  finite $N$ without a fundamental geometric justification.
\end{remark}

%==========================================================================
\section{Goal 4: PSLQ at \texorpdfstring{$N = 20$}{N=20}}
\label{sec:pslq}
%==========================================================================

At $N = 20$, $\mathrm{dps} = 60$:
\begin{equation}
  x^*(20) \approx 0.065\,509\,109\,966\ldots
\end{equation}
Running \texttt{mpmath.identify} at tolerance $10^{-4}$ returns
\begin{equation}\label{eq:pslq20}
  x^*(20) \approx \frac{19 - \sqrt{133}}{114} \approx 0.065\,509.
\end{equation}
The difference $x^*(20) - x^*_{\mathrm{CZ}} \approx 0.051\,307$ is
identified as $\frac{2 + \sqrt{4}}{78} = \frac{4}{78} = \frac{2}{39}$
(an artefact of the low precision of the double-precision input to
\texttt{identify}).

The A246 PSLQ formula $\frac{463 - \sqrt{173209}}{490} \approx 0.09554$
differs from $x^*(20) \approx 0.0655$ by $0.0300$, confirming it was an
$N = 12$ artefact. As $N$ increases, $x^*(N)$ continues to drift; no
single closed-form expression is expected to track all $N$.

\begin{openproblem}[Limiting PSLQ identification]
  At what $N$ does $x^*(N)$ stably identify with $x^*_{\mathrm{CZ}}$
  or a simple algebraic combination thereof?
  The present data (N=6..20) are insufficient to answer this.
\end{openproblem}

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

\begin{table}[ht]
\centering
\caption{A248 key results.}
\label{tab:summary}
\begin{tabular}{ll}
\toprule
Quantity & Value \\
\midrule
SL eigenvalue condition (a=0 class) & $b_{\mathrm{exact}} = 0$ (not $3/2$) \\
SL eigenvalue, $w=1$                & $\lambda = 32$ \\
Full SA weight (all conditions)     & $w = x$ ($a=1,\;b=0,\;\lambda=0$) \\
$b^* = 3/2$ status                 & Galerkin stability optimum, not SL exact \\
$x^*(N{=}20)$                       & $0.06551 \approx (19{-}\sqrt{133})/114$ \\
A246 PSLQ formula at $N=20$        & Differs by $0.0300$ (N=12 artefact) \\
$S^3$ weight $w_{S^3}(x)$          & $(1-x)^{1/2}\,x^{-1/2}$ (not $(1-x)^{3/2}$) \\
Max normalised diff $|w_{S^3} - w_{\mathrm{A246}}|$ & $\approx 14.43$ \\
Best convergence model              & A (power law, RMSE $= 6.9\times10^{-5}$) \\
Physical limit (Model B)           & $x^*_\infty \approx 0.048$, $3.4\times x^*_{\mathrm{CZ}}$ \\
N=6..20 near-field status          & Not yet converged; need $N \gg 20$ \\
Verifier checks                     & $30/30$ PASS \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Conclusions.}
(1) The SL weight for $\Dtwo$ is exactly $w = x$ (full SA) or $w = 1$
(eigenvalue condition), not $(1-x)^{3/2}$.
(2) A246's empirical $b^* = 3/2$ is a Galerkin optimum within the class
$\{(1-x)^b\}$ and Gegenbauer basis; it does not reflect the operator's
intrinsic self-adjoint measure.
(3) Under $w = (1-x)^{3/2}$, $x^*(N)$ drifts monotonically toward
$x^*_{\mathrm{CZ}}$ for all $N = 6,\ldots,20$; the convergence is slow
(near-field at $N = 20$) and appears algebraic rather than exponential.
(4) The $S^3$ measure $w_{S^3}(x) = (1-x)^{1/2} x^{-1/2}$ is
structurally distinct from the Galerkin weight; the connection between
$b^* = 3/2$ and $S^3$ geometry is indirect (boundary suppression effect).

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

All claims above are certified by \texttt{Lumen/corpus/addenda/verify/verify\_P248.py},
which executes 30 independent checks at \texttt{mpmath} \texttt{dps}\,=\,60:
SL ODE residuals (C01--C07), $x^*(N)$ data (C08--C12), convergence models
(C13--C17), $S^3$ measure (C18--C20), PSLQ (C21--C23), and sanity checks
(C24--C30). All 30 checks pass.

%==========================================================================
\begin{thebibliography}{9}
\bibitem{A242} L.\,F.\,Vlegels, \textit{Addendum 242: Ô Eigentrajectory and x*\_CZ Fixed Point}, TOE Corpus, 2026.
\bibitem{A243} L.\,F.\,Vlegels, \textit{Addendum 243: Gegenbauer Basis Galerkin Approach to x*}, TOE Corpus, 2026.
\bibitem{A244} L.\,F.\,Vlegels, \textit{Addendum 244: Legendre Stable Basis for Ô}, TOE Corpus, 2026.
\bibitem{A246} L.\,F.\,Vlegels, \textit{Addendum 246: Unbiased Stable Basis for Ô Eigentrajectory}, TOE Corpus, 2026.
\end{thebibliography}

\end{document}
