% Addendum 246: Unbiased Stable Basis for the Ô Eigentrajectory
% Four strategies: D²_B4 eigenbasis, commutator/simultaneous diag,
% adaptive hybrid Gegenbauer, Sturm-Liouville weight construction.
% L. F. Vlegels, Independent Researcher
% 2026-05-24

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\title{Addendum A246: Unbiased Stable Basis\\
for the $\hat{O}$ Eigentrajectory\\[4pt]
\large D$^2_{B^4}$ Eigenbasis, Commutator Rotation,\\
Adaptive Hybrid, and Sturm--Liouville Weight}

\author{L. F. Vlegels\\Independent Researcher}
\date{2026-05-24}

\begin{document}
\maketitle

\begin{abstract}
Addendum A244 identified a fundamental truncation--bias trade-off
in the $x^*$ eigentrajectory of the Gegenbauer operator
$\hat{O}_{\rm geg} = D^2_{B^4} + \Delta_{S^3} + \gamma T_{\rm geg} + \alpha\rho(x)$:
bases that stabilize $x^*$ across truncation levels $N$ do so
by suppressing the $T_{\rm geg} = \Delta_{S^3} \circ D^2_{B^4}$ coupling,
shifting the stable fixed point away from the target
$x^*_\mathrm{CZ} \approx 0.01420$.
This addendum tests four strategies designed to simultaneously
achieve low truncation range \emph{and} preserve $T_{\rm geg}$-coupling.
\textbf{Strategy~1} (D$^2_{B^4}$ eigenbasis via monomial diagonalization)
achieves range $0.184$ at $x^*(N{=}12) = 0.544$ --- unstable.
\textbf{Strategy~2} (perturbative simultaneous diagonalization) is
immediately ruled non-perturbative: $\|[D^2_{B^4}, \Delta_{S^3}]\|_F
= 2.28 \times 10^9 \gg \|\Delta_{S^3}\|_F = 156.8$ (ratio $\approx
1.45 \times 10^7$); the rotation worsens instability (range $0.670$).
\textbf{Strategy~3} (adaptive hybrid Gegenbauer basis seeded by
the $N{=}8$ ground state) achieves range $0.079$ at
$x^*(N{=}12) = 0.895$ --- tighter but biased high.
\textbf{Strategy~4} (optimal Jacobi weight $w(x) = (1-x)^{3/2}$,
found by minimizing Galerkin asymmetry of $D^2_{B^4}$ over a
Jacobi family $x^a(1-x)^b$) achieves range
$\mathbf{0.04951} < 0.05$ at $x^*(N{=}12) = 0.09554$.
This marginally resolves the trade-off: the range target is met
and $x^* = 0.09554$ is substantially closer to $x^*_\mathrm{CZ}$
than the Legendre value $0.387$ (A244).
The PSLQ oracle suggests $x^* \approx (463 - \sqrt{173209})/490$.
All results are verified by \texttt{verify\_P246.py} (30/30 checks,
\texttt{mpmath} dps${}=60$).
\end{abstract}

\tableofcontents
\newpage

% ─────────────────────────────────────────────────────────────────────────────
\section{Background and Open Problem}
\label{sec:background}

\subsection{The operator and its constants}

The Gegenbauer operator angular sector studied in A238--A244 is
\begin{equation}
  \hat{O}_{\rm geg} = D^2_{B^4} + \Delta_{S^3} + \gamma T_{\rm geg} + \alpha\rho(x),
  \label{eq:Ohat}
\end{equation}
where
\begin{align}
  D^2_{B^4}   &= 16x^2\partial^4 + 96x\partial^3 + 96\partial^2, \label{eq:D2B4}\\
  \Delta_{S^3} &= \operatorname{diag}\bigl(-n(n+2)\bigr)_{n=0}^{N-1}, \label{eq:DS3}\\
  T_{\rm geg}  &= \Delta_{S^3} \circ D^2_{B^4}, \label{eq:T}\\
  \rho(x)      &= 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x, \label{eq:rho}
\end{align}
with constants
\begin{equation}
  \Omega = 4\pi^3+\pi^2+\pi \approx 137.036, \quad
  \alpha = \Omega^{-1}, \quad
  \gamma = \tfrac{3}{4}.
\end{equation}
The commutator-zero target is
$x^*_\mathrm{CZ} = (\pi-1)/(48\pi) \approx 0.01420$.

\subsection{The inherited trade-off from A244}

A244 established:
\begin{itemize}
  \item In the Gegenbauer (Chebyshev-$U$) basis the $x^*$ eigentrajectory
    swings by $\Delta x^* \approx 0.39$ over $N \in \{6,8,10,12\}$ because
    $T_{\rm geg}$-coupling entries grow $O(N^6)$.
  \item The Legendre basis $P^{(0,0)}$ achieves range $9.06\times 10^{-4}$
    but converges to $x^* \approx 0.387$ --- the truncation suppresses
    the $T_{\rm geg}$-coupling that would pull $x^*$ toward $x^*_\mathrm{CZ}$.
  \item \textbf{Fundamental trade-off}: any basis that diagonalizes
    $\Delta_{S^3}$ exactly gains spectral separation for free but inherits
    $O(N^6)$ off-diagonal growth from $D^2_{B^4}$; any basis that absorbs
    the growth pays a bias in $x^*$.
\end{itemize}

\subsection{Goal of this addendum}

Find a basis $\{\varphi_n\}_{n=0}^{N-1}$ such that both $\Delta_{S^3}$
\emph{and} $D^2_{B^4}$ have bounded Galerkin matrix elements, so that
$T_{\rm geg} = \Delta_{S^3} \cdot D^2_{B^4}$ is also bounded.
Four strategies are evaluated.

% ─────────────────────────────────────────────────────────────────────────────
\section{Commutator Analysis}
\label{sec:commutator}

Before testing Strategy 2 we quantify how far $D^2_{B^4}$ and
$\Delta_{S^3}$ are from commuting.

\begin{proposition}[Non-perturbative non-commutativity]
  In the Gegenbauer basis at $N = 10$,
  \begin{equation}
    \|[D^2_{B^4},\,\Delta_{S^3}]\|_F = 2.275 \times 10^9,
    \quad
    \|\Delta_{S^3}\|_F = 156.8,
    \quad
    \text{ratio} \approx 1.45 \times 10^7.
  \end{equation}
  Simultaneous diagonalization by a rotation of size
  $\varepsilon \sim \|[D^2_{B^4},\Delta_{S^3}]\| /
  (\text{eigenvalue gap of }D^2_{B^4})$ is not perturbative.
\end{proposition}

\begin{proof}
  Direct Galerkin matrix computation (see \texttt{verify\_P246.py}, C05--C06).
\end{proof}

\begin{remark}
The D$^2_{B^4}$ eigenvalues in the Galerkin Gegenbauer representation
at $N=14$ span
$\{-1.85 \times 10^8,\,-2.31 \times 10^7,\,1.40\times 10^2,\ldots\}$,
confirming extreme ill-conditioning of the Gegenbauer representation
of $D^2_{B^4}$.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Strategy 1: D$^2_{B^4}$ Eigenbasis}
\label{sec:strategy1}

\subsection{Construction}

Build the Galerkin $D^2_{B^4}$ matrix in the monomial basis
$\{\varphi_k^{(\mathrm{m})} = x^k\}_{k=0}^{19}$ (size $N_\mathrm{mono} = 20$)
under the inner product $\langle f,g\rangle_w = \int_0^1 f(x)g(x)w(x)\,dx$
with the S$^3$ Haar weight $w(x) = \sqrt{x(1-x)}$.

The action of $D^2_{B^4}$ on a monomial is
\begin{equation}
  D^2_{B^4}\,x^k = c_k\,x^{k-2}, \quad
  c_k = 16k(k-1)(k-2)(k-3) + 96k(k-1)(k-2) + 96k(k-1),
\end{equation}
giving the Galerkin stiffness matrix $S_{jk} = \int_0^1 x^j (D^2_{B^4}\,x^k)\,w\,dx$
and mass matrix $M_{jk} = \int_0^1 x^{j+k}w\,dx$.
The Galerkin $D^2_{B^4}$ representation is $\mathbf{D}^2 = M^{-1}S$.

Diagonalize $\frac{1}{2}(\mathbf{D}^2 + (\mathbf{D}^2)^\top)$ and
sort eigenvectors by ascending $|\lambda|$.
The first $N$ eigenvectors (as monomial coefficients) define basis
functions $\varphi_k^{(1)}(x) = \sum_{j=0}^{19} V_{jk}\,x^j$,
orthonormalized under $\langle\cdot,\cdot\rangle_w$.

\subsection{Results}

\begin{table}[h]
\centering
\begin{tabular}{ccc}
\toprule
$N$ & $x^*$ (Strategy 1) & Comment \\
\midrule
6  & 0.5584 & \\
8  & 0.3743 & \\
10 & 0.4787 & \\
12 & 0.5443 & \\
\midrule
\multicolumn{2}{c}{Range (max$-$min)} & $0.1841$ \\
\bottomrule
\end{tabular}
\caption{Strategy 1 (D$^2_{B^4}$ eigenbasis): $x^*$ at four truncation levels.}
\label{tab:s1}
\end{table}

\begin{remark}
The D$^2_{B^4}$ eigenbasis does not diagonalize $\Delta_{S^3}$ ---
it merely re-expresses D$^2_{B^4}$ in a different representation.
The $T_{\rm geg}$-coupling remains large and the instability persists,
though the range $0.184$ is an improvement over the Gegenbauer $0.39$.
The monomial-basis eigenvalue spread is $>10^6$,
confirming that diagonalizing D$^2_{B^4}$ alone does not tame the
off-diagonal $T_{\rm geg}$ growth.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Strategy 2: Perturbative Simultaneous Diagonalization}
\label{sec:strategy2}

\subsection{Construction}

Given the result of Section~\ref{sec:commutator} (commutator ratio $\approx 10^7$),
a perturbative simultaneous diagonalization is known to fail before the
computation begins.
Nevertheless we implement it for completeness:
\begin{enumerate}
  \item Diagonalize D$^2_{B^4}$ in the Gegenbauer basis: $V^\top \mathbf{D}^2 V = \Lambda$.
  \item Compute $\tilde\Delta = V^\top \Delta_{S^3} V$ (the $\Delta_{S^3}$ matrix
        in the D$^2_{B^4}$ eigenbasis).
  \item Apply a first-order rotation $R \approx I + A$ where
        $A_{mn} = \varepsilon\,\tilde\Delta_{mn}/(\lambda_m^{D^2} - \lambda_n^{D^2})$
        for $m \neq n$, with $\varepsilon = 0.1$.
  \item Re-orthonormalize via QR decomposition.
  \item Build $\hat{O}$ in the new basis and compute $x^*$.
\end{enumerate}

\subsection{Results}

\begin{table}[h]
\centering
\begin{tabular}{ccc}
\toprule
$N$ & $x^*$ (Strategy 2) & Comment \\
\midrule
6  & 0.5364 & \\
8  & 0.1930 & \\
10 & 0.7120 & \\
12 & 0.8629 & \\
\midrule
\multicolumn{2}{c}{Range (max$-$min)} & $0.6699$ \\
\bottomrule
\end{tabular}
\caption{Strategy 2 (simultaneous diagonalization): $x^*$ at four truncation levels.}
\label{tab:s2}
\end{table}

\begin{remark}
As predicted, the rotation amplifies rather than tames the instability.
The range $0.670$ exceeds even the Gegenbauer $0.39$.
Simultaneous diagonalization of D$^2_{B^4}$ and $\Delta_{S^3}$ is
provably non-perturbative and cannot be patched by a small rotation.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Strategy 3: Adaptive Hybrid Gegenbauer Basis}
\label{sec:strategy3}

\subsection{Construction}

The hypothesis is that including the $N{=}8$ ground-state eigenfunction
$\psi_0$ in the basis removes the low-mode instability.
Procedure:
\begin{enumerate}
  \item Solve the $N{=}8$ Gegenbauer eigenvalue problem for $\hat{O}$;
        sort real eigenvectors by ascending eigenvalue.
  \item Take the first $n_\mathrm{seed} = 6$ eigenvectors as seed functions
        (expressed as Gegenbauer combinations on the quadrature grid).
  \item Pad with the higher Gegenbauer modes $U_6,\ldots,U_{N-1}$.
  \item Apply Gram--Schmidt orthonormalization under $\langle\cdot,\cdot\rangle_w$.
\end{enumerate}

Orthonormality check at $N = 12$:
$\|Q^\top Q - I\|_F = 3.07 \times 10^{-14} \ll 10^{-10}$.

\begin{table}[h]
\centering
\begin{tabular}{ccc}
\toprule
$N$ & $x^*$ (Strategy 3) & Comment \\
\midrule
6  & 0.8305 & \\
8  & 0.8157 & \\
10 & 0.8429 & \\
12 & 0.8952 & \\
\midrule
\multicolumn{2}{c}{Range (max$-$min)} & $0.0795$ \\
\bottomrule
\end{tabular}
\caption{Strategy 3 (adaptive hybrid): $x^*$ at four truncation levels.}
\label{tab:s3}
\end{table}

\begin{remark}
The hybrid basis stabilizes the range to $0.079$ --- below the Gegenbauer
instability but above the target $0.05$.
However, it shifts $x^*$ to $\approx 0.89$, far from both the Gegenbauer
result and $x^*_\mathrm{CZ}$.
Including the ground-state eigenfunction as a seed vector does constrain
the ground state at small $N$, but because the $N{=}8$ ground state
already lies at $x^* \approx 0.83$, the hybrid basis inherits that bias
and propagates it to larger $N$.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Strategy 4: Sturm--Liouville Weight}
\label{sec:strategy4}

\subsection{Motivation}

A self-adjoint operator has real eigenvalues and a complete orthogonal
eigenbasis.
The natural question is: does there exist a weight $w^*(x) > 0$ such
that $D^2_{B^4}$ is formally self-adjoint under
$\langle f,g\rangle_{w^*} = \int_0^1 f(x)g(x)\,w^*(x)\,dx$?
For a fourth-order operator the Sturm--Liouville condition is a
complicated fourth-order ODE for $w^*$.
Rather than solving this ODE exactly, we search the Jacobi family
$w_{a,b}(x) = x^a(1-x)^b$ for the pair $(a,b)$ that minimizes the
asymmetry of the Galerkin $D^2_{B^4}$ matrix:
\begin{equation}
  (a^*, b^*) = \arg\min_{a,b\in\mathcal{G}}
  \|\mathbf{D}^2_{a,b} - (\mathbf{D}^2_{a,b})^\top\|_F,
\end{equation}
where $\mathcal{G} = \{0, 0.25, 0.5, 0.75, 1.0, 1.5\}^2$ and
$\mathbf{D}^2_{a,b}$ is the $N{=}8$ Gegenbauer Galerkin $D^2_{B^4}$
matrix computed under inner product with weight $w_{a,b}$.

\subsection{Result of the search}

\begin{proposition}[Optimal Jacobi weight]
  The minimizer within $\mathcal{G}^2$ is
  \begin{equation}
    (a^*, b^*) = (0,\,3/2), \qquad
    w^*(x) = (1-x)^{3/2},
  \end{equation}
  with residual asymmetry $\|\mathbf{D}^2 - (\mathbf{D}^2)^\top\|_F
  = 4.165 \times 10^5$.
  The weight is positive on all of $(0,1)$.
\end{proposition}

\begin{remark}
  The residual asymmetry is large in absolute terms, confirming that no
  Jacobi weight makes $D^2_{B^4}$ exactly self-adjoint.
  The Jacobi search is an approximation to the true Sturm--Liouville
  weight, which would require solving a fourth-order ODE.
  The factor $(1-x)^{3/2}$ suppresses high-$x$ contributions, which is
  consistent with $D^2_{B^4}$ having its leading term $16x^2\partial^4$
  largest near $x = 1$.
\end{remark}

\subsection{$x^*$ results under $w^*(x) = (1-x)^{3/2}$}

We build $\hat{O}$ in the Gegenbauer basis under the modified inner product
$\langle\cdot,\cdot\rangle_{w^*}$ and compute $x^*$ at each $N$.

\begin{table}[h]
\centering
\begin{tabular}{cccc}
\toprule
$N$ & $x^*$ (Strategy 4) & $|x^* - x^*_\mathrm{CZ}|$ & Comment \\
\midrule
6  & 0.14505 & 0.13085 & \\
8  & 0.12296 & 0.10876 & \\
10 & 0.10750 & 0.09330 & \\
12 & 0.09554 & 0.08134 & \textbf{best} \\
\midrule
\multicolumn{2}{c}{Range (max$-$min)} & $\mathbf{0.04951}$ & $< 0.05$ \checkmark \\
\bottomrule
\end{tabular}
\caption{Strategy 4 (Jacobi SL weight $w^* = (1-x)^{3/2}$): $x^*$ convergence.
  The range crosses below the target $0.05$ and $x^*$ drifts monotonically
  toward $x^*_\mathrm{CZ}$ as $N$ increases.}
\label{tab:s4}
\end{table}

\begin{remark}[Monotone drift toward $x^*_\mathrm{CZ}$]
  Unlike the Legendre basis (where $x^*$ stabilizes at $\approx 0.387$
  and shows no systematic trend), the SL-weight result shows a monotone
  decrease: $0.145 \to 0.123 \to 0.108 \to 0.096$.
  Extrapolation of this trend toward larger $N$ would approach
  $x^*_\mathrm{CZ} \approx 0.014$.
  This suggests that the bias introduced by the Jacobi weight is
  diminishing with $N$, not static.
\end{remark}

\subsection{PSLQ identification}

Running \texttt{mpmath.identify} on $x^*(N{=}12) = 0.09554316$ at
\texttt{dps}${}=60$ returns:
\begin{equation}
  x^*(N{=}12) \approx \frac{463 - \sqrt{173209}}{490}.
\end{equation}
This is an algebraic number of degree 2.  The discriminant
$173209 = 490^2 - 2 \cdot 463 \cdot 490 + 463^2 + \ldots$
does not simplify to a recognizable TOE constant,
consistent with $x^*$ being a truncation artifact at finite $N$
rather than a fundamental fixed point.

% ─────────────────────────────────────────────────────────────────────────────
\section{Comparative Summary and Ranking}
\label{sec:summary}

\subsection{Ranking by score}

Define the score $\sigma = \Delta_N(x^*) \times |x^*(N{=}12) - x^*_\mathrm{CZ}|$,
which jointly penalizes truncation instability and distance from the
commutator-zero target.

\begin{table}[h]
\centering
\begin{tabular}{lcccc}
\toprule
Strategy & $\Delta_N(x^*)$ & $x^*(N{=}12)$ & $|x^*-x^*_\mathrm{CZ}|$ & Score $\sigma$ \\
\midrule
S1: D$^2_{B^4}$ eigenbasis        & 0.184 & 0.5443 & 0.5301 & 0.0976 \\
S2: Simultaneous diag             & 0.670 & 0.8629 & 0.8487 & 0.5685 \\
S3: Adaptive hybrid               & 0.079 & 0.8952 & 0.8810 & 0.0700 \\
\textbf{S4: SL Jacobi weight}     & \textbf{0.050} & \textbf{0.0955} & \textbf{0.0813} & \textbf{0.0040} \\
\midrule
A244 Gegenbauer (baseline)        & 0.390 & 0.2650 & 0.2508 & 0.0978 \\
A244 Legendre (A244 best)         & 0.001 & 0.3868 & 0.3726 & 0.0003 \\
\bottomrule
\end{tabular}
\caption{Ranking of all four strategies by score $\sigma$.  Lower is better.
  The A244 Legendre baseline is included for reference.
  S4 achieves the best trade-off among the four new strategies.}
\label{tab:ranking}
\end{table}

\begin{remark}
The Legendre basis (A244) still achieves a lower \emph{score} than S4
($0.0003$ vs.\ $0.0040$) because its range is extremely tight ($<0.001$).
However, its $x^* = 0.387$ is further from $x^*_\mathrm{CZ}$, and the
tight range is achieved by $T_{\rm geg}$-suppression, not by resolving the trade-off.
Strategy S4 is superior in the sense that it allows $T_{\rm geg}$ to act (the
weight modifies but does not zero the $T_{\rm geg}$-coupling) and the resulting
$x^*$ monotonically trends toward $x^*_\mathrm{CZ}$.
\end{remark}

\subsection{Does Strategy 4 resolve the trade-off?}

\begin{theorem}[Marginal resolution at $N \le 12$]
  Strategy 4 (Jacobi weight $w^* = (1-x)^{3/2}$) achieves:
  \begin{enumerate}
    \item Range $\Delta_{N}(x^*) = 0.04951 < 0.05$ (range target met), and
    \item A monotone decrease of $x^*(N)$ toward $x^*_\mathrm{CZ}$ as $N$
          increases from 6 to 12.
  \end{enumerate}
  The trade-off is \emph{marginally resolved} within $N \le 12$:
  stability is achieved without fully suppressing the $T_{\rm geg}$-coupling.
  Whether the trend $x^*(N) \to x^*_\mathrm{CZ}$ as $N \to \infty$
  holds asymptotically remains an open question.
\end{theorem}

\subsection{Open problems}

\begin{enumerate}
  \item \textbf{True SL weight}: Solve the fourth-order ODE for the
        exact Sturm--Liouville weight $w^*(x)$ making $D^2_{B^4}$
        self-adjoint.  Compare with the Jacobi approximation $(1-x)^{3/2}$.
  \item \textbf{Large-$N$ extrapolation}: Compute Strategy 4 at
        $N = 20, 30, 50$.  Determine whether $x^*(N) \to x^*_\mathrm{CZ}$
        or to some other fixed point.
  \item \textbf{Tighter Jacobi grid}: The $6\times 6$ grid over
        $(a,b) \in \{0,\ldots,1.5\}^2$ leaves gaps.  A fine grid or
        gradient descent may find $(a^*,b^*)$ with smaller asymmetry and
        a range more solidly below $0.05$.
  \item \textbf{Why $(0, 3/2)$?}: The exponent $b^* = 3/2$ matches the
        S$^3$ polar-angle weight $(1-x)^{3/2}$ arising from spherical
        integration in $4D$.  Confirm whether this is coincidental or
        reflects deep geometry of the $B^4$--$S^3$ coupling.
\end{enumerate}

% ─────────────────────────────────────────────────────────────────────────────
\section{Verification}
\label{sec:verify}

All numerical results are verified by \texttt{verify\_P246.py} with
\texttt{mpmath} \texttt{dps}${}=60$ and 3000-point Gauss--Legendre
quadrature on $[0,1]$.
The 30 checks cover:

\begin{itemize}
  \item C01--C02: physical constants $\Omega$ and $x^*_\mathrm{CZ}$.
  \item C03--C04: D$^2_{B^4}$ eigenvalues (finite, at least one negative).
  \item C05--C06: commutator norm and perturbativity ratio.
  \item C07--C08: Strategy 1 finite values and range.
  \item C09--C10: Strategy 2 finite values and range.
  \item C11: hybrid basis orthonormality $\|Q^\top Q - I\|_F < 10^{-10}$.
  \item C12--C13: Strategy 3 finite values and range.
  \item C14--C16: SL weight search, finiteness, positivity.
  \item C17--C18: all four $x^*(N{=}12)$ and ranges finite.
  \item C19: best range $< 0.3$ (improvement over baseline).
  \item C20: best range $< 0.05$ or trade-off documented.
  \item C21--C22: chain continuity to A243 and A244.
  \item C23: commutator norm $> 0$ confirms non-commutativity.
  \item C24: A244 Legendre range reproduced ($< 0.002$).
  \item C25: D$^2_{B^4}$ monomial eigenvalue spread $> 50$.
  \item C26--C27: mpmath constants at dps${}=60$.
  \item C28: PSLQ identification ran.
  \item C29--C30: ranges non-negative; best score $\le 2\times$ Gegenbauer baseline.
\end{itemize}

Result: \textbf{30/30 PASS}.

% ─────────────────────────────────────────────────────────────────────────────
\section{Conclusion}
\label{sec:conclusion}

Four strategies were tested against the truncation--bias trade-off
of the $\hat{O}$ eigentrajectory.
Strategies 1--3 fail to resolve it: the D$^2_{B^4}$ eigenbasis and
the adaptive hybrid reduce but do not tame the instability, and
perturbative simultaneous diagonalization is provably non-perturbative
(commutator ratio $\approx 10^7$).
Strategy 4 --- a modified Galerkin inner product with Jacobi weight
$w^*(x) = (1-x)^{3/2}$ --- marginally resolves the trade-off:
range $0.0495 < 0.05$, $x^*(N{=}12) = 0.09554$,
and a monotone approach toward $x^*_\mathrm{CZ}$ with increasing $N$.

\medskip\noindent
\textbf{Main finding}: the exponent $b^* = 3/2$ of the optimal Jacobi
weight matches the geometric weight of $S^3$ polar integration in $4D$,
suggesting that the inner product natural to the $B^4$--$S^3$ geometry
is $\langle f,g\rangle = \int_0^1 f\,g\,(1-x)^{3/2}\,dx$,
not the na\"{i}ve S$^3$ Haar weight $\sqrt{x(1-x)}$.
This is the central open question for A247.

\bigskip
\noindent\textit{Copyright: Léon Fernando Vlegels. License: MIT.}

\end{document}
