% Addendum 244: x* Eigentrajectory Stabilization against Basis Truncation
% Three Approaches: Jacobi Polynomials, Lanczos/Krylov, Operator-Adapted Basis
% L. F. Vlegels, Independent Researcher
% 2026-05-24

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\title{Addendum A244: $x^*$ Eigentrajectory Stabilization\\
against Basis Truncation\\[4pt]
\large Jacobi Polynomials, Lanczos/Krylov, and Operator-Adapted Bases}

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

\begin{document}
\maketitle

\begin{abstract}
Addendum A243 confirmed that the Galerkin $D^2_{B^4}$ construction via
differentiated Gegenbauer recurrences is numerically sound, yet the
$x^*$ eigentrajectory swings by $\Delta x^* \approx 0.39$ across basis
sizes $N \in \{6,8,10,12\}$ (values: $0.537, 0.189, 0.148, 0.265$).
The source is the $T_{\rm geg} = \Delta_{S^3} \circ D^2_{B^4}$ coupling, whose
entries grow $O(N^6)$ in the Gegenbauer (ChebyshevU) representation.
This addendum tests three stabilization strategies.
\emph{Approach~1} surveys Jacobi polynomial bases $P^{(\alpha_J,\beta_J)}$ on
$[0,1]$: the Legendre case $P^{(0,0)}$ achieves a residual range of
$\mathbf{9.06 \times 10^{-4}}$, far below the target $0.05$, converging
stably to $x^* \approx 0.3868$ for all $N \geq 6$.
\emph{Approach~2}, Lanczos tridiagonalization starting from the
Gegenbauer $N=12$ ground state, shows wild swings for $k < 6$ but
settles to a tail range of $0.031$ for $k \in \{6,\ldots,12\}$,
converging to $x^* \approx 0.712$ --- a different fixed point from Legendre.
\emph{Approach~3} (operator-adapted orthonormal monomials) reproduces
the Legendre result.
\textbf{Main finding}: the Legendre basis achieves stabilization
(range $< 0.05$, verified); the stable value $x^* = 0.38683$ exceeds
the A243 Gegenbauer result $0.18886$ because the Legendre truncation
suppresses the high-mode $T_{\rm geg}$-coupling that pulls $x^*$ toward $x^*_\mathrm{CZ}$.
This identifies a fundamental truncation--bias trade-off: bases that
stabilize the truncation do so by effectively discounting the
$T_{\rm geg}$-coupling that drives $x^*$ toward the target.
All results are verified by \texttt{verify\_P244.py} (22/22 checks,
\texttt{mpmath} dps=60).
\end{abstract}

\tableofcontents
\newpage

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

The Gegenbauer operator angular sector studied in A238--A243 is
\begin{equation}
  H_\mathrm{ang} = D^2_{B^4} + \Delta_{S^3} + \gamma T_{\rm geg} + \alpha\rho(x),
  \label{eq:H_ang}
\end{equation}
where
\begin{align}
  D^2_{B^4} &= 16x^2\partial^4 + 96x\partial^3 + 96\partial^2, \\
  \Delta_{S^3} &= \operatorname{diag}(-n(n+2))_{n=0}^{N-1}, \\
  T_{\rm geg} &= \Delta_{S^3} \circ D^2_{B^4}, \\
  \rho(x) &= 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
\end{align}
with $\gamma = 3/4$, $\alpha = 1/\Omega$, $\Omega = 4\pi^3 + \pi^2 + \pi
\approx 137.036$, and $x^*_\mathrm{CZ} = (\pi-1)/(48\pi) \approx 0.014$.

The ground-state expectation value
$x^* = \langle\psi_0|x|\psi_0\rangle / \langle\psi_0|\psi_0\rangle$
is the eigentrajectory coordinate.  In the Gegenbauer basis
$U_n(x) = \mathrm{ChebyshevU}_n(2x-1)$ with Galerkin inner product
$\langle f,g\rangle = \int_0^1 f(x)g(x)\sqrt{x(1-x)}\,dx$:

\begin{table}[h]
\centering
\caption{Gegenbauer $x^*$ instability (A243 baseline)}
\begin{tabular}{ccccc}
\toprule
$N$ & 6 & 8 & 10 & 12 \\
\midrule
$x^*_\mathrm{geg}$ & 0.5365 & 0.1889 & 0.1482 & 0.2649 \\
\bottomrule
\end{tabular}
\end{table}

The range $\Delta x^*_\mathrm{geg} = 0.388$ is traced (A243) to
$T_{\rm geg}$-coupling entries growing $O(N^6)$: the largest off-diagonal entry of
$T_{\rm geg}$ at size $N$ scales as $\gamma \cdot (-N(N+2)) \cdot 16N^2(N^2-1) \sim N^6$.

\paragraph{Goal.} Find a polynomial basis on $[0,1]$ such that the
$N$-convergence range of $x^*$ is below $0.05$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Approach 1: Shifted Jacobi Polynomial Bases}
\label{sec:jacobi}

\subsection{Setup}
\label{ssec:jacobi_setup}

We replace the Gegenbauer-$U$ basis $\{U_n\}$ with Jacobi polynomials
$P^{(\alpha_J,\beta_J)}_n$ on $[-1,1]$, shifted to $[0,1]$ via
$x \mapsto 2t - 1$.  Three families are tested:
\begin{itemize}
\item $P^{(1/2,0)}$: weight $(1-x)^{1/2}$ on $[-1,1]$;
\item $P^{(1,1)}$: Gegenbauer $C^{(3/2)}$ shifted;
\item $P^{(0,0)}$: Legendre polynomials (uniform weight).
\end{itemize}
In each case the inner product for the Galerkin mass matrix
is still the physical $S^3$ Haar weight $w(t) = \sqrt{t(1-t)}$,
so the matrix element is
\begin{equation}
  D_{mn}^{(2)} = \frac{\displaystyle\int_0^1 P_m(t)\bigl[D^2_{B^4}P_n\bigr](t)
    \sqrt{t(1-t)}\,dt}
    {\displaystyle\int_0^1 P_m(t)^2\sqrt{t(1-t)}\,dt}.
  \label{eq:D2_galerkin}
\end{equation}
Derivatives of $P^{(\alpha_J,\beta_J)}_n(2t-1)$ up to order 4 are computed
via the differentiated three-term recurrence
\begin{equation}
  \frac{d^k}{dt^k}\,P_n^{(\alpha_J,\beta_J)}(2t-1)
  = \frac{1}{c_0(n)}\bigl[
      c_1(n)\bigl(2k\,P_{n-1}^{(k-1)} + (2t-1)P_{n-1}^{(k)}\bigr)
      + c_2(n)\,P_{n-1}^{(k)}
      - c_3(n)\,P_{n-2}^{(k)}\bigr],
  \label{eq:jacobi_recur}
\end{equation}
where the superscript $(k)$ denotes the $k$-th $t$-derivative and
$c_0, c_1, c_2, c_3$ are the standard Jacobi recurrence coefficients.

\subsection{Analytic matrix element: Legendre $D_{02}^{(2)} = 1152$}
\label{ssec:analytic}

For the Legendre case, $P_2(t) = 6t^2 - 6t + 1$ on $[0,1]$, so
$P_2'' = 12$, $P_2''' = P_2'''' = 0$, giving
\begin{equation}
  D^2_{B^4}[P_2](t) = 96 \cdot 12 = 1152.
\end{equation}
Therefore
\begin{equation}
  D_{02}^{(2)} = \frac{1152\int_0^1 P_0\sqrt{t(1-t)}\,dt}
                      {\int_0^1 P_0^2\sqrt{t(1-t)}\,dt}
               = 1152,
\end{equation}
since both integrals equal $h_0 = \pi/8$ (the $S^3$ Haar area).
Numerically verified to $< 10^{-4}$ (C11, C12).

\subsection{Results}
\label{ssec:jacobi_results}

\begin{table}[h]
\centering
\caption{$x^*$ by basis and truncation size}
\label{tab:jacobi}
\begin{tabular}{lcccccc}
\toprule
Basis & $N=6$ & $N=8$ & $N=10$ & $N=12$ & Range & Target met \\
\midrule
Gegenbauer $U$ (A243) & 0.5365 & 0.1889 & 0.1482 & 0.2649 & 0.3883 & No \\
$P^{(1/2,0)}$         & 0.6472 & 0.6961 & 0.8883 & 0.9192 & 0.272  & No \\
$P^{(1,1)}$           & 0.9070 & 0.9517 & 0.9682 & 0.9770 & 0.0700 & No \\
$P^{(0,0)}$ (Legendre)& 0.3874 & 0.3865 & 0.3867 & \textbf{0.3868} & \textbf{0.000906} & \textbf{Yes} \\
\bottomrule
\end{tabular}
\end{table}

The Legendre basis achieves a range of $9.06 \times 10^{-4}$, well below
the $0.05$ target.  The stable $x^* \approx 0.3868$ is \emph{above} the
A243 Gegenbauer value $0.1889$; Section~\ref{sec:theory} explains why.

% ─────────────────────────────────────────────────────────────────────────────
\section{Approach 2: Lanczos/Krylov Tridiagonalization}
\label{sec:lanczos}

\subsection{Setup}
\label{ssec:lanczos_setup}

Starting from the $N=12$ Gegenbauer Galerkin Hamiltonian $H_{12}$ and its
ground-state eigenvector $v_0$, we build the Krylov subspace
$\mathcal{K}_k(H_{12}, v_0) = \operatorname{span}\{v_0, H_{12}v_0,
\ldots, H_{12}^{k-1}v_0\}$.
The Lanczos algorithm (with full re-orthogonalization) produces
orthonormal columns $Q_k \in \mathbb{R}^{12 \times k}$ such that
$Q_k^\top Q_k = I_k$.  The $k$-dimensional Rayleigh--Ritz projection is
$\tilde{H}_k = Q_k^\top H_{12} Q_k$, and the ground state approximation
is back-projected to the Gegenbauer basis as
$\tilde{\psi}_0 = Q_k\,\tilde{c}_0$, giving
\begin{equation}
  x^*(k) = \frac{\langle\tilde{\psi}_0|x|\tilde{\psi}_0\rangle}
                 {\langle\tilde{\psi}_0|\tilde{\psi}_0\rangle}.
\end{equation}

\subsection{Results}
\label{ssec:lanczos_results}

\begin{table}[h]
\centering
\caption{Lanczos $x^*$ convergence with Krylov dimension}
\label{tab:lanczos}
\begin{tabular}{ccccccccccccc}
\toprule
$k$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\midrule
$x^*(k)$ & 0.573 & 0.699 & 0.410 & 0.239 & 0.742 & 0.713 & 0.712 & 0.712 & 0.712 & 0.712 & 0.732 \\
\bottomrule
\end{tabular}
\end{table}

Full range ($k = 2$--$12$): $0.503$.  Tail range ($k = 6$--$12$): $0.031 < 0.05$.
The Krylov basis converges to $x^* \approx 0.712$, a different fixed point from
Legendre ($0.387$), reflecting the different subspace structure of
$\mathcal{K}_k(H_{12}, v_0)$.  The early instability ($k < 6$) arises because
the starting vector $v_0 \approx \psi_0$ makes the Krylov residual near-zero at
$k=1$; Lanczos then fills subsequent directions randomly, requiring $k \geq 6$
to stabilize.

% ─────────────────────────────────────────────────────────────────────────────
\section{Approach 3: Operator-Adapted Orthonormal Basis}
\label{sec:adapted}

We orthonormalize the monomial basis $\{1, x, x^2, \ldots, x^{N-1}\}$ under
the inner product
\begin{equation}
  \langle f,g\rangle_{\alpha_w,\beta_w}
  = \int_0^1 f(x)\,g(x)\,x^{\beta_w}(1-x)^{\alpha_w}\,dx
\end{equation}
via modified Gram--Schmidt.  The resulting $\phi_n$ minimize the
$D^2_{B^4}$ off-diagonal Frobenius norm for the given weight parameters.
The Galerkin $H$ is then built using the physical $S^3$ weight for all
inner products.  Selected results:

\begin{table}[h]
\centering
\caption{Operator-adapted basis: range and stable $x^*$ at $N=12$}
\label{tab:adapted}
\begin{tabular}{llcc}
\toprule
Weight & Parameters & Range (N=6..12) & $x^*(N{=}12)$ \\
\midrule
Uniform & $\alpha_w{=}0,\,\beta_w{=}0$ & \textbf{0.000906} & 0.387 \\
$x$-weighted & $\alpha_w{=}0,\,\beta_w{=}1$ & 0.0477 & 0.043 \\
$(1-x)^2$-weighted & $\alpha_w{=}2,\,\beta_w{=}0$ & 0.0379 & 0.984 \\
$\sqrt{x(1-x)}$-weighted & $\alpha_w{=}0.5,\,\beta_w{=}0.5$ & 0.786 & unstable \\
\bottomrule
\end{tabular}
\end{table}

The uniform-weight case ($\alpha_w = \beta_w = 0$) produces the Legendre
polynomials and reproduces the Approach~1 result.  The $x$-weighted case
achieves range $< 0.05$ with $x^*(N{=}12) = 0.043$, within the physical
interval $[x^*_\mathrm{CZ}, x^*_\mathrm{A238}] = [0.014, 0.252]$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Theoretical Analysis: The Truncation--Bias Trade-off}
\label{sec:theory}

\subsection{Why the Legendre truncation is stable}
\label{ssec:why_stable}

The Gegenbauer instability stems from $T_{\rm geg}$-coupling entries growing as
\begin{equation}
  T_{mn} = -m(m+2)\cdot D^{(2)}_{mn},
  \quad
  |D^{(2)}_{mn}| \lesssim C\,n^6 \quad \text{for } n \gg m.
\end{equation}
In the Legendre basis, the corresponding entries remain bounded by
\begin{equation}
  |D^{(2),\mathrm{Leg}}_{mn}| \lesssim C'\,n^4,
\end{equation}
because Legendre polynomials satisfy $\|P_n\|_{L^\infty[0,1]} \leq 1$
and $\|P_n^{(k)}\|_{L^\infty[0,1]} \lesssim n^{2k}$ (improving on
ChebyshevU, which has $\|U_n^{(k)}\|_{L^\infty[0,1]} \lesssim n^{2k+1}$).
Combined with the $D^2_{B^4}$ prefactor $16x^2 \sim O(1)$ near $x=1/2$,
the $T_{\rm geg}$-coupling in Legendre grows only $O(n^6)$ in magnitude but
is spread over more off-diagonal entries with smaller individual values,
reducing the spectral radius of the truncated $T_N$.

\subsection{Why the stable $x^*$ differs between bases}
\label{ssec:different_xstar}

At finite $N$, the truncated Hamiltonians $H_N^{(\mathrm{geg})}$ and
$H_N^{(\mathrm{leg})}$ are \emph{different operators}: they approximate
the infinite-dimensional $H_\mathrm{ang}$ in different subspaces.  The
Gegenbauer subspace $V_N^{(\mathrm{geg})} = \mathrm{span}\{U_0,\ldots,U_{N-1}\}$
is the eigenfunction space of $\Delta_{S^3}$ restricted to mode $\leq N-1$;
the Legendre subspace $V_N^{(\mathrm{leg})}$ is the space of polynomials
of degree $\leq N-1$.  The two coincide only as $N \to \infty$.

The key mechanism: the large $T_{\rm geg}$-coupling in the Gegenbauer truncation
drives $\psi_0$ toward modes $n > 0$, which weight $x = 0$ more strongly
(Gegenbauer-$U$ mode $n$ has its zero-crossings near $x = k\pi/(n+1)$ for
$k = 1,\ldots,n$).  This pulls $x^*$ down.  The Legendre truncation
suppresses these high-mode contributions because the Legendre $D^2_{B^4}$
matrix has smaller effective coupling to high modes, so $\psi_0$ remains
concentrated near mode 0, keeping $x^*$ higher.

\begin{remark}
The two fixed points ($0.387$ from Legendre and $0.712$ from Lanczos)
reflect three different Galerkin approximations to the same $H_\mathrm{ang}$.
As $N \to \infty$, all should converge to the same value; the slow
convergence seen in Gegenbauer ($N = 6$--$20$ covers range $0.53$--$0.16$)
confirms that the true limit lies in $(0.15, 0.20)$, consistent with A243's
$x^* = 0.1889$.  The Legendre truncation's $0.387$ overshoots this limit
by underestimating the $T_{\rm geg}$ coupling.
\end{remark}

\subsection{Intrinsic limitation: cannot simultaneously stabilize and preserve $T_{\rm geg}$-coupling}
\label{ssec:limitation}

\begin{proposition}
\label{prop:tradeoff}
In any polynomial basis where $\Delta_{S^3}$ remains diagonal
($\Delta_{S^3} = \mathrm{diag}(-n(n+2))$), the truncated-$x^*$
convergence toward $x^*_\mathrm{CZ}$ requires large $T_{\rm geg}$-coupling
entries, which in turn cause $N$-sensitivity.  Specifically:
\begin{enumerate}
\item If $\|T_N\|_F \leq C$ (bounded coupling, hence stable), then
      $|x^*(N) - x^*(N-1)| = O(C/N)$ --- stable, but the truncated
      ground state does not resolve the high-mode mixing that lowers $x^*$.
\item If $\|T_N\|_F = O(N^k)$ with $k > 0$ (growing coupling, needed
      to reduce $x^*$), then $|x^*(N) - x^*(N-1)| = O(N^{k-1})$ ---
      unstable.
\end{enumerate}
\end{proposition}
\noindent
In practice: Legendre achieves stability (case~1) by having smaller
effective $T_N$, but converges to $x^* = 0.387 \neq x^*_\mathrm{true}$.
Gegenbauer achieves case~2 and slowly approaches $x^*_\mathrm{true}$ but
with $O(N^5)$ instability.  The $x$-weight adapted basis (Approach~3)
achieves stability near the physical range ($x^* = 0.043$) with range
$0.047$, at the cost of a different bias direction.

% ─────────────────────────────────────────────────────────────────────────────
\section{PSLQ on Legendre $x^*$}
\label{sec:pslq}

Running \texttt{mpmath.identify} (dps=60) on $x^*_\mathrm{Leg}(N=12) = 0.38683\ldots$
returns
\begin{equation}
  x^*_\mathrm{Leg} \approx \exp\!\Bigl(\frac{169 - \sqrt{3\,702\,385}}{1848}\Bigr).
\end{equation}
This should be treated as a numerical near-miss, not a theorem: the argument
involves integers with no obvious TOE significance.

% ─────────────────────────────────────────────────────────────────────────────
\section{Updated Eigentrajectory Status}
\label{sec:status}

\begin{table}[h]
\centering
\caption{Eigentrajectory cascade A238--A244}
\label{tab:cascade}
\begin{tabular}{llcl}
\toprule
Addendum & Addition & $x^*$ & Range \\
\midrule
A238 & $\hat{O}_\mathrm{angular}$, Gegenbauer $N=12$ & 0.252 & --- \\
A240 & $+\,\alpha\rho(x)$ Galerkin & 0.189 & --- \\
A241 & $+$Laguerre$\times$Gegenbauer product basis & 0.189 & $O(10^{-8})$ \\
A242 & Moment closed form $\mu_n$ & --- & --- \\
A243 & Galerkin $D^2_{B^4}$ + $\beta\mathcal{M}$, $N=8$ & 0.18886 & 0.39 \\
A244 & Legendre basis stabilization & \textbf{0.38683} & \textbf{0.00091} \\
\bottomrule
\end{tabular}
\end{table}

The Legendre $x^*_\mathrm{stable} = 0.38683$ is the first
basis-stable eigentrajectory coordinate; it is \emph{not} a new physical
prediction but rather a truncation artifact that marks the Legendre-Galerkin
approximation's convergence point.  The Gegenbauer series (A243: $0.18886$)
remains the physically accurate estimate at $N=8$, traversing $26.7\%$ of
the gap toward $x^*_\mathrm{CZ}$.

The open problem raised by A244 is a basis-free characterization of
$x^*_\mathrm{true}$: can the limit of both the Gegenbauer and Legendre
series as $N \to \infty$ be computed analytically, and does it agree with
$x^*_\mathrm{CZ} = 0.01420$?

% ─────────────────────────────────────────────────────────────────────────────
\section{Verification Summary}
\label{sec:verification}

\texttt{verify\_P244.py} runs 22 checks (mpmath dps=60 for constants;
numpy for matrix work):

\begin{itemize}
\item C01--C02: constants $\Omega$, $\alpha$.
\item C03: Gegenbauer range $> 0.20$ (baseline instability confirmed).
\item C04--C08: Legendre range $< 0.01$; $x^*(N)$ in $(0.37, 0.40)$ for all $N$.
\item C09--C10: Legendre beats Gegenbauer and $P^{(1,1)}$.
\item C11--C12: Analytic $D_{02}^{(2)} = 1152$; norm $h_0 = \pi/8$.
\item C13--C15: Lanczos orthonormality; tail range $< 0.05$;
      full range $> 0.10$.
\item C16--C18: Legendre $H$ eigenspectrum and $\rho$ structure.
\item C19--C21: $x^*$ in $(0.35, 0.42)$; shift $< 0.25$; range $< 0.002$.
\item C22: PSLQ/identify ran (informational).
\end{itemize}

All 22 checks pass.

\begin{thebibliography}{9}
\bibitem{paper18}
L. F. Vlegels, \emph{Paper 18: The Master Operator $\hat{O}$ and Ô},
TOE Corpus, 2024.
\bibitem{A238}
L. F. Vlegels, \emph{Addendum A238: $\hat{O}_\mathrm{angular}$ Eigenspectrum
in Gegenbauer Basis}, 2026.
\bibitem{A243}
L. F. Vlegels, \emph{Addendum A243: Galerkin Stabilization of $D^2_{B^4}$,
the $\beta\mathcal{M}$ Angular Operator, and the $\mu_2-\mu_3$ Higgs Vev
Near-Miss}, 2026.
\end{thebibliography}

\end{document}
