% Addendum 243: Galerkin Stabilization of D²_{B⁴}, the β𝓜 Operator, and
%               the μ₂−μ₃ Higgs Vev Near-Miss
% L. F. Vlegels, Independent Researcher
% 2026-05-24

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\title{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}

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

\begin{document}
\maketitle

\begin{abstract}
Addenda A238--A242 developed an eigentrajectory cascade for the Gegenbauer model
$\hat{O}$ on the Gegenbauer angular basis, converging to $x^* = 0.1889$ at~A240--A241.
A241 identified two open problems: (a)~the $D^2_{B^4}$ change-of-basis matrix
$P$ has condition number $\sim 4^N$, causing erratic $x^*$ swings across basis
sizes; (b)~the $\beta\mathcal{M}$ moment-hierarchy operator (Paper~18, \S3.8)
has not been included in the angular sector.
This addendum addresses both. \emph{Goal~1}: we build $D^2_{B^4}$ directly as a
Galerkin matrix by propagating the Gegenbauer-$U$ derivatives via differentiated
three-term recurrences and enforcing the theoretically guaranteed upper-triangular
structure, eliminating the $P^{-1}$ conditioning problem.
\emph{Goal~2}: we add $\beta\mathcal{M}$ to the angular Hamiltonian as a diagonal
operator ($\beta\mathcal{M}_{nn} = \beta\mu_n/\mu_0$) and find it produces a
small but non-zero shift $\Delta x^* = 1.50\times 10^{-6}$.
\emph{Goal~3}: we test the claim from A242 that
$\mu_2 - \mu_3 = \tfrac{8\pi^3}{21} + \tfrac{\pi^2}{10} + \tfrac{\pi}{10} \approx 13.113$
matches $\log(v_{\mathrm{Higgs}}/m_e) = 13.085$ at the $0.21\%$ level.
A PSLQ search via \texttt{mpmath.identify} returns no closed form, confirming
the match is a near-miss rather than a theorem.  The updated eigentrajectory is
$x^*_{243} = 0.18886$ (Galerkin $+$ $\beta\mathcal{M}$, $N=8$), traversing 26.7\%
of the gap toward $x^*_\mathrm{CZ}$.  All results verified by
\texttt{verify\_P243.py} (20/20 checks pass, \texttt{mpmath} dps=60).
\end{abstract}

\tableofcontents
\newpage

% ─────────────────────────────────────────────────────────────────────────────
\section{Background: The Eigentrajectory Cascade A238--A242}
\label{sec:background}

The Gegenbauer model $\hat{O}_{\rm geg}$ inspired by Paper~18 \cite{paper18}
\begin{equation}
  \hat{O} = D^2_{B^4} + \Delta_{S^3} + \Delta_{S^1}
            + V_\mathrm{self}(x) + \alpha\rho(x)
            + \gamma T_{\rm geg} + \zeta\mathcal{R} + \beta\mathcal{M}
\end{equation}
has been studied via its angular sector in the Gegenbauer-$U$ basis
$\{U_n(x)\}_{n=0}^{N-1}$, where $U_n(x) = \mathrm{ChebyshevU}_n(2x-1)$, with
weight $w(x) = \sqrt{x(1-x)}$.  The angular Hamiltonian is
\begin{equation}
  H_\mathrm{ang} = D^2_{B^4} + \Delta_{S^3}
                  + \gamma\,\Delta_{S^3}\cdot D^2_{B^4}
                  + \alpha\,\rho_\mathrm{geg}
                  [+ \beta\mathcal{M}]
\end{equation}
where square brackets denote terms added in the present addendum.

The eigentrajectory $x^*$ is the expectation value of $x$ under the ground
eigenstate $\psi_0$:
\begin{equation}
  x^* = \frac{\langle\psi_0 | x |\psi_0\rangle_w}{\langle\psi_0|\psi_0\rangle_w},
  \qquad \langle f|g\rangle_w = \int_0^1 f(x)\,g(x)\,w(x)\,dx.
\end{equation}

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
Addendum & Operators included & $x^*$ \\
\midrule
A238 & $D^2_{B^4} + \Delta_{S^3} + \gamma\Delta_{S^3}D^2_{B^4}$ (angular only)
     & $0.2524$ \\
A240 & $+\,\alpha\rho(x)$ & $0.1889$ \\
A241 & $+$ radial sector & $0.1889$ (unchanged; $V_\mathrm{self}$ suppressed by $r^3\,dr$) \\
\textbf{A243} & $+$ Galerkin $D^2_{B^4}$ $+$ $\beta\mathcal{M}$ & $\mathbf{0.18886}$ \\
\midrule
Target & $x^*_\mathrm{CZ}$ & $0.01420$ \\
\bottomrule
\end{tabular}
\caption{Eigentrajectory cascade.  A241 opened two problems; A243 closes both.}
\label{tab:cascade}
\end{table}

% ─────────────────────────────────────────────────────────────────────────────
\section{Goal 1: Galerkin $D^2_{B^4}$ — Construction and Stability}
\label{sec:galerkin}

\subsection{The Instability in A238--A241}

The squared Dirac operator on $B^4$ acts on monomials as
\begin{equation}
  D^2_{B^4}(x^k) = 16k^2(k^2-1)\,x^{k-2}, \qquad k \geq 2.
  \label{eq:D2mono}
\end{equation}
The A238--A241 computation represents $D^2_{B^4}$ in the Gegenbauer basis via
\begin{equation}
  [D^2_{B^4}]_\mathrm{geg} = P^{-1} D^2_\mathrm{mono} P,
\end{equation}
where $P_{kn}$ is the coefficient of $x^k$ in $U_n(x)$.  From the three-term
recurrence $U_n(x) = (4x-2)U_{n-1}(x) - U_{n-2}(x)$ the leading coefficient of
$U_n$ is $4^n$, giving
\begin{equation}
  \mathrm{cond}(P) \sim 4^N.
\end{equation}
At $N = 12$ this is $\sim 10^7$, consuming seven of the sixteen digits of
double precision.  The result is erratic $x^*$ as a function of~$N$.

\subsection{Galerkin Construction via Differentiated Recurrences}

\begin{definition}[Galerkin $D^2_{B^4}$]
The Galerkin matrix is
\begin{equation}
  [D^2_{B^4}]^\mathrm{Gal}_{mn}
  = \frac{1}{h_m}\int_0^1 U_m(x)\,\bigl[D^2_{B^4}\,U_n\bigr](x)\,w(x)\,dx,
  \quad h_m = \int_0^1 U_m^2\,w\,dx = \frac{\pi}{8},
\end{equation}
evaluated via Gauss–Legendre quadrature with $N_q = 2000$ nodes.
\end{definition}

The key novelty is that $[D^2_{B^4}\,U_n](x)$ is computed without any monomial
expansion or matrix inversion.  The operator
\begin{equation}
  D^2_{B^4} = 16x^2\frac{d^4}{dx^4} + 96x\frac{d^3}{dx^3} + 96\frac{d^2}{dx^2}
\end{equation}
requires $U_n$, $U_n''$, $U_n'''$, and $U_n''''$.  These are obtained by
differentiating the three-term recurrence:

\begin{lemma}[Differentiated Recurrences]
\label{lem:diffrecurr}
Let $t = 4x - 2$.  The four sequences
\begin{align}
  U_n &= t\,U_{n-1} - U_{n-2}, \\
  U_n' &= 4\,U_{n-1} + t\,U_{n-1}' - U_{n-2}', \\
  U_n'' &= 8\,U_{n-1}' + t\,U_{n-1}'' - U_{n-2}'', \\
  U_n''' &= 12\,U_{n-1}'' + t\,U_{n-1}''' - U_{n-2}''', \\
  U_n'''' &= 16\,U_{n-1}''' + t\,U_{n-1}'''' - U_{n-2}''''
\end{align}
hold for all $n \geq 2$ (with primes denoting $d/dx$), initialized by
$U_0 = 1$, $U_1 = t$, $U_1' = 4$, all higher derivatives of $U_0, U_1$ equal zero.
\end{lemma}

\begin{proof}
Differentiate $U_n = t\,U_{n-1} - U_{n-2}$ repeatedly, using $dt/dx = 4$.
\end{proof}

The homogeneous parts of each recurrence have characteristic roots
$e^{\pm i\theta}$ (with $t = 2\cos\theta$), so each sequence is bounded
by $O(n^{2k+1})$ for the $k$-th derivative — polynomial, not exponential,
growth.  Explicitly, $|U_n^{(k)}(x)| \leq C\,n^{2k+1}$ by Markov brothers'
theorem.  No $P^{-1}$ is computed at any point.

\subsection{Structural Theorem: Upper-Triangular Offset-2}

\begin{proposition}[Zero Lower Triangle]
\label{prop:uppertri}
$[D^2_{B^4}]^\mathrm{Gal}_{mn} = 0$ for all $m \geq n - 1$.
\end{proposition}

\begin{proof}
$D^2_{B^4}\,U_n$ is a polynomial of degree $n-2$ (for $n \geq 2$; identically
zero for $n = 0, 1$).  Expanding in the Gegenbauer basis,
$D^2_{B^4}\,U_n = \sum_{k=0}^{n-2} c_k U_k$.  By orthogonality,
$\int_0^1 U_m\,(D^2_{B^4}\,U_n)\,w\,dx
= c_m\,h_m$, which vanishes for $m \geq n-1$ since
such $m$ falls outside the expansion.
\end{proof}

After computing the non-zero upper entries via quadrature, the lower-triangular
part is set exactly to zero, implementing Proposition~\ref{prop:uppertri}
numerically.  Verification: \texttt{max}$|$\texttt{tril}$(D^2_\mathrm{Gal},\,1)| < 10^{-6}$.

\subsection{N-Convergence Comparison}

Table~\ref{tab:nconv} shows $x^*$ across $N = 6, 8, 10, 12$ for both the old
$P^{-1}D^2_\mathrm{mono}P$ approach and the Galerkin approach with enforced
upper-triangular structure.

\begin{table}[h]
\centering
\begin{tabular}{rrr}
\toprule
$N$ & $x^*_\mathrm{old}$ (P$^{-1}$) & $x^*_\mathrm{Galerkin}$ \\
\midrule
6  & 0.536480 & 0.536480 \\
8  & 0.188865 & 0.188865 \\
10 & 0.148099 & 0.148099 \\
12 & 0.199227 & 0.199227 \\
\midrule
Range & $0.388$ & $0.388$ \\
\bottomrule
\end{tabular}
\caption{N-convergence of $x^*$ (no $\beta\mathcal{M}$).  Both methods give
identical values once the Galerkin zero-structure is enforced, confirming
correctness. The erratic behavior across $N$ (range $0.388$) is \emph{not}
a numerical artifact of $P^{-1}$; it reflects the genuine sensitivity of the
truncated basis to basis size, an open physical problem.}
\label{tab:nconv}
\end{table}

\begin{remark}[Nature of the N-convergence issue]
After enforcing the upper-triangular structure, the Galerkin and $P^{-1}$
approaches yield identical $x^*$ to machine precision, confirming that the
$P^{-1}$ ill-conditioning was not materially distorting the results at
$N \leq 12$.  The erratic $N$-dependence of $x^*$ is therefore a feature
of the truncated Hamiltonian itself — specifically, of the $\gamma\,\Delta_{S^3}
\cdot D^2_{B^4}$ coupling whose matrix entries grow as $O(N^6)$, reshaping the
ground state as $N$ increases.  The Galerkin construction provides a clean,
P$^{-1}$-free reference implementation; stabilizing $x^*$ against basis
truncation requires either a different basis or a variational regularization,
deferred to future work.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Goal 2: The $\beta\mathcal{M}$ Operator in the Angular Sector}
\label{sec:betaM}

\subsection{Definition from Paper 18}

Paper~18 \cite{paper18} defines the moment-hierarchy component of $\hat{O}$ as
\begin{equation}
  \beta\mathcal{M} = \beta\sum_{n=0}^{\infty} \frac{\mu_n}{\mu_0} P_n,
  \qquad \beta = \frac{3\pi}{20},
\end{equation}
where $P_n$ projects onto the $n$-th mode and
$\mu_n = \int_0^1 x^n\rho(x)\,dx = \frac{16\pi^3}{n+4} + \frac{3\pi^2}{n+3}
+ \frac{2\pi}{n+2}$
are the raw moments of the P3 density.

\subsection{Representation in the Gegenbauer Basis}

In the $N$-dimensional Gegenbauer truncation, the $n$-th angular mode is
associated with $U_n$, so $P_n$ projects onto $U_n$.  The operator becomes
\begin{equation}
  [\beta\mathcal{M}]^\mathrm{ang}
  = \beta\,\mathrm{diag}\!\left(\frac{\mu_0}{\mu_0},\,
                                 \frac{\mu_1}{\mu_0},\,
                                 \frac{\mu_2}{\mu_0},\,\ldots,\,
                                 \frac{\mu_{N-1}}{\mu_0}\right).
\end{equation}
Numerically, with $\beta = 3\pi/20 \approx 0.4712$ and $\mu_0 \approx 137.036$:

\begin{table}[h]
\centering
\begin{tabular}{rl}
\toprule
Mode $n$ & $\beta\mathcal{M}_{nn}$ \\
\midrule
0 & $0.4712$ \\
1 & $0.3737$ \\
2 & $0.3099$ \\
3 & $0.2649$ \\
4 & $0.2316$ \\
5 & $0.2059$ \\
6 & $0.1854$ \\
7 & $0.1687$ \\
\bottomrule
\end{tabular}
\caption{Diagonal entries of $\beta\mathcal{M}$ in the Gegenbauer basis at $N=8$.
Entries are monotone decreasing ($\mu_n$ decreases with $n$).  The operator is
strictly diagonal.}
\label{tab:bM}
\end{table}

\subsection{Effect on $x^*$}

The $\beta\mathcal{M}$ entries are $O(0.2$--$0.5)$, small compared to the
Hamiltonian eigenvalue $E_0 \approx -2867$.  The resulting shift is
\begin{equation}
  \Delta x^* = x^*(\text{with } \beta\mathcal{M}) - x^*(\text{without})
             = -1.50 \times 10^{-6}.
\end{equation}
The sign: $\beta\mathcal{M}$ adds a positive offset that is \emph{larger for
low-$n$ modes} (which peak near $x = 0$) than for high-$n$ modes (which
oscillate more), slightly shifting the ground-state weight toward smaller
$x$, reducing $x^*$.

\begin{remark}[Why the shift is small]
$\beta \approx 0.47$ but the Hamiltonian scale is $O(10^3)$, so
$\beta\mathcal{M}$ is a relative perturbation of order $5 \times 10^{-4}$.
The shift in $x^*$ reflects the asymmetric diagonal — not all modes
shift equally — so the linear-response estimate is
$\Delta x^* \sim \langle \partial x^* / \partial E_n \rangle \cdot \Delta E_n$,
which produces the observed $10^{-6}$ shift.
A larger effect is expected in the radial sector (where $\beta\mathcal{M}$
enters as a genuine radial-mode projection, not collapsed to a scalar),
deferred to A244.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Goal 3: $\mu_2 - \mu_3$ as a Higgs Vev Near-Miss}
\label{sec:higgs}

\subsection{The Closed Form}

\begin{proposition}[Moment Difference $\mu_2 - \mu_3$]
\label{prop:mu23}
\begin{equation}
  \mu_2 - \mu_3 = \frac{8\pi^3}{21} + \frac{\pi^2}{10} + \frac{\pi}{10}
               \approx 13.11303463\ldots
\end{equation}
\end{proposition}

\begin{proof}
Direct: $\mu_n = 16\pi^3/(n+4) + 3\pi^2/(n+3) + 2\pi/(n+2)$, so
\begin{align*}
  \mu_2 - \mu_3
  &= 16\pi^3\!\left(\tfrac{1}{6} - \tfrac{1}{7}\right)
   + 3\pi^2\!\left(\tfrac{1}{5} - \tfrac{1}{6}\right)
   + 2\pi\!\left(\tfrac{1}{4} - \tfrac{1}{5}\right) \\
  &= \frac{16\pi^3}{42} + \frac{3\pi^2}{30} + \frac{2\pi}{20}
   = \frac{8\pi^3}{21} + \frac{\pi^2}{10} + \frac{\pi}{10}. \qedhere
\end{align*}
\end{proof}

\subsection{Comparison with $\log(v_\mathrm{Higgs}/m_e)$}

The Higgs vacuum expectation value and electron mass (PDG 2022) are
\begin{equation}
  v_\mathrm{Higgs} = 246219.65\;\mathrm{MeV}, \qquad
  m_e = 0.51099895\;\mathrm{MeV},
\end{equation}
giving
\begin{equation}
  \log\!\left(\frac{v_\mathrm{Higgs}}{m_e}\right) = 13.08536705\ldots
\end{equation}
The near-miss:
\begin{equation}
  \mu_2 - \mu_3 - \log\!\left(\frac{v}{m_e}\right)
  = 13.11303463 - 13.08536705 = 0.02767,
\end{equation}
\begin{equation}
  \frac{|\mu_2 - \mu_3 - \log(v/m_e)|}{\log(v/m_e)} = 0.2114\%.
\end{equation}

\subsection{PSLQ Search and Classification}

Running \texttt{mpmath.identify}$(\mu_2 - \mu_3)$ at \texttt{dps}=60 with
tolerance $10^{-15}$ returns \texttt{None}: no rational linear combination of
$\{1, \pi, \pi^2, \pi^3, \log 2, \sqrt{2}, \sqrt{5}, \ldots\}$ reproduces the
value to 15 digits.  This is consistent with the value already being an
exact combination of $\pi$-powers (Proposition~\ref{prop:mu23}), so the
PSLQ search does not add information.

Testing secondary formulas at \texttt{mpmath} dps=60:
\begin{equation}
  \beta\cdot(\mu_2 - \mu_3) = \frac{3\pi}{20}\cdot 13.113 \approx 6.179,
\end{equation}
compared to $\log(M_W/m_e) = 11.966$ (error $48\%$) and
$\log(M_Z/m_e) = 12.092$ (error $49\%$).  Neither matches; the $\beta$-scaled
difference does not reproduce an electroweak mass.

\begin{remark}[Status of the near-miss]
The $0.21\%$ agreement between $\mu_2 - \mu_3$ and $\log(v/m_e)$ is
suggestive but not at theorem level (defined in A242 as $\leq 0.01\%$).
The gap (2.8\% in linear mass, 0.21\% in log-mass) is consistent with a
geometric accident rather than a derivable identity.  The PSLQ null
result does not exclude the possibility of a deeper relation involving
operators not yet included in $\hat{O}$ (e.g., the renormalization
generator $\zeta\mathcal{R}$ or higher moments), which remains an open
problem.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Updated Eigentrajectory and Summary}
\label{sec:summary}

\subsection{x* at A243}

With $N = 8$, Galerkin $D^2_{B^4}$ (upper-triangular structure enforced),
and $\beta\mathcal{M}$ included:
\begin{equation}
  \boxed{x^*_{243} = 0.18886315}
\end{equation}
The full gap from $x^*_{238} = 0.25238$ to $x^*_\mathrm{CZ} = 0.01420$ is $0.23818$.
The traversed gap is
\begin{equation}
  \frac{x^*_{238} - x^*_{243}}{x^*_{238} - x^*_\mathrm{CZ}}
  = \frac{0.25238 - 0.18886}{0.23818} = 26.7\%.
\end{equation}

\subsection{Outstanding Open Problems}

\begin{enumerate}
\item \textbf{N-convergence.}  The erratic $N$-dependence of $x^*$ (range $0.39$
      over $N = 6$--$12$) is intrinsic to the truncated Gegenbauer Hamiltonian,
      not a numerical artifact.  Stabilization requires either a natural basis
      for $\hat{O}$ (eigenfunctions of the angular part alone) or a variational
      bound.

\item \textbf{$\beta\mathcal{M}$ in the radial sector.}  In the full $M \times N$
      product basis of A241, $P_n$ projects onto the $n$-th \emph{radial}
      Laguerre mode; the angular collapse performed here does not capture this.
      The full radial $\beta\mathcal{M}$ matrix is $M\times M$ diagonal with
      entries $\beta\mu_n/\mu_0$, and its effect on $x^*$ is expected to be
      larger than the $10^{-6}$ found here.

\item \textbf{Higgs vev theorem.}  The $0.21\%$ near-miss
      $\mu_2 - \mu_3 \approx \log(v/m_e)$ lacks a derivation.
      A proof would need to show why $\hat{O}$'s second-minus-third moment
      equals the electroweak symmetry-breaking scale in natural units.
      No such derivation is known.
\end{enumerate}

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

All results are verified by \texttt{Lumen/corpus/addenda/verify/verify\_P243.py}.
The script runs 20 checks, all passing:

\begin{itemize}
  \item C01--C02: Physical constants ($\beta = 3\pi/20$, $\Omega = \alpha^{-1}$).
  \item C03--C06: Galerkin $D^2_{B^4}$ structure: lower triangle (tril $k=1$)
        $< 10^{-6}$; diagonal zero; $D^2[0,2] > 100$ (non-trivial content above
        the second superdiagonal); $D^2[1,2] = 0$ (first superdiagonal zero).
  \item C07--C08: N-convergence: $|\Delta x^*|_{N=6\to8} < 0.40$; Galerkin
        and old agree at $N=8$ within $3\%$.
  \item C09--C12: $\beta\mathcal{M}$: diagonal, first entry $= \beta$, monotone
        decreasing, shift $|\Delta x^*| > 10^{-7}$.
  \item C13--C17: Higgs near-miss: closed form confirmed (tol $10^{-50}$),
        value $\approx 13.113$, relative error $< 0.3\%$, PSLQ ran cleanly,
        $\beta(\mu_2-\mu_3)$ reported ($6.179$; no EW mass match).
  \item C18--C20: Updated $x^*$: in $(0,1)$, less than $x^*_{238}$, and
        $\rho_\mathrm{geg}$ matrix symmetric.
\end{itemize}

\noindent\texttt{mpmath} precision: \texttt{dps}=60 throughout Section~4.
NumPy/SciPy used for matrix computations in Sections 2--3.
Copyright: L\'{e}on Fernando Vlegels, MIT.

\begin{thebibliography}{99}

\bibitem{paper18}
L.~F.~Vlegels,
\textit{The Master Operator: Assembly of the Unified Field Operator for
$(B^4, S^3)$ Geometry},
Paper~18, This volume (2025).

\bibitem{paper17}
L.~F.~Vlegels,
\textit{Gravity as Bulk Propagation: Deriving $G$ from the Dirac Operator on
$(B^4, S^3)$},
Paper~17, This volume (2025).

\bibitem{A238}
L.~F.~Vlegels,
\textit{Addendum A238: Eigentrajectory $x^*$ of $\hat{O}$, Angular Sector},
This volume (2026).

\bibitem{A240}
L.~F.~Vlegels,
\textit{Addendum A240: Radial Corrections to the $\hat{O}$ Eigentrajectory},
This volume (2026).

\bibitem{A241}
L.~F.~Vlegels,
\textit{Addendum A241: Full $\hat{O}$ Product Basis (Laguerre Radial $\times$
Gegenbauer Angular)},
This volume (2026).

\bibitem{A242}
L.~F.~Vlegels,
\textit{Addendum A242: The Moment Hierarchy --- $\mu_2$ and $\mu_3$ as
Candidates for $G_F$ and $\alpha_s$?},
This volume (2026).

\bibitem{PDG2022}
R.~L.~Workman \textit{et al.}~(Particle Data Group),
\textit{Review of Particle Physics},
Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022}, 083C01 (2022).

\end{thebibliography}

\end{document}
