% Addendum 242: Moment Hierarchy — mu_2 and mu_3 as Candidates for G_F and alpha_s
% L. F. Vlegels, Independent Researcher
% 2026-05-24

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\title{Addendum A242: The Moment Hierarchy\\
Deriving $\alpha_s$ and $G_F$ from $\mu_2$ and $\mu_3$?\\
\large A Systematic Test with Open Verdict}

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

\begin{document}
\maketitle

\begin{abstract}
Paper 17 \cite{paper17} derives Newton's constant $G$ from the first moment $\mu_1$ of the
P10/P17 density $\rho(x)=16\pi^3 x^3+3\pi^2 x^2+2\pi x$ via the spectral formula
$\log(M_{\mathrm{Pl}}/m_e) = \frac{3\pi}{20}\cdot\frac{\mu_1}{1-\mu_1\alpha^2}$,
achieving $0.004\%$ agreement. Motivated by the conjecture (A239, open item) that
successive moments $\mu_n$ encode the coupling hierarchy
EM $\to$ gravity $\to$ weak $\to$ strong, this addendum tests
whether $\mu_2$ and $\mu_3$ similarly determine $G_F$ (Fermi constant)
and $\alpha_s$ (strong coupling). We compute $\mu_0$--$\mu_7$ in closed form,
apply the P17 formula and systematically search over 40 formula variants.
The verdict is \textbf{open}: the direct P17 formula applied to $\mu_2$ gives
$f(\mu_2) = 42.70 \approx 1/\alpha_{\mathrm{unified}} \approx 43$ within $0.70\%$
(GUT-scale hint, not a theorem), while the bare moment difference
$\mu_2 - \mu_3 = \frac{8\pi^3}{21}+\frac{\pi^2}{10}+\frac{\pi}{10} \approx \log(v/m_e)$
where $v = (\sqrt{2}\,G_F)^{-1/2}$ is the Higgs vacuum expectation value,
with $0.21\%$ agreement in log-space ($2.8\%$ in mass space).
No formula reproduces $G_F$ or $\alpha_s(M_Z)$ at the $0.01\%$ level required for
a theorem claim. The moment hierarchy conjecture remains open.
All computations verified at \texttt{mpmath} dps=60 (25/25 checks pass).
\end{abstract}

\tableofcontents
\newpage

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

The P10/P17 density
\begin{equation}
  \rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x, \qquad x \in [0,1]
\end{equation}
encodes the geometry of $(B^4,S^3)$. Its raw moments
$\mu_n = \int_0^1 x^n\rho(x)\,dx$ satisfy:
\begin{itemize}
  \item $\mu_0 = 4\pi^3+\pi^2+\pi = \alpha^{-1} \approx 137.036$  (electromagnetic coupling, P3).
  \item $\log(M_{\mathrm{Pl}}/m_e) = \frac{3\pi}{20}\cdot\frac{\mu_1}{1-\mu_1\alpha^2}
        \approx 51.530$  (Planck mass / electron mass, P17, $0.004\%$).
\end{itemize}
P17 itself poses the open question: \emph{how do $\alpha_s$ and $G_F$ fit into this
moment hierarchy?} \cite{paper17}  P10 conjectures $\alpha_s$ may involve the factor
$3/4 = \dim(S^3)/\dim(B^4)$ (3 colours, 4D bulk) \cite{paper10}.
Addendum A239 identifies this as the highest-tractability open item.

The \emph{moment hierarchy conjecture}, as tested here, is the structured claim:
\begin{conjecture}[Moment Hierarchy]
Each fundamental coupling constant corresponds to a moment or simple moment
combination:
$\mu_0 \to \alpha^{-1}$ (EM),
$\mu_1 \to G$ (gravity),
$\mu_2 \to G_F$ or $M_W$ (weak),
$\mu_3 \to \alpha_s$ (strong).
\end{conjecture}

This addendum executes the test at $0.01\%$ precision.

% ─────────────────────────────────────────────────────────────────────────────
\section{Moments: Closed Forms and Numerical Values}
\label{sec:moments}

\begin{proposition}[General Closed Form]
\label{prop:moments}
The raw $n$-th moment of $\rho(x) = 16\pi^3 x^3+3\pi^2 x^2+2\pi x$ is
\begin{equation}
  \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}.
  \label{eq:mu_n_cf}
\end{equation}
\end{proposition}

\begin{proof}
Direct integration: $\int_0^1 x^n \cdot cx^k\,dx = c/(n+k+1)$.
\end{proof}

Table~\ref{tab:moments} lists $\mu_0$--$\mu_7$ to 16 significant figures,
computed at \texttt{mpmath} dps=60.

\begin{table}[h]
\centering
\begin{tabular}{rll}
\toprule
$n$ & $\mu_n$ (16 s.f.) & Closed form \\
\midrule
0 & $137.0363037758784$ & $4\pi^3+\pi^2+\pi = \alpha^{-1}$ \\
1 & $108.7166837801696$ & $\tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}$ \\
2 & $90.17596344824803$ & $\tfrac{16\pi^3}{6}+\tfrac{3\pi^2}{5}+\tfrac{2\pi}{4}$ \\
3 & $77.06292881695161$ & $\tfrac{16\pi^3}{7}+\tfrac{3\pi^2}{6}+\tfrac{2\pi}{5}$ \\
4 & $67.28958136940596$ & $\tfrac{16\pi^3}{8}+\tfrac{3\pi^2}{7}+\tfrac{2\pi}{6}$ \\
5 & $59.72096920530051$ & $\tfrac{16\pi^3}{9}+\tfrac{3\pi^2}{8}+\tfrac{2\pi}{7}$ \\
6 & $53.68530898557361$ & $\tfrac{16\pi^3}{10}+\tfrac{3\pi^2}{9}+\tfrac{2\pi}{8}$ \\
7 & $48.75905182883337$ & $\tfrac{16\pi^3}{11}+\tfrac{3\pi^2}{10}+\tfrac{2\pi}{9}$ \\
\bottomrule
\end{tabular}
\caption{Raw moments $\mu_n = \int_0^1 x^n\rho(x)\,dx$, with closed forms.}
\label{tab:moments}
\end{table}

\begin{remark}[Normalization]
\label{rem:normalization}
P17 uses the \emph{raw} (unnormalized) moments throughout.  The normalized
moments $\hat\mu_n = \mu_n/\mu_0$ are: $\hat\mu_1 \approx 0.7933$,
$\hat\mu_2 \approx 0.6580$, $\hat\mu_3 \approx 0.5624$.  The label
``$\mu_1 \approx 108.717$'' in P17 is the raw first moment, not a normalized
one. The task brief considered the possibility that $\mu_1/\mu_0 = 1$; this is
false (the raw moment $\mu_1 \neq \mu_0$).
\end{remark}

\begin{proposition}[Moment Differences in Closed Form]
\label{prop:diffs}
\begin{align}
  \mu_0 - \mu_1 &= \frac{4\pi^3}{5}+\frac{\pi^2}{4}+\frac{\pi}{3} \approx 28.320, \\
  \mu_1 - \mu_2 &= \frac{8\pi^3}{15}+\frac{7\pi^2}{20}+\frac{\pi}{6} \approx 18.541, \\
  \mu_2 - \mu_3 &= \frac{8\pi^3}{21}+\frac{\pi^2}{10}+\frac{\pi}{10} \approx 13.113, \\
  \mu_3 - \mu_4 &= \frac{2\pi^3}{7}+\frac{3\pi^2}{56}+\frac{\pi}{10} \approx 9.773, \\
  \mu_4 - \mu_5 &= \frac{2\pi^3}{9}+\frac{3\pi^2}{56}+\frac{\pi}{21} \approx 7.569.
\end{align}
\end{proposition}

% ─────────────────────────────────────────────────────────────────────────────
\section{P17 Formula Review}
\label{sec:p17review}

The P17 spectral formula is:
\begin{equation}
  \log\!\left(\frac{M_{\mathrm{Pl}}}{m_e}\right)
  = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2},
  \label{eq:p17}
\end{equation}
where $\alpha = 1/\mu_0$, $\beta = 3\pi/20 = \dim(S^3)\cdot\pi\,/\,(\dim(B^4)\cdot V_4)$,
and the denominator $1 - \mu_1\alpha^2$ represents self-lensing (depth-weighted
double refraction through the bulk). We write $f(\mu_n)$ for this functional:
\begin{equation}
  f(\mu_n) = \beta \cdot \frac{\mu_n}{1 - \mu_n\alpha^2}.
\end{equation}

\textbf{Numerical verification.}
\begin{align}
  \mu_1 &= 108.716683780\ldots \\
  \mu_1\alpha^2 &= 108.717 \times (1/137.036)^2 = 0.005790 \\
  f(\mu_1) &= \frac{3\pi}{20}\cdot\frac{108.717}{0.994210} = 51.52985 \\
  \log(M_{\mathrm{Pl}}/m_e) &= \log(1.22089 \times 10^{19}\ \text{GeV}\,/\, 5.110 \times 10^{-4}\ \text{GeV}) = 51.52784 \\
  \text{Error} &= 0.0039\%.
\end{align}

The physical picture: $\mu_0$ measures the boundary $S^3$ phase (EM), while $\mu_1$
depth-weights the bulk $B^4$ propagation (gravity). The formula~\eqref{eq:p17}
holds to $0.004\%$ with zero free parameters.

% ─────────────────────────────────────────────────────────────────────────────
\section{Conjecture Test: $\mu_2$, $\mu_3$ and Higher Interactions}
\label{sec:test}

\subsection{The P17 formula applied to $\mu_2$ and $\mu_3$}

Direct computation gives:
\begin{align}
  f(\mu_2) &= \frac{3\pi}{20}\cdot\frac{90.176}{1 - 90.176\,\alpha^2}
            = \frac{3\pi}{20}\cdot\frac{90.176}{0.99520} = 42.699, \\[4pt]
  f(\mu_3) &= \frac{3\pi}{20}\cdot\frac{77.063}{1 - 77.063\,\alpha^2}
            = \frac{3\pi}{20}\cdot\frac{77.063}{0.99590} = 36.465.
\end{align}

We compare these against every relevant Standard Model observable:

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
Observable & Value & Error from $f(\mu_2)=42.699$ \\
\midrule
$\log(M_{\mathrm{Pl}}/m_e)$ & 51.528 & 17.1\% \\
$\log(M_W/m_e)$ & 11.966 & $>100\%$ \\
$\log(M_Z/m_e)$ & 12.092 & $>100\%$ \\
$1/\alpha_s(M_Z)$ & 8.482 & $>100\%$ \\
$1/\alpha_{\mathrm{unified}} \approx 43$ & $\sim 43$ & \textbf{0.70\%} \\
\bottomrule
\end{tabular}
\caption{$f(\mu_2) = 42.699$ compared to Standard Model observables.
The best match is $1/\alpha_{\mathrm{unified}}$ at the approximate GUT scale.}
\label{tab:f_mu2}
\end{table}

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
Observable & Value & Error from $f(\mu_3)=36.465$ \\
\midrule
$\log(M_{\mathrm{Pl}}/m_e)$ & 51.528 & 29.2\% \\
$\log(M_Z/m_e)$ & 12.092 & $>100\%$ \\
$1/\alpha_s(M_Z)$ & 8.482 & $>100\%$ \\
$1/\alpha_W(M_Z)$ & 29.6 & 18.9\% \\
\bottomrule
\end{tabular}
\caption{$f(\mu_3)=36.465$ compared to Standard Model observables. No close match.}
\label{tab:f_mu3}
\end{table}

\textbf{Conclusion for the direct formula:} The naive extension $\mu_n \to f(\mu_n)$
does not reproduce $G_F$ or $\alpha_s(M_Z)$ at any acceptable precision.
The only suggestive result is $f(\mu_2) \approx 1/\alpha_{\mathrm{unified}}$
at the GUT scale within $0.70\%$ --- but this is approximate (the unified coupling
is model-dependent, ranging from $1/42$ to $1/44$ in $SU(5)$ variants)
and falls well short of the $0.01\%$ threshold for a theorem claim.

\subsection{Alternative formulas and systematic search}

We tested over 40 formula variants, including: alternative $\beta$ coefficients
($\pi/20$, $\pi/24$, $\pi/30$, $1/8$, etc.), alternative denominators
($1-\mu_n\alpha^k$ for $k=1,2,3,4$), moment differences $\mu_i - \mu_j$,
moment ratios, products, and $\beta(\mu_i-\mu_j)/(1-(\mu_i-\mu_j)\alpha^2)$.
The best results within $2\%$ of any Standard Model observable are:

\begin{table}[h]
\centering
\begin{tabular}{llll}
\toprule
Formula & Value & Observable & Error \\
\midrule
$f(\mu_2)$ & 42.699 & $1/\alpha_{\mathrm{unified}} \approx 43$ & 0.70\% \\
$\mu_2/\mu_1 \cdot \log(M_{\mathrm{Pl}}/m_e)$ & 42.740 & $1/\alpha_{\mathrm{unified}} \approx 43$ & 0.60\% \\
$\mu_2-\mu_3$ & 13.113 & $\log(v/m_e) = 13.085$ & 0.21\% \\
$\beta(\mu_0-\mu_1)/(1-(\mu_0-\mu_1)\alpha^2)$ & 13.365 & $\log(1/\sqrt{G_F}/m_e)=13.259$ & 0.81\% \\
$\pi/20 \cdot \mu_3/(1-\mu_3\alpha^2)$ & 12.155 & $\log(M_Z/m_e) = 12.092$ & 0.52\% \\
$f(\mu_1)-f(\mu_2)$ & 8.830 & $1/\alpha_s(M_Z) = 8.482$ & 4.11\% \\
\bottomrule
\end{tabular}
\caption{Best-matching formula variants, sorted by proximity to any known observable.}
\label{tab:bestmatches}
\end{table}

% ─────────────────────────────────────────────────────────────────────────────
\section{Results}
\label{sec:results}

\subsection{The Higgs VEV candidate: $\mu_2 - \mu_3$}

The most structurally clean near-miss is the bare moment difference:
\begin{equation}
  \mu_2 - \mu_3 = \frac{8\pi^3}{21}+\frac{\pi^2}{10}+\frac{\pi}{10}
                = 13.11303463\ldots
\end{equation}
compared to the logarithm of the Higgs vacuum expectation value:
\begin{equation}
  \log\!\left(\frac{v}{m_e}\right) = \log\!\left(\frac{246.22\ \text{GeV}}{0.511\ \text{MeV}}\right)
  = 13.08537\ldots
\end{equation}
where $v = (\sqrt{2}\,G_F)^{-1/2} = 246.22$ GeV is defined directly from $G_F$.
The fractional error in log-space is $\mathbf{0.21\%}$, corresponding to
$v_{\mathrm{implied}} = e^{\mu_2-\mu_3}\cdot m_e = 253.1$ GeV versus
$v_{\mathrm{measured}} = 246.2$ GeV ($2.8\%$ in mass space).

This is the strongest G\raisebox{.2ex}{\tiny F}-related proximity result in the
moment hierarchy. However, $0.21\%$ is 50 times larger than the $0.004\%$
precision of the P17 gravity formula. The structural question --- why the
\emph{difference} $\mu_2 - \mu_3$ and not a single moment, and why no
self-lensing factor --- has no clear answer from the current theory.
We state this as an open observation, not a theorem.

\subsection{The GUT-scale candidate: $f(\mu_2)$}

The P17 formula applied to $\mu_2$ gives:
\begin{equation}
  f(\mu_2) = \frac{3\pi}{20}\cdot\frac{\mu_2}{1-\mu_2\alpha^2} = 42.699.
\end{equation}
Grand unified theories predict a common coupling at $M_{\mathrm{GUT}} \sim 10^{15}$--$10^{16}$ GeV,
with $\alpha_{\mathrm{unified}}^{-1} \approx 42$--$44$ depending on the model.
The standard $SU(5)$ value at the unification scale is approximately
$1/\alpha_{\mathrm{unified}} \approx 43.1$.
Agreement with $f(\mu_2)$ is $0.70\%$, which is tantalizing but requires
the GUT scale as an external input and falls short of the $0.01\%$ standard.

\subsection{Summary table}

\begin{table}[h]
\centering
\begin{tabular}{lllll}
\toprule
Moment & Formula & Value & Target & Verdict \\
\midrule
$\mu_0$ & direct & $137.036$ & $\alpha^{-1}=137.036$ & \textbf{closed (P3)} \\
$\mu_1$ & $f(\mu_1)$ & $51.530$ & $\log(M_{\mathrm{Pl}}/m_e)=51.528$ & \textbf{closed (P17), 0.004\%} \\
$\mu_2$ & $f(\mu_2)$ & $42.699$ & $1/\alpha_{\mathrm{unified}}\approx 43$ & open, 0.70\% \\
$\mu_2-\mu_3$ & bare diff & $13.113$ & $\log(v/m_e)=13.085$ & open, 0.21\% \\
$G_F$ & any & --- & $\log(1/\sqrt{G_F}/m_e)=13.259$ & \textbf{open} \\
$\alpha_s(M_Z)$ & any & --- & $1/\alpha_s=8.482$ & \textbf{open} \\
\bottomrule
\end{tabular}
\caption{Moment hierarchy: closed results and open problems.}
\label{tab:hierarchy}
\end{table}

% ─────────────────────────────────────────────────────────────────────────────
\section{Alternative Formula Families}
\label{sec:alternatives}

\subsection{Factored betas}

The P17 coefficient $\beta = 3\pi/20 = \frac{\dim(S^3)\cdot\pi}{\dim(B^4)\cdot V_4}$
carries geometric content. We tested whether $\mu_2$ or $\mu_3$ might connect
to the weak or strong interactions via a modified coefficient.

For $\log(M_Z/m_e) = 12.092$, the required coefficient is:
\begin{equation}
  \beta_Z = \frac{12.092}{\mu_2/(1-\mu_2\alpha^2)} = \frac{12.092}{90.611} = 0.1334
  = \frac{\pi}{23.54}.
\end{equation}
This does not correspond to a geometrically motivated integer.
The closest natural candidate, $\pi/24 = \dim(S^3)\pi/(8\dim(B^4))$, gives
$11.86$, off by $0.88\%$.

For $\alpha_s(M_Z) = 0.1179$, inverting yields $1/\alpha_s = 8.482$. No
single-moment formula with a natural coefficient achieves this within $2\%$.
The nearest formula involving $\mu_n$ only, $f(\mu_1) - f(\mu_2) = 8.830$,
is off by $4.1\%$.

\subsection{The $\mu_0 - \mu_1$ gap}

The formula applied to the first moment-gap:
\begin{equation}
  \beta\,\frac{\mu_0 - \mu_1}{1-(\mu_0-\mu_1)\alpha^2} = 13.365
\end{equation}
is the second-closest candidate for $\log(1/\sqrt{G_F}/m_e) = 13.259$,
with $0.81\%$ error. This uses $\mu_0 - \mu_1 = 4\pi^3/5+\pi^2/4+\pi/3 = 28.320$,
the boundary-to-bulk spectral gap, which may have geometric significance.
Not a theorem at current precision.

\subsection{Running coupling structure}

The $\log(\mu_n/\mu_{n+1})$ sequence shows structure:
\begin{align}
  \log(\mu_0/\mu_1) &= 0.231, \quad
  \log(\mu_1/\mu_2) = 0.186, \quad
  \log(\mu_2/\mu_3) = 0.157.
\end{align}
The decreasing steps are consistent with logarithmic running, suggesting
$\mu_n \sim c\cdot e^{-\gamma n}$ for some $\gamma$.
Whether this logarithmic structure encodes the RG running of couplings
is an open structural question.

% ─────────────────────────────────────────────────────────────────────────────
\section{Open Problems}
\label{sec:open}

The following are stated precisely for future work:

\begin{enumerate}
  \item \textbf{G\raisebox{.2ex}{\tiny F} closure.} Find a formula involving moments
    $\mu_n$ and the coupling $\alpha$ that reproduces
    $\log(1/\sqrt{G_F}/m_e) = 13.259$ within $0.01\%$, with a
    geometrically motivated coefficient. The current best attempt
    $\mu_2 - \mu_3 = 13.113$ has $0.21\%$ error in log-space ($2.8\%$ in $v$).

  \item \textbf{$\alpha_s$ closure.}  Find a formula reproducing
    $1/\alpha_s(M_Z) = 8.482$ within $0.01\%$. This may require identifying
    the natural scale at which to evaluate $\alpha_s$ (the formula may target
    $\alpha_s(M_{\mathrm{GUT}})$ rather than $\alpha_s(M_Z)$).

  \item \textbf{$\mu_2$ physical interpretation.}  The P17 conjecture
    (Conj.~7.1 in \cite{paper17}) proposes $\mu_2$ encodes the cosmological
    constant.  This should be tested explicitly.

  \item \textbf{Self-lensing structure for differences.}  Why does
    $\mu_2 - \mu_3$ match $\log(v/m_e)$ bare (without self-lensing), while
    $\mu_1$ requires the full $f(\mu_1)$ with the $1/(1-\mu_1\alpha^2)$ factor?
    A structural explanation is needed.

  \item \textbf{GUT scale from $f(\mu_2)$.}  Is the near-miss $f(\mu_2) \approx 43$
    a genuine prediction of the unification scale, or coincidence?
    A derivation of $M_{\mathrm{GUT}}$ from the moment sequence would be required.
\end{enumerate}

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

The moment hierarchy conjecture is not confirmed at theorem precision.
The moment structure of the P10/P17 density \emph{does} produce tantalizing
proximity results: $f(\mu_2) = 42.70$ is within $0.70\%$ of the GUT-scale
unified coupling, and $\mu_2 - \mu_3 = 13.113$ is within $0.21\%$ of
$\log(v/m_e)$ where $v = (\sqrt{2}\,G_F)^{-1/2}$ is the Higgs VEV.  Neither
reaches the $0.01\%$ bar required for a theorem claim.

The two closed results in the hierarchy remain $\mu_0 = \alpha^{-1}$ (exact)
and $f(\mu_1) = \log(M_{\mathrm{Pl}}/m_e)$ at $0.004\%$. All
five attempts to extend the hierarchy to the weak and strong sectors remain
open. The systematic record of what was tried and what precision was achieved
is documented in Table~\ref{tab:bestmatches} and in \texttt{verify\_P242.py}
(25/25 checks pass).

\begin{thebibliography}{9}
\bibitem{paper17}
L.~F.~Vlegels,
\textit{Gravity as Bulk Propagation: Deriving $G$ from the Dirac Operator on $(B^4,S^3)$},
This volume, Paper 17 (2025).

\bibitem{paper10}
L.~F.~Vlegels,
\textit{The Three-Fourths Boundary Correction: Simplex-Sphere-Hypercube Equilibrium},
This volume, Paper 10 (2025).

\bibitem{addA239}
L.~F.~Vlegels,
\textit{Addendum A239: Gravity/Cosmology Tractability Map},
This volume (2026).
\end{thebibliography}

\end{document}
