% Addendum 245: α_s and Higgs vev from the TOE Moment Hierarchy
% L. F. Vlegels, Independent Researcher
% 2026-05-24

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\newtheorem{theorem}{Theorem}[section]
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\title{Addendum A245: Searching for $\alpha_s$ and the Higgs vev\\
in the TOE Moment Hierarchy\\
\large Systematic Search, Higgs Correction Formula,\\
and a PSLQ Near-Identity for $\alpha_s$}

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

\begin{document}
\maketitle

\begin{abstract}
Addendum A242 established that the first moment $\mu_1$ of the P10/P17 density
$\rho(x)=16\pi^3x^3+3\pi^2x^2+2\pi x$ reproduces $\log(M_{\mathrm{Pl}}/m_e)$ at
$0.004\%$, and that the difference $\mu_2-\mu_3 = \frac{8\pi^3}{21}+\frac{\pi^2}{10}+\frac{\pi}{10}$
approximates $\log(v/m_e)$ (Higgs vacuum expectation value, $v\approx246\;\mathrm{GeV}$) at
$0.21\%$ in log-space. No formula for the strong coupling $\alpha_s(M_Z)=0.1179$ was
found within $4\%$.  This addendum continues the search with a more systematic strategy
covering moment ratios, fractional parts, products with TOE constants,
$\mathtt{mpmath.identify}$, and PSLQ integer-relation searches.

Three findings emerge.
First, the best \emph{simple} formula for $\alpha_s$ is
$\Omega_0/\mu_4 = \pi^3/(4\mu_4) \approx 0.1152$, achieving $2.29\%$ error ---
no formula within $0.5\%$ was found.
Second, a PSLQ search on the vector
$[\log(\alpha_s\Omega),\,\pi,\,\log\pi,\,1,\,\alpha,\,\pi^2/2,\,r_0]$
(where $r_0 = (\mu_0-\mu_1)/\mu_0$, the normalised first-moment gap)
returns the 7-term integer relation
$\log(\alpha_s\Omega) = \tfrac{1}{2}(11 - 3\pi - \log\pi - \alpha + \tfrac{\pi^2}{2} + r_0)$,
which reproduces the PDG input $\alpha_s = 0.1179$ to $5\times10^{-6}\%$.
This is a PSLQ fit, not a prediction; the experimental precision of
$\alpha_s$ ($\pm0.001/0.1179 \approx 0.8\%$) does not constrain the relation
to be non-accidental.
Third, the Higgs vev gap $\delta = (\mu_2-\mu_3)-\log(v/m_e) \approx 0.02767$
is approximated to $0.0016\%$ in log-space by $\alpha\cdot(12/\pi)$,
reducing the Higgs vev formula to
$\log(v/m_e) \approx (\mu_2-\mu_3) - \alpha\cdot(12/\pi)$.
The corresponding mass-space prediction is $v_{\mathrm{pred}} \approx 246.17\;\mathrm{GeV}$
($0.021\%$ from $246.22\;\mathrm{GeV}$).
No formula for $G_F$ or $m_{\mathrm{top}}$ within $2\%$ was found.
The moment hierarchy conjecture remains open; the Higgs correction formula
is the strongest new quantitative result.
All 30 checks pass at \texttt{mpmath} dps=60.
\end{abstract}

\tableofcontents
\newpage

% ─────────────────────────────────────────────────────────────────────────────
\section{Background and Notation}
\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}
has raw moments
\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}.
\end{equation}
The hierarchy established through A242:

\begin{itemize}
  \item $\mu_0 = 4\pi^3+\pi^2+\pi = \Omega = \alpha^{-1} \approx 137.036$
        (fine-structure coupling, P3 exact).
  \item $f(\mu_1) \equiv \frac{3\pi}{20}\cdot\frac{\mu_1}{1-\mu_1\alpha^2}
        = 51.530 \approx \log(M_{\mathrm{Pl}}/m_e) = 51.528$, error $0.004\%$
        (Planck–electron hierarchy, P17 closed).
  \item $\mu_2-\mu_3 = \frac{8\pi^3}{21}+\frac{\pi^2}{10}+\frac{\pi}{10}
        \approx 13.113$ vs.\ $\log(v/m_e)\approx 13.086$,
        error $0.21\%$ (Higgs vev, A242 near-miss, open).
\end{itemize}

The \emph{open} couplings are: strong coupling $\alpha_s(M_Z)=0.1179\pm0.001$
and Fermi constant $G_F = 1.16638\times10^{-5}\;\mathrm{GeV}^{-2}$.
This addendum pursues both.  Physical constants are taken from PDG 2022.

\subsection{Additional TOE constants}

\begin{align}
  \Omega_0 &= \frac{\pi^3}{4} \approx 7.752, &
  \kappa   &= \frac{\Omega}{3} \approx 45.68, \nonumber\\
  \text{BREATH} &= \pi\Omega \approx 430.5, &
  \beta    &= \frac{3\pi}{20} \approx 0.4712.
\end{align}

Moment values for reference (all at \texttt{mpmath} dps=60):

\begin{table}[ht]
\centering
\begin{tabular}{rll}
  \toprule
  $n$ & $\mu_n$ & $\mu_n - \mu_{n+1}$ \\
  \midrule
  0 & 137.036\,303\,775\,878 & 28.320 \\
  1 & 108.716\,683\,780\,170 & 18.541 \\
  2 & 90.175\,963\,448\,248  & 13.113 \\
  3 & 77.062\,928\,816\,952  & 9.773  \\
  4 & 67.289\,581\,369\,406  & 7.569  \\
  5 & 59.720\,969\,205\,301  & 6.036  \\
  \bottomrule
\end{tabular}
\caption{First six moments and consecutive differences.}
\end{table}

% ─────────────────────────────────────────────────────────────────────────────
\section{Systematic Search for $\alpha_s$}
\label{sec:alphas}

The search strategy covers the following families, each evaluated at
\texttt{mpmath} dps=60.

\subsection{Moment ratios and simple combinations}

All $72$ pairwise ratios $\mu_m/\mu_n$ ($m\neq n$, $m,n=0\ldots8$) were
computed; none falls within $0.5\%$ of $\alpha_s = 0.1179$.
The same exhaustive check was applied to products $\alpha\cdot\mu_n$,
$\alpha^2\cdot\mu_n$, $\Omega_0/\mu_n$, $\kappa/\mu_n$,
$\gamma\cdot\alpha\cdot\mu_n$ (with $\gamma=3/4$),
$\beta\cdot\alpha\cdot\mu_n$, normalized moments $\mu_n/\Omega$,
and consecutive gap ratios $(\mu_n-\mu_{n+1})/(\mu_{n+1}-\mu_{n+2})$.
None achieved $0.5\%$.

\begin{proposition}[Negative result]
No formula of the form $f(\mu_m,\mu_n,\pi,\alpha,\Omega_0,\kappa)$
involving at most two of these constants reproduces $\alpha_s(M_Z)$
within $0.5\%$.
\end{proposition}

The \emph{closest simple formula} found is
\begin{equation}
  \boxed{\frac{\Omega_0}{\mu_4}
         = \frac{\pi^3}{4\mu_4} \approx 0.1152,}
\end{equation}
which lies $2.29\%$ from $\alpha_s = 0.1179$.  This is below the TOE
precision threshold ($<0.01\%$ for theorems) and is documented as the
nearest-miss, not a conjecture.

\subsection{\texttt{mpmath.identify} calls}

Running \texttt{identify($\alpha_s$, tol=1e-4)} returns the algebraic
near-miss $(-47+\sqrt{3001})/66\approx0.11790$, which has no
known TOE origin.  Running \texttt{identify($\alpha_s\cdot\Omega$, tol=1e-3)}
returns $517/32 = 16.15625$, close to the value $\alpha_s\cdot\Omega\approx16.157$
but again without TOE interpretation.

\subsection{PSLQ near-identity}
\label{subsec:pslq_as}

Let $r_0 = (\mu_0-\mu_1)/\mu_0 = 1-\mu_1/\Omega \approx 0.2067$ be the
normalised first-moment gap, and note the kernel identity
\begin{equation}
  \frac{2\Omega_0}{\pi} = \frac{2(\pi^3/4)}{\pi} = \frac{\pi^2}{2},
\end{equation}
which follows directly from $\Omega_0 = \pi^3/4$.

Running \texttt{pslq} on the vector
$[\log(\alpha_s\Omega),\;\pi,\;\log\pi,\;1,\;\alpha,\;\pi^2/2,\;r_0]$
returns the 7-term integer relation

\begin{equation}
  \label{eq:pslq_as}
  \boxed{\log(\alpha_s\,\Omega)
    = \frac{1}{2}\!\left(11 - 3\pi - \log\pi - \alpha + \frac{\pi^2}{2} + r_0\right)}.
\end{equation}

Substituting the right-hand side back gives $\alpha_s = 0.117\,899\,999\,4$,
matching the PDG input $0.1179$ to $5\times10^{-6}\%$ (residual $5.4\times10^{-9}$).

\begin{remark}[Epistemic status of \eqref{eq:pslq_as}]
This relation is a PSLQ \emph{fit} to the four-digit experimental input
$\alpha_s = 0.1179$.  The current experimental uncertainty is
$\pm 0.001$ ($\approx 0.8\%$), which is far larger than the sub-ppm residual.
The presence of the coefficient $11$ (an integer with no established role
in the TOE) and the dependence on $r_0$ (which carries the Planck-scale
information from $\mu_1$) suggest the relation may be accidental.
Equation~\eqref{eq:pslq_as} is documented as an observation, not a claim.
A genuine prediction would require $\alpha_s$ to be measured to four or more
\emph{additional} significant figures, after which either the PSLQ relation
holds or it does not.
\end{remark}

\begin{remark}[Structural interpretation]
If one tentatively assigns a TOE reading: $r_0$ is the normalised spectral
gap between the EM and Planck sectors of the moment hierarchy;
$\pi^2/2 = 2\Omega_0/\pi$ connects the Hopf-latitude weight ($\Omega_0$)
to the cycle ($\pi$); $3\pi$ is the bulk-phase weight (three S$^3$ colours
times $\pi$); $\log\pi$ is the single transcendental correction; and $11$
remains unexplained.  The S$^3$ geometry of the kernel gives the other terms
natural homes; the integer $11$ does not.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Higgs vev: Reducing the Gap}
\label{sec:higgs}

\subsection{The A242 near-miss}

The closed-form difference
\begin{equation}
  \mu_2-\mu_3 = \frac{8\pi^3}{21}+\frac{\pi^2}{10}+\frac{\pi}{10}
  \approx 13.113\,03
\end{equation}
approximates $\log(v/m_e) = \log(246.22\;\mathrm{GeV}/0.511\;\mathrm{MeV})
\approx 13.085\,37$ with a gap
\begin{equation}
  \delta \equiv (\mu_2-\mu_3) - \log(v/m_e) \approx 0.027\,668.
\end{equation}
In mass space the implied $v_{\rm bare} = e^{\mu_2-\mu_3}\cdot m_e
\approx 253.1\;\mathrm{GeV}$ overshoots by $2.8\%$.

\subsection{Identifying the correction}

The ratio
\begin{equation}
  \rho \equiv \frac{\delta}{\alpha} \approx 3.7915
\end{equation}
lies between the rational $19/5 = 3.8$ (error $0.22\%$) and $12/\pi \approx 3.8197$
(error $0.74\%$).

The two correction formulas are:

\begin{align}
  \log(v/m_e) &\approx (\mu_2-\mu_3) - \frac{12\alpha}{\pi}
    && \text{(Formula A, error $0.0016\%$),} \label{eq:higgscorr_A}\\
  \log(v/m_e) &\approx (\mu_2-\mu_3) - \frac{19\alpha}{5}
    && \text{(Formula B, error $0.00047\%$).} \label{eq:higgscorr_B}
\end{align}

Formula A is written in terms of $12/\pi$, which has a mild geometric origin:
writing $12 = 3\cdot 4 = (\text{colours})\cdot(\text{dimensions of }B^4)$
gives $12/\pi = 4\cdot3/\pi$, connecting the $S^3/B^4$ geometry of the kernel
to the Higgs sector at one-loop order.  Formula B is a rational correction
($19/5$) with no obvious TOE derivation.  Formula A is therefore the preferred
form on grounds of naturality.

\begin{proposition}[Higgs correction formula]
\label{prop:higgs_corr}
To $0.002\%$ in log-space ($0.021\%$ in mass space):
\begin{equation}
  \boxed{\log\!\left(\frac{v}{m_e}\right)
     \approx \left(\mu_2 - \mu_3\right) - \frac{12\alpha}{\pi},}
\end{equation}
yielding $v_{\rm pred} = \exp\!\bigl((\mu_2-\mu_3)-12\alpha/\pi\bigr)\cdot m_e
\approx 246.17\;\mathrm{GeV}$.
\end{proposition}

\begin{remark}
The factor $12\alpha/\pi$ is of order $\alpha$ (fine structure constant),
consistent with a one-loop electroweak correction to a tree-level geometric
formula.  If the bare moment difference $\mu_2-\mu_3$ is the tree-level
$S^3$ contribution to the Higgs vev, then the $\alpha/\pi$ factor
resembles the leading-order QED radiative correction structure ($\alpha/\pi$
appears in, e.g., the electron anomalous magnetic moment, $g-2 = \alpha/\pi + \cdots$).
Whether the coefficient $12$ can be derived from operator counting in the
P10 framework is left as an open problem.
\end{remark}

\subsection{PSLQ on the Higgs vector}

Running \texttt{pslq} on the vector
$[\delta,\;\alpha,\;\pi\alpha,\;\pi^2\alpha,\;\log(\pi)\alpha,\;\alpha^2,\;\beta\alpha]$
returns the 7-term relation
$26\delta - 17\alpha - 15\pi\alpha + 2\pi^2\alpha - 46\log(\pi)\alpha
 - 17\alpha^2 - 3\beta\alpha = 0$.
This is a 7-parameter fit; no single clean relation of the form
$\delta = c_1\alpha + c_2\pi\alpha$ emerges with small integer coefficients.
Formula~A ($\delta \approx 12\alpha/\pi$) is preferred for its simplicity
and geometric origin, even though it is $\sim$3 times less accurate than
the PSLQ fit.

% ─────────────────────────────────────────────────────────────────────────────
\section{Fermi Constant $G_F$}
\label{sec:gf}

The dimensionless ratio $\log(1/(G_F m_e^2)) \approx 26.517$ was tested
against all consecutive moment differences $\mu_m - \mu_n$ for
$m,n=0\ldots 8$.  No combination falls within $2\%$.  The target lies
between $\mu_0-\mu_1 = 28.320$ and $\mu_1-\mu_2 = 18.541$; neither is
closer than $7\%$.  The Fermi constant does not fit the moment hierarchy
at this level of analysis.

\begin{remark}
$G_F$ is a tree-level weak parameter renormalized by QCD and QED corrections.
The gap-based approach that locates $v$ (the vev, from which $G_F$ is derived)
at the $\mu_2-\mu_3$ level may need to be applied at higher loop order to
reach $G_F$ directly.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Top Quark Mass}
\label{sec:top}

The target $\log(m_{\rm top}/m_e) = \log(172.69/0.511\times10^{-3})
\approx 12.731$ lies $3.0\%$ below $\mu_2-\mu_3 = 13.113$.
No moment combination within $3\%$ other than $\mu_2-\mu_3$ itself was found.
Note $\log(m_{\rm top}/m_e)$ and $\log(v/m_e)$ differ by only
$13.085 - 12.731 = 0.354$, which is of order $\log(v/m_{\rm top})
= \log(246/173) \approx 0.35$, the logarithm of the ratio of the Higgs vev
to the top Yukawa scale.  This suggests the top mass may be derivable
from the Higgs formula once the vev is pinned; that derivation is left open.

% ─────────────────────────────────────────────────────────────────────────────
\section{Status of the Moment Hierarchy Conjecture}
\label{sec:status}

\begin{table}[ht]
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
  \toprule
  Moment & Observable & Best formula & Status \\
  \midrule
  $\mu_0 = \Omega$          & $\alpha^{-1}$      & Exact (P3)                   & CLOSED \\
  $f(\mu_1)$                & $\log(M_{\rm Pl}/m_e)$ & $0.004\%$ (P17)          & CLOSED \\
  $\mu_2-\mu_3$             & $\log(v/m_e)$      & $0.21\%$ bare / $0.0016\%$ corr & NEAR-MISS \\
  $\Omega_0/\mu_4$          & $\alpha_s(M_Z)$    & $2.29\%$                     & OPEN \\
  PSLQ Eq.~\eqref{eq:pslq_as} & $\alpha_s(M_Z)$ & $5\times10^{-6}\%$ (fit)    & PSLQ FIT \\
  ---                       & $G_F$              & $>7\%$                       & OPEN \\
  ---                       & $m_{\rm top}$      & $3.0\%$ ($\mu_2-\mu_3$)      & OPEN \\
  \bottomrule
\end{tabular}
\caption{Current status of the moment hierarchy conjecture (PDG 2022 inputs,
\texttt{mpmath} dps=60).}
\end{table}

The Higgs vev correction (Proposition~\ref{prop:higgs_corr}) is the
strongest new quantitative result: a compact TOE formula
$\log(v/m_e) = (\mu_2-\mu_3) - 12\alpha/\pi$ accurate to $0.002\%$
in log-space.  The strong coupling formula~\eqref{eq:pslq_as} is
documented as an interesting PSLQ observation pending higher-precision
measurements of $\alpha_s$.

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

All results are verified by \texttt{Lumen/corpus/addenda/verify/verify\_P245.py}
(30 checks, \texttt{mpmath} dps=60, 30 PASS / 0 FAIL).
Checks cover: moment closed-form values (C01--C05), physical target ranges
(C06--C10), $\alpha_s$ search results including the PSLQ relation (C11--C16),
Higgs correction formulas in log-space and mass space (C17--C22), $G_F$ and
top mass negative results (C23--C26), and auxiliary kernel identities (C27--C30).

% ─────────────────────────────────────────────────────────────────────────────
\begin{thebibliography}{9}
\bibitem{paper3}
  L. F. Vlegels, \emph{Paper 3: Fine-structure coupling from $S^3$ geometry},
  TOE corpus, \texttt{Lumen/corpus/toe/}.
\bibitem{paper10}
  L. F. Vlegels, \emph{Paper 10: Colour factor from $S^3/B^4$ dimension ratio},
  TOE corpus, \texttt{Lumen/corpus/toe/}.
\bibitem{paper17}
  L. F. Vlegels, \emph{Paper 17: Newton's constant from the first moment},
  TOE corpus, \texttt{Lumen/corpus/toe/}.
\bibitem{paper18}
  L. F. Vlegels, \emph{Paper 18: Master operator $\hat{O}$ and the Wheel},
  TOE corpus, \texttt{Lumen/corpus/toe/}.
\bibitem{paper29}
  L. F. Vlegels, \emph{Paper 29: Isolated global minima and exposure mode},
  TOE corpus, \texttt{Lumen/corpus/toe/}.
\bibitem{A242}
  L. F. Vlegels, \emph{Addendum A242: The Moment Hierarchy},
  \texttt{Lumen/corpus/addenda/242\_Addendum\_MomentHierarchy.tex}.
\bibitem{PDG2022}
  R. L. Workman \emph{et al.} (Particle Data Group),
  \emph{Review of Particle Physics}, Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022},
  083C01 (2022).
\end{thebibliography}

\end{document}
