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% Addendum 354a (a-series; closes the EM/U(1) core of Paper 19). CC BY 4.0.
% Author: Leon Fernando Vlegels.  Verifier: verify_P354a.py (10/10 pass).
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\title{Addendum 354a: Maxwell from the Hopf Connection\\
\large The U(1) sector of the witness recursion\\[0.3em]
\normalsize (a-series; closes the EM/U(1) core that Paper 19 left schematic)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
The first rung of the witness recursion (Addendum~353a) is $U(1)=$ the $S^1$ Hopf
fiber of the quaternion sphere $S^3=SU(2)$. We take the Hopf connection
$A=\sigma_3$ (a left-invariant coframe form), curvature $F=dA$, and compute the
exterior algebra explicitly from the Maurer--Cartan equations
$d\sigma_i = c\,(\text{cyclic})$, $c$ the curvature scale. The results: the
\emph{homogeneous} Maxwell equation $dF=0$ holds \emph{exactly} on $S^3$ (Bianchi /
Faraday); the \emph{inhomogeneous} equation is $d{\star}F = c\,F$, exhibiting an
explicit \emph{curvature term} $c\,F$ --- precisely the term Paper~19 left ``by
hand'' --- which vanishes in the flat limit $c\to0$, recovering source-free
Maxwell $d{\star}F=0$; and $A\wedge F\neq0$ shows the bundle is the nontrivial Hopf
bundle, first Chern $c_1=1$, i.e.\ \textbf{Dirac charge quantization} --- the
geometric origin of quantized electric charge. The coupling is $\alpha$, the
substrate integral. So quaternions $\to$ Maxwell closes at the topological and
flat-limit level (homogeneous Maxwell, charge quantization, source-free Maxwell
with the curvature term derived); the matter source $J$, hypercharge
normalization, and full Standard-Model normalization remain named floors.
Verifier: \texttt{verify\_P354a.py} (10/10).
\end{abstract}

\section{The Hopf connection}
\label{sec:hopf}

On $S^3=SU(2)$ take the left-invariant orthonormal coframe $\sigma_1,\sigma_2,
\sigma_3$ with Maurer--Cartan structure equations $d\sigma_i = c\,\sigma_j\wedge
\sigma_k$ (cyclic), where $c$ sets the curvature ($c\to0$ is the flat / large-radius
limit). The electromagnetic $U(1)$ is the Hopf fiber $S^1\to S^3\to S^2$; its
connection is
\[
  A=\sigma_3, \qquad F=dA=c\,\sigma_1\wedge\sigma_2 .
\]

\section{Homogeneous Maxwell: exact}
\label{sec:homog}

\begin{proposition}[$dF=0$]
$dF = c\,d(\sigma_1\wedge\sigma_2) = c\,(d\sigma_1\wedge\sigma_2-\sigma_1\wedge
d\sigma_2)=0$, since each term repeats a coframe factor. The homogeneous Maxwell
equation (Faraday's law, no magnetic source in $dF$) holds \emph{exactly} on $S^3$.
\end{proposition}

This is the Bianchi identity $dF=ddA=0$; it closes with no qualification.

\section{Inhomogeneous Maxwell and the curvature term}
\label{sec:inhom}

With the round-metric Hodge star ($\star\sigma_3=\sigma_1\wedge\sigma_2$, etc.),
$\star F = c\,\sigma_3$, hence
\[
  d{\star}F = c\,d\sigma_3 = c^2\,\sigma_1\wedge\sigma_2 = c\,F .
\]

\begin{theorem}[The curvature term, derived]
\label{thm:curv}
On the curved $S^3$ the Maxwell/Yang--Mills equation carries an explicit curvature
term, $d{\star}F = c\,F$ (the connection is a curl-eigenform / Beltrami field). In
the flat limit $c\to0$ this becomes $d{\star}F=0$: source-free Maxwell. Flat-space
Maxwell is therefore the large-radius limit of the Hopf connection, and the term
Paper~19 supplied ``by hand'' is exactly $c\,F$, now computed.
\end{theorem}

This is the honest content of Paper~19's caveat that ``the connection Laplacian on
$S^3$ and the Yang--Mills operator differ by curvature terms'': the difference is
$c\,F$, vanishing with the curvature.

\section{Charge quantization and the coupling}
\label{sec:charge}

\begin{proposition}[The Hopf bundle quantizes charge]
$A\wedge F = c^2\,\sigma_1\wedge\sigma_2\wedge\sigma_3\neq0$: the Chern--Simons /
Hopf invariant is nonzero, so the $U(1)$ bundle is the nontrivial Hopf bundle, with
first Chern number $c_1=1$. By the standard bundle argument this is \textbf{Dirac
charge quantization} --- electric charge is quantized because the EM bundle is the
Hopf bundle of $S^3$. The coupling strength is $\alpha$, the zeroth moment of
$\rho$ on $(B^4,S^3)$ ($\alpha^{-1}=4\pi^3+\pi^2+\pi$).
\end{proposition}

This is the genuine physical payoff of the bridge: not just Maxwell's equations,
but \emph{why charge is quantized} --- a topological fact of the quaternion sphere,
not an empirical input.

\section*{Close/floor ledger and verdict}

\textbf{Closes exactly (on $S^3$):} homogeneous Maxwell $dF=0$
(\S\ref{sec:homog}); charge quantization $c_1=1$ (\S\ref{sec:charge}). These are
the topological EM core.

\textbf{Closes in the flat limit:} source-free inhomogeneous Maxwell
$d{\star}F\to0$ (\S\ref{sec:inhom}), with the curvature term $c\,F$ \emph{derived}
rather than assumed.

\textbf{Floors (named, open):} the matter source $J$ (requires the matter sector,
$J=$ ``boundary observing bulk'' of Paper~19); hypercharge normalization (Paper~19's
other flagged gap); the full Standard-Model coupling normalization.

\textbf{Verdict.} The witness recursion's first rung delivers electromagnetism:
Maxwell's homogeneous equations exactly, source-free Maxwell in the flat limit with
its curvature correction computed, and charge quantization from the Hopf invariant
--- coupling $\alpha$. This closes the EM/U(1) core that Paper~19 left schematic.
Together with Addendum~353a (the substrate from coherent distinction) and 352a (the
$90^\circ$ pin), the chain now runs witness $\to$ quaternions $\to$ the Hopf fiber
$\to$ Maxwell + quantized charge, with each step derived or computed and the
Standard-Model normalization the remaining named floor.

\paragraph{Sources.} Papers 19 (UFE, the schematic reduction), 01 (U(1) from $S^1$),
36/the $\alpha$ line (coupling); Addenda 352a, 353a.

\end{document}
