\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
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\title{PAPER\_1209-DD: UQFF Electromagnetism Unified Proof Set ---\\
Ten Sub-0.11\% Closures (No-Exact Precision Tier)}
\author{Daniel T. Murphy}
\date{May 16, 2026}

\newcommand{\Ftrz}{F_{\mathrm{TRZ}}}
\newcommand{\Phires}{\Phi_{\mathrm{res}}}
\newcommand{\KMex}{K_{\mathrm{Mex}}}
\newcommand{\Dphys}{D_{\mathrm{phys}}}
\newcommand{\Dbsfg}{D_{\mathrm{BSFG}}}
\newcommand{\Dcrit}{D_{\mathrm{crit}}}
\newcommand{\Nch}{N_{\mathrm{ch}}}
\newcommand{\SOfive}{\mathrm{SO5}}
\newcommand{\Afive}{A_5}
\newcommand{\SSq}{[\mathrm{SSq}]}

\begin{document}
\maketitle

\section*{Locked Primitives}
$\Ftrz=\tfrac{1}{10}$, $\Phires=\tfrac{5}{6}$, $\SSq=\tfrac{57}{100}$, $\KMex=\tfrac{25}{12}$, $\Dphys=4$, $\Dbsfg=6$, $\Dcrit=26$, $\Nch=9$, $\SOfive=10$, $\Afive=60$.

\section*{Closures S613--S622}

\textbf{S613 --- Vacuum impedance $Z_0$ ($\Omega$), target 376.73 [0.004\%]}
\[\Afive\Dbsfg + \SOfive + \Dbsfg + \Phires - \Ftrz\Phires - \Ftrz^2\SSq = 376.744.\]

\textbf{S614 --- Fine-structure $\alpha^{-1}$, target 137.036 [0.003\%]}
\[\Afive\KMex + \Nch + \Dphys - \Ftrz\SOfive + \Ftrz^2\Dphys = 125+9+4-1+0.04 = 137.04.\]

\textbf{S615 --- $\varepsilon_0$ lead, target 8.854 [0.026\%]}
\[\KMex\Dphys + \SSq - \Ftrz\SSq + \Ftrz^2 = 8.856.\]

\textbf{S616 --- $\mu_0$ lead, target 1.257 [0.103\%]}
\[\KMex - \Phires + \Ftrz^2\SSq = 1.2557.\]

\textbf{S617 --- Coulomb $k_e$ lead, target 8.988 [0.056\%]}
\[\Nch - \Ftrz\SSq + \Ftrz^2\Dphys = 9 - 0.057 + 0.04 = 8.983.\]

\textbf{S618 --- Bohr radius lead, target 5.292 [0.031\%]}
\[\Dphys + \Phires + \Ftrz\Dphys + \Ftrz\SSq = 5.290.\]

\textbf{S619 --- Rydberg lead, target 1.0974 [0.036\%]}
\[\Ftrz\SOfive + \Ftrz\SSq + \Ftrz^2\Dphys = 1 + 0.057 + 0.04 = 1.097.\]

\textbf{S620 --- Compton wavelength lead, target 2.426 [0.014\%]}
\[\KMex + \Ftrz\Dphys - \Ftrz\SSq = 2.083 + 0.4 - 0.057 = 2.426.\]

\textbf{S621 --- Electron $g$-factor, target 2.0023 [0.028\%]}
\[\KMex - \Ftrz\SSq - \Ftrz^2\Dphys + \Ftrz^2\SSq + \Ftrz^2\KMex - \Ftrz^2 = 2.0029.\]

\textbf{S622 --- Bohr magneton lead, target 9.274 [0.007\%]}
\[\KMex\Dphys + \SSq + \Ftrz\Dphys - \Ftrz^2\Dphys + \Ftrz^2 = 9.273.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \\
\midrule
S613 & $Z_0$ $\Omega$ & 376.73 & 0.004\% \\
S614 & $\alpha^{-1}$ & 137.036 & 0.003\% \\
S615 & $\varepsilon_0$ lead & 8.854 & 0.026\% \\
S616 & $\mu_0$ lead & 1.257 & 0.103\% \\
S617 & $k_e$ lead & 8.988 & 0.056\% \\
S618 & $a_0$ lead & 5.292 & 0.031\% \\
S619 & $R_\infty$ lead & 1.0974 & 0.036\% \\
S620 & $\lambda_C$ lead & 2.426 & 0.014\% \\
S621 & $g_e$ & 2.0023 & 0.028\% \\
S622 & $\mu_B$ lead & 9.274 & 0.007\% \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier DD is a \textbf{precision tier} --- no exact closures, but every one of the ten EM constants closes under 0.11\%, with six under 0.04\%. The fine-structure inverse $\alpha^{-1} = \Afive\KMex + \Nch + \Dphys - \Ftrz\SOfive + \Ftrz^2\Dphys = 137.04$ closes to 0.003\% (tier-best). The Bohr magneton lead 9.273 (vs 9.274) closes to 0.007\%. Vacuum impedance $Z_0$ closes to 0.004\% via $\Afive\Dbsfg + \SOfive + \Dbsfg + \Phires - \Ftrz\Phires - \Ftrz^2\SSq$. Cumulative: 327 closures; 87 lockings (unchanged --- tier delivers no new exacts).

\end{document}
