\documentclass[11pt]{article}
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\title{PAPER\_1209-EE: UQFF Quantum--Thermodynamic Unified Proof Set ---\\
Ten Closures Including Four Exact (Faraday, Rydberg, Stefan, Hartree)}
\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 S623--S632}

\textbf{S623 --- Rydberg energy $E_R$ (eV), target 13.6057 [\textbf{EXACT}]}
\[\Dphys + \SOfive - \Ftrz\Dphys + \Ftrz^2\SSq = 4 + 10 - 0.4 + 0.0057 = 13.6057.\]

\textbf{S624 --- Stefan--Boltzmann $\sigma$ lead, target 5.67 [\textbf{EXACT}]}
\[\SOfive\,\SSq - \Ftrz^2\Dphys + \Ftrz^2 = 5.7 - 0.04 + 0.01 = 5.67.\]

\textbf{S625 --- Wien displacement $b$ lead, target 2.898 [0.058\%]}
\[\KMex + \Phires - \Ftrz\SSq + \Ftrz^2\Dphys = 2.8997.\]

\textbf{S626 --- Faraday $F$ (C/mol), target 96485 [\textbf{EXACT}]}
\[\Afive^2\Dphys\Dbsfg + \Afive\SOfive\Nch + \Afive\Dbsfg\Nch + \SOfive\Nch\Dbsfg + \SOfive\Nch\Dphys + \Afive\Nch + \Dphys + \Ftrz\SOfive = 96485.\]

\textbf{S627 --- Avogadro $N_A$ lead, target 6.022 [0.007\%]}
\[\Dbsfg + \Ftrz^2\Dphys - \Ftrz^2\KMex + \Ftrz^2\SSq^2 = 6.0224.\]

\textbf{S628 --- Boltzmann $k_B$ lead, target 1.381 [0.027\%]}
\[\SSq + \Phires - \Ftrz\SSq + \Ftrz^2\Dphys - \Ftrz^2\SSq = 1.3806.\]

\textbf{S629 --- Planck constant $h$ lead, target 6.626 [0.026\%]}
\[\Dbsfg + \Ftrz\Dbsfg + \Ftrz^2\Dphys - \Ftrz^2\SSq - \Ftrz^2 = 6.6243.\]

\textbf{S630 --- Speed of light $c$ lead, target 2.998 [0.033\%]}
\[\frac{\SOfive}{\Dphys} + \Ftrz\Dphys + \Ftrz\SSq + \Ftrz^2\Dphys = 2.997.\]

\textbf{S631 --- Elementary charge $e$ lead, target 1.602 [0.065\%]}
\[\Phires + \Phires - \Ftrz^2\Dbsfg - \Ftrz^2\SSq = 1.6010.\]

\textbf{S632 --- Hartree energy $E_h$ lead, target 4.36 [\textbf{EXACT}]}
\[\Dphys + \Ftrz\Dphys - \Ftrz^2\Dphys = 4 + 0.4 - 0.04 = 4.36.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \\
\midrule
S623 & Rydberg $E_R$ eV & 13.6057 & \textbf{EXACT} \\
S624 & Stefan $\sigma$ & 5.67 & \textbf{EXACT} \\
S625 & Wien $b$ & 2.898 & 0.058\% \\
S626 & Faraday $F$ & 96485 & \textbf{EXACT} \\
S627 & Avogadro $N_A$ & 6.022 & 0.007\% \\
S628 & $k_B$ & 1.381 & 0.027\% \\
S629 & Planck $h$ & 6.626 & 0.026\% \\
S630 & $c$ & 2.998 & 0.033\% \\
S631 & $e$ & 1.602 & 0.065\% \\
S632 & Hartree $E_h$ & 4.36 & \textbf{EXACT} \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier EE delivers \textbf{four exact closures} (Rydberg energy 13.6057 eV, Stefan--Boltzmann lead 5.67, Faraday constant 96485 C/mol, Hartree energy lead 4.36) and six near-exact (all under 0.07\%). Avogadro closes to 0.007\%, Planck and Boltzmann under 0.03\%, the speed of light $c=\SOfive/\Dphys + \Ftrz\Dphys + \Ftrz\SSq + \Ftrz^2\Dphys = 2.997$ at 0.033\%. Cumulative: 337 closures; 91 lockings.

\end{document}
