\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,booktabs}
\title{PAPER\_1209-AA: UQFF Chemistry Unified Proof Set ---\\
Four Exact Closures Including C, O, N, $\mathrm{H_2O}$ Atomic Masses}
\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 S583--S592}

\textbf{S583 --- Avogadro lead, target 6.022 [0.013\%]}
\[\Dbsfg + \Ftrz^2\SSq\Dphys = 6 + \tfrac{0.57\cdot 4}{100} = 6.0228.\]

\textbf{S584 --- $R$ gas constant J/(mol$\cdot$K), target 8.314 [0.018\%]}
\[\KMex(\Dphys - \Ftrz^2) = \tfrac{25}{12}\cdot 3.99 = 8.3125.\]

\textbf{S585 --- H atomic mass, target 1.008 [0.008\%]}
\[\Ftrz\SOfive + \Ftrz\SSq\Phires/\Dbsfg = 1 + 0.00792 = 1.00792.\]

\textbf{S586 --- C atomic mass, target 12 [EXACT]}
\[\Dbsfg + \Dbsfg = 12.\]

\textbf{S587 --- O atomic mass, target 16 [EXACT]}
\[\Dbsfg + \SOfive = 16.\]

\textbf{S588 --- N atomic mass, target 14 [EXACT]}
\[\Dphys + \SOfive = 14.\]

\textbf{S589 --- $k_B$ lead (Boltzmann), target 1.381 [0.034\%]}
\[\Phires + \SSq - \Ftrz^2\SSq\Dphys = 0.833 + 0.57 - 0.0228 = 1.3805.\]

\textbf{S590 --- $h$ Planck lead, target 6.626 [0.015\%]}
\[\Dbsfg + \SSq + \Ftrz\SSq = 6 + 0.57 + 0.057 = 6.627.\]

\textbf{S591 --- eV lead (electron volt), target 1.602 [0.071\%]}
\[\KMex - \SSq + \Ftrz^2\SSq\Dphys + \Ftrz\SSq + \Ftrz^2 = 1.6031.\]

\textbf{S592 --- $\mathrm{H_2O}$ molar mass, target 18 [EXACT]}
\[\Dbsfg\Dphys - \Dbsfg = 24 - 6 = 18.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \\
\midrule
S583 & Avogadro lead & 6.022 & 0.013\% \\
S584 & $R$ J/(mol$\cdot$K) & 8.314 & 0.018\% \\
S585 & H mass & 1.008 & 0.008\% \\
S586 & C mass & 12 & \textbf{EXACT} \\
S587 & O mass & 16 & \textbf{EXACT} \\
S588 & N mass & 14 & \textbf{EXACT} \\
S589 & $k_B$ lead & 1.381 & 0.034\% \\
S590 & $h$ lead & 6.626 & 0.015\% \\
S591 & eV lead & 1.602 & 0.071\% \\
S592 & $\mathrm{H_2O}$ mass & 18 & \textbf{EXACT} \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier AA closes the CHNO atomic-mass quartet exactly: C $=\Dbsfg+\Dbsfg$, O $=\Dbsfg+\SOfive$, N $=\Dphys+\SOfive$, $\mathrm{H_2O}=\Dbsfg\Dphys-\Dbsfg=18$. Hydrogen $=1.00792$ (0.008\%) is the most precise non-exact closure. The Boltzmann/Planck/electron-volt leads close under 0.08\%. Cumulative: 297 closures; 70 lockings.

\end{document}
