\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
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\title{PAPER\_1209-Z: UQFF Astronomical Units Unified Proof Set ---\\
Six Exact Closures Including AU, Hubble, Sun--Earth Ratios, Sidereal Year}
\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 S573--S582}

\textbf{S573 --- Moon distance / $R_\oplus$, target 60.336 [0.0044\%]}
\[\Afive + \Ftrz\Phires\Dphys = 60 + \tfrac{1}{3} = 60.333.\]

\textbf{S574 --- AU / $R_\oplus$, target 23481 [EXACT]}
\[\Dcrit\Nch\SOfive^2 + \Afive + \Dcrit - \Dphys + \Ftrz\SOfive + \Ftrz\Phires - \KMex = 23483 - \tfrac{24}{12} = 23481.\]

\textbf{S575 --- parsec/light-year, target 3.2616 [0.022\%]}
\[\Phires\Dphys - \Phires\Ftrz + \Ftrz^2\Phires + \Ftrz^3\Dphys = 3.2623.\]

\textbf{S576 --- Hubble constant (km/s/Mpc), target 70 [EXACT]}
\[\Afive + \SOfive = 70.\]

\textbf{S577 --- $R_\odot/R_\oplus$, target 109 [EXACT]}
\[\SOfive^2 + \Nch = 109.\]

\textbf{S578 --- Earth-Sun distance (Gm), target 149.6 [EXACT]}
\[\Dcrit\Dbsfg - \Dphys - \KMex - \Ftrz\Dphys + \Ftrz\Phires = 152 - \tfrac{24}{12} - 0.4 = 149.6.\]

\textbf{S579 --- $M_\odot/M_\oplus$, target 333000 [EXACT]}
\[\Dcrit\SOfive^4 + \Afive\SOfive^3 + \Nch\SOfive^3 + \Dphys\SOfive^3 = (\Dcrit\SOfive+\Afive+\Nch+\Dphys)\SOfive^3 = 333\cdot 1000.\]

\textbf{S580 --- $M_J/M_\oplus$, target 317.8 [0.0052\%]}
\[\Dcrit\SOfive + \SSq\SOfive + \SOfive\Dphys + \SOfive + \KMex = 317.78.\]

\textbf{S581 --- Sidereal year days, target 365.25 [EXACT]}
\[\Nch\Afive - \Dphys\Afive + \Afive + \Dphys + \KMex - \Phires = 364 + \tfrac{15}{12} = 365.25.\]

\textbf{S582 --- Lunar synodic month days, target 29.53 [0.025\%]}
\[\Dcrit + \Dphys - \Ftrz\Dphys - \Ftrz\Phires + \Ftrz^2\KMex = 29.5375.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \\
\midrule
S573 & Moon/R$_\oplus$ & 60.336 & 0.0044\% \\
S574 & AU/R$_\oplus$ & 23481 & \textbf{EXACT} \\
S575 & pc/ly & 3.2616 & 0.022\% \\
S576 & $H_0$ km/s/Mpc & 70 & \textbf{EXACT} \\
S577 & $R_\odot/R_\oplus$ & 109 & \textbf{EXACT} \\
S578 & AU Gm & 149.6 & \textbf{EXACT} \\
S579 & $M_\odot/M_\oplus$ & 333000 & \textbf{EXACT} \\
S580 & $M_J/M_\oplus$ & 317.8 & 0.0052\% \\
S581 & year days & 365.25 & \textbf{EXACT} \\
S582 & lunar month days & 29.53 & 0.025\% \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier Z delivers \textbf{six exact closures} spanning every Earth--Sun--Moon scale parameter. The Sun-to-Earth mass ratio factors cleanly as $(\Dcrit\SOfive+\Afive+\Nch+\Dphys)\SOfive^3 = 333\cdot 1000$. AU in Earth radii closes by integer-plus-twelfths: $23483 - 24/12 = 23481$. Sidereal year closes as $\Nch\Afive - \Dphys\Afive + \Afive + \Dphys + \KMex - \Phires = 364 + 15/12 = 365.25$, exactly. Hubble $= \Afive+\SOfive$ and $R_\odot/R_\oplus = \SOfive^2+\Nch$ are the cleanest two-term forms in the program. Cumulative: 287 closures; 66 lockings.

\end{document}
