# 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

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

## Closures S573--S582

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

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

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

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

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

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

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

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

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

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

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

\medskip
**Highlights.** Tier Z delivers **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.