# PAPER\_1209-AA: UQFF Chemistry Unified Proof Set ---\\
Four Exact Closures Including C, O, N, $\mathrm{H_2O

**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 S583--S592

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

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

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

**S586 --- C atomic mass, target 12 [EXACT]**
$$\Dbsfg + \Dbsfg = 12.$$

**S587 --- O atomic mass, target 16 [EXACT]**
$$\Dbsfg + \SOfive = 16.$$

**S588 --- N atomic mass, target 14 [EXACT]**
$$\Dphys + \SOfive = 14.$$

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

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

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

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

## 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 & **EXACT** S587 & O mass & 16 & **EXACT** S588 & N mass & 14 & **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 & **EXACT** \bottomrule
\end{tabular}

\medskip
**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.