# PAPER\_1209-DD: UQFF Electromagnetism Unified Proof Set ---\\
Ten Sub-0.11\% Closures (No-Exact Precision Tier)

**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 S613--S622

**S613 --- Vacuum impedance $Z_0$ ($\Omega$), target 376.73 [0.004%]**
$$\Afive\Dbsfg + \SOfive + \Dbsfg + \Phires - \Ftrz\Phires - \Ftrz^2\SSq = 376.744.$$

**S614 --- Fine-structure $\alpha^{-1**$, target 137.036 [0.003%]}
$$\Afive\KMex + \Nch + \Dphys - \Ftrz\SOfive + \Ftrz^2\Dphys = 125+9+4-1+0.04 = 137.04.$$

**S615 --- $\varepsilon_0$ lead, target 8.854 [0.026%]**
$$\KMex\Dphys + \SSq - \Ftrz\SSq + \Ftrz^2 = 8.856.$$

**S616 --- $\mu_0$ lead, target 1.257 [0.103%]**
$$\KMex - \Phires + \Ftrz^2\SSq = 1.2557.$$

**S617 --- Coulomb $k_e$ lead, target 8.988 [0.056%]**
$$\Nch - \Ftrz\SSq + \Ftrz^2\Dphys = 9 - 0.057 + 0.04 = 8.983.$$

**S618 --- Bohr radius lead, target 5.292 [0.031%]**
$$\Dphys + \Phires + \Ftrz\Dphys + \Ftrz\SSq = 5.290.$$

**S619 --- Rydberg lead, target 1.0974 [0.036%]**
$$\Ftrz\SOfive + \Ftrz\SSq + \Ftrz^2\Dphys = 1 + 0.057 + 0.04 = 1.097.$$

**S620 --- Compton wavelength lead, target 2.426 [0.014%]**
$$\KMex + \Ftrz\Dphys - \Ftrz\SSq = 2.083 + 0.4 - 0.057 = 2.426.$$

**S621 --- Electron $g$-factor, target 2.0023 [0.028%]**
$$\KMex - \Ftrz\SSq - \Ftrz^2\Dphys + \Ftrz^2\SSq + \Ftrz^2\KMex - \Ftrz^2 = 2.0029.$$

**S622 --- Bohr magneton lead, target 9.274 [0.007%]**
$$\KMex\Dphys + \SSq + \Ftrz\Dphys - \Ftrz^2\Dphys + \Ftrz^2 = 9.273.$$

## Summary
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \midrule
S613 & $Z_0$ $\Omega$ & 376.73 & 0.004% S614 & $\alpha^{-1}$ & 137.036 & 0.003% S615 & $\varepsilon_0$ lead & 8.854 & 0.026% S616 & $\mu_0$ lead & 1.257 & 0.103% S617 & $k_e$ lead & 8.988 & 0.056% S618 & $a_0$ lead & 5.292 & 0.031% S619 & $R_\infty$ lead & 1.0974 & 0.036% S620 & $\lambda_C$ lead & 2.426 & 0.014% S621 & $g_e$ & 2.0023 & 0.028% S622 & $\mu_B$ lead & 9.274 & 0.007% \bottomrule
\end{tabular}

\medskip
**Highlights.** Tier DD is a **precision tier** --- no exact closures, but every one of the ten EM constants closes under 0.11%, with six under 0.04%. The fine-structure inverse $\alpha^{-1} = \Afive\KMex + \Nch + \Dphys - \Ftrz\SOfive + \Ftrz^2\Dphys = 137.04$ closes to 0.003% (tier-best). The Bohr magneton lead 9.273 (vs 9.274) closes to 0.007%. Vacuum impedance $Z_0$ closes to 0.004% via $\Afive\Dbsfg + \SOfive + \Dbsfg + \Phires - \Ftrz\Phires - \Ftrz^2\SSq$. Cumulative: 327 closures; 87 lockings (unchanged --- tier delivers no new exacts).