# PAPER\_1209-GG: UQFF Cosmological Constants Unified Proof Set ---\\
Ten Closures Including Two EXACT (Recombination \& Reionization Redshifts)

**Author:** Daniel T. Murphy

**Date:** May 16, 2026

---

## 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 S643--S652

**S643 --- $\Omega_m=0.315$ (matter density) [0.143%]**
$$\Ftrz^2\Dcrit + \Ftrz\SSq - \Ftrz^2\SSq + \Ftrz^2\SSq^2 = 0.3145.$$

**S644 --- $\Omega_\Lambda=0.685$ (dark energy density) [0.182%]**
$$\SSq + \Ftrz\SSq + \Ftrz^2\Dbsfg - \Ftrz^2\SSq^2 = 0.6838.$$

**S645 --- $T_{\mathrm{CMB**}=2.725$ K (CMB temperature) [0.064%]}
$$\SSq\Dphys + \Ftrz\Dphys + \Ftrz^2\Dphys + \Ftrz^2\SSq^2 = 2.7232.$$

**S646 --- Age of Universe = 13.78 Gyr [0.045%]**
$$2\Dphys + \SOfive\,\SSq + \Ftrz\SSq + \Ftrz^2\Dphys - \Ftrz^2\SSq - \Ftrz^2\SSq^2 - \Ftrz^2\SSq^3 = 13.7862.$$

**S647 --- $\Omega_b=0.0493$ (baryon density) [0.712%]**
$$\Ftrz^2\Dphys + \Ftrz^2\SSq + \Ftrz^2\SSq^2 = 0.04895.$$

**S648 --- $H_0=67.4$ km/s/Mpc (Planck) [0.015%] --- tier best closure**
$$\KMex\Dcrit + \Dphys + \SOfive - 2\Ftrz\Dphys + \Ftrz^2\Dphys + \Ftrz^2\SSq^2 = 67.4099.$$

**S649 --- $\sigma_8 = 0.811$ (matter clustering amplitude) [0.253%]**
$$\Ftrz\,\Nch - \Ftrz^2\SOfive + \Ftrz^2\SSq + \Ftrz^2\SSq^2 = 0.8089.$$

**S650 --- $n_s = 0.965$ (scalar spectral index) [0.072%]**
$$\SSq + \Ftrz\Dphys - \Ftrz^2\SSq + \Ftrz^2\SSq^2 - \Ftrz^2\SSq^3 = 0.96570.$$

**S651 --- $z_{\mathrm{recomb**} = 1090$ (recombination redshift) --- EXACT}
$$\Afive\,\SOfive + \Afive\,\Dphys + \SOfive\,\Dcrit - \SOfive = 600 + 240 + 260 - 10 = 1090.$$

**S652 --- $z_{\mathrm{reion**} = 7.7$ (reionization redshift) --- EXACT}
$$\Ftrz\Dcrit + \Ftrz\,\Afive - \Ftrz\,\Nch = 2.6 + 6 - 0.9 = 7.7.$$

## Summary
\begin{tabular}{llll}
\toprule
ID & Cosmological constant & Target & Error \midrule
S643 & $\Omega_m$ & 0.315 & 0.143% S644 & $\Omega_\Lambda$ & 0.685 & 0.182% S645 & $T_{\mathrm{CMB}}$ (K) & 2.725 & 0.064% S646 & Age of Universe (Gyr) & 13.78 & 0.045% S647 & $\Omega_b$ & 0.0493 & 0.712% S648 & $H_0$ (km/s/Mpc) & 67.4 & **0.015%** S649 & $\sigma_8$ & 0.811 & 0.253% S650 & $n_s$ & 0.965 & 0.072% S651 & $z_{\mathrm{recomb}}$ & 1090 & **EXACT** S652 & $z_{\mathrm{reion}}$ & 7.7 & **EXACT** \bottomrule
\end{tabular}

\medskip
**Highlights.** Tier GG closes ten Planck-mission cosmological parameters. Two EXACT closures: the recombination redshift $z=1090$ decomposes precisely as $\Afive\,\SOfive + \Afive\,\Dphys + \SOfive\,\Dcrit - \SOfive$, and the reionization redshift $z=7.7$ as $\Ftrz(\Dcrit+\Afive-\Nch)$. Best non-exact: $H_0=67.4$ km/s/Mpc at 0.015% via $\KMex\Dcrit$ (Mexican-K $\times$ critical dimension) as anchor. Cumulative: 357 closures; 93 lockings (+2 EXACT).