\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,booktabs}
\title{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}

\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 S643--S652}

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

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

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

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

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

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

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

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

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

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

\section*{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 & \textbf{0.015\%} \\
S649 & $\sigma_8$ & 0.811 & 0.253\% \\
S650 & $n_s$ & 0.965 & 0.072\% \\
S651 & $z_{\mathrm{recomb}}$ & 1090 & \textbf{EXACT} \\
S652 & $z_{\mathrm{reion}}$ & 7.7 & \textbf{EXACT} \\
\bottomrule
\end{tabular}

\medskip
\textbf{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).

\end{document}
