\documentclass[11pt]{article}
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\title{PAPER\_1209-FF: UQFF Mathematical Constants Unified Proof Set ---\\
Ten Rational Closures of Classical Irrationals (best 0.019\%)}
\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}
$\Ftrz=\tfrac{1}{10}$, $\Phires=\tfrac{5}{6}$, $\SSq=\tfrac{57}{100}$, $\KMex=\tfrac{25}{12}$, $\Dphys=4$, $\Dbsfg=6$, $\Dcrit=26$, $\Nch=9$, $\SOfive=10$, $\Afive=60$.

\textbf{Note.} Classical mathematical constants ($\pi, e, \varphi, \gamma, \sqrt{2}, \sqrt{3}, \sqrt{5}, \ln 2, \ln 10, \zeta(3)$) are \emph{irrational}: no finite rational combination of UQFF primitives can be exact. This tier delivers ten high-precision rational closures, the best at 0.019\% ($\ln 10$).

\section*{Closures S633--S642}

\textbf{S633 --- $\pi$, target 3.14159 [0.031\%]}
\[\Phires\Dphys - \Ftrz^2\SOfive - \Ftrz^2\Dphys - \Ftrz\SSq - \Ftrz^2\SSq + \Ftrz^2 = 3.1406.\]

\textbf{S634 --- $e$ (Euler), target 2.71828 [0.050\%]}
\[\KMex + \SSq + \Ftrz\SSq + \Ftrz^2\Dphys - \Ftrz^2\,\SOfive/\Dphys - \Ftrz^2\SSq = 2.7196.\]

\textbf{S635 --- $\varphi$ (golden ratio), target 1.61803 [0.101\%]}
\[\Phires + \Phires - \Ftrz\SSq + \Ftrz^2 = 1.6197.\]

\textbf{S636 --- $\gamma$ (Euler--Mascheroni), target 0.57721 [0.262\%]}
\[\Ftrz\Nch\SSq + \Ftrz^2\SOfive\SSq + \Ftrz^2\SSq = 0.5757.\]

\textbf{S637 --- $\sqrt{2}$, target 1.41421 [0.105\%]}
\[\SSq + \Ftrz\Dphys + \Ftrz\Dphys + \Ftrz^2\SSq + \Ftrz^2\Dphys = 1.4157.\]

\textbf{S638 --- $\sqrt{3}$, target 1.73205 [0.023\%]}
\[\SSq + 3\,\Ftrz\Dphys + \Ftrz^2\SSq - \Ftrz^2\Dphys - \Ftrz^2\SSq^2 = 1.7324.\]

\textbf{S639 --- $\sqrt{5}$, target 2.23607 [0.045\%]}
\[\KMex + \Ftrz\SSq + \Ftrz^2\Dbsfg + \Ftrz^2\Dphys - \Ftrz^2\SSq^2 = 2.2371.\]

\textbf{S640 --- $\ln 2$, target 0.69315 [0.070\%]}
\[\Phires - 2\,\Ftrz\SSq - \Ftrz^2\Dbsfg + \Ftrz^2\Dphys - \Ftrz^2\SSq = 0.6936.\]

\textbf{S641 --- $\ln 10$, target 2.30259 [0.019\%] --- tier best}
\[\KMex + 2\,\Ftrz\SSq + \Ftrz^2\Dbsfg + \Ftrz^2\Dphys + \Ftrz^2\SSq = 2.3030.\]

\textbf{S642 --- $\zeta(3)$ (Apéry), target 1.20206 [0.033\%]}
\[2\,\SSq + \Ftrz^2\Dbsfg + \Ftrz^2\SSq - \Ftrz^2\SSq^2 = 1.2024.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Constant & Target & Error \\
\midrule
S633 & $\pi$ & 3.14159 & 0.031\% \\
S634 & $e$ & 2.71828 & 0.050\% \\
S635 & $\varphi$ & 1.61803 & 0.101\% \\
S636 & $\gamma$ & 0.57721 & 0.262\% \\
S637 & $\sqrt{2}$ & 1.41421 & 0.105\% \\
S638 & $\sqrt{3}$ & 1.73205 & 0.023\% \\
S639 & $\sqrt{5}$ & 2.23607 & 0.045\% \\
S640 & $\ln 2$ & 0.69315 & 0.070\% \\
S641 & $\ln 10$ & 2.30259 & \textbf{0.019\%} \\
S642 & $\zeta(3)$ & 1.20206 & 0.033\% \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier FF closes ten of the most famous irrational constants of mathematics using only the 11 locked UQFF primitives. Five of the ten close under 0.05\%; the tier best is $\ln 10$ at 0.019\%, followed by $\sqrt{3}$ at 0.023\% and $\pi$ at 0.031\%. Cumulative: 347 closures; 91 lockings (no new exacts --- irrational targets cannot admit rational closure).

\end{document}
