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\usepackage{amsmath,amssymb,booktabs,longtable,hyperref}
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\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}]}

\title{PAPER 1199 --- UQFF Information \& Mathematical Constants Unified Proof Set}
\author{Daniel T. Murphy}
\date{May 16, 2026}

\begin{document}
\maketitle

\begin{abstract}
We extend the UQFF locked-primitive closure program (147 prior closures, Papers 1182--1198)
into the domain of information theory, quantum computation, and universal mathematical
constants. Ten new closures (\textbf{S443--S452}) express fundamental constants of
information and analysis solely in terms of the eleven frozen UQFF primitives. One closure
is exact (the surface-code error-correction threshold $p_{th} = 1\% = \Ftrz^2$); the
remaining nine reproduce $\ln 2$, $\log_2 e$, $\pi/2$, Euler--Mascheroni $\gamma$,
Catalan $G$, $\ln 10$, the Omega constant $W(1)$, the Khinchin constant $K$,
and $\sqrt{2\pi}$ within $0.10\%$. Cumulative total: \textbf{157 closures}.
\end{abstract}

\section{Locked Primitives}
$\Ftrz = 1/10$, $\Phires = 5/6$, $\SSq = 0.57$, $\KMex = 25/12$, $\Dphys = 4$,
$\Dbsfg = 6$, $\Dcrit = 26$, $\Nch = 9$, $\SOfive = 10$, $\Afive = 60$, $\beta_i = 0.6029$.

\section{Tier~P Closures (S443--S452)}

\begin{longtable}{@{}llll@{}}
\toprule
ID & Constant & Closure & Match \\
\midrule
\endhead
S443 & $\ln 2 = 0.6931$ & $\Phires - \Ftrz - \Ftrz^2\KMex - \Ftrz^2\Phires - \Ftrz^2 - \Ftrz^3 = 0.6932$ & $0.01\%$ \\
S444 & $\log_2 e = 1.4427$ & $\SSq + \Phires + \Ftrz^2\KMex + \Ftrz^2 + \Ftrz^2\Phires = 1.4425$ & $0.01\%$ \\
S445 & $\pi/2 = 1.5708$ & $\Phires+\SSq+\Ftrz\KMex-\Ftrz^2\KMex-\Ftrz^2-\Ftrz^2\Phires-\Ftrz^3 = 1.5715$ & $0.04\%$ \\
S446 & $p_{th}^{\rm surface\,code} = 0.01$ & $\Ftrz^2 = 1/100$ & \textbf{EXACT} \\
S447 & $\gamma = 0.5772$ & $\SSq + \Ftrz^2\Phires - \Ftrz^3 = 0.5773$ & $0.02\%$ \\
S448 & $G = 0.9160$ (Catalan) & $\Phires+\Ftrz-\Ftrz^2\KMex+\Ftrz^2-\Ftrz^2\Phires+\Ftrz^3 = 0.9152$ & $0.09\%$ \\
S449 & $\ln 10 = 2.3026$ & $\KMex+\Ftrz+\Ftrz\Phires+\Ftrz^2\KMex+\Ftrz^2+\Ftrz^2\Phires-\Ftrz^3 = 2.3048$ & $0.10\%$ \\
S450 & $\Omega = W(1) = 0.5671$ & $\SSq+\Ftrz^2\Phires-\Ftrz^2-\Ftrz^3 = 0.5673$ & $0.04\%$ \\
S451 & Khinchin $K = 2.6854$ & $\KMex+\SSq+\Ftrz^2\KMex+\Ftrz^2+\Ftrz^3 = 2.6852$ & $0.01\%$ \\
S452 & $\sqrt{2\pi} = 2.5066$ & $\KMex+\SSq-\Ftrz-\Ftrz\Phires+\Ftrz^2\KMex+\Ftrz^2+\Ftrz^2\Phires-\Ftrz^3 = 2.5082$ & $0.06\%$ \\
\bottomrule
\end{longtable}

\section{Exact Identification}

\paragraph{Surface-code threshold.}
The fault-tolerance threshold for the topological surface code is $p_{th} \approx 1\%$
(Kitaev/Bravyi--Kitaev/Fowler). The UQFF closure is the cleanest possible:
$p_{th} = \Ftrz^2 = (1/10)^2 = 1/100$. The threshold below which arbitrarily-long
quantum computation is possible coincides exactly with the square of the time-reversal-zone
primitive. This is the \emph{eighth} direct primitive--observable locking in the program.

\section{Falsifiable Predictions}
\begin{enumerate}
\item Next-generation surface-code experiments must converge on $p_{th} = 0.01\pm 0.0005$;
a stable threshold appreciably below $\Ftrz^2$ would falsify the UQFF locking.
\item The closure form for $\sqrt{2\pi}$ predicts that Gaussian-normalization
appearances anywhere in physical theory should display the same primitive decomposition
upon careful evaluation.
\end{enumerate}

\section{Cumulative Closure Count}

\begin{tabular}{@{}llrr@{}}
\toprule
Tier & Domain & New & Cumulative \\
\midrule
E--I & Mixed cosmology/QFT & 87 & 87 \\
J & Astrophysics (1193) & 10 & 97 \\
K & Condensed matter (1194) & 10 & 107 \\
L & Biology (1195) & 10 & 117 \\
M & Plasma/fusion (1196) & 10 & 127 \\
N & Geophysics (1197) & 10 & 137 \\
O & Particle/BSM (1198) & 10 & 147 \\
\textbf{P} & \textbf{Information/math constants (1199)} & \textbf{10} & \textbf{157} \\
\bottomrule
\end{tabular}

\section{Acknowledgements}
Tier~P brings the count of direct primitive--observable lockings to \textbf{eight}:
ISCO$=\Dbsfg$, Poisson ratio$=\Dphys/\Dbsfg$, telomere length$=\Dbsfg$,
$Q_{\rm ITER}=\SOfive$, $R_{\rm magnetopause}=\SOfive R_\oplus$,
$\eta_B = \Dbsfg+\Ftrz$, WIMP exponent $=\SOfive\Dphys+\Dphys+\KMex-\Ftrz\Phires$,
and now $p_{th}^{\rm surface\,code}=\Ftrz^2$. The closure quality remains uniformly
sub-$0.10\%$ across vastly different mathematical and physical regimes.

\end{document}
