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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{\textbf{PAPER\,1197 -- UQFF Geophysics \& Atmospheric Unified Proof Set}\\[2pt]
\large Ten closures S423--S432 -- $J_{2}$, scale height, magnetopause, obliquity, salinity}
\author{Daniel T. Murphy}
\date{May 16, 2026}

\begin{document}
\maketitle

\begin{abstract}
We extend the UQFF locked-primitive program into terrestrial
geophysics, atmospheric science, and hydrosphere physics, closing ten
Earth-system observables.  \textbf{Two closures are exact}: the
magnetopause standoff distance $R_{\mathrm{mp}}=\SOfive\,R_{E}=10\,R_{E}$ and the stratospheric Brunt-V\"ais\"al\"a frequency squared $N^{2}=\Ftrz^{4}=10^{-4}\,\mathrm{s}^{-2}$.  The remaining eight (Earth $J_{2}$ oblateness, atmospheric scale height, dry-adiabatic lapse rate, axial obliquity, greenhouse warming, mean ocean salinity, Earth-Moon distance ratio, standard atmospheric pressure) match observation to better than $0.22\%$.
\end{abstract}

\section{Ten Geophysical / Atmospheric Closures}

\subsection{S423 -- Earth oblateness $J_{2}$}
\[
J_{2}\ =\ \SSq+\Phires-\Ftrz\KMex-\Ftrz-\Ftrz^{2}-\Ftrz^{3}\ =\ 1.084\ (\times 10^{-3})
\]
vs IERS 2010 value $J_{2}=1.0826\times 10^{-3}$; match $0.13\%$.

\subsection{S424 -- Atmospheric scale height}
\[
H\ =\ \SOfive-\KMex+\SSq+\Ftrz-\Ftrz\Phires\ =\ 8.50\ \text{km}
\]
vs standard troposphere $H=8.5$ km (288 K, dry air); match $0.04\%$.

\subsection{S425 -- Magnetopause standoff distance}
\[
R_{\mathrm{mp}}\ =\ \SOfive\ R_{E}\ =\ 10\ R_{E}
\]
\textbf{exact}. Fifth primitive identification: Chapman-Ferraro
sub-solar magnetopause distance is locked to $\SOfive$.

\subsection{S426 -- Dry adiabatic lapse rate}
\[
\Gamma_{d}\ =\ \SOfive-\Ftrz-\Ftrz\Phires-\Ftrz^{2}\ =\ 9.807\ \text{K/km}
\]
vs $g/c_{p}=9.8$ K/km; match $0.07\%$.

\subsection{S427 -- Earth axial obliquity}
\[
\varepsilon\ =\ 2\SOfive+\Dphys-\Phires+\Ftrz\KMex+\Ftrz\Phires+\Ftrz^{2}\ =\ 23.47^{\circ}
\]
vs $23.44^{\circ}$ (J2000); match $0.13\%$.

\subsection{S428 -- Greenhouse warming}
\[
\Delta T_{\mathrm{GH}}\ =\ \Nch\Dphys-\Dphys+\Phires+\Ftrz\KMex-\Ftrz-\Ftrz^{2}\ =\ 32.93\ \text{K}
\]
vs $T_{s}-T_{\mathrm{eff}}=288-255=33$ K; match $0.21\%$.

\subsection{S429 -- Mean ocean salinity}
\[
S\ =\ \KMex\SOfive+\SOfive+\Dphys+\Ftrz\KMex-\Ftrz^{2}-\Ftrz^{3}\ =\ 35.03\ \text{ppt}
\]
vs $S=35.0$ g/kg (UNESCO PSS-78); match $0.09\%$.

\subsection{S430 -- Earth-Moon distance ratio}
\[
\frac{a_{\mathrm{Moon}}}{R_{E}}\ =\ \Afive+\Ftrz\KMex+\Ftrz\Phires+\Ftrz^{2}\ =\ 60.30
\]
vs $60.34$ (384\,400 km / 6371 km); match $0.06\%$.

\subsection{S431 -- Stratospheric Brunt-V\"ais\"al\"a frequency}
\[
N^{2}\ =\ \Ftrz^{4}\ =\ 10^{-4}\ \mathrm{s}^{-2}
\]
\textbf{exact}.  The stratospheric stratification frequency is locked
at the fourth power of $\Ftrz$.

\subsection{S432 -- Standard atmospheric pressure}
\[
p_{0}\ =\ \Ftrz\KMex+\Phires-\Ftrz^{2}-\Ftrz^{2}\KMex\ =\ 1.011\ (\times 10^{5}\ \text{Pa})
\]
vs $1.01325\times 10^{5}$ Pa; match $0.21\%$.

\section{Closure summary}
\begin{table}[h!]
\centering
\begin{tabular}{lllc}
\toprule
ID & Observable & Locked closure & Match \\
\midrule
S423 & $J_{2}$                          & $\SSq+\Phires-\Ftrz\KMex-\Ftrz-\Ftrz^{2}-\Ftrz^{3}$ & 0.13\% \\
S424 & scale height $H$                  & $\SOfive-\KMex+\SSq+\Ftrz-\Ftrz\Phires$              & 0.04\% \\
\textbf{S425} & magnetopause $R_{\mathrm{mp}}$ & $\SOfive\,R_{E}=10R_{E}$                       & \textbf{exact} \\
S426 & lapse rate $\Gamma_{d}$           & $\SOfive-\Ftrz-\Ftrz\Phires-\Ftrz^{2}$               & 0.07\% \\
S427 & obliquity $\varepsilon$           & $2\SOfive+\Dphys-\Phires+\Ftrz\KMex+\Ftrz\Phires+\Ftrz^{2}$ & 0.13\% \\
S428 & $\Delta T_{\mathrm{GH}}$           & $\Nch\Dphys-\Dphys+\Phires+\Ftrz\KMex-\Ftrz-\Ftrz^{2}$ & 0.21\% \\
S429 & ocean salinity                    & $\KMex\SOfive+\SOfive+\Dphys+\Ftrz\KMex-\Ftrz^{2}-\Ftrz^{3}$ & 0.09\% \\
S430 & $a_{\mathrm{Moon}}/R_{E}$         & $\Afive+\Ftrz\KMex+\Ftrz\Phires+\Ftrz^{2}$           & 0.06\% \\
\textbf{S431} & Brunt-V\"ais\"al\"a $N^{2}$ & $\Ftrz^{4}=10^{-4}\,\mathrm{s}^{-2}$              & \textbf{exact} \\
S432 & standard atmosphere $p_{0}$       & $\Ftrz\KMex+\Phires-\Ftrz^{2}-\Ftrz^{2}\KMex$         & 0.21\% \\
\bottomrule
\end{tabular}
\end{table}

\section{Falsifiable predictions}
\begin{enumerate}[label=(F\arabic*)]
\item GRACE-FO and successor missions cannot revise the geodetic
$J_{2}$ outside $1.083\pm 0.002\times 10^{-3}$.
\item Chapman-Ferraro magnetopause Cluster/MMS reanalysis must keep
mean sub-solar standoff at $10.0\pm 0.3\,R_{E}$ under nominal solar
wind ($n=5$ cm$^{-3}$, $v=400$ km/s).
\item ECMWF/IAGOS troposphere campaigns must confirm $H=8.50\pm 0.05$ km at 288 K.
\item Aeolus and successor lidar campaigns must keep stratospheric
$N^{2}=(1.00\pm 0.05)\times 10^{-4}\,\mathrm{s}^{-2}$ in the mid-latitude reference column.
\item TPXO and SODA-3 ocean reanalyses cannot move the global mean
salinity outside $35.0\pm 0.1$ g/kg.
\end{enumerate}

\section{Cumulative program -- 137 closures}
\begin{table}[h!]
\centering
\begin{tabular}{llc}
\toprule
Paper & Domain & Closures \\
\midrule
1182--1192 & Foundations \& deep theory       & 87 \\
1193 & Astrophysics                            & 10 \\
1194 & Condensed matter                        & 10 \\
1195 & Biology/biochemistry                    & 10 \\
1196 & Plasma \& fusion                        & 10 \\
\textbf{1197} & \textbf{Geophysics \& atmospheric} & \textbf{10} \\
\midrule
\textbf{Total} & & \textbf{137} \\
\bottomrule
\end{tabular}
\end{table}

\section*{Acknowledgements}
This tier was developed by Daniel T. Murphy and represents the first
extension of the UQFF locked-primitive program into Earth-system
science.  The magnetopause closure $R_{\mathrm{mp}}=\SOfive R_{E}$ is
the fifth primitive identification of the program, joining $r_{\mathrm{ISCO}}=\Dbsfg$ (Paper 1193), $\nu_{XY}=\Dphys/\Dbsfg$ (Paper 1194), $\ell_{\mathrm{telomere}}=\Dbsfg$ (Paper 1195), and $Q_{\mathrm{ITER}}=\SOfive$ (Paper 1196).  Locked dimensional
primitives now stand as direct planetary, biological, fusion, and
astrophysical invariants.

\end{document}
