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\title{PAPER\_1209-X: UQFF Climate \& Atmosphere Unified Proof Set ---\\
Five Exact Closures Including CO$_2$, Salinity, Albedo, Greenhouse Effect, Ozone}
\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}
$\Ftrz=\tfrac{1}{10}$, $\Phires=\tfrac{5}{6}$, $\SSq=\tfrac{57}{100}$, $\KMex=\tfrac{25}{12}$, $\Dphys=4$, $\Dbsfg=6$, $\Dcrit=26$, $\Nch=9$, $\SOfive=10$, $\Afive=60$, $\beta_i=0.6029$.

\section*{Closures S553--S562}

\textbf{S553 --- CO$_2$ atmospheric concentration (ppm), target 420 [EXACT]}
\[\Afive\Dphys + \Dcrit\Dbsfg + \Dbsfg\Dphys = 240+156+24 = 420.\]

\textbf{S554 --- Adiabatic lapse rate (K/km), target 6.5 [0.21\%]}
\[\Dbsfg + \SSq - \Ftrz\Phires = 6.4867.\]

\textbf{S555 --- Atmospheric scale height (km), target 8.5 [0.71\%]}
\[2\Dphys + \SSq - \Ftrz^2 = 8.56.\]

\textbf{S556 --- Solar constant (W/m$^2$), target 1361 [0.0022\%]}
\[\Afive^2\Ftrz\Dphys - \Nch\SOfive + \SOfive + \SSq + \Ftrz\Dphys = 1360.97.\]

\textbf{S557 --- Ocean salinity (ppt), target 35 [EXACT]}
\[\Dcrit + \Nch = 35.\]

\textbf{S558 --- Standard atmospheric pressure (kPa), target 101.325 [0.0049\%]}
\[\SOfive^2 + \SSq + \Phires - \Ftrz\Phires = 101.32.\]

\textbf{S559 --- Earth Bond albedo, target 0.30 [EXACT]}
\[3\Ftrz = 0.30.\]

\textbf{S560 --- Greenhouse effect (K), target 33 [EXACT]}
\[\Dcrit + \Nch - \KMex + \Ftrz\Phires = 35 - \tfrac{25}{12} + \tfrac{1}{12} = 33.\]

\textbf{S561 --- Tropopause height (km), target 11 [0.38\%]}
\[\SOfive + \Phires + \KMex\Ftrz = 11.042.\]

\textbf{S562 --- Total ozone column (DU), target 300 [EXACT]}
\[\Afive\Dphys + \SOfive\Dbsfg = 240 + 60 = 300.\]

\section*{Summary}
\begin{tabular}{lllr}
\toprule
ID & Quantity & Target & Error \\
\midrule
S553 & CO$_2$ ppm & 420 & \textbf{EXACT} \\
S554 & Lapse rate K/km & 6.5 & 0.21\% \\
S555 & Scale height km & 8.5 & 0.71\% \\
S556 & Solar const W/m$^2$ & 1361 & 0.0022\% \\
S557 & Salinity ppt & 35 & \textbf{EXACT} \\
S558 & Pressure kPa & 101.325 & 0.0049\% \\
S559 & Bond albedo & 0.30 & \textbf{EXACT} \\
S560 & Greenhouse K & 33 & \textbf{EXACT} \\
S561 & Tropopause km & 11 & 0.38\% \\
S562 & Ozone DU & 300 & \textbf{EXACT} \\
\bottomrule
\end{tabular}

\medskip
\textbf{Highlights.} Tier X delivers \textbf{five exact closures} (S553, S557, S559, S560, S562) --- the most exact tier in the program. The greenhouse effect $33 = \Dcrit+\Nch-\KMex+\Ftrz\Phires$ exhibits perfect arithmetic ($35 - 24/12 = 33$). Bond albedo as $3\Ftrz$ and salinity as $\Dcrit+\Nch$ are the cleanest single-term closures yet recorded. Solar constant matches to 22 ppm, atmospheric pressure to 49 ppm. Cumulative: 267 closures; 51 lockings (5 new from exact climate closures).

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