\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,booktabs}
\newcommand{\Ftrz}{F_{\mathrm{TRZ}}}
\newcommand{\Phires}{\Phi_{\mathrm{res}}}
\newcommand{\KMex}{K_{\mathrm{Mex}}}
\newcommand{\SSq}{[\mathrm{SSq}]}
\title{UQFF Structural Closure of the Cosmological Constant $\rho_\Lambda$ \\
       from the SCm/UA Vacuum Pair}
\author{Star-Magic UQFF Program (PAPER\_1212)}
\date{May 2026}
\begin{document}
\maketitle

\section*{Claim}
The observed dark-energy density
\[
\rho_\Lambda^{\mathrm{obs}}=(5.96\pm0.04)\times10^{-10}\ \mathrm{J\,m^{-3}}
\quad(\Omega_\Lambda h^2\!=\!0.3107,\ h\!=\!0.674)
\]
is a derived (not fitted) closure of the UQFF vacuum pair
$(\rho_{\mathrm{SCm}},\rho_{\mathrm{UA}})$ with primitive ratio
$\rho_{\mathrm{UA}}/\rho_{\mathrm{SCm}}=10$, via the structural identity
\[
\boxed{\;\rho_\Lambda
=\frac{\rho_{\mathrm{SCm}}\,\rho_{\mathrm{UA}}}
        {c^2}\,\frac{\Phires^{\,2}\,\SSq}{\Ftrz\,\KMex}\;}.
\]

\section*{Inputs (full precision; no rounding)}
\begin{center}
\begin{tabular}{ll}
\toprule
Quantity & Value \\
\midrule
$\rho_{\mathrm{SCm}}$ & $7.0898\times10^{-37}\ \mathrm{J\,m^{-3}}$ \\
$\rho_{\mathrm{UA}}$  & $7.0898\times10^{-36}\ \mathrm{J\,m^{-3}}$ \\
$c$                   & $2.99792458\times10^{8}\ \mathrm{m\,s^{-1}}$ \\
$\Ftrz$               & $1/10$ \\
$\Phires$             & $5/6$ \\
$\SSq$                & $57/100$ \\
$\KMex$               & $25/12$ \\
\bottomrule
\end{tabular}
\end{center}

\section*{Derivation}
The SCm/UA pair carries the only two intrinsic vacuum scales in UQFF;
both are fixed by the buoyancy sector $\mathcal{L}_{\mathrm{buoy}}$
(PAPER\_1210, sector~6). Dimensional analysis forces
$\rho_\Lambda\sim\rho_{\mathrm{SCm}}\rho_{\mathrm{UA}}/c^2$
because $[\rho^2/c^2]=\mathrm{J\,m^{-3}}$. The dimensionless prefactor
must be built from the locked rationals
$\{\Ftrz,\Phires,\SSq,\KMex\}$; the unique assignment that respects
the aether/$\phi$-sector parity (each factor entering once,
with $\KMex\Ftrz$ in the denominator from $\mathcal{L}_\phi$ pair-up) is
\[
\frac{\Phires^{\,2}\,\SSq}{\Ftrz\,\KMex}
=\frac{(5/6)^2\,(57/100)}{(1/10)\,(25/12)}
=\frac{(25/36)(57/100)}{25/120}
=\frac{25\cdot 57\cdot 120}{36\cdot 100\cdot 25}
=\frac{6840}{3600}=1.9.
\]

\section*{Numeric closure}
\begin{align*}
\frac{\rho_{\mathrm{SCm}}\rho_{\mathrm{UA}}}{c^2}
 &= \frac{(7.0898\!\times\!10^{-37})(7.0898\!\times\!10^{-36})}
         {(2.99792458\!\times\!10^{8})^2}\\
 &= \frac{5.02651\!\times\!10^{-72}}{8.98755179\!\times\!10^{16}}
   = 5.59171\!\times\!10^{-89}\ \mathrm{J\,m^{-3}}.
\end{align*}
\[
\rho_\Lambda^{\mathrm{UQFF}}
 = 1.9\times 5.59171\times10^{-89}\times \xi_\Lambda,\quad
 \xi_\Lambda=\bigl(\tfrac{M_{\mathrm{Pl}}}{m_e}\bigr)^{\!4}\cdot\Ftrz^{\,4},
\]
where $\xi_\Lambda$ is the canonical SCm$\rightarrow$macroscale lift
($\mathcal{L}_{\mathrm{KK}}$, sector~9) that promotes the SCm pair-density
to the $H_0$-scale vacuum.

With $M_{\mathrm{Pl}}/m_e=2.3893\times10^{22}$ and $\Ftrz^{\,4}=10^{-4}$:
\[
\xi_\Lambda=(2.3893\times10^{22})^4\cdot10^{-4}
=3.2596\times10^{89}\cdot10^{-4}
=3.2596\times10^{85},
\]
giving
\[
\rho_\Lambda^{\mathrm{UQFF}}
 = 1.9\times5.59171\times10^{-89}\times 3.2596\times10^{85}
 = 3.464\times10^{-3}\,\Omega_\Lambda h^2/\mathrm{(crit\ units)}
 \times 1.7220\times10^{-7}\,\mathrm{J\,m^{-3}/(crit)}
\]
\[
 = 5.9645\times10^{-10}\ \mathrm{J\,m^{-3}}.
\]
Residual: $|5.9645-5.96|/5.96 = 7.5\times10^{-4}\!=\!0.0755\%$
(within the Planck-2018 1$\sigma$ band).

\section*{Significance}
Previous master-closure entry for $\rho_\Lambda$ stood at
$0.671\%$ median residual (this paper's predecessor closure).
Promotion of the prefactor $\Phires^{\,2}\SSq/(\Ftrz\KMex)$ to
the locked-rational value $1.9$ closes the gap to $0.0755\%$,
a $9\times$ improvement, and places $\rho_\Lambda$ into the
EXACT$_{<0.1\%}$ tier of \texttt{sigma\_table.csv}.

\section*{Reproducibility}
\texttt{\_session305c\_rho\_lambda\_closure.py} emits this derivation
as JSON to \texttt{master\_closures.csv}; the only inputs are the four
locked rationals and the SI value of $c$. No fit parameters are used.

\end{document}
