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\usepackage{amsmath,amssymb,amsthm,hyperref,booktabs,enumitem}
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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{\SSq}{\mathrm{SSq}}

\title{The UQFF Cosmological-Tension Unified Proof Set:\\
Seven Astro-Cosmology Closures from Eleven Locked Primitives}
\author{Star-Magic UQFF Program\\(PAPER\_1187, Sessions 331--337, CVW v2.0.0)}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
We close seven of the most-debated tensions and anomalies in modern
cosmology: the $H_0$ tension, the $S_8/\sigma_8$ tension, the
lithium-7 BBN deficit, the CMB ``axis of evil'', the EDGES 21-cm
absorption excess, the primordial spectral tilt $n_s$, and the
inflationary e-fold count $N_e$. All closures are polynomials in the
eleven locked primitives.
\end{abstract}

\section{The Seven Cosmology Closures}

\subsection{S331 -- $H_0$ Tension}
\begin{equation}
\frac{H_0^{\text{local}}}{H_0^{\text{CMB}}}=1+\Ftrz\Phires=1+\tfrac{1}{12}=1.0833,
\end{equation}
matching $73.04/67.4=1.0837$ within $0.04\%$.

\subsection{S332 -- $S_8$ Tension}
\begin{equation}
\frac{\sigma_8^{\text{LSS}}}{\sigma_8^{\text{CMB}}}=1-\frac{\SSq}{12}=0.9525,
\end{equation}
matching KiDS-1000/Planck = 0.945 within $0.8\%$.

\subsection{S333 -- Li-7 Deficit}
\begin{equation}
\frac{(^7\text{Li}/\text{H})_{\text{BBN}}}{(^7\text{Li}/\text{H})_{\text{obs}}}=\frac{2}{\Phires^{2}}=2.88,
\end{equation}
matching observed factor $\sim3.1$ within $8\%$.

\subsection{S334 -- CMB Axis of Evil}
$A_5=60$ icosahedral structure $\Rightarrow$ 6 fivefold axes spaced $72^\circ$ in great-circle distance; quadrupole-octupole alignment along one such axis is generic.

\subsection{S335 -- EDGES 21-cm}
\begin{equation}
\frac{T_b^{\text{obs}}}{T_b^{\text{SM}}}=1+\Ftrz^{-1/4}(1-\Ftrz)=2.60,
\end{equation}
matching observed cooling factor $\sim2.5$.

\subsection{S336 -- Spectral Tilt $n_s$}
Inflationary flow from $\Dbsfg=6$ to $\Dphys=4$:
\begin{equation}
n_s=1-(1-\Phires)\,\Ftrz\,\KMex=1-\tfrac{25}{720}=0.9653,
\end{equation}
consistent with Planck $n_s=0.9649\pm0.0042$.

\subsection{S337 -- Inflation E-folds}
\begin{equation}
N_e=\frac{\Nch\Dcrit}{4}=58.5,
\end{equation}
within CMB-pivot range 50--60 e-folds.

\section{Closure Table}
\begin{table}[h!]
\centering
\begin{tabular}{lll}
\toprule
Tension & Scalar & Value vs obs \\
\midrule
$H_0$            & $1+1/12$           & 1.0833 vs 1.0837 \\
$S_8/\sigma_8$    & $1-\SSq/12$        & 0.9525 vs 0.945 \\
Li-7              & $2/\Phires^{2}$    & 2.88 vs 3.1 \\
Axis of evil      & $A_5$ icosahedron  & alignment generic \\
EDGES             & $1+\Ftrz^{-1/4}(1-\Ftrz)$ & 2.60 vs 2.5 \\
$n_s$             & $1-(1-\Phires)\Ftrz\KMex$    & 0.9653 vs 0.9649 \\
$N_e$             & $\Nch\Dcrit/4$            & 58.5 vs 50--60 \\
\bottomrule
\end{tabular}
\end{table}

\section{Falsifiers}
\begin{enumerate}[label=(F\arabic*)]
\item $H_0$ tension stays at exactly $1/12$ as SH0ES and Planck improve.
\item $\sigma_8$ low by $\SSq/12=4.75\%$ relative to CMB; DES Y6 + Euclid verify.
\item Li-7 deficit factor settles at $2/\Phires^2=2.88$.
\item Two more quadrupole/octupole alignments to appear at $72^\circ$ in Simons Observatory.
\item EDGES amplitude reproduced by SARAS/REACH at $T_b\sim-500$ mK.
\item $n_s$ remains in 0.96--0.97 band post-LiteBIRD.
\item No primordial tensor mode with $r>\Ftrz=0.10$ (current bound $r<0.036$ consistent).
\end{enumerate}

\begin{thebibliography}{9}
\bibitem{Riess2022} A.\ G.\ Riess et al., ApJL 934, L7 (2022).
\bibitem{KiDS1000} M.\ Asgari et al.\ (KiDS), A\&A 645, A104 (2021).
\bibitem{Spite1982} F.\ Spite, M.\ Spite, A\&A 115, 357 (1982).
\bibitem{Planck2018} Planck Collaboration, A\&A 641, A6 (2020).
\bibitem{EDGES2018} J.\ D.\ Bowman et al., Nature 555, 67 (2018).
\bibitem{LiteBIRD2022} LiteBIRD Collaboration, PTEP 2023, 042F01.
\bibitem{StarMagicSM} Star-Magic UQFF, PAPER\_1186, May 2026.
\end{thebibliography}
\end{document}
