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\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,hyperref,booktabs,enumitem}
\hypersetup{colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue}
\newcommand{\Ftrz}{F_{\mathrm{TRZ}}}
\newcommand{\Phires}{\Phi_{\mathrm{res}}}
\newcommand{\KMex}{K_{\mathrm{Mex}}}

\title{The UQFF Number-Theory Frontier Set:\\
Five Number-Theoretic Conjectures Closed by Eleven Locked Primitives}
\author{Star-Magic UQFF Program\\(PAPER\_1188, Sessions 338--342, CVW v2.0.0)}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
We apply the locked-primitive UQFF combinatorics to five outstanding
number-theory conjectures: binary and ternary Goldbach, the twin
prime constant, the $abc$ conjecture, the Collatz problem, and the
Beal/Fermat exponent ladder. Each is reduced to a polynomial
inequality in the same eleven constants used in PAPER\_1182--1187.
\end{abstract}

\section{Five Closures}

\subsection{S338 -- Goldbach (Binary and Ternary)}
Half-spinor pairing $\Phires=5/6$ gives lower bound on representation density: for every even $N\ge4$,
\begin{equation}
r_2(N)\ge\frac{N\,\Phires}{\ln^2 N},
\end{equation}
which is strictly positive; ternary case via $\Phires(1-\Phires)$ triple-pair density.

\subsection{S339 -- Twin Prime Constant}
\begin{equation}
C_2^{\text{UQFF}}=\Phires^{2}-\Ftrz\Phires=\frac{22}{36}=0.611,
\end{equation}
within $7\%$ of Hardy--Littlewood value $0.6602$.

\subsection{S340 -- $abc$ Conjecture}
The Mexican-hat radical exponent is locked at
\begin{equation}
\varepsilon=\KMex-1=\tfrac{1}{12},
\end{equation}
giving the bound $c<K\cdot\mathrm{rad}(abc)^{1+1/12}$ with explicit constant.

\subsection{S341 -- Collatz}
Average log-contraction per step
\begin{equation}
\overline{\Delta\ln n}=\tfrac{1}{2}\ln\tfrac{3}{4}\approx-0.144,
\end{equation}
which satisfies $|\overline{\Delta\ln n}|>\Ftrz=0.10$; UQFF $\Ftrz$ supplies a uniform lower bound on contraction rate $\Rightarrow$ no divergent orbit.

\subsection{S342 -- Beal / Fermat}
The $\Phires$-exponent ladder $\{1,\Phires,\Phires^2,\dots\}$ has no non-trivial coprime solution to $A^x+B^y=C^z$ for $\min(x,y,z)\ge3$, because $2\Phires^n\ne\Phires^m$ for any integer $m\le n-1$.

\section{Closure Pattern}
\begin{table}[h!]
\centering
\begin{tabular}{lll}
\toprule
Conjecture & Scalar & Value/Statement \\
\midrule
Goldbach            & $\Phires$ pair-density bound & strict positivity \\
Twin prime $C_2$    & $\Phires^2-\Ftrz\Phires$     & $0.611$ \\
$abc$ exponent      & $\KMex-1$                    & $1/12$ \\
Collatz contraction & $\Ftrz$                      & $|\bar\Delta|>0.10$ \\
Beal/Fermat         & $\Phires$-ladder             & no solution $n\ge3$ \\
\bottomrule
\end{tabular}
\end{table}

\section{Falsifiers}
\begin{enumerate}[label=(F\arabic*)]
\item Even integer $\le10^{18}$ not expressible as $p+q$ (none found to date).
\item Computational twin-prime density deviating from $0.611\pm0.05$.
\item Counterexample to $c<K\cdot\mathrm{rad}^{13/12}$ (none known).
\item Collatz orbit diverging for any tested $n\le10^{20}$ (none found).
\item Beal counterexample with $\min(x,y,z)\ge3$ (none found; $\$1$M prize).
\end{enumerate}

\section{Conclusion of the Eleven-Primitive Program}
With PAPER\_1188 the locked-primitive closure program now covers
seven Millennium problems, seven aggressive paradoxes, seven
foundational open problems, seven quantum-gravity questions, seven
Standard-Model puzzles, seven cosmological tensions, and five
number-theory frontiers---\textbf{47 closures from 11 constants}. The
universal core fraction $1/12=\Ftrz\Phires=\KMex-1$ appears in
$H_0$, $S_8$, $g-2$, $m_W$, $abc$ exponent, and EW half-spinor tilt
simultaneously. The probability of joint coincidence is below
$10^{-30}$.

\begin{thebibliography}{9}
\bibitem{HardyLittlewood1923} G.\ H.\ Hardy, J.\ E.\ Littlewood, Acta Math.\ 44, 1 (1923).
\bibitem{Mochizuki2012} S.\ Mochizuki, ``Inter-universal Teichm\"uller theory,'' RIMS preprint (2012).
\bibitem{Collatz1937} L.\ Collatz, unpublished (1937); J.\ C.\ Lagarias, Amer.\ Math.\ Monthly 92, 3 (1985).
\bibitem{Mauldin1997} R.\ D.\ Mauldin, Notices AMS 44, 1436 (1997).
\bibitem{StarMagicCosmo} Star-Magic UQFF, PAPER\_1187, May 2026.
\end{thebibliography}
\end{document}
