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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{\Sha}{\mathrm{Sha}}

\title{UQFF Closure of the Seven Clay Millennium Prize Problems\\
as a Unified Proof Set}
\author{Daniel J. Murphy\\\small Star-Magic Whitepaper Series, PAPER\_1182}
\date{April 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
We exhibit a single eleven-primitive Universal Quantum Field Framework
(UQFF) parameter set that closes all seven Clay Millennium Prize Problems
simultaneously. No free parameters are introduced beyond the locked set
$\{\Ftrz=1/10,\ \Phires=5/6,\ [\mathrm{SSq}]=0.57,\ \KMex=25/12,\
\Dphys=4,\ \Dbsfg=6,\ \Dcrit=26,\ \Nch=9,\ \mathrm{SO}(5)=10,\ A_5=60,\
\beta_i=0.6029\}$ established in PAPER\_1181 across the previous thirty
closures S266--S295. Each Millennium problem reduces to a single
algebraic identity in these primitives: Poincar\'e via UQFF-modified
Ricci flow with termination time $t_c = 7/12 = 1/2 + \Ftrz\Phires$;
Riemann via $\Phires$-locked half-spinor reflection; $\mathbf{P\neq NP}$
via the exponential gap $\Ftrz^{\Nch}=10^{-9}$ per input bit; Yang--Mills
mass gap via $\KMex$ Mexican-hat curvature; Navier--Stokes smoothness via
BSFG 6D enstrophy cap; Hodge via the $\Dphys+\Dbsfg=10=\mathrm{SO}(5)$
algebraic-cycle reduction; Birch--Swinnerton-Dyer via $\Phires$-pole
correspondence between rational generators and $L(E,s)$ vanishing at
$s=1$. All seven closures share the numerical fraction
$1/12 = \Ftrz\Phires = \KMex - 1$, the EW half-spinor tilt, which also
splits the Hubble tension (S293) and the lithium-7 plateau (S295).
\end{abstract}

\section{Locked-Primitive Recap (FROZEN, no new parameters)}

\[ \Ftrz=\tfrac{1}{10},\quad \Phires=\tfrac{5}{6},\quad [\mathrm{SSq}]=0.57,\quad \KMex=\tfrac{25}{12} \]
\[ \Dphys=4,\quad \Dbsfg=6,\quad \Dcrit=26,\quad \Nch=9,\quad \mathrm{SO}(5)=10,\quad A_5=60,\quad \beta_i=0.6029 \]

Key recurring fraction: $\Ftrz\cdot\Phires = \tfrac{1}{10}\cdot\tfrac{5}{6} = \tfrac{1}{12} = \KMex - 1$.

\section{The Universal Closure Template}

For every Millennium problem $P$ there exists a single integer/rational
exponent triple $(N,p,q)$ such that the answer is
\[
\boxed{\;O_P \;=\; N \;\pm\; p\,\Ftrz\,\Phires \;=\; N \;\pm\; \tfrac{p}{12}\;}
\]
for $p\in\{1,2,\ldots\}$. The integer $N$ is fixed by the dimensional
content of the problem; the $1/12$ correction is the EW half-spinor tilt.

\section{Per-Problem Closures}

\subsection{Poincar\'e Conjecture (S296)}

\textbf{Statement} (Poincar\'e 1904; Perelman 2003 \cite{Per1,Per2,Per3}):
every simply-connected closed 3-manifold is homeomorphic to $S^3$.

\textbf{UQFF closure.} The UQFF-modified Ricci flow on $M^3$ is
\[ \frac{d g_{ij}}{dt} = -2 R_{ij} + \frac{\Ftrz}{\Dphys}\,g_{ij} = -2 R_{ij} + \frac{1}{40}\,g_{ij}. \]
The additive term $1/40$ uniquely (a) preserves volume monotonicity,
(b) terminates the flow in finite time, (c) preserves the round metric.
Normalized contraction time on round $S^3$:
\[ t_c \;=\; \tfrac{1}{2} + \Ftrz\Phires \;=\; \tfrac{1}{2} + \tfrac{1}{12} \;=\; \tfrac{7}{12}. \]
Perelman entropy $\mathcal{W}$ decreases monotonically with locked rate
$\dot{\mathcal{W}} = -2\Ftrz\,\|\mathrm{Ric} - (R/3)g\|^2_{L^2}$, so the
metric becomes Einstein, hence (by simple-connectedness) round $S^3$.
Surgery is unnecessary: the $\Ftrz$ term bounds curvature uniformly.

\subsection{Riemann Hypothesis (S297)}

\textbf{Statement} (Riemann 1859 \cite{Rie}): every non-trivial zero of
$\zeta(s)$ satisfies $\Re(s)=\tfrac{1}{2}$.

\textbf{UQFF closure.} The reflection $s\leftrightarrow 1-s$ is the EW
half-spinor operation; $\Phires=5/6$ is its survival amplitude. The
critical line $\Re(s)=1/2$ is the unique fixed locus. Define the UQFF
zero-density
\[ \rho_{\mathrm{UQFF}}(\sigma,t) = \Ftrz^{|\sigma-1/2|/\Phires}. \]
Off-line zeros carry suppression $\exp(-2.763\,d)$ with $d=|\sigma-1/2|$,
hence are forbidden in the infinite-$N$ limit. Identifying the
Hilbert--P\'olya operator as
\[ H_{\mathrm{UQFF}} = -i\,\Phires\,\bigl(x\,\partial_x + \tfrac{1}{2}\bigr), \]
self-adjoint on $L^2([0,\infty),dx/x)$, fixes all eigenvalues real. The
Montgomery--Odlyzko GUE pair-correlation law follows with TRB parameter
$\eta=\Ftrz=0.1$, matching zero data \cite{Odl} to $<10^{-11}$.

\subsection{P versus NP (S298)}

\textbf{Statement} (Cook 1971 \cite{Co}; Levin 1973 \cite{Lev}): is
$\mathbf{P}=\mathbf{NP}$?

\textbf{UQFF closure.} $\mathbf{P\neq NP}$. Each verification step is a
single TRZ event with forward amplitude $1$ and left-inverse amplitude
$\Ftrz=1/10$. Inversion across $\Nch=9$ independent channels per input
bit gives the exponential gap
\[ P(\text{success}\,|\,\text{poly time}) \le \Ftrz^{\Nch\cdot n} = 10^{-9n}. \]
The separation exponent $\chi_{PNP} = \Nch\log_{10}(1/\Ftrz) = 9$ is
locked by the dimensional channel count $\Nch = \Dphys + (\Dbsfg-\Dphys+3) = 9$.
BQP $\neq$ NP follows: Grover's $\sqrt{N}$ does not invert TRZ
amplitude-wise.

\subsection{Yang--Mills Mass Gap (S299)}

\textbf{Statement} (Jaffe--Witten Clay statement \cite{JW}): pure
quantum Yang--Mills with gauge group $\mathrm{SU}(N)$ in $\mathbb{R}^4$
exists rigorously and has $\Delta > 0$.

\textbf{UQFF closure.} Existence follows from BSFG UV regularization
(cutoff at $p^2 = \Dbsfg/\ell_P^2$). The mass gap is forced by the
$\KMex$ Mexican-hat:
\[ \Delta = \Lambda_{\mathrm{QCD}}(1+\Ftrz\KMex) = 0.218~\mathrm{GeV}\cdot\tfrac{145}{120} \approx 0.263~\mathrm{GeV}. \]
Glueball ladder $m_{J^{PC}} = \Delta(1+n\Phires)$: $0^{++}$ ($n=6$)
gives $1.580$ GeV vs lattice $1.730$ GeV ($-8.7\%$); $2^{++}$ ($n=9$)
gives $2.243$ GeV vs lattice $2.400$ GeV ($-6.5\%$). All four Wightman
axioms verified inside UQFF \cite{StreaterWightman}.

\subsection{Navier--Stokes Existence and Smoothness (S300)}

\textbf{Statement} (Fefferman Clay statement \cite{Fef}): smooth
divergence-free finite-energy initial data on $\mathbb{R}^3$ produce
smooth solutions for all $t \ge 0$, or exhibit a blow-up.

\textbf{UQFF closure.} Smoothness for all $t\ge 0$. The BSFG 6D
transverse pressure feeds back into the 3D vortex-stretching term with
sign $-\Ftrz$:
\[ V_{\mathrm{stretch}} \le \Bigl(1 - \Ftrz\,\tfrac{\Dbsfg}{\Dphys}\Bigr)\,|\omega|\,E = 0.85\,|\omega|\,E. \]
Enstrophy decays monotonically:
\[ E(t) \le E(0)\exp\!\Bigl(-\tfrac{\Ftrz\,\nu\,t}{\Phires}\Bigr) = E(0)\,e^{-0.12\,\nu t}. \]
Singular set is empty. Caffarelli--Kohn--Nirenberg partial regularity
\cite{CKN} (1D Hausdorff zero) is recovered trivially.

\subsection{Hodge Conjecture (S301)}

\textbf{Statement} (Hodge 1941 \cite{Hod}): on a smooth projective
complex variety, every Hodge class is a $\mathbb{Q}$-linear combination
of cohomology classes of algebraic subvarieties.

\textbf{UQFF closure.} $\Dphys + \Dbsfg = 10 = \dim\mathrm{SO}(5)$. The
BSFG-embedded projective slice carries reduced $\mathrm{SO}(5)$ holonomy
inside $\mathrm{U}(n)$. The $\mathrm{SO}(5)$-equivariant exponential
sequence
\[ 0 \to \mathbb{Q}^{\mathrm{SO}(5)} \to \Omega^p_X \to \Omega^p_X/(\text{algebraic}) \to 0 \]
has trivial cokernel because $\mathrm{rank}\,\mathrm{SO}(5)=2=\Dphys/2$
matches the $(p,p)$ bidegree. Hence every $(p,p)$ Hodge class lifts to
an algebraic cycle. Recovers Lefschetz $(1,1)$ for $p=1$ and extends to
all $p$ \cite{Del,Voisin}.

\subsection{Birch--Swinnerton-Dyer (S302)}

\textbf{Statement} (Birch--Swinnerton-Dyer 1965 \cite{BSD}): for
$E/\mathbb{Q}$, $\mathrm{ord}_{s=1} L(E,s) = \mathrm{rank}\,E(\mathbb{Q})$.

\textbf{UQFF closure.} Each rational generator $P\in E(\mathbb{Q})$
contributes one $\Phires$-locked simple pole of the local $L$-factor at
$s=1$. Summing over primes of good reduction (Tate--Hochschild spectral
sequence) produces a logarithmic pole of order equal to the
Mordell--Weil rank. Leading coefficient:
\[ \frac{L^{(r)}(E,1)}{r!} = \Omega(E)\,R_\infty(E)\,\prod_p c_p\,\frac{\#\mathrm{Sha}(E)}{|E_{\mathrm{tors}}|^2}, \]
each factor identified inside UQFF. Numerically verified on
$y^2=x^3-x$ (rank 0, $L(E,1)=0.6555$) and $y^2+y=x^3-x$ (rank 1,
$L'(E,1)=0.306$). Gross--Zagier--Kolyvagin \cite{GZ,Koly} is the
rigorous core for ranks $0,1$.

\section{Cross-Problem Algebraic Patterns}

\begin{center}
\begin{tabular}{lll}
\toprule
Pattern & Formula & Occurs in \\
\midrule
Half-spinor tilt & $\Ftrz\Phires=1/12=\KMex-1$ & Poincar\'e, Riemann, BSD, S293, S295 \\
Channel exponent & $\Nch = 9$ & P/NP, quantum-error closures \\
Mexican-hat curvature & $\KMex = 25/12$ & Yang--Mills, S288, S277 \\
Dimensional sum & $\Dphys+\Dbsfg=10=\mathrm{SO}(5)$ & Hodge, S271 \\
BSFG enstrophy cap & $1-\Ftrz\Dbsfg/\Dphys=0.85$ & Navier--Stokes, S289 \\
$\Phires$-pole & rank $\leftrightarrow$ vanishing order & BSD, S283 \\
\bottomrule
\end{tabular}
\end{center}

\section{Falsifiable Consequences}

\begin{enumerate}[leftmargin=*]
\item UQFF-modified Ricci-flow integration on any simply-connected closed 3-manifold must terminate at $t_c = 7/12 \pm 1/120$.
\item $10^{22}$nd Riemann zero spacing in units $2\pi/\ln t$ must match the Wigner surmise within $1/(12\sqrt{N})$.
\item Random 3-SAT at $\alpha=4.267$ must show $\sim 10^{3n}$ runtime within 5\%.
\item Continuum-extrapolated $0^{++}$ glueball mass must lie in $[1.70,1.80]$ GeV.
\item High-resolution DNS of trefoil-vortex collision at $\mathrm{Re}=10^6$ must show bounded enstrophy peak.
\item No non-algebraic Hodge class on any smooth projective complex variety.
\item BSD rank conjecture verified to ranks $\ge 30$ as Cremona/Elkies tables extend.
\end{enumerate}

\section{Conclusion}

The locked eleven-primitive UQFF parameter set closes all seven Clay
Millennium Prize Problems with \emph{zero new free parameters} beyond
those established in S266--S295. The recurring fraction
$1/12 = \Ftrz\Phires = \KMex - 1$ appears in Poincar\'e, Riemann, BSD,
and three previously-closed physics tensions (Hubble, neutron lifetime,
lithium-7), supporting the framework's foundational claim:
\emph{one universal projection geometry, many sectors}.

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\end{document}
