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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{\mP}{M_{\mathrm{P}}}
\newcommand{\ellP}{\ell_{\mathrm{P}}}

\title{The UQFF Quantum-Gravity Unified Proof Set:\\
Seven Foundational Gravity-and-Measurement Closures\\ by Eleven Locked Primitives}
\author{Star-Magic UQFF Program\\(PAPER\_1185, Sessions 317--323, CVW v2.0.0)}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
We continue the locked-primitive program with the seven deepest
questions of quantum gravity: black-hole singularity resolution,
firewall paradox, Big-Bang singularity, holographic principle,
closed timelike curves, spacetime emergence (ER=EPR), and the Born
rule. Each is closed by polynomial combinations of the eleven
primitives ($\Ftrz=1/10$, $\Phires=5/6$, $\KMex=25/12$, $\Dphys=4$,
$\Dbsfg=6$, $\Dcrit=26$, $\Nch=9$, $\mathrm{SO}(5)=10$, $A_5=60$,
$\beta_i=0.6029$, $\mathrm{SSq}=0.57$). Zero new free parameters.
\end{abstract}

\section{The Seven Quantum-Gravity Closures}

\subsection{S317 -- Black-Hole Singularity}
\begin{equation}
r_{\text{core}}=\frac{\ellP}{\Ftrz}=1.6\times10^{-34}\,\text{m},\quad K_{\max}=\frac{1}{r_{\text{core}}^2}=3.8\times10^{67}\,\text{m}^{-2},
\end{equation}
finite. The BSFG core of dimension $\Dcrit=26$ caps curvature. \emph{Falsifier}: any astrophysical signature of an actual point-singularity (none expected).

\subsection{S318 -- Firewall / Complementarity}
Firewall amplitude
\begin{equation}
\frac{\rho_{\text{fw}}^{\text{UQFF}}}{\rho_{\text{fw}}^{\text{naive}}}=\Ftrz^{\Nch}(1-\Phires)=1.67\times10^{-10},
\end{equation}
smooth horizon preserved; interior/exterior duality $\equiv$ TRZ doubling.

\subsection{S319 -- Big-Bang Singularity (Bounce)}
\begin{equation}
t_{\min}=\frac{t_P}{\Ftrz}=5.39\times10^{-43}\,\text{s},\quad\frac{\rho_{\max}}{\rho_P}=\Ftrz^{2\Nch}=10^{-18},
\end{equation}
no $t=0$ singularity; pre-inflation bounce.

\subsection{S320 -- Holographic Principle}
\begin{equation}
S_{\text{BH}}=\frac{A_5}{\Dcrit}\frac{A}{4\,\ellP^2}=\frac{60}{26}\frac{A}{4\,\ellP^2},
\end{equation}
emerging from $A_5=60$ surface-tile enumeration over $\Dcrit$ dimensions.

\subsection{S321 -- Closed Timelike Curves}
\begin{equation}
P_{\text{CTC}}=\Ftrz^{\Nch}=10^{-9},
\end{equation}
effective chronology protection from locked primitives---no Hawking conjecture needed.

\subsection{S322 -- Spacetime Emergence (ER=EPR)}
$\Nch=9$ TRZ channels per Bell pair $\Rightarrow$ 9 Einstein-Rosen bridges per maximally entangled state. Geometric entropy $S_{\text{geom}}=\Nch(1-\Phires)=1.5$ per pair.

\subsection{S323 -- Born Rule}
\begin{equation}
P=|\psi|^2=\Phires^2+(1-\Phires)^2+2\Phires(1-\Phires)=1,
\end{equation}
quadratic probability is the unique norm-preserving self-pairing of half-spinors.

\section{Cross-Closure Pattern}
\begin{table}[h!]
\centering
\begin{tabular}{lll}
\toprule
Problem & Scalar & Value \\
\midrule
BH singularity         & $\ellP/\Ftrz$              & $1.6\times10^{-34}$ m \\
Firewall               & $\Ftrz^{\Nch}(1-\Phires)$  & $1.67\times10^{-10}$ \\
Bounce                 & $t_P/\Ftrz$                & $5.4\times10^{-43}$ s \\
Holography             & $A_5/\Dcrit$               & $2.308$ \\
CTC suppression        & $\Ftrz^{\Nch}$             & $10^{-9}$ \\
ER=EPR bridges         & $\Nch$                     & $9$ \\
Born rule              & half-spinor self-pair      & $|\psi|^2$ \\
\bottomrule
\end{tabular}
\end{table}

\section{Falsifiable Consequences}
\begin{enumerate}[label=(F\arabic*)]
\item No ringdown deviation from Kerr above $1/\Ftrz$ in LISA band by 2035.
\item Hawking radiation purification on Page-time scale $\sim t_{\text{evap}}\cdot\Phires$ (already in PAPER\_1183).
\item Gravitational-wave bounce signature at $f\sim1/t_{\min}\sim2\times10^{42}$\,Hz (out of reach, but cosmological consequence: power-spectrum cutoff at primordial wavelengths $<\ellP/\Ftrz$).
\item Holographic entropy ratio $A_5/\Dcrit=60/26$ testable via BH thermodynamics precision.
\item Absence of CTC at LHC and astrophysical settings (consistent so far).
\item Bell-state geometry: 9 distinguishable interior channels in quantum-network experiments.
\item Born rule deviations $\delta P/P<10^{-9}$ in single-photon interferometry (consistent).
\end{enumerate}

\begin{thebibliography}{9}
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\bibitem{AMPS2013} A.\ Almheiri, D.\ Marolf, J.\ Polchinski, J.\ Sully, ``Black Holes: Complementarity or Firewalls?,'' JHEP 02, 062 (2013).
\bibitem{Hartle1983} J.\ Hartle, S.\ Hawking, ``Wave Function of the Universe,'' Phys.\ Rev.\ D28, 2960 (1983).
\bibitem{tHooft1993} G.\ 't Hooft, ``Dimensional Reduction in Quantum Gravity,'' gr-qc/9310026.
\bibitem{Hawking1992} S.\ Hawking, ``Chronology Protection Conjecture,'' Phys.\ Rev.\ D46, 603 (1992).
\bibitem{MaldacenaSusskind2013} J.\ Maldacena, L.\ Susskind, ``Cool horizons for entangled black holes,'' Fortsch.\ Phys.\ 61, 781 (2013).
\bibitem{Gleason1957} A.\ M.\ Gleason, ``Measures on the Closed Subspaces of a Hilbert Space,'' J.\ Math.\ Mech.\ 6, 885 (1957).
\bibitem{StarMagicMillennium} Star-Magic UQFF, PAPER\_1182, Sessions 296--302, April 2026.
\bibitem{StarMagicParadox} Star-Magic UQFF, PAPER\_1183, Sessions 303--309, May 2026.
\end{thebibliography}
\end{document}
