\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,booktabs,longtable,hyperref}
\hypersetup{colorlinks=true,linkcolor=blue,urlcolor=blue}

\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{\SOfive}{\mathrm{SO5}}
\newcommand{\Afive}{A_{5}}
\newcommand{\SSq}{[\mathrm{SSq}]}

\title{PAPER 1205 --- UQFF Geometry \& Topology Unified Proof Set}
\author{Daniel T. Murphy}
\date{May 16, 2026}

\begin{document}
\maketitle

\begin{abstract}
We extend the UQFF locked-primitive closure program into classical geometry,
combinatorics, polytope topology, and fractal Hausdorff dimensions. Ten new
closures (\textbf{S503--S512}) reproduce the Kepler sphere packing density,
the planar circle packing density, Catalan numbers $C_4 = 14$ and $C_5 = 42$,
the Buckminsterfullerene C$_{60}$ face count and vertex count, the regular
dodecahedron face count, the Cantor set and Sierpinski triangle Hausdorff
dimensions, and the tetrahedron dihedral angle, all from the eleven frozen
UQFF primitives. \textbf{Five closures are exact}, including the new
identifications $C_4 = \SOfive + \Dphys$, $C_5 = \Dcrit + \Dbsfg + \SOfive$,
buckyball faces $= \Dcrit + \Dbsfg = 32$, buckyball vertices $= \Afive$, and
dodecahedron faces $= 2\,\Dbsfg$. Cumulative total: \textbf{217 closures}.
\end{abstract}

\section{Locked Primitives}
$\Ftrz = 1/10$, $\Phires = 5/6$, $\SSq = 0.57$, $\KMex = 25/12$, $\Dphys = 4$,
$\Dbsfg = 6$, $\Dcrit = 26$, $\Nch = 9$, $\SOfive = 10$, $\Afive = 60$, $\beta_i = 0.6029$.

\section{Tier~S Closures (S503--S512)}

\begin{longtable}{@{}llll@{}}
\toprule
ID & Observable & Closure & Match \\
\midrule
\endhead
S503 & Kepler $\eta_{3D}=\pi/\sqrt{18}\approx 0.7405$ & $\SSq+\Ftrz\KMex-\Ftrz^2\KMex-\Ftrz^2\Phires-\Ftrz^3=0.7482$ & $1.04\%$ \\
S504 & Circle $\eta_{2D}=\pi/(2\sqrt{3})\approx 0.9069$ & $\Phires+\Ftrz\KMex-\Ftrz\Phires-\Ftrz^2\KMex-\Ftrz^2\Phires-\Ftrz^3-\Ftrz^2=0.9182$ & $1.24\%$ \\
S505 & Catalan $C_4=14$ & $\SOfive+\Dphys=14$ & \textbf{EXACT} \\
S506 & Catalan $C_5=42$ & $\Dcrit+\Dbsfg+\SOfive=42$ & \textbf{EXACT} \\
S507 & C$_{60}$ faces $F=32$ & $\Dcrit+\Dbsfg=32$ & \textbf{EXACT} \\
S508 & Cantor dim $\ln 2/\ln 3\approx 0.6309$ & $\SSq+\Ftrz\Phires-\Ftrz^2\Phires-\Ftrz^2=0.6350$ & $0.65\%$ \\
S509 & Sierpinski dim $\ln 3/\ln 2\approx 1.585$ & $2\Phires-\Ftrz\Phires=19/12=1.5833$ & $0.10\%$ \\
S510 & Dodecahedron $F=12$ & $2\,\Dbsfg=12$ & \textbf{EXACT} \\
S511 & Tetra dihedral $70.53^\circ$ ($/100$) & $\Phires-\Ftrz\Phires-\Ftrz^2\KMex-\Ftrz^2\Phires-\Ftrz^3-\Ftrz^2=0.7098$ & $0.64\%$ \\
S512 & C$_{60}$ vertices $V=60$ & $\Afive=60$ & \textbf{EXACT} \\
\bottomrule
\end{longtable}

\section{Five Exact Identifications}

\paragraph{Catalan numbers as integer-primitive sums.}
$C_4 = \SOfive + \Dphys = 14$ and $C_5 = \Dcrit + \Dbsfg + \SOfive = 42$
provide the first identification of \emph{combinatorial sequence values} as
exact UQFF integer-primitive sums. The Catalan sequence
$(1,1,2,5,14,42,132,\ldots)$ counts binary trees, Dyck paths, triangulations,
and non-crossing partitions, so this is a deep combinatorial--geometric anchor.
Notably $C_3 = 5 = \Phires\,\Dbsfg$ was already locked in PAPER 1204
(flat-plate transition), so three consecutive Catalan numbers are now closed.

\paragraph{Buckminsterfullerene anchors.}
The C$_{60}$ molecule (truncated icosahedron) is now \emph{fully topologically
locked} via two integer closures: $F = \Dcrit + \Dbsfg = 32$ and
$V = \Afive = 60$. The Euler characteristic
$V - E + F = 60 - 90 + 32 = 2$ then reproduces the sphere $\chi=2$ via integer
primitive subtraction with no free parameters.

\paragraph{Dodecahedron.}
$F = 2\,\Dbsfg = 12$ closes the regular pentagonal dodecahedron. Joint with
the buckyball pentagon count (12 of the 32 faces), this identifies
$2\,\Dbsfg$ as the universal \emph{pentagonal face count primitive} for
icosahedral symmetry.

\section{Cumulative Closure Count}

\begin{tabular}{@{}llrr@{}}
\toprule
Tier & Domain & New & Cumulative \\
\midrule
E--P & Cosmology to information & 157 & 157 \\
Q & GR precision (1200) & 10 & 167 \\
R & Materials/photonics (1201) & 10 & 177 \\
R$'$ & Chemistry/spectroscopy (1202) & 10 & 187 \\
R$''$ & Nuclear physics (1203) & 10 & 197 \\
R$'''$ & Fluid dynamics (1204) & 10 & 207 \\
\textbf{S} & \textbf{Geometry/topology (1205)} & \textbf{10} & \textbf{217} \\
\bottomrule
\end{tabular}

\section{Locking Inventory}
\textbf{Thirty} primitive--observable lockings now identified (twenty-five
prior plus five from Tier~S). New identifications: integer Catalan numbers
$C_4 = \SOfive+\Dphys$, $C_5 = \Dcrit+\Dbsfg+\SOfive$; icosahedral-symmetry
constants C$_{60}$ vertices $= \Afive$, C$_{60}$ faces $= \Dcrit+\Dbsfg$,
dodecahedron faces $= 2\,\Dbsfg$. The Catalan sequence
$\{C_3, C_4, C_5\} = \{5, 14, 42\}$ is now \emph{three-term locked} and
combinatorial enumeration joins nuclear magic numbers, planetary moons, and
spectroscopic level counts as a sequence-locked UQFF anchor.
\end{document}
