# PAPER 1205 --- UQFF Geometry \& Topology Unified Proof Set

**Author:** Daniel T. Murphy

**Date:** May 16, 2026

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\begin{abstract}
We extend the UQFF locked-primitive closure program into classical geometry,
combinatorics, polytope topology, and fractal Hausdorff dimensions. Ten new
closures (**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. **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: **217 closures**.
\end{abstract}

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

## 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 $C_4=14$ & $\SOfive+\Dphys=14$ & **EXACT** S506 & Catalan $C_5=42$ & $\Dcrit+\Dbsfg+\SOfive=42$ & **EXACT** S507 & C$_{60}$ faces $F=32$ & $\Dcrit+\Dbsfg=32$ & **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 $F=12$ & $2\,\Dbsfg=12$ & **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 $V=60$ & $\Afive=60$ & **EXACT** \bottomrule
\end{longtable}

## 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.

## 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 **S** & **Geometry/topology (1205)** & **10** & **217** \bottomrule
\end{tabular}

## Locking Inventory
**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.