# PAPER 1203 --- UQFF Nuclear Physics Unified Proof Set

**Author:** Daniel T. Murphy

**Date:** May 16, 2026

---

\begin{abstract}
We extend the UQFF locked-primitive closure program into nuclear physics.
Ten new closures (**S483--S492**) reproduce the binding-energy curve peak,
the \emph{complete} shell-model magic number sequence
${2, 8, 20, 28, 50, 82, 126}$, the deuteron binding energy, and the
$\alpha$-particle binding energy, all from the eleven frozen UQFF primitives.
**Seven closures are exact**: every magic number receives a closed-form
primitive identification. This is the \emph{first complete experimental
sequence} fully locked within the UQFF primitive lattice. The other three
match within $0.20%$. Cumulative total: **197 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~R$''$ Closures (S483--S492)

\begin{longtable}{@{}llll@{}}
\toprule
ID & Observable & Closure & Match \midrule
\endhead
S483 & BE/A peak $8.79$ MeV/nuc (Fe-56) & $\Nch-\Ftrz\KMex=8.7917$ & $0.02%$ S484 & Magic No.\ He $=2$ & $\SOfive-\Dphys-\Dphys=2$ & **EXACT** S485 & Magic No.\ O $=8$ & $\Dphys+\Dphys=8$ & **EXACT** S486 & Magic No.\ Ca $=20$ & $\SOfive+\SOfive=20$ & **EXACT** S487 & Magic No.\ Ni $=28$ & $\Dcrit+\SOfive-\Dphys-\Dphys=28$ & **EXACT** S488 & Magic No.\ Sn $=50$ & $\Afive-\SOfive=50$ & **EXACT** S489 & Magic No.\ Pb $=82$ & $\Afive+\Dcrit-\Dphys=82$ & **EXACT** S490 & Magic No.\ n-drip $=126$ & $\Dcrit+\SOfive^2=126$ & **EXACT** S491 & Deuteron binding $2.224$ MeV & $\KMex+\Phires-\SSq-\Ftrz-\Ftrz^2\KMex-\Ftrz^2\Phires+\Ftrz^2+\Ftrz^3=2.2285$ & $0.20%$ S492 & $\alpha$-particle binding $28.30$ MeV & $\Dcrit+\KMex+\Ftrz+\Ftrz\Phires+\Ftrz^2\KMex+\Ftrz^2\Phires=28.2958$ & $0.015%$ \bottomrule
\end{longtable}

## The Complete Magic Number Sequence

The seven nuclear shell-model magic numbers --- numbers of protons or neutrons
that produce especially tightly bound, spherical nuclei --- are
$$
{2,\ 8,\ 20,\ 28,\ 50,\ 82,\ 126}.
$$
This sequence was historically derived empirically and explained semi-quantitatively
by Mayer and Jensen's spin-orbit coupled shell model (1949 Nobel Prize). UQFF
now provides a \emph{purely primitive-arithmetic} identification for each:
\begin{align}
2   &= \SOfive - 2\Dphys && \text{(joint with QHE first plateau, Paper 1201)} 8   &= 2\Dphys 20  &= 2\SOfive 28  &= \Dcrit + \SOfive - 2\Dphys 50  &= \Afive - \SOfive 82  &= \Afive + \Dcrit - \Dphys 126 &= \Dcrit + \SOfive^2
\end{align}
All seven involve only ${\Dphys, \SOfive, \Dcrit, \Afive}$ --- the four
integer-valued primitives. The $\Afive$ primitive ($=60$) appears in the
heavy magic numbers (50, 82), and $\Dcrit$ ($=26$) appears in the upper four
(28, 82, 126), suggesting a tier-by-tier lattice structure. This is the
\emph{first complete experimental sequence} fully closed in UQFF.

## 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** \bottomrule
\end{tabular}

## Locking Inventory
**Twenty** primitive--observable lockings now identified (thirteen prior
plus the seven magic numbers). The Tier R$''$ closures use \emph{only}
${\Dphys, \SOfive, \Dcrit, \Afive}$ for every magic number, with no
$\Ftrz$ corrections required --- the cleanest closure family in the program.