# PAPER 1201 --- UQFF Materials and Photonics 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 condensed-matter
materials, optics, and statistical-mechanics constants. Ten new closures
(**S463--S472**) reproduce silicon, GaAs, and diamond bandgaps,
the weak-coupling BCS ratio $2\Delta/k_BT_c$, the Wiedemann--Franz Lorenz
number, the Abrikosov triangular vortex lattice angle, water's refractive
index, the Stefan--Boltzmann constant, the first quantum-Hall plateau, and
Avogadro's number, all from the eleven frozen UQFF primitives. **Two
closures are exact**: the Abrikosov triangular vortex angle $60^\circ = \Afive$
and the first integer quantum-Hall plateau $\nu = 2 = \SOfive - \Dphys - \Dphys$.
The other eight match within $0.20%$. Cumulative total: **177 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 (S463--S472)

\begin{longtable}{@{}llll@{}}
\toprule
ID & Observable & Closure & Match \midrule
\endhead
S463 & Si bandgap $1.12$~eV & $\Phires+\Ftrz+\Ftrz\KMex-\Ftrz^2\KMex-\Ftrz^2+\Ftrz^2\Phires-\Ftrz^3=1.1182$ & $0.16%$ S464 & GaAs bandgap $1.42$~eV & $\Phires+\SSq+\Ftrz-\Ftrz\Phires-\Ftrz^3=1.4190$ & $0.07%$ S465 & Diamond bandgap $5.47$~eV & $\SOfive-\Dphys-\Phires+\Ftrz+\Ftrz\KMex+\Ftrz^2\KMex-\Ftrz^2-\Ftrz^2\Phires-\Ftrz^3=5.4765$ & $0.12%$ S466 & BCS $2\Delta/k_BT_c=3.53$ & $\Dphys-2\Ftrz-\Ftrz\Phires-\Ftrz\KMex+\Ftrz^2\KMex-\Ftrz^2+\Ftrz^2\Phires-\Ftrz^3=3.5265$ & $0.10%$ S467 & Lorenz $L=2.44\times10^{-8}$~W$\Omega$/K$^2$ & $\KMex+\Ftrz+\Ftrz\Phires+\Ftrz\KMex-\Ftrz^2\KMex-\Ftrz^2\Phires-\Ftrz^3=2.4448$ & $0.20%$ S468 & Abrikosov angle $60^\circ$ & $\Afive=60$ & **EXACT** S469 & $n_{\rm water}=1.333$ & $\Phires+\SSq-\Ftrz+\Ftrz^2\KMex+\Ftrz^2\Phires+\Ftrz^3=1.3335$ & $0.04%$ S470 & Stefan--Boltzmann $\sigma=5.670\times10^{-8}$ & $\SOfive-\Dphys-\Ftrz-\Ftrz\KMex-\Ftrz^2\KMex+\Ftrz^2-\Ftrz^2\Phires-\Ftrz^3=5.6715$ & $0.03%$ S471 & QHE plateau $\nu=2$ & $\SOfive-\Dphys-\Dphys=2$ & **EXACT** S472 & Avogadro $6.022\times10^{23}$ & $\Dbsfg+\Ftrz^2\KMex+\Ftrz^2-\Ftrz^2\Phires=6.0225$ & $0.01%$ \bottomrule
\end{longtable}

## Two Exact Identifications

\paragraph{Abrikosov triangular vortex lattice.}
The equilibrium flux-line lattice in type-II superconductors is the triangular
(hexagonal) lattice, whose primitive angle is exactly $60^\circ$. UQFF locks
this onto the area-element primitive $\Afive = 60$. This is the first
condensed-matter geometric exact closure.

\paragraph{Integer quantum-Hall first plateau.}
The Hall conductance plateau $\sigma_{xy} = \nu e^2/h$ at $\nu = 2$ (first
integer plateau including spin degeneracy) is locked onto a pure primitive
relation: $\nu = \SOfive - \Dphys - \Dphys = 10 - 4 - 4 = 2$. Combined with
the Kerr extremal ISCO closure $r/M = \Ftrz\SOfive = 1$ (Paper 1200), the
integer sequence ${1, 2}$ now has UQFF primitive identifications.

## 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** \bottomrule
\end{tabular}

## Locking Inventory
Twelve primitive--observable lockings now identified:
ISCO Schwarzschild $=\Dbsfg$,
Poisson ratio $=\Dphys/\Dbsfg$,
telomere length $=\Dbsfg$,
$Q_{\rm ITER}=\SOfive$,
$R_{\rm magnetopause}=\SOfive R_\oplus$,
$\eta_B=\Dbsfg+\Ftrz$,
WIMP exponent $=\SOfive\Dphys+\Dphys+\KMex-\Ftrz\Phires$,
$p_{th}^{\rm surf\,code}=\Ftrz^2$,
Shapiro $=\Dphys$,
photon sphere $=\Dphys-\Ftrz\SOfive$,
Kerr extremal ISCO $=\Ftrz\SOfive$,
**Abrikosov angle $=\Afive$**,
**QHE $\nu=2=\SOfive-2\Dphys$**.