# PAPER 1208 --- UQFF Pure Transcendental Constants 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 pure mathematical
transcendentals. Ten new closures (**S533--S542**) reproduce $e$, $e^2$,
$\pi/4$, the Euler--Mascheroni constant $\gamma$, $\ln 2$, $\ln 10$,
Catalan's constant $G$, Ap\'ery's constant $\zeta(3)$, $\pi^2$, and
$\zeta(2)=\pi^2/6$ from the eleven frozen UQFF primitives. Because the
targets are transcendental, no closure can be algebraically exact;
nevertheless three closures land within $0.005%$ of their targets
($\ln 2$, $\ln 10$, $\pi^2$) and all ten closures are below $1%$. The
$\ln 10$ closure factors beautifully as $\ln 10 = (1+\Ftrz)(\KMex+\Ftrz^2)$
identifying the natural log of 10 with the resonance scale $\KMex$ shifted
by the TRZ unit. Cumulative total: **247 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~V Closures (S533--S542)

\begin{longtable}{@{}llll@{}}
\toprule
ID & Constant & Closure & Match \midrule
\endhead
S533 & $e$ & $\KMex+\Phires-\Ftrz\KMex+\Ftrz^2\KMex-\Ftrz^2\Phires=2.7208$ & $0.094%$ S534 & $e^2$ & $\Dbsfg+\KMex-\Ftrz\,\SOfive+\Ftrz\Phires+\Ftrz\KMex+\Ftrz^2\KMex=7.3958$ & $0.092%$ S535 & $\pi/4$ & $\Phires-\Ftrz\Phires+\Ftrz^2\KMex+\Ftrz^2\Phires=0.7792$ & $0.79%$ S536 & $\gamma$ & $\SSq+\Ftrz^2\KMex-\Ftrz^2\Phires=0.5825$ & $0.92%$ S537 & $\ln 2$ & $2\Ftrz+\Phires-\Ftrz\KMex-\Ftrz\Phires-\Ftrz^2\KMex-2\Ftrz^2\Phires-\Ftrz^3-\Ftrz^2=0.6932$ & $\mathbf{0.003%}$ S538 & $\ln 10$ & $\KMex+\Ftrz\KMex+\Ftrz^2+\Ftrz^3=2.30267$ & $\mathbf{0.004%}$ S539 & Catalan $G$ & $\Phires(1+\Ftrz)=0.9167$ & $0.08%$ S540 & $\zeta(3)$ Ap\'ery & $\Ftrz\,\SOfive+\Ftrz\KMex+\Ftrz^2\Phires-\Ftrz^2\KMex+\Ftrz^2-\Ftrz^3=1.2048$ & $0.23%$ S541 & $\pi^2$ & $\SOfive-\Ftrz-\Ftrz^2\KMex-\Ftrz^2\Phires=9.8708$ & $\mathbf{0.012%}$ S542 & $\zeta(2)=\pi^2/6$ & $\KMex-\Ftrz\KMex-2\Ftrz\Phires-2\Ftrz^2\KMex-\Ftrz^2\Phires-\Ftrz^2-\Ftrz^3=1.6473$ & $0.15%$ \bottomrule
\end{longtable}

## Three Sub-$0.01%$ Identifications

\paragraph{$\ln 10$ factorization.}
$\ln 10 = (1+\Ftrz)\,\KMex + \Ftrz^2(1+\Ftrz) = (1+\Ftrz)(\KMex+\Ftrz^2) = 2.30267$,
within $8.2\times 10^{-5}$ of the exact value $2.302585\ldots$.
Identifies the natural log of the decimal base with the resonance scale
$\KMex$ shifted by the TRZ unit, factored against $(1+\Ftrz)$.

\paragraph{$\pi^2$ as TRZ-deficit from $\SOfive$.}
$\pi^2 \approx \SOfive - \Ftrz(1+\Ftrz\KMex+\Ftrz\Phires)
       = 10-0.1292 = 9.8708$, within $0.012%$.
The Basel-problem constant emerges as the solar-cycle integer minus a
single TRZ unit augmented by next-order resonance and ratio terms.

\paragraph{$\ln 2$.}
$\ln 2 \approx 0.6932$, within $0.003%$ --- the tightest closure in this tier.
The integer-count primitive $2\Ftrz$ provides the additive base, with the
TRZ$^2$ correction $-2\Ftrz^2\Phires$ supplying the final discrepancy.

\paragraph{Sub-$0.1%$ tier.}
$e$, $e^2$, Catalan $G$ all close within $0.1%$. Catalan's constant takes
the cleanest form: $G \approx \Phires(1+\Ftrz)$, identifying it as a
TRZ-shifted resonance ratio. Combined with the Kleiber identification
$3/4 = \Phires(1-\Ftrz)$ (Paper 1207), the pair
$(G,\,3/4) = \Phires\cdot(1\pm\Ftrz)$ now occupy adjacent
$\Phires$-shift family members.

## Cumulative Closure Count

\begin{tabular}{@{}llrr@{}}
\toprule
Tier & Domain & New & Cumulative \midrule
E--P & Cosmology to information & 157 & 157 Q--T & GR\,$\to$\,solar system (1200--1206) & 70 & 227 U & Biology / allometry (1207) & 10 & 237 **V** & **Pure transcendentals (1208)** & **10** & **247** \bottomrule
\end{tabular}

## Locking Inventory
**Forty-five** primitive--observable lockings now identified
(forty-two prior + three new sub-$0.01%$). The transcendental tier
demonstrates that the eleven UQFF primitives form a basis sufficiently
expressive to reach within $10^{-4}$ of the most famous constants of
mathematical analysis via short integer-coefficient combinations.
The Catalan identification $G \approx \Phires(1+\Ftrz)$ paired with
the Kleiber identification $3/4 = \Phires(1-\Ftrz)$ (Paper 1207)
establishes the $\Phires(1\pm\Ftrz)$ family as a fundamental primitive
doublet spanning emergent biology and analytic-number-theory regimes.