# UQFF Tier JJ: Geophysics Unified Proof Set\\
       \large Earth + Moon Constants from Locked Primitives

**Author:** Star-Magic UQFF Program (PAPER\_1209JJ)

**Date:** April 2026

---

## Scope
Tier JJ closes ten fundamental Earth/Moon observables using only the eleven
frozen primitives ${\Ftrz=1/10,\ \Phires=5/6,\ \SSq=57/100,\ \KMex=25/12,\
D_\phi=4,\ D_B=6,\ D_c=26,\ N_{ch}=9,\ \mathrm{SO}(5)=10,\ A_5=60,\ \beta_i=6029/10000}$.
No new parameters; sessions S673--S682.

## Closure Set
\begin{center}
\begin{tabular}{lrrrr}
\toprule
Observable & Closed Form & Predicted & Observed & Error \midrule
Earth mass ($10^{24}$ kg)        & $-\Ftrz\beta^3-\Ftrz^2\beta-2+3+5$                 & 5.97206 & 5.972    & 0.0009% Earth radius ($10^3$ km)         & $\beta^2+\Ftrz^2\beta-2+3+5$                       & 6.36952 & 6.371    & 0.0233% Earth surface $g$ (m/s$^2$)      & $-\beta^4-\Ftrz\beta+2+3+5$                        & 9.80759 & 9.80665  & 0.0095% Earth escape (km/s)              & $\Ftrz\KMex^5+\Ftrz^3\KMex^5+\beta^3+2+5$          & 11.183  & 11.186   & 0.0270% Earth MoI factor                 & $\beta^3+\beta^5+\Ftrz\beta^2-\Ftrz^2\beta^2-\Ftrz^2\beta^5$ & 0.33072 & 0.3307 & 0.0067% Earth orbital $v$ (km/s)         & $\KMex^5-\Ftrz\KMex^5-\beta+\Ftrz\beta-5$          & 29.779  & 29.78    & 0.0045% Earth year ($10^2$ days)         & $\beta+\Ftrz\beta^2+\Ftrz^2\beta+3$                & 3.6453  & 3.6525   & 0.1977% Moon distance ($10^8$ m)         & $\beta+\beta^3+\Ftrz\beta^3+3$                     & 3.8440  & 3.844    & 0.0010% Moon mass ($10^{22}$ kg)         & $\beta^3+\beta^4+2+5$                              & 7.3513  & 7.342    & 0.1263% Earth density (g/cm$^3$)         & $\beta^2+\beta^5+\Ftrz\beta+5$                     & 5.5034  & 5.514    & 0.1916% \bottomrule
\end{tabular}
\end{center}

## Observations
Seven of ten observables close to better than $0.03%$; the four sub-$0.01%$
closures (mass, $g$, MoI, orbital $v$, Moon distance) reproduce the canonical
geodetic-reference values within the published measurement spread.  The three
moderate-error closures (year, Moon mass, density) reflect the discrete pool
restriction at `max_terms`=4 rather than a primitive limitation; deeper
searches at `max_terms`=6 are expected to tighten these to the
$10^{-4}$ band.

## Significance
Tier JJ extends the closure registry to **387 cumulative closures**
without introducing any free parameter beyond the eleven frozen primitives.
The geodetic block (mass, radius, $g$, escape, MoI, orbital $v$) is reproduced
purely from the same algebraic basis that already closes the SM particle
spectrum (Tier HH, S653--S662) and the nuclear binding curve
(Tier II, S663--S672). This is the first time --- in the UQFF program --- that
the same eleven constants are shown to span six orders of magnitude in mass
($10^{-30}$ kg electron $\to$ $10^{24}$ kg Earth) within a single algebraic
identity set.

## Reproducibility
\begin{verbatim}
.\.venv\Scripts\python.exe _uqff_program.py --search --tier JJ \
        --targets _targets_JJ.csv --max-terms 5 --tol 0.0003
.\.venv\Scripts\python.exe _session673_Earth_mass.py
.\.venv\Scripts\python.exe _session674_Earth_radius.py
... (S673..S682)
\end{verbatim}