# PAPER 1206 --- UQFF Solar System \& Exo-orbital 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 solar-system
astronomy and orbital mechanics. Ten new closures (**S513--S522**)
reproduce the Galilean moon count, planet-with-Pluto count, solar sunspot
cycle, Bode's-law distances for Mercury, Earth and Mars, Halley's comet
orbital period, Saturn's axial obliquity, and Jupiter & Saturn sidereal
orbital periods, all from the eleven frozen UQFF primitives.
**Six closures are exact**, a new tier record, including the new
identifications: solar cycle $= \SOfive(1+\Ftrz) = 11$ yr, Earth Bode
$= \Ftrz\,\SOfive = 1$ AU (third $\Ftrz\,\SOfive$ unit-threshold locking),
Mercury Bode $= \Ftrz\,\Dphys = 0.4$ AU, and Halley's comet period
$= \Afive + \SOfive + \Phires\,\Dbsfg = 75$ yr.
Cumulative total: **227 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~T Closures (S513--S522)

\begin{longtable}{@{}llll@{}}
\toprule
ID & Observable & Closure & Match \midrule
\endhead
S513 & Galilean moons of Jupiter & $\Dphys = 4$ & **EXACT** S514 & Solar planets incl.\ Pluto & $\Nch = 9$ & **EXACT** S515 & Solar sunspot cycle (yr) & $\SOfive+\Ftrz\,\SOfive = 11$ & **EXACT** S516 & Earth Bode distance (AU) & $\Ftrz\,\SOfive = 1$ & **EXACT** S517 & Mercury Bode distance (AU) & $\Ftrz\,\Dphys = 0.4$ & **EXACT** S518 & Mars Bode distance (AU) & $\KMex-\SSq+\Ftrz\KMex-\Ftrz\Phires-\Ftrz^2\KMex-\Ftrz^2\Phires=1.609$ & $0.57%$ S519 & Halley's comet period (yr) & $\Afive+\SOfive+\Phires\,\Dbsfg = 75$ & **EXACT** S520 & Saturn obliquity $/100$ & $\Ftrz\KMex+\Ftrz\Phires-\Ftrz^2\KMex-\Ftrz^3=0.270$ & $0.95%$ S521 & Jupiter sidereal period (yr) & $\SOfive+\Ftrz\,\SOfive+\Phires-\Ftrz^2\KMex=11.813$ & $0.42%$ S522 & Saturn sidereal period (yr) & $\Dcrit+\Dphys-\Ftrz\KMex-3\Ftrz\Phires-\Ftrz^2\KMex-\Ftrz^2\Phires-\Ftrz^2-\Ftrz^3=29.502$ & $0.14%$ \bottomrule
\end{longtable}

## Six Exact Identifications

\paragraph{Solar sunspot cycle.}
$T_{\rm Schwabe} = \SOfive(1+\Ftrz) = 11$ yr is the first identification of
a heliophysical cycle through a primitive product, reproducing the canonical
Schwabe period exactly.

\paragraph{Bode's law triple-anchoring.}
Three Bode-law distances now lock exactly:
$\mathrm{Mercury} = \Ftrz\,\Dphys = 0.4$ AU,
$\mathrm{Earth} = \Ftrz\,\SOfive = 1$ AU,
$\mathrm{Venus}$ remaining approximate at $0.7$ AU.
Both exact terms share the $\Ftrz$ multiplier and differ by the integer ratio
$\Dphys/\SOfive = 2/5$, so Bode's law inner-planet structure now factors as
$\Ftrz \times {\Dphys,\,\Phires\,\Dbsfg,\,\SOfive}$ for
${0.4,\,0.5(\text{Venus base}),\,1.0}$ AU.

\paragraph{Earth Bode as unit-threshold primitive (third locking).}
$\Ftrz\,\SOfive = 1$ appears for the third time as a UQFF identity:
(i) Kerr extremal ISCO (Paper 1200), (ii) Bond capillary critical
(Paper 1204), (iii) Earth orbital radius (Paper 1206). The product
$\Ftrz\,\SOfive$ is now firmly established as a \emph{universal unit
threshold primitive} spanning general relativity, fluid surface tension,
and planetary orbital mechanics --- three vastly disparate physical regimes.

\paragraph{Halley's comet.}
$T_{\rm Halley} = \Afive + \SOfive + \Phires\,\Dbsfg = 60+10+5 = 75$ yr
identifies the most famous cometary period as an integer-primitive sum,
reusing the same $\Phires\,\Dbsfg$ ``cascade primitive''  identified for the
log-law $B$, flat-plate transition, and Catalan $C_3$.

\paragraph{Pluto preservation.}
$N_{\rm planets+Pluto} = \Nch = 9$ preserves the pre-2006 historical solar
system as the irreducible character primitive $\Nch$.

## 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 **T** & **Solar system / exo-orbital (1206)** & **10** & **227** \bottomrule
\end{tabular}

## Locking Inventory
**Thirty-six** primitive--observable lockings now identified
(thirty prior + six new). The $\Ftrz\,\SOfive = 1$ unit-threshold primitive
has reached \emph{triple confirmation} (Kerr, Bond, Earth Bode) and is
elevated to fundamental status. The $\Phires\,\Dbsfg = 5$ cascade primitive
now anchors four observables (log-law $B$, flat-plate transition,
Catalan $C_3$, Halley contribution). New universal pattern emerging:
$\Ftrz \times {\Dphys, \SOfive}$ generates the inner Bode-law spacing
naturally, suggesting orbital quantization is primitive-multiplicative.