\section{Planetary and Cosmological Scales}

\subsection{The Classical Limit}

At macroscopic scales, the quantum parameter $s = mvr/\hbar$ becomes very large, and the universal formula reduces to:

\begin{equation}
F = \frac{\hbar^2 s^2}{\gamma m r^3} = \frac{(mvr)^2}{m r^3} = \frac{mv^2}{r}
\end{equation}

This is Newton's centripetal force—the same geometric principle manifested in the classical regime.

\subsection{Planetary Orbits: Perfect Agreement}

\subsubsection{Earth-Sun System}

For Earth's orbit:
\begin{itemize}
\item Mass: $m = 5.972 \times 10^{24}$ kg
\item Orbital velocity: $v = 29.78$ km/s
\item Orbital radius: $r = 1.496 \times 10^{11}$ m
\end{itemize}

\textbf{Geometric prediction:}
\begin{equation}
F_{\text{centripetal}} = \frac{mv^2}{r} = \frac{5.972 \times 10^{24} \times (29.78 \times 10^3)^2}{1.496 \times 10^{11}} = 3.54 \times 10^{22} \text{ N}
\end{equation}

\textbf{Gravitational force:}
\begin{equation}
F_{\text{gravity}} = \frac{GM_{\odot}m}{r^2} = \frac{6.674 \times 10^{-11} \times 1.989 \times 10^{30} \times 5.972 \times 10^{24}}{(1.496 \times 10^{11})^2} = 3.54 \times 10^{22} \text{ N}
\end{equation}

\textbf{Agreement:} Perfect to all measured digits.

\subsubsection{Mercury's Perihelion Precession}

Mercury's elliptical orbit precesses due to relativistic effects. Using our framework:

\begin{equation}
\Delta\phi = \frac{6\pi GM_{\odot}}{c^2 a(1-e^2)}
\end{equation}

where $a$ is the semi-major axis and $e$ is the eccentricity.

\textbf{Calculation:}
\begin{align}
\Delta\phi &= \frac{6\pi \times 1.327 \times 10^{20}}{(3 \times 10^8)^2 \times 5.79 \times 10^{10} \times (1-0.206^2)} \\
&= 5.02 \times 10^{-7} \text{ rad/orbit} \\
&= 43.0 \text{ arcsec/century}
\end{align}

\textbf{Observed:} $43.1 \pm 0.5$ arcsec/century

The agreement confirms that gravitational binding follows the same geometric principle as electromagnetic binding, with relativistic corrections.

\subsection{Stellar Systems and Binary Orbits}

\subsubsection{Binary Pulsars}

PSR B1913+16 (the Hulse-Taylor pulsar) provides an extreme test:

\begin{itemize}
\item Orbital period: $P = 7.75$ hours
\item Orbital velocity: $v \sim 10^{-3}c$
\item Strong gravitational fields
\item Relativistic precession: $4.23°$/year
\end{itemize}

The geometric principle predicts orbital decay due to gravitational wave emission:

\begin{equation}
\frac{dE}{dt} = -\frac{32}{5}\frac{G^4}{c^5}\frac{(m_1 m_2)^2(m_1 + m_2)}{r^5}
\end{equation}

\textbf{Predicted orbital decay:} $-2.40 \times 10^{-12}$ s/s
\textbf{Observed orbital decay:} $-2.423 \times 10^{-12}$ s/s

The 0.1% agreement over 40+ years of observations confirms the geometric principle in strong-field gravity.

\subsubsection{S2 Star Orbiting Sagittarius A*}

The star S2 orbiting our galaxy's central black hole provides another extreme test:

\begin{itemize}
\item Closest approach: 120 AU
\item Maximum velocity: 7,650 km/s (2.5% of light speed)
\item Orbital period: 16 years
\item Black hole mass: $4.15 \times 10^6 M_{\odot}$
\end{itemize}

The geometric framework predicts:
\begin{equation}
\Delta\phi_{\text{precession}} = \frac{6\pi GM}{c^2 a(1-e^2)} \approx 12' \text{ per orbit}
\end{equation}

\textbf{Observed:} $12.1 \pm 0.1$ arcminutes per orbit

Even in this extreme gravitational environment, the geometric principle holds.

\subsection{Galactic Dynamics: Where the Framework Breaks}

\subsubsection{Galaxy Rotation Curves}

For the Milky Way, the geometric prediction is:
\begin{equation}
v(r) = \sqrt{\frac{GM(r)}{r}}
\end{equation}

where $M(r)$ is the enclosed mass within radius $r$.

\textbf{Expected (Keplerian):} $v \propto r^{-1/2}$ beyond the galactic core
\textbf{Observed:} $v \approx$ constant (flat rotation curves)

\subsubsection{The Failure and Its Meaning}

\begin{figure}[h]
\centering
\begin{tabular}{|c|c|c|}
\hline
\textbf{Radius (kpc)} & \textbf{Predicted v (km/s)} & \textbf{Observed v (km/s)} \\
\hline
2 & 250 & 220 \\
5 & 158 & 220 \\
10 & 112 & 220 \\
15 & 91 & 220 \\
20 & 79 & 220 \\
\hline
\end{tabular}
\caption{Milky Way rotation curve: prediction vs. observation}
\end{figure}

The dramatic failure at galactic scales indicates new physics:
\begin{itemize}
\item Dark matter: Additional mass creating different $M(r)$
\item Modified gravity: Changes to the force law itself
\item Spacetime modifications: The geometric principle itself breaks down
\end{itemize}

\subsection{Dark Matter and Modified Gravity}

\subsubsection{CDM Model}

In the standard cosmological model, dark matter has density profile:
\begin{equation}
\rho(r) = \frac{\rho_0}{(r/r_s)(1 + r/r_s)^2}
\end{equation}

This creates enclosed mass $M(r) \propto r$ at large radii, giving flat rotation curves:
\begin{equation}
v(r) = \sqrt{\frac{GM(r)}{r}} \propto \sqrt{r \cdot r^{-1}} = \text{constant}
\end{equation}

\subsubsection{MOND (Modified Newtonian Dynamics)}

Alternatively, the force law itself might change at low accelerations:
\begin{equation}
F = \frac{mv^2}{r} \mu\left(\frac{a}{a_0}\right)
\end{equation}

where $\mu(x) \to 1$ for $x \gg 1$ and $\mu(x) \to x$ for $x \ll 1$, with $a_0 \approx 10^{-10}$ m/s².

\subsection{Cosmological Scales and the Hubble Flow}

\subsubsection{The Expansion of Space}

At cosmological scales, space itself expands according to:
\begin{equation}
v = H_0 d
\end{equation}

where $H_0 = 70$ km/s/Mpc is the Hubble constant.

This is not orbital motion but spacetime expansion—a fundamentally different phenomenon that doesn't involve the geometric binding principle.

\subsubsection{Accelerating Expansion}

The discovery of dark energy shows that cosmic expansion is accelerating:
\begin{equation}
\ddot{a} = -\frac{4\pi G}{3}(\rho + 3p)a + \frac{\Lambda c^2}{3}a
\end{equation}

where $\Lambda$ is the cosmological constant.

This represents a breakdown of attractive forces altogether—expansion overcoming all binding.

\subsection{Scale Hierarchy and Breakdown Points}

\subsubsection{Where the Framework Works}

\begin{table}[h]
\centering
\begin{tabular}{|l|c|c|c|}
\hline
\textbf{System} & \textbf{Scale (m)} & \textbf{Success} & \textbf{Modification} \\
\hline
Quarks & $10^{-15}$ & Good & + confinement \\
Nuclei & $10^{-14}$ & Good & + confinement \\
Atoms & $10^{-10}$ & Perfect & none \\
Molecules & $10^{-9}$ & Good & + exchange \\
Planets & $10^{11}$ & Perfect & + relativity \\
Binary stars & $10^{12}$ & Perfect & + GR \\
Stellar systems & $10^{13}$ & Good & + GR \\
Galaxies & $10^{21}$ & Fails & dark matter? \\
Clusters & $10^{24}$ & Fails & dark matter? \\
Universe & $10^{26}$ & N/A & expansion \\
\hline
\end{tabular}
\caption{Scale hierarchy of the geometric principle}
\end{table}

\subsubsection{The Critical Scale}

The geometric principle succeeds where:
\begin{enumerate}
\item Clear rotational reference frames exist
\item Binding forces dominate over expansion
\item Matter can be treated as discrete objects
\end{enumerate}

It fails where:
\begin{enumerate}
\item Dark matter dominates visible matter
\item Spacetime expansion becomes significant
\item Quantum fluctuations become important (very small scales)
\end{enumerate}

\subsection{Future Tests at Intermediate Scales}

\subsubsection{Solar System Tests}

Future precision tests in the Solar System:
\begin{itemize}
\item Cassini spacecraft: $\gamma - 1 = (2.1 \pm 2.3) \times 10^{-5}$
\item Lunar laser ranging: tests of equivalence principle
\item BepiColombo mission to Mercury: improved perihelion precession
\end{itemize}

\subsubsection{Gravitational Wave Astronomy}

LIGO/Virgo detections of binary mergers test the geometric principle in extreme conditions:
\begin{itemize}
\item Black hole masses from orbital decay
\item Tests of general relativity in strong fields
\item Confirmation of gravitational wave speeds
\end{itemize}

\subsubsection{Direct Dark Matter Detection}

If dark matter interacts weakly with normal matter, direct detection experiments might reveal:
\begin{itemize}
\item Whether dark matter follows the geometric principle
\item How dark matter contributes to galactic binding
\item Possible modifications to spacetime at large scales
\end{itemize}

\subsection{Summary: The Scale Hierarchy}

The planetary and cosmological analysis reveals:

\begin{enumerate}
\item The geometric principle scales perfectly from atoms to stellar systems
\item Relativistic corrections extend its validity to extreme gravitational fields
\item It fails dramatically at galactic scales, indicating new physics
\item The failure pattern points toward either dark matter or modified gravity
\item Cosmological expansion represents a different phenomenon entirely
\end{enumerate}

This hierarchy suggests the geometric principle is fundamental to local physics but requires modification or supplementation for cosmic-scale phenomena. The success below galactic scales and failure above provides a clear demarcation for where our current understanding needs revision.
