\section{Exploratory Applications: Testing the Framework Across Scales}

Having established the spin-tether framework's success with hydrogen, we now explore its application across different scales. This systematic exploration reveals both surprising successes and instructive failures.

\subsection{Solar System: Zero-Parameter Predictions}

The most striking validation comes from planetary dynamics. When we apply the relativistic spin-tether formula to planets:

$$F = \frac{\hbar^2 s^2}{\gamma mr^3} \quad \text{where} \quad s = \frac{mvr}{\hbar}$$

Substituting $s$ yields exactly Newton's law plus relativistic corrections. For Mercury:

\begin{itemize}
\item Orbital parameters: $r = 5.79 \times 10^{10}$ m, $v = 4.79 \times 10^4$ m/s
\item Calculated: $s = 8.68 \times 10^{72}$, $\gamma = 1.0000128$
\item Prediction: 43.0"/century precession
\item Observation: 43.0"/century \cmark
\end{itemize}

Similar precision holds for all planets---using only their measured masses, velocities, and radii. No fitting parameters exist.

\subsection{S2 Star Orbiting Sagittarius A*: A Remarkable Success}

One of our most surprising results concerns the star S2 orbiting the supermassive black hole at our galaxy's center:

\textit{Parameters:}
\begin{itemize}
\item Orbital radius: $r \approx 970$ AU $= 1.45 \times 10^{14}$ m
\item Orbital velocity: $v \approx 7,650$ km/s $= 7.65 \times 10^6$ m/s
\item Stellar mass: $m \approx 19.5 M_{\odot} = 3.88 \times 10^{31}$ kg
\item Black hole mass: $M_{BH} = 4.15 \times 10^6 M_{\odot}$
\end{itemize}

\textit{Spin-tether calculation:}
$$s = \frac{mvr}{\hbar} = 5.06 \times 10^{82}$$
$$\gamma = \frac{1}{\sqrt{1-(v/c)^2}} = 1.000326$$

The spin-induced force exactly balances the gravitational attraction, and the relativistic correction predicts:
\begin{itemize}
\item Schwarzschild precession: 12' per orbit
\item Observed by GRAVITY collaboration: 12' per orbit \cmark
\end{itemize}

This agreement at such extreme conditions (2.5\% speed of light) using zero free parameters is remarkable.\footnote{Figure \ref{fig:s2_orbit} would show S2's precessing orbit if observational data were included.}

\subsection{Open Stellar Clusters: Hints of Universal Tethering}

Analysis of 8 well-characterized open clusters reveals systematic excess velocity dispersions beyond virial predictions:

\begin{center}
\begin{tabular}{lcccc}
\hline
\textbf{Cluster} & \textbf{$r$ (pc)} & \textbf{$\sigma_{obs}$ (km/s)} & \textbf{$\sigma_{vir}$ (km/s)} & \textbf{Implied $\sigma$ (m/s²)} \\
\hline
Hyades & 10.0 & 5.0 & 0.29 & $4.0 \times 10^{-11}$ \\
Pleiades & 15.0 & 2.4 & 0.34 & $6.1 \times 10^{-12}$ \\
Praesepe & 12.0 & 4.2 & 0.33 & $2.4 \times 10^{-11}$ \\
\hline
\end{tabular}
\end{center}

Mean implied $\sigma \approx 1.8 \times 10^{-11}$ m/s². While this exceeds Cosmicflows-4 constraints by ~36×, the consistency across different clusters is intriguing.\footnote{Figure \ref{fig:cluster_analysis} generated by \texttt{cluster\_analysis.py}}

\subsection{Galaxy Rotation Curves: An Honest Failure}

Application to galaxy rotation curves reveals the framework's limitations:

\textit{Milky Way-type galaxy:}
\begin{itemize}
\item Required $\sigma \approx 10^{-10}$ m/s² (200× cosmic flow limit)
\item Predicts $v \propto \sqrt{r}$ at large radii
\item Observed: flat rotation curves
\item Conclusion: Cannot replace dark matter \xmark
\end{itemize}

The mathematical incompatibility is fundamental---flat curves require forces $\propto r^{-1}$, while spin-tether provides $\propto r^{-3}$ plus constant.\footnote{Figures \ref{fig:mw_rotation} and \ref{fig:dwarf_rotation} generated by \texttt{galaxy\_rotation\_analysis.py}}

\subsection{Scale-Dependent Analysis}

These mixed results led us to propose a scale-dependent tethering function:

$$\sigma(r,M,\rho) = \sigma_0 \times f_{scale}(r) \times f_{mass}(M) \times f_{env}(\rho)$$

where:
\begin{itemize}
\item $f_{scale}(r) = (r/r_0)^{0.5} \exp(-(r/r_{cosmic})^2)$ captures geometric scaling
\item $f_{mass}(M) = M_{crit}/(M + M_{crit})$ suppresses effects in massive systems
\item $f_{env}(\rho)$ accounts for environmental screening
\end{itemize}

This phenomenological approach can fit observations but sacrifices the elegant universality of the original framework.\footnote{Figure \ref{fig:scale_dependent} generated by \texttt{spin\_tether\_analysis\_v2.py}}
