\section{Related Work and Theoretical Context}

Analogies between classical and quantum phenomena have a long history in physics. Bohmian mechanics \cite{Bohm1952} attempts to give particles definite trajectories guided by a pilot wave, blending classical-like paths with quantum outcomes. Similarly, prior works have drawn parallels between fundamental forces at different scales \cite{Holdom2017,Panpanich2018}.

Modified gravity theories like MOND \cite{Milgrom1983} have attempted to explain galactic dynamics without dark matter by modifying Newton's laws at low accelerations ($a_0 \sim 1.2 \times 10^{-10}$ m/s$^2$). Subsequent developments \cite{Bekenstein2004,Famaey2012} have explored relativistic extensions of these ideas. Our approach differs by adding a new force term rather than modifying existing laws, though as we will show, it faces similar challenges in explaining galaxy rotation curves.

Recent observations have provided unprecedented tests of gravity in extreme regimes. The GRAVITY collaboration's tracking of star S2 orbiting Sagittarius A* \cite{Gravity2018,Gravity2020} has confirmed general relativistic effects with remarkable precision. Similarly, Gaia's astrometric data \cite{GaiaDR3} offers new opportunities to test modified gravity theories at stellar cluster scales.

\section{Atoms are Balls: Multi-Element Verification}

\subsection{The Core Insight}

Current quantum mechanics treats atoms as two-dimensional systems with angular momentum quantum numbers. But what if atoms are actually three-dimensional spinning spheres—balls, not circles? This simple conceptual shift leads to profound mathematical consequences.

\subsection{Universal Formula for Atomic Binding}

For any atom treated as a 3D spinning sphere, the binding force emerges from rotational geometry:

$$F_{\text{spin}} = \frac{\hbar^2 s^2}{mr^3}$$

where $s = mvr/\hbar$ is calculated from the electron's actual motion. We'll demonstrate this works not just for hydrogen, but across the periodic table.