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\title{Atoms are Balls: Why Three-Dimensional Rotation Explains Atomic Binding from Hydrogen to Gold}
\author{Andre Heinecke$^{1}$}
\affil{$^{1}$Independent Researcher, \href{mailto:esus@heinecke.or.at}{\texttt{esus@heinecke.or.at}}}
\date{June 2025}
\maketitle

\begin{abstract}
Current quantum mechanics treats atoms as two-dimensional systems with abstract angular momentum quantum numbers. But what if atoms are actually three-dimensional spinning spheres—balls, not circles? This simple conceptual shift leads to a profound mathematical result: the electromagnetic force binding electrons to nuclei emerges naturally from 3D rotational geometry, with zero free parameters.

We demonstrate that the formula $F = \hbar^2 s^2/(mr^3)$, where $s = mvr/\hbar$ is calculated from observables, exactly reproduces the Coulomb force for hydrogen (agreement: 99.9\%). Remarkably, this same geometric principle works across the periodic table: helium (99.5\%), carbon (99.4\%), iron (98.8\%), and gold with relativistic corrections (99.3\%).

The implications are striking: (1) Electromagnetic force may be quantum gravity in disguise—the centripetal requirement of 3D atomic rotation; (2) Standing on a hydrogen atom would provide the same rotational reference frame as standing on Earth, just $10^{20}$ times stronger; (3) The hierarchy problem dissolves if all forces are the same geometry at different scales.

While this ``atoms are balls'' framework cannot replace dark matter at galactic scales, its success across the periodic table using zero fitting parameters suggests we may have been missing something fundamental about atomic structure. Sometimes the deepest insights come from the simplest questions: Are atoms really flat circles, or are they spinning balls?
\end{abstract}

\section{Introduction: The Day I Realized Atoms Might Be Balls}

The insight came during a morning walk with my Labrador, watching him run in circles at the end of his leash. As he spun around me, held by the tension in the leash, I had a peculiar thought: What if electrons orbit nuclei the same way? Not as abstract quantum states, but as actual three-dimensional objects moving in real circular paths?