\section{Theoretical Framework: Spacetime from Spinning Balls}

\subsection{Space is Intrinsic, Time is Relational}

The fundamental insight underlying this work is the recognition that spacetime emerges differently for its spatial and temporal components:

\textbf{Spatial reference emerges from the 3D rotation you're part of:}
\begin{itemize}
\item The spin axis defines north/south
\item Centripetal acceleration defines up/down (your "weight")
\item The rotation direction defines east/west
\item Your own chirality defines left/right
\end{itemize}

\textbf{Temporal reference requires external observation:}
\begin{itemize}
\item On Earth, we need the sun, moon, or stars to tell time
\item An isolated spinning system has no intrinsic time
\item Time emerges from comparing cycles between systems
\end{itemize}

This distinction is crucial: space is intrinsic to rotation, time is relational between rotations.

\subsection{Requirements for Spatial Existence}

To exist in three-dimensional space—to have a definite "where"—a system must provide a spatial reference frame. This requires:

\begin{enumerate}
\item \textbf{A rotation axis}: Defining a primary spatial direction
\item \textbf{A binding force}: Creating "up" and "down" through acceleration
\item \textbf{A rotation direction}: Distinguishing the sense of motion
\item \textbf{Three-dimensional extent}: 2D rotations cannot create 3D reference frames
\end{enumerate}

Only three-dimensional rotating objects satisfy all requirements. A 2D circle spinning in abstract space provides no real spatial reference.

\subsection{Why Atoms Must Be Three-Dimensional}

Current quantum mechanics models atoms as 2D systems with angular momentum quantum numbers. But consider:

\textbf{If atoms were truly 2D:}
\begin{itemize}
\item No real spin axis → no spatial orientation
\item No surface to "stand on" → no up/down reference
\item Abstract rotation → no connection to real 3D space
\item No spatial reference frame → cannot exist in spacetime
\end{itemize}

\textbf{But atoms demonstrably:}
\begin{itemize}
\item Exist at definite positions in 3D space
\item Form directional bonds creating 3D molecules
\item Interact with 3D electromagnetic fields
\item Build our three-dimensional world
\end{itemize}

Therefore, atoms MUST be three-dimensional spinning objects—balls providing spatial reference frames through rotation.

\subsection{The Centripetal Force of Existence}

Once we recognize atoms as 3D balls, the nature of atomic binding becomes clear. Just as standing on Earth requires centripetal force (gravity) to maintain your reference frame, existing on an atomic "surface" requires centripetal force.

For circular motion at radius $r$ with velocity $v$:
$$F_{\text{centripetal}} = \frac{mv^2}{r}$$

In quantum mechanics, the velocity is constrained by the uncertainty principle. For the ground state:
$$v \sim \frac{\hbar}{mr}$$

Substituting:
$$F_{\text{centripetal}} = \frac{m(\hbar/mr)^2}{r} = \frac{\hbar^2}{mr^3}$$

This is our fundamental formula—not derived from electromagnetic theory but from the pure geometry of 3D rotation.

\subsection{The Mathematical Identity}

For any atom treated as a 3D spinning sphere, the binding force must be:

$$F = \frac{\hbar^2}{\gamma m r^3}$$

where $\gamma$ accounts for relativistic effects in heavy atoms. This formula:
\begin{itemize}
\item Contains NO free parameters
\item Includes NO quantum numbers
\item Represents pure 3D rotational geometry
\item Is the "weight" on an atomic surface
\end{itemize}

We will show this exactly equals the Coulomb force—not approximately, but as a mathematical identity.

\section{The Atoms are Balls Framework}

\subsection{Core Principles}

\begin{enumerate}
\item \textbf{Atoms are 3D balls}: Not 2D abstractions but physical rotating spheres
\item \textbf{Spatial frames from rotation}: Each atom provides its own reference frame
\item \textbf{Forces are geometric}: What we call "forces" are centripetal requirements
\item \textbf{One principle, many scales}: The same geometry from quarks to planets
\end{enumerate}

\subsection{The Universal Formula}

At every scale where 3D objects rotate:

$$F = \frac{\text{rotation-dependent factor}}{mr^3} \times \text{scale corrections}$$

\begin{itemize}
\item \textbf{Atomic scale}: $F = \hbar^2/(\gamma m r^3)$ (quantum regime)
\item \textbf{Macroscopic scale}: $F = (mvr)^2/(mr^3) = mv^2/r$ (classical regime)
\item \textbf{Nuclear scale}: Additional binding terms for confined systems
\end{itemize}

\subsection{Why This Works}

The framework succeeds because it recognizes a fundamental truth: to exist in spacetime requires having a spatial reference frame, and such frames only emerge from 3D rotation. The "forces" we observe are simply the centripetal requirements for maintaining these frames at different scales.

This isn't a model or approximation—it's recognizing what forces actually ARE.