# PART II ## The Rydberg Series One of the greatest achievements of early quantum theory was predicting the *discrete* energy levels of the hydrogen atom, known as the *Rydberg series*. This series describes the specific colors (frequencies) of light emitted by excited hydrogen atoms, formalized empirically by Johannes Rydberg in 1888. Long before the internal structure of the atom was understood, experiments showed that glowing hydrogen does not emit a continuous rainbow of light. Instead, it emits light only at very specific, sharply defined colors. Rydberg found that these frequencies follow a simple mathematical pattern involving integers $n = 1, 2, 3, \dots$. The emitted photon energies scale with the inverse square of these integers: $$ E_n \propto \frac{1}{n^2}. $$ The standard explanation, developed by Bohr and later Schrödinger, ties this scaling to the electrostatic interaction between an electron and a proton, reminiscent of a planetary view. In this picture, larger $n$ corresponds to the electron occupying an orbit farther from the nucleus. However, the formula itself contains no reference to distance, radius, or geometry—only to the integer $n$. ## The Geometry of Quantization We can interpret the $1/n^2$ factor not as a change in spatial size, but as a reorganization of energy into progressively finer structure. An analogy for this reorganization can be drawn on a napkin. Imagine a flat rectangular sheet of paper. Draw a set of straight lines from edge to edge, crossing in the middle. These lines divide the napkin into a grid. If we draw $n$ lines in each of the two independent directions, the surface is partitioned into a grid of distinct cells: $$ n \times n = n^2 \text{ cells}. $$ Before any lines are drawn, the surface corresponds to the ground state ($n=1$), associated with energy $E_1$. As we increase the number of subdivisions $n$, the total surface area remains the same, but it is partitioned into smaller and smaller regions. If the total energy of the configuration is conserved and distributed uniformly across the surface, then as the number of cells ($n^2$) increases, the energy density associated with each individual cell decreases as: $$ E_n \propto \frac{E_1}{n^2}. $$ In this abstract but constrained way, we recover the Rydberg scaling without invoking wavefunctions, operators, or planetary orbits. Atomic energy levels are simply global subdivisions of a conserved surface. ## The Torus To understand the physical basis of this grid, we must look at the topology of confinement. Return to the flat napkin. First, identify and glue together one pair of opposite edges. The flat napkin becomes a tube. Lines that originally ended on one edge now reappear continuously on the opposite edge. Next, take this tube and identify its two circular ends. Gluing these ends together produces a closed surface with no boundary—a **Torus**. Any lines drawn on the original napkin become closed paths on the torus. However, they form closed loops only if they match their own position when crossing an identified edge. This requirement ensures global continuity of the grid. In a source-free Maxwell universe, electromagnetic fields on this surface must satisfy these continuity conditions along the two independent cycles of the torus: along the axis of the tube and around the ring. This imposes a discretization condition on the wavelength. Along a closed loop of length $L$, the field must satisfy: $$ n \lambda = L. $$ These are the same conditions that produce standing waves on a string, now applied to a closed surface with two independent winding numbers. ## Energy Reorganization In this view, the Rydberg series does not describe an electron moving to a larger orbit in space. It describes the electromagnetic field reorganizing itself into progressively finer standing-wave patterns. Increasing $n$ corresponds to increasing the number of global windings on the surface. More windings impose more nodes on the same conserved topology. Transitions between levels are therefore related to the difference in **cell sizes** (or effective tube widths) between two subdivisions. To move from level $n$ to level $m$, the system must supply exactly the energy difference required to "patch" the geometry from one tube width to another: $$ \Delta E = E_1 \left( \frac{1}{n^2} - \frac{1}{m^2} \right). $$ The photon is the packet of energy that facilitates this topological patching. The ground state ($n=1$) is unique. It represents the configuration where the torus is composed of a single coherent cell—the state where the flux tube is pulled as tight as topologically possible. As we shall see, the geometric limit of this "tightness" is what determines the coupling constant of the universe. ## Charge as Topology Finally, we must account for the appearance of electric charge. In a source-free universe, $$ \nabla \cdot \mathbf{E} = 0 $$ everywhere. No electric field originates from a point. How, then, does a particle appear to have charge? Consider the standing wave on the torus. The field lines wrap around the two independent cycles, characterized by the winding numbers $(m,n)$. These windings represent closed circulations of electromagnetic energy. At any local patch of the surface, the field lines entering and leaving balance so that the net flux vanishes. However, the global circulation does not vanish. The pair $(m,n)$ characterizes a vortex-like flow of energy along the two directions of the torus. Now, enclose this configuration within a spherical surface of radius $r$ much larger than the torus itself. The total electromagnetic circulation (the "topological charge") is a conserved quantity fixed by the winding numbers. As this fixed quantity is projected through a sphere whose area grows as $4\pi r^2$, the observed field intensity necessarily falls off as: $$ \text{Intensity} \propto \frac{1}{r^2}. $$ This reproduces the phenomenology of charge. In this view, charge is not a primitive substance added to the universe. It is an effective, topological quantity: the far-field signature of closed electromagnetic circulation. What we measure as Coulomb force is simply the geometric dilution of this conserved topology over distance. ## Matter In this framework, matter is not a substrate distinct from the field. Matter is a self-sustained electromagnetic field configuration whose internal couplings generate delayed response, confinement, and stability. As we shall see, all effective material properties—mass, charge, inertia, and spectral structure—are emergent consequences of source-free Maxwell dynamics in a Maxwell Universe.