Wave Coherence and the Toroidal Hydrogen Atom
Deriving Atomic Spectra from Williamson–van der Mark Toroidal Photon Topology Without Coulomb Potentials
By Stevie Horton | February 2026
Hydrogen is the simplest atom and the most precisely measured system in physics. Its spectral lines — the Lyman, Balmer, Paschen, and higher series — are known to extraordinary precision. The standard quantum mechanical treatment reproduces these measurements beautifully. By any empirical standard, the theory works.
And yet the foundations of that theory rest on assumptions that are never derived — only postulated. The electron is a point particle with an intrinsic property called “charge.” That charge generates a force. The force gets inserted into a wave equation as a potential, and quantized energy levels fall out. It works. But what is charge? The theory doesn’t say. What is the electron? A mathematical singularity with no spatial extent. Why does the Rydberg constant have its value? Because five “fundamental constants” happen to combine in a particular way. Why do those constants have those values? Silence.
This paper proposes an alternative derivation of hydrogen’s spectral structure that requires no charges, no forces, no potentials, and no point particles. It requires only waves in a medium, and the topology of those waves.
The Electron as Confined Photon
In 1997, J.G. Williamson and M.B. van der Mark proposed that the electron can be modeled as a photon confined to a toroidal topology — a circularly polarized electromagnetic wave traveling in a closed loop. The electron isn’t a particle possessing properties. It’s a self-sustaining coherent wave structure whose observed properties emerge from geometry.
Mass arises because the confined photon carries energy, and confined energy resists acceleration. The electron’s mass is simply mₑ = ħωₑ/c². Mass isn’t fundamental — it’s derived from wave confinement.
Charge emerges from the phase relationship between the toroid’s internal wave and its external coupling field. Two toroidal structures in phase constructively interfere (attraction); out of phase, they destructively interfere (repulsion). Quantization of charge follows from the requirement for stable standing wave patterns — only integer topological winding numbers produce stable configurations.
Spin-½ arises from the toroidal topology itself. A single topological twist means the wave must complete two full revolutions (720°) to return to its starting phase. This is the geometric origin of fermionic behavior, including the Pauli exclusion principle: two identical single-twist toroids can’t occupy the same coherence state because their antisymmetric phase structure forces the coherence function to zero.
The proton, in this framework, is a more complex composite topological structure — three confined wave modes (quarks in standard language, higher-order toroidal knots in ours). Its near-field extends into the surrounding medium, creating a region of modified coherence properties — analogous to a refractive index gradient, but for coherence rather than light speed.
Deriving Hydrogen’s Spectrum
Two Coupled Toroids
Hydrogen isn’t a charged particle orbiting another charged particle. It’s two toroidal wave structures achieving resonant coupling — the electron toroid finding stable standing wave configurations in the near-field of the proton toroid. The “atomic orbitals” are the set of resonant modes where the electron toroid can maintain coherent phase-locking with the proton toroid.
Quantization (n = 1, 2, 3, …) falls out from the requirement that the phase must close on itself — exactly like Williamson’s original closure condition for the electron itself, but applied to the coupled system.
The Dual Phase Closure Condition
For the electron toroid to maintain a stable coupled mode, two conditions must be satisfied simultaneously:
Internal closure — the electron’s own toroidal wave must complete its 720° twist and return to phase. This is what makes it an electron. This condition is fixed.
External closure — the electron toroid’s orbital coupling with the proton toroid must also close on itself. The wave pattern traced by the electron structure around the proton structure must return to the same phase after each complete cycle.
The internal closure is fixed. But the external closure has multiple solutions. The electron toroid can couple at different radial distances, and at each distance, the phase accumulated per orbit is different. The stable modes are those where the total accumulated phase is an integer multiple of 2π.
Where 1/n² Comes From
In spherical geometry, the phase accumulation per orbit goes as 1/r, because the circumference scales with r while the wave’s interaction amplitude with the proton toroid falls off spherically. The energy of each mode — the coherence coupling strength — scales as the square of the phase density, giving 1/r². Since the allowed radii go as n (integer standing wave modes), the energy levels go as 1/n².
No Coulomb potential invoked. The 1/n² spacing comes purely from:
- Spherical geometry of two coupled toroidal wave structures
- Phase closure requirement (quantization)
- Amplitude falloff inherent to spherical waves
The same result that Bohr obtained by postulating angular momentum quantization, and that Schrödinger obtained by solving a wave equation with a Coulomb potential, emerges here from wave topology alone.
For the full derivation — tracing the chain from Coulomb’s measurement through Gauss’s theorem, Newton’s shell theorem, and Williamson’s toroidal topology — see Where 1/n² Comes From.
The Rydberg Constant Reframed
In standard QM, the Rydberg constant is built from five supposedly fundamental quantities mashed together. In the wave coherence framework, it decomposes into three — all derivable from the medium’s properties and toroidal topology:
The electron toroid’s fundamental frequency (ωₑ) — the internal circulation rate of the confined photon. This replaces “electron mass” since mass is confined wave energy.
The coupling geometry (α) — the fine structure constant (~1/137) becomes the ratio of the electron toroid’s interaction cross-section to its orbital coherence length. It’s a geometric property of how efficiently two toroidal wave structures exchange phase information — how “leaky” the toroid is.
The medium propagation speed (c) — not “the speed of light in vacuum” but the fundamental propagation rate of the wave medium itself.
The key prediction: α should be derivable from first principles of toroidal geometry. It’s not a free parameter of nature — it’s a consequence of the single-twist toroid’s shape.
The Medium: Properties and Evidence
If hydrogen’s structure emerges from two Williamson toroids coupling through a medium, then the medium must have specific properties we can work backward to identify.
Toroidal confinement support. The medium isn’t passive or homogeneous — it has structure that permits topological closure. Think of water supporting vortices.
Non-dispersive at toroidal frequencies. The stability of electrons means the medium propagates waves at the toroid’s operating frequency without loss — a superconducting condition for specific coherent modes.
Partial entrainment by mass. Regions of high wave energy density (massive objects) partially modify the local medium state. The medium near Earth isn’t identical to the medium in deep space.
Isotropic propagation in local rest frame. The medium propagates waves at c in all directions in its local frame, but that frame varies depending on nearby mass.
What the Experiments Actually Show
The standard narrative — “Michelson-Morley proved there is no aether” — is a simplification that obscures the actual data.
Michelson-Morley (1887) found a small but nonzero fringe shift. It was smaller than predicted for a fully stationary aether and was interpreted as null. But the raw data show systematic trends that were never fully explained.
Michelson-Gale (1925) detected light-speed variations along the rotational axis of the Earth — a positive result consistent with a medium partially entrained by Earth’s rotation. This experiment has been largely forgotten.
Dayton Miller (1906–1933) performed over 326,000 interferometer turns with 16 readings each — more than 5.2 million measurements — using the largest and most sensitive interferometer ever constructed of its type, with a 64-meter round-trip light path. He consistently observed a fringe shift of 0.12 ± 0.01, incompatible with zero, corresponding to a drift of approximately 9 km/s. The effect correlated with sidereal period and increased at altitude — exactly what a partially-entrained medium would produce.
Einstein knew what was at stake: “I believe that I have really found the relationship between gravitation and electricity, assuming that the Miller experiments are based on a fundamental error. Otherwise, the whole relativity theory collapses like a house of cards.”
The scientific community chose to treat Miller’s result as systematic error rather than engage with its implications.
The Sagnac Effect (1913–present) isn’t just an experimental result — it’s working technology. Every ring laser gyroscope in every aircraft, every fiber optic gyroscope in every submarine, operates because light speed is different depending on direction of travel relative to rotation. If no medium existed, these instruments wouldn’t work.
The standard response is that general relativity explains Sagnac through frame-dragging — but that’s a mathematical description of exactly what a medium would do, while insisting no medium is involved. The math is equivalent. The ontology differs. The wave coherence framework simply identifies the medium that the mathematics already describe.
Testable Predictions
The wave coherence derivation agrees with standard QM on most observables — it must, because the data is the data. But it diverges in four specific edge cases.
Prediction 1: Lamb Shift Reinterpretation. Standard QM attributes the 2S½/2P½ splitting to virtual particle vacuum fluctuations. The coherence framework predicts it arises from asymmetry in the toroidal near-field — S-states interact with the proton toroid’s internal structure differently than P-states. The shift should correlate with proton toroidal geometry, not cutoff-dependent vacuum energy.
Prediction 2: Maximum Stable Quantum Number. At very high n, the external phase closure becomes so loose that the electron toroid becomes vulnerable to decoherence from the medium itself. There should be a maximum stable n that depends on local medium coherence, not just ionization energy.
Prediction 3: Topology-Dependent Isotope Shifts. The proton and deuteron have different internal topologies. The coherence model predicts coupling shifts beyond what reduced mass corrections explain — measurable with current precision spectroscopy.
Prediction 4: Coherence-Dependent Transition Rates. Spontaneous emission rates should be subtly different in regions of high electromagnetic coherence (laser cavities, near superconductors, tuned resonant cavities) versus free space — beyond what the Purcell effect accounts for. This may be testable with existing equipment.
Geometric Correspondences: From Atoms to Architecture
The angular relationships governing toroidal wave coupling appear across scales. While correspondences aren’t proof, they suggest a deeper geometric principle at work.
The 36° Connection
The Pyramid of Khafre at Giza was built on a 3-4-5 triangle, producing a slope of 53.13°. The complement of that angle is 36.87° — within a degree of 36°, the twist angle of DNA’s double helix.
In canonical B-DNA, each base pair rotates 36° relative to its predecessor, completing 360° every 10 base pairs and forming a decagonal cross-section with ten-fold rotational symmetry. That 36° is exactly half the pentagon’s interior rotation of 72°, placing DNA’s fundamental geometry in the domain of the golden ratio.
The Golden Ratio in DNA
B-DNA is saturated with φ. The major groove to minor groove ratio is approximately 21 to 13 angstroms — consecutive Fibonacci numbers approaching φ. The length-to-width ratio of one helical turn approximates φ. The cross-section is decagonal: two pentagons, one rotated 36° from the other.
The protein alpha helix has 3.6 residues per turn. Microtubules assemble from 13 protofilaments — a Fibonacci number. Viral capsids use icosahedral symmetry built on golden ratio geometry. The golden angle of 137.5° governs phyllotaxis throughout the plant kingdom.
The Chain
| Scale | Structure | Angle/Number | Connection |
|---|---|---|---|
| Atomic | Williamson toroid (electron) | α ≈ 1/137 | Toroidal coupling geometry |
| Atomic | Hydrogen spectral series | 1/n² spacing | Spherical phase closure |
| Molecular | B-DNA double helix | 36° per base pair | Decagonal/pentagonal geometry |
| Molecular | Protein alpha helix | 3.6 residues/turn | Same angular family |
| Biological | Phyllotaxis | 137.5° golden angle | Fibonacci/φ optimization |
| Architectural | Pyramid of Khafre | 53.13° (complement 36.87°) | 3-4-5 triangle |
If the wave coherence framework is correct, these aren’t coincidental numerical curiosities. They’re expressions of the same coherence-optimal geometry operating at every scale — from the topology that defines the electron, through the phase closure that quantizes atomic spectra, to the molecular architecture that stores biological information.
Implications for Locality
The framework redefines nonlocality. In standard QM, entanglement is spooky action at a distance. In the coherence framework, two entangled particles are toroidal structures that were once a single coupled wave system. They share phase coherence through the medium — which never broke, because the medium is continuous. What we call nonlocality is just the medium carrying the phase relationship. No more mysterious than two ends of a vibrating string being correlated.
The measurement problem dissolves too. “Collapse” is a decoherence event — a specific topological moment where an external wave structure (the detector) couples to the system and breaks the phase closure of one mode, forcing coherence into a definite configuration.
Bell inequality violations that rule out local hidden variables don’t rule out a coherence medium. The medium itself is the nonlocal element — not as action at a distance, but as a continuous substrate maintaining phase relationships.
Conclusion and Program
Hydrogen’s spectral structure — the 1/n² energy levels, the Rydberg constant, the quantum numbers — can be derived from the resonant coupling of two Williamson–van der Mark toroidal wave structures in a coherent medium, without point particles, intrinsic charges, Coulomb potentials, or the Schrödinger equation. The derivation requires only waves, topology, and phase closure.
The experimental record regarding the medium — Michelson-Gale, Dayton Miller’s persistent signal, the operational Sagnac effect, GPS corrections — is consistent with a partially mass-entrained coherence medium and does not support the strong claim that no medium exists.
The geometric correspondences between toroidal coupling angles (α, 36°, φ) and those found in DNA, biological structures, and historical architecture suggest that coherence-optimal geometry may be a universal organizing principle — one that nature employs from atomic spectra to the storage of genetic information.
Three priorities going forward:
First, a rigorous derivation of the fine structure constant from single-twist Williamson toroid topology. Second, experimental testing of coherence-dependent transition rates using existing precision spectroscopy. Third, extension of the coupled-toroid model to multi-electron atoms and molecular bonding — building toward a complete coherence-based chemistry.
Stevie Horton is the Marketing & Sales-Ops Director at BTB Brands and an independent researcher in wave coherence theory, toroidal photon topology, and alternative physics frameworks.