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Where 1/n² Comes From

Why hydrogen's energy levels scale as 1/n² — traced from Coulomb's measurement through Gauss's theorem and Newton's shell theorem to Williamson's toroidal topology. No new physics, just geometry.

wave coherencetoroidal electronhydrogen spectrumWilliamsonCoulombGaussNewtontopologyinverse square law

Where 1/n² Comes From

Companion to Wave Coherence and the Toroidal Hydrogen Atom

By Stevie Horton | February 2026

The Honest Question

Before presenting the derivation, we should address the objection a careful reader will already be forming: if the energy levels scale as 1/n², and Coulomb’s law also produces 1/n² energy levels in the standard treatment, how is this not just Coulomb’s law repackaged in topological language?

The answer requires us to look at what Coulomb actually measured, what Gauss explained about why Coulomb’s measurement had to come out the way it did, and what Newton proved about the uniqueness of that result. These are three of the most established results in physics. The only new ingredient we add is what produces the field divergence in the first place — and for that, we stand on Williamson and van der Mark’s 1997 model. Everything else follows from geometry that physicists have trusted for over two centuries.

Coulomb’s Measurement

In 1785, Charles-Augustin de Coulomb published a series of experiments using a torsion balance to measure the force between charged objects. He found that the force decreased with the square of the distance: double the separation, quarter the force. This was an empirical observation — a measurement, not an explanation. Coulomb could tell you what the force did. He could not tell you why it did it. The inverse square relationship was treated as a fundamental law of nature, and for most practical purposes, it still is.

But an empirical law invites a deeper question: is the inverse square relationship a special property of electric charge, or does it arise from something more fundamental?

Gauss’s Explanation

In 1835, Carl Friedrich Gauss provided the answer. His divergence theorem — known in electrostatics as Gauss’s law — demonstrated that the inverse square falloff is not a special property of charge. It is a geometric consequence of flux conservation in three-dimensional space.

The argument is simple enough to follow without equations. Imagine any source producing a conserved quantity that radiates outward in all directions — it doesn’t matter what the quantity is. Now imagine enclosing that source in a spherical shell. All of the flux must pass through the shell, because the quantity is conserved. If you double the radius of the shell, its surface area increases by a factor of four (because the surface area of a sphere is 4πr²). But the total flux passing through hasn’t changed — the same amount is still being emitted by the source. So the flux per unit area — the field strength at any point on the shell — must decrease by a factor of four.

This is the inverse square law. It has nothing to do with charge. It has nothing to do with electrostatics specifically. It is a consequence of two things only: the source is localized, and space has three dimensions. Any conserved quantity emanating from a localized source in three-dimensional space must produce a 1/r² field. There is no other possibility consistent with flux conservation and spherical symmetry.

Coulomb measured 1/r². Gauss explained that he had to measure 1/r², because he was measuring a conserved flux in three-dimensional space.

Newton’s Uniqueness Proof: The Shell Theorem

Isaac Newton proved something even more powerful in the Principia (1687), though its full significance is often underappreciated. The shell theorem states that the net gravitational (or electrostatic) field inside a uniform spherical shell is exactly zero, and that outside the shell, the field is identical to that of a point source at the center.

The remarkable part — and the part most relevant to our argument — is that Newton proved this result is unique to the inverse square law. If the field fell off as 1/r³, or 1/r^1.5, or any power other than exactly 1/r², the interior field would not vanish. There would be residual forces inside the shell. The clean separation between interior and exterior — the ability to treat any spherically symmetric source as if it were a point — depends entirely on the exponent being exactly 2.

This is not a minor mathematical convenience. It is why the hydrogen problem is solvable in the first place. The electron experiences the proton’s field as if all the source were concentrated at a single point, because the field falls off as 1/r², and only because it falls off as 1/r². Newton’s shell theorem guarantees this.

The Williamson Connection

Now we can state precisely what Williamson and van der Mark contributed, and what we are building upon.

In their 1997 paper, Williamson and van der Mark showed that a photon confined in periodic boundary conditions of one wavelength, with toroidal topology, produces a net field divergence — an inward-directed (or outward-directed) electric field everywhere outside the rotation horizon. This divergence arises not from an intrinsic property called “charge” but from the commensurability of the photon’s field rotation with its orbital path around the toroid. The topology — specifically, the single-twist closure in a non-simply-connected space — forces the field to point consistently inward. They calculated the resulting apparent charge at approximately 0.91e, noting that it is independent of the size of the object and depends only on the topology and the internal field distribution.

This is the one genuinely new physical claim in our derivation: that the source of the divergence is topological. Everything that follows is established physics.

Once a localized, conserved divergence exists — regardless of its origin — Gauss’s theorem takes over. The flux is conserved. The space is three-dimensional. The field must fall off as 1/r². And Newton’s shell theorem guarantees that the electron toroid, at any distance r from the proton toroid, experiences the coupling field exactly as if the proton’s entire topological divergence were concentrated at a point.

We are not introducing a new force law. We are not postulating an inverse square relationship. We are showing that the inverse square relationship is the inevitable geometric consequence of a localized topological divergence in three-dimensional space — the same consequence that Gauss identified in 1835, applied to a source that Williamson identified in 1997.

From 1/r² to 1/n²

With the 1/r² coupling established on purely geometric grounds, the path to hydrogen’s spectrum is straightforward.

Williamson and van der Mark’s original closure condition requires that the confined photon’s phase closes on itself after traversing the toroidal path — periodic boundary conditions of exactly one wavelength. This is what makes the electron a stable structure. It is a fixed condition with a single solution.

We extend this principle to the coupled system. When the electron toroid enters the near-field of the proton toroid, it must find configurations where both closure conditions are satisfied simultaneously: its own internal phase closure (which is fixed and defines it as an electron) and an external phase closure with the proton’s coupling field. The external closure requires that the phase accumulated over a complete orbital cycle returns to its starting value — that is, the total accumulated phase must be an integer multiple of 2π.

This external condition, unlike the internal one, admits multiple solutions. At different radial distances, the phase accumulated per orbit differs. The stable modes — the ones where the wave pattern closes on itself — occur at discrete radii where the accumulated phase equals exactly n × 2π, with n = 1, 2, 3, and so on.

Since the allowed radii scale with n (each successive standing wave mode fits one more complete wavelength into the orbital path), and the coupling energy at each radius scales as the square of the field strength (which goes as 1/r²), the energy of the nth mode scales as 1/n².

Every step in this chain is either established physics or a direct extension of Williamson’s published model:

  • Topological divergence → localized source: Williamson and van der Mark (1997)
  • Localized source in 3D → 1/r² field: Gauss’s theorem (1835)
  • 1/r² uniquely permits point-source equivalence: Newton’s shell theorem (1687)
  • Phase closure of single toroid → stable electron: Williamson and van der Mark (1997)
  • Phase closure of coupled system → quantized modes: Our extension of Williamson’s principle
  • Quantized modes + 1/r² coupling → 1/n² energy levels: Geometric consequence

No Coulomb potential was invoked. No charge was assumed as fundamental. The same spectral structure 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 and three-dimensional geometry alone.

This is a companion article to Wave Coherence and the Toroidal Hydrogen Atom, which presents the full derivation of hydrogen’s spectral structure from toroidal photon topology.

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.