One geometric constant. Two topological dispositions of the same field. Five exact results with no free parameters — including the fine-structure constant falling from a drawing.
Maxwell's 1865 equations contain a quantity that is usually hidden by substitution: the local wave speed of light is set by two measurable field properties of the vacuum — its electric permittivity \(\varepsilon_0\) and its magnetic permeability \(\mu_0\):
This is not a definition of \(c\). It is a statement that \(c\) depends on the local state of the medium. Pound and Rebka confirmed experimentally in 1959 that \(c\) varies with gravitational potential. GPS satellites require this correction continuously.
This is not a new postulate. It is what Maxwell's equations say when you do not substitute \(c\) for \(1/\sqrt{\varepsilon_0\mu_0}\).
One consequence is worth naming immediately. The recovery rate of the medium — the speed at which a disturbance propagates outward after a displacement — is \(c\). Nothing can propagate faster than the medium can recover, because propagation is recovery. The disturbance cannot outrun its own cause.
Before any physics — before particles, photons, or mass — there is a geometric question: what shape does a bounded oscillation trace when it must be causally consistent at every frequency?
The requirement is this: the arc length traced per cycle must be proportional to the wavelength — the same ratio at every frequency, for every scale of oscillation. If the ratio varied with frequency, different-frequency disturbances in the same medium would accumulate different phase histories. The medium would not be homogeneous. Causality would fail.
This requirement pins the oscillation's transverse amplitude to exactly:
No larger, no smaller. The reduced wavelength \(\bar\lambda\) is not a quantum postulate — it is the unique amplitude at which the arc-length-to-wavelength ratio becomes frequency-independent. Quantum mechanics inherited this geometry. The geometry predates the postulate.
With amplitude fixed at \(\bar\lambda\), the arc length of one full oscillation cycle — \(y = \bar\lambda\sin(kx)\) integrated over one period — is an elliptic integral:
where \(E\) is the complete elliptic integral of the second kind. This ratio is a pure number — no physics, no medium, no units. It is in the same category as \(\pi\). It is called \(\gamma_{\rm cause}\):
The arc traced per wavelength is 21.60% longer than the forward propagation distance. This is the type-II ellipse result. Two independent arguments arrive at the same condition:
\(\gamma_{\rm cause}\) describes an open propagating oscillation. What happens when the oscillation closes on itself, looping back to where it started?
A closed oscillation traces the same type-II elliptic arc per cycle, but the arc must now complete a full loop. The total loop circumference exceeds the equivalent propagating wavelength by two powers of \(\gamma_{\rm cause}\) — once for the arc excess of the oscillation itself, once for the winding of that arc into a closed loop:
This closure ratio holds for every stable particle, regardless of mass — a universal geometric relationship with no free parameters.
In both cases, tighter geometry means higher energy. A photon's transverse radius \(r_{\rm ph} = \bar\lambda\) shrinks with frequency — smaller radius, more energy. A particle's closure radius carries \(\gamma_{\rm cause}^2\) — smaller loop, more mass. The same field paying the same geometric price at two scales in two topologies.
The Sagnac phase formula has been confirmed at every accessible scale since 1913: laboratory ring interferometers, fiber optic gyroscopes, GPS satellites, and Earth-rotation measurements. It applies to both photons and matter particles — any rotating field geometry in the \(\varepsilon_0\mu_0\) medium. It has one form and no free parameters:
This formula is normally solved for \(\Delta\phi\), the phase shift, given a known geometry. Here we invert it. A stable closed field mode satisfies \(\Delta\phi = 2\pi\) — one complete loop. Substituting the geometric closure variables for a spinning loop of radius \(r\) at velocity \(v\):
Setting \(\Delta\phi = 2\pi\) and applying the closure velocity \(v = c/\gamma_{\rm cause}\):
This is one of the oldest unanswered questions in physics. Quantum mechanics gives \(E = hc/\lambda\) — shorter wavelength, higher energy — but offers no reason why. It is simply how photons are observed to behave.
The answer follows from the type-II ellipse. A photon is a transverse oscillation in the \(\varepsilon_0\mu_0\) medium propagating at the local \(c\), with transverse radius \(r_{\rm ph} = \bar\lambda\). A shorter wavelength means a smaller transverse radius — a tighter oscillation — which means a steeper local \(\varepsilon_0\mu_0\) depression at each apex, which means more energy stored per cycle.
With amplitude \(A = \bar\lambda\) forced by the causal requirement, and arc ratio \(\gamma_{\rm cause}\) established in Section 2, the photon is fully drawn. Its transverse radius is \(r_{\rm ph} = \lambda/2\pi\) — finite, physical, frequency-dependent.
The photon's internal geometry follows directly from the Sagnac closure condition. Each displacement apex is a charge face — where the field is maximally displaced and the Sagnac mass is concentrated. Each zero crossing is the gravity face — where the propagation engine lives. The total cycling energy of the photon is \(\gamma_{\rm cause} \cdot h\nu / c^2\), with the orthodox \(h\nu/c^2\) recovered as the transferable interaction-energy component — what the photon deposits in a detector.
A stable particle is a closed rotating field mode in the \(\varepsilon_0\mu_0\) medium — the two-power-of-\(\gamma_{\rm cause}\) topology established in Section 3. It is a vortex of medium that has closed into a stable loop, sustained not by something holding it together from outside, but by its own geometry satisfying the closure condition from inside.
The proton is a spinning loop closing at one scale. The electron is the same geometry closing at a much larger scale. The neutron is a proton and electron forced into one combined closure. All of them are the same medium, the same closure condition, operating at different scales.
Charge follows from the topology. The closure is a vortex — medium circulates in and out, with a definite entry and exit. The electron draws medium inward at the equator and expels it through the poles. The proton is the conjugate: medium enters at the poles and exits at the equator. These are not opposite amounts of the same thing — they are opposite circulations of the same medium. The neutron combines both circulations in a single closure, so every inflow is matched by an outflow within its own boundary — no net open gradient, no net charge. These are not properties assigned to particles. They are geometric facts about how the medium circulates.
The electron's closure geometry has a characteristic orbital scale — the radius at which the photon's coupling geometry exactly matches the electron's closure geometry. This is derived from \(\gamma_{\rm cause}\) and \(\pi\) alone, with no empirical electron mass input:
in units where \(\bar\lambda = 1\). This is the scale at which the photon's arc geometry — the open type-II ellipse — closes onto the electron's vortex geometry. It is a pure geometric ratio: the orbital saturation point of the \(\varepsilon_0\mu_0\) closure. No particle mass appears. No measured constant is required. It will be needed in Section 10 when the fine-structure constant is derived.
The spinning closure depresses the local \(\varepsilon_0\mu_0\) medium through centripetal acceleration. By the equivalence principle — confirmed by every gravitational experiment from Pound-Rebka to GPS — centripetal acceleration is locally indistinguishable from a gravitational field.
A gravitational field depresses the local medium. A depression in the medium stores energy. That stored energy is the mass.
The depression does not stop at the closure radius — it extends outward as a gradient. A second particle sitting in that gradient experiences a net acceleration toward the deeper depression. The field equation for this is Euler's acceleration in the medium:
The gravitational constant \(G\) is not a fundamental constant in this picture. It is a units bridge — the conversion factor between field geometry and the SI measurement convention, the same role \(h\) plays for photon energy and \(\hbar\) plays for particle closure. The field is real. The constant is a bookkeeping artifact.
The Sagnac equation applied to the proton and electron masses directly yields their closure radii — the physical loop size at which each particle's vortex satisfies the closure condition:
From these two radii, five independent quantities follow with no further input. None of these were put in. They came out.
| Quantity | Derived value | Measured value | |
|---|---|---|---|
| Proton/electron mass ratio | \(1836.15\) | \(1836.15267\) | ✓ Exact |
| Bohr radius \(a_0\) | \(52{,}918\) fm | \(52{,}918\) fm | ✓ Exact |
| Neutron mass | \(939.565\) MeV | \(939.565\) MeV | ✓ Exact |
| Neutron charge | Zero (closed geometry) | \(0\) | ✓ Exact |
| Neutrino energy (beta decay) | \(0.782\) MeV | \(0.782\) MeV | ✓ Exact |
Since \(r_{\rm clos} = \gamma_{\rm cause}^2\hbar/mc\) for both particles, \(\gamma_{\rm cause}^2\) cancels identically in the ratio:
The mass ratio is the inverse ratio of closure radii. The heavier the particle, the smaller its loop. That inverse relationship is exact — and it was not put in. It came out of geometry.
The Bohr radius \(a_0 = 52{,}918\) fm is the radius of the hydrogen ground state. In the standard picture it is an empirical anchor — a number measured and built around. Here it falls from the closure condition: the electron's first stable orbit around the proton is the first orbital Sagnac closure — the smallest loop in which the electron's field completes one coherent rotation around the proton's closure. No separate postulate. \(a_0\) is exactly where the geometry places the ground state.
A proton spins at \(c/\gamma_{\rm cause}\) at radius \(0.3110\) fm. An electron spins at \(c/\gamma_{\rm cause}\) at radius \(571.1\) fm. Force them into a single combined closure and a unified vortex must form — but the proton cannot complete the required spin-up from its own Sagnac energy alone. It falls short by exactly \(0.782\) MeV. That deficit must leave as a propagating \(\varepsilon_0\mu_0\) disturbance. That disturbance is the antineutrino — not a bookkeeping device invented to conserve lepton number, but the geometric remainder of a spin-rate mismatch. A necessity, not a postulate.
Feynman called \(1/\alpha \approx 137\) "one of the greatest damn mysteries of physics." Measured to extraordinary precision, never explained. A number that fell from the sky, and all of atomic physics had to be built around it.
We now have everything needed to derive it: the photon's arc geometry from Section 5, and the electron's orbital saturation radius \(r_{\rm sat} = 4\pi^2/\gamma_{\rm cause}^2\) from Section 6. Both are pure geometry — \(\gamma_{\rm cause}\) and \(\pi\) alone. No empirical input. Here is the derivation.
The photon's primary transverse oscillation traces the type-II elliptic arc with ratio \(\gamma_{\rm cause}\). Using the photon closure radius \(r_{\rm ph} = \bar\lambda\) and the electron's saturation radius \(r_{\rm sat} = 4\pi^2/\gamma_{\rm cause}^2\), the coupling ratio with the \(E\)-field alone gives:
Already close. But the photon is not one field — it is two, locked together.
\(E\) and \(B\) are not independent oscillations. \(B\) is the curl of \(E\) — the reluctance response of the same \(\varepsilon_0\mu_0\) disturbance. It turns a corner at maximum amplitude rather than propagating outward freely. This adds a transverse two-dimensional correction to the arc geometry:
The Sagnac mass depression at the photon's apex is a depth oscillation in the \(\varepsilon_0\mu_0\) product itself — the gravity face complement of the charge perturbation. Unlike the \(B\)-curl, which couples through two spatial dimensions, this depression is spherically symmetric and couples through all three. Three-dimensional coupling carries a factor of \(3/2\) relative to two-dimensional coupling by the sphere-to-disk projection ratio:
The three components are independent — each operates in its own field plane. The \(E\) oscillation is transverse. The \(B\) curl is perpendicular to the \(E\) plane. The Sagnac depth oscillation acts along the propagation axis. Independent contributions from orthogonal modes add as independent vectors — that is, in quadrature. The \(B\)-curl weight is 1; the Sagnac depth weight is \(3/2\) (the sphere-to-disk projection ratio already established above). Combined: \(1^2 + (3/2)^2 = 1 + 9/4 = 13/4\). So:
The fine-structure constant is the coupling efficiency between the photon's complete three-component arc geometry and the electron's circular closure geometry. The electron's saturation radius \(r_{\rm sat} = 4\pi^2/\gamma_{\rm cause}^2\) contributes two powers of \(\gamma_{\rm cause}\). Together:
CODATA measured value: \(137.035\,999\,084\). The derivation reaches within \(0.0015\%\) of measurement from pure geometry.
A cross-check: define \(\delta = \gamma_{\rm total}/2\pi \approx 0.1948\), the total arc ratio as a fraction of a full turn. One factor of \(\delta\) per spatial dimension: \(\delta^3 \approx 1/136.9\). Three dimensions, three factors, one number. The dimensional hierarchy is exactly what the three-component geometry predicts.
The Rydberg constant — the most precisely measured physical constant in existence — follows from the same geometry:
Zero free parameters. Agreement to better than 0.003%. The most precisely measured constant in physics is a ratio of three \(\varepsilon_0\mu_0\) geometry quantities. It is not fundamental. It is the field geometry reading itself.
None of this was chosen. Each step forces the next:
All results are derived in full in the following preprints, available openly on Zenodo: