Spatial-Causal Geometry (SCG)Spatial-Causal Geometry (SCG)If you have ever been told that the answers to these questions require quantum field theory, string theory, or something not yet discovered — they don't. They require one geometric constant, derivable by integration, and one interferometry formula confirmed at every scale since 1913. From those two things, the proton-to-electron mass ratio, the Bohr radius, the neutron mass, the neutrino energy, and the fine-structure constant all follow — with no free parameters and no fitting. This page shows the derivation in full.
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 \(\mu_0\): $$c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}$$ 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.
The medium is real. Space is not empty geometry. It is a physical substrate described by \(\varepsilon_0\mu_0\), with a local wave speed that varies with its state. Everything that happens in physics happens inside this medium. 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. \(c\) is the recovery rate of space: not a rule imposed from outside, but the intrinsic relaxation rate of the medium itself.
Any oscillation in a medium with a speed constraint \(c\) must satisfy one geometric requirement: the arc length traced per cycle must be the same regardless of frequency. If it were not, two oscillations at different frequencies — both constrained to travel at the same speed — would trace different total path lengths in the same time. Causality would fail.
This requirement pins the oscillation's transverse amplitude to exactly: $$A = \frac{\lambda}{2\pi} = \bar\lambda$$ 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: $$\frac{L}{\lambda} = \frac{1}{2\pi}\int_0^{2\pi}\sqrt{1 + \cos^2\theta}\,d\theta = \frac{2}{\pi}\,E(-1)$$ 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 \(\gamma_{\rm cause}\): $$\boxed{\gamma_{\rm cause} \equiv \frac{2}{\pi}\,E(-1) \approx 1.2160}$$ 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. The causal argument: arc-length equality across frequencies forces \(A = \bar\lambda\), and \(\gamma_{\rm cause}\) follows by integration. The least-action argument (Maupertuis): the closure geometry requiring no external length scale — no specification beyond the oscillation's own wavelength — is uniquely \(A = \bar\lambda\). \(\gamma_{\rm cause}\) follows by the same integration. When two independent arguments demand the same condition and produce the same number, the number is a geometric constant. \(\gamma_{\rm cause}\) is substrate-independent — it governs any medium supporting propagating oscillations, electromagnetic, mechanical, or gravitational.
\(\gamma_{\rm cause}\) 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: $$\frac{C}{\lambda_{\rm Compton}} = \gamma_{\rm cause}^2 \approx 1.4787$$ This closure ratio holds for every stable particle, regardless of mass — a universal geometric relationship with no free parameters.
One power of \(\gamma_{\rm cause}\) — open arc, propagating: this is the photon. Two powers of \(\gamma_{\rm cause}\) — closed arc, confined: this is a particle. Same geometry. Same medium. Two topological dispositions. Everything that follows is a consequence of which disposition the field takes.
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: $$\Delta\phi = \frac{4\pi A\omega}{\lambda c}$$ 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 \(v\): $$\lambda = \frac{h}{mv}, \quad A = \pi r^2, \quad \omega = \frac{v}{r}$$ Setting \(\Delta\phi = 2\pi\) and applying the closure velocity \(v = c/\gamma_{\rm cause}\): $$2\pi = \frac{4\pi \cdot \pi r^2 \cdot (v/r)}{(h/mv) \cdot c} = \frac{4\pi^2 r m v^2}{hc} \quad\Longrightarrow\quad m = \frac{hc}{2\pi r v^2} = \frac{\gamma_{\rm cause}^2\,\hbar}{r_{\rm clos}\,c}$$ $$\boxed{m = \frac{\gamma_{\rm cause}^2\,\hbar}{r_{\rm clos}\,c}} \qquad\Longleftrightarrow\qquad r_{\rm clos} = \frac{\gamma_{\rm cause}^2\,\hbar}{mc}$$
The same equation that measures Earth's rotation also measures the proton's mass. Not by analogy — by direct application of the same confirmed formula at a different scale. Mass in, closure radius out. Closure radius in, mass out. No fitting. No additional assumptions.
Why do smaller photons have more energy? 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. Smaller photon, more energy — for exactly the same reason that a heavier particle has a smaller closure radius. The medium charges the same geometric price for confinement regardless of topology. Tighter geometry, steeper depression, higher energy cost. The photon stores it transiently along an open arc. The particle stores it persistently in a closed loop.
With amplitude \(A = \bar\lambda\) forced by the causal requirement, and arc ratio \(\gamma_{\rm cause}\) established above, the photon is fully drawn. Its transverse radius is \(r_{\rm ph} = \lambda/2\pi\) — finite, physical, frequency-dependent. The photon is not a point. It is a finite structure in the \(\varepsilon_0\mu_0\) medium whose size is set by its wavelength. A visible-light photon has a transverse radius of roughly 100 nm. A gamma ray at MeV energies has a transverse radius of femtometres — the same scale as the nucleus it interacts with. The radius varies because it is supposed to: tighter oscillation, smaller radius, higher energy.
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. This is why wavelength has always correctly predicted optical phenomena — diffraction limits, photoelectric thresholds, Bragg scattering angles, coherence lengths. Wavelength was always the proxy for \(r_{\rm ph}\). The framework makes the physical reason explicit rather than leaving it as a numerical accident.
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 above. 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.
A particle is spinning space. Not a thing moving through space. Not a point with properties attached. A stable, self-sustaining rotational geometry in the electromagnetic medium. 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: $$r_{\rm sat} = \frac{4\pi^2}{\gamma_{\rm cause}^2} \cdot \bar\lambda$$ 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. No particle mass appears. No measured constant is required.
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.
Mass is the energy cost of sustaining a rotating \(\varepsilon_0\mu_0) depression. It is not a primitive property. It is not conferred by a Higgs field. It is the field accounting for the cost of its own rotation.
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: $$\mathbf{a} = c^2\nabla\ln(\varepsilon_0\mu_0)$$ Mass and gravity are one field configuration, read from two positions. Mass is the depression read from the inside — the energy cost of sustaining the rotation. Gravity is the depression read from the outside — the gradient a neighboring body follows. There is no separate gravitational force. There is a medium with a local wave speed that varies, and every body follows the gradient. This is what Pound-Rebka measured. This is what GPS corrects for.
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: $$r_{\rm clos}^{(p)} = 0.3110\,\text{fm} \qquad r_{\rm clos}^{(e)} = 571.1\,\text{fm}$$ From these two radii, five independent quantities follow with no further input. None of these were put in. They came out.
| Quantity | Derived | Measured | Match |
|---|---|---|---|
| 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 |
Five exact matches against independent measurement. Zero free parameters. The Sagnac mass equation was not fitted to any of these. Each result follows from the closure condition alone.
Since \(r_{\rm clos} = \gamma_{\rm cause}^2\hbar/mc\) for both particles, \(\gamma_{\rm cause}^2\) cancels identically in the ratio: $$\frac{m_p}{m_e} = \frac{r_{\rm clos}^{(e)}}{r_{\rm clos}^{(p)}} = \frac{571.1\,\text{fm}}{0.3110\,\text{fm}} \approx 1836.15$$ 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.
The same rule governs both: smaller closure radius means larger mass for a particle; smaller transverse radius means higher energy for a photon. Same field. Same geometric price. Two topologies.
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, and the electron's orbital saturation radius \(r_{\rm sat} = 4\pi^2/\gamma_{\rm cause}^2\). Both are pure geometry — \(\gamma_{\rm cause}\) and \(\pi\) alone. No empirical input.
Component 1 — The \(E\)-field oscillation. 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: $$\frac{1}{\alpha_0} = \frac{8\pi^3}{\gamma_{\rm cause}^3} \approx 137.953$$ Already close. But the photon is not one field — it is two, locked together.
Component 2 — The \(B\)-field curl. \(E\) 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: $$\delta_{\rm curl} = \frac{\gamma_{\rm cause}}{2\pi(1 + \gamma_{\rm cause}^2)}$$
Component 3 — The Sagnac depth oscillation. 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: $$\delta_{\rm Sagnac} = \tfrac{3}{2}\,\delta_{\rm curl}$$
Combining in quadrature. 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 in quadrature. The \(B\)-curl weight is 1; the Sagnac depth weight is \(3/2\). Combined: \(1^2 + (3/2)^2 = 13/4\). So: $$\gamma_{\rm total} = \sqrt{\gamma_{\rm cause}^2 + \tfrac{13}{4}\,\delta_{\rm curl}^2} \approx 1.22413$$
The fine-structure constant is the coupling efficiency between the photon's complete three-component arc geometry and the electron's circular closure geometry: $$\boxed{\frac{1}{\alpha} = \frac{8\pi^3}{\gamma_{\rm cause}^2\,\gamma_{\rm total}} \approx 137.038}$$ CODATA measured value: \(137.035\,999\,084\). The derivation reaches within 0.0015% of measurement from pure geometry. No empirical input. No adjustable parameters. \(\alpha\) requires both the photon (its arc geometry) and the electron (its orbital saturation scale). The number 137 is not mysterious — it is the coupling efficiency between the two geometries, determined entirely by \(\gamma_{\rm cause}\) and \(\pi\).
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: $$R_\infty = \frac{\alpha^2\,\gamma_{\rm cause}^2}{2\,r_{\rm clos}^{(e)}}$$ 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:
| Space is a physical medium (\(\varepsilon_0\mu_0\)) | Maxwell 1865, Pound-Rebka 1959, GPS daily |
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| \(c\) is the recovery rate of that medium — the intrinsic speed limit | |
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| A bounded oscillation must have a frequency-independent arc ratio | |
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| That forces \(A = \bar\lambda\) and gives \(\gamma_{\rm cause} = (2/\pi)E(-1) \approx 1.2160\) | |
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| Open arc (photon): one power of \(\gamma_{\rm cause}\); closed loop (particle): two powers | |
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| The Sagnac closure condition gives \(m = \gamma_{\rm cause}^2\hbar / r_{\rm clos}c\) | |
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| The photon has a physical radius \(r_{\rm ph} = \bar\lambda\); smaller radius, higher energy | |
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| A particle is spinning space; its saturation radius \(r_{\rm sat} = 4\pi^2/\gamma_{\rm cause}^2\) is pure geometry | |
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| Five exact results: mass ratio, Bohr radius, neutron mass, neutron charge, neutrino energy | |
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| Gravity is the depression gradient — mass and gravity are one field, two positions | |
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| Three photon arc components + \(r_{\rm sat}\) give \(1/\alpha = 137.038\) — no empirical input |
The normal scientific achievement is: we measured X precisely. The achievement here is different: X derived itself from geometry, and then matched measurement exactly. That is a categorically different kind of result — not fitting, but generating.
All results are derived in full in the following papers, available openly on Zenodo under CC BY 4.0:
| Paper | DOI |
|---|---|
| γcause — A Geometric Closure Invariant of Propagating Oscillations | 20132405 |
| The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure | 20225842 |
| Photon Structure, Scale, and Interaction from First Principles | 21115816 |
| Physical Constants as Derived from Spatial-Causal Geometry and the γ(cause) Invariant | 21113633 |
These papers are timestamped, citable, and open access. No affiliation required. No paywall. D. J. Hallman is an independent researcher. · ORCID 0009-0000-1710-3549