What if reality obeys a hidden structure?

Only two fields exist, electromagnetism and spacetime. Everything we call a particle is a pattern their interaction holds in place.

Where it begins

The proton is usually explained by quarks: particles bound by colour charge that no experiment has ever isolated on its own. But then what is the proton? This theory asks whether two fields can do that work alone.

Physics counts four fundamental forces. Here, what we call the weak, strong, and gravitational fields are themselves misnomers. Only the spacetime and EM fields exist, and their longitudinal and other interactions produce effects we measure as gravitational, strong and weak.

If a particle is a pattern the two fields hold in place, its size and its weight are two readings of one quantity. The mass and the radius are the same thing shown differently. A heavier particle is a tighter one.

The muon and the tau share the electron's charge and spin at about 207 and 3,477 times its mass. Higher level leptons don't just have more twist. They have obtained something that allows that twist to exist. There's a phase transition up the lepton chain between the stable rungs, and when it decays, something comes out of it.

What it stands on

Maxwell's field tensor is the foundation. Maxwell's equations already put energy, momentum, pressure and stress into the field itself. Light pushes on what it strikes, and light whose total momentum cancels has rest mass: a mirrored box of light weighs E/c² more than the empty box. What linear Maxwell theory lacks is a way to hold a wavepacket in place. Something has to give the energy a return path.

Jussi Lindgren, with Jukka Liukkonen and Kovacs, has spent years deriving physics from spacetime geometry. Their papers obtain the wave equations of quantum mechanics from optimisation over spacetime, and treat electromagnetism as a purely geometric theory in which the metric carries the electromagnetic potential. UMFT starts from that identification of geometry and potential as one object, and adds return, winding and boundary dynamics of its own. Those supply the return path.

Two numbers from three and ten

Two integers organise the calculation. Three is the number of directions in a particle's rest space. Ten is the number of independent components of a symmetric tensor in four dimensions. In UMFT they weight the primitive and composed returns of a charged closure, and the same pair turns up in two places.

The first is the fine-structure constant, α, the strength of electromagnetism, which physics measures and has never derived. A return graph weighted 3 and 10 has compliance 13/45. Taken around a full turn, 4π² divided by that compliance is 136.66, and the nearest odd spherical boundary mode is 137. The half-winding divides a phase of π equally between two orientations, so its action is (π/2)² + (π/2)² = π²/2. With a junction term and a toroidal coupling to the neighbouring mode, the eigenvalue settles at

\[ \Lambda=137+\frac{\pi^2/2-1/(4\Lambda)}{\Lambda}+\frac{v^2}{\Lambda-135}, \] \[ v=\frac{q}{4\pi^2},\qquad q=\frac{\rho}{1+\sqrt{1-\rho^2}},\qquad \rho=\sqrt{\frac{8}{\pi\Lambda}}, \] \[ \alpha^{-1}=\Lambda=137.035999156. \]

CODATA's 2022 value is 137.035999177 ± 0.000000021.

The second is the tau's mass. Measured in square roots of mass, the step from muon to tau is ten-thirds of the step from electron to muon, loaded by the same α:

\[ m_\tau=\left[\sqrt{m_\mu}+\frac{10}{3}\sqrt{1-\frac{\alpha}{2\pi^2}}\,\left(\sqrt{m_\mu}-\sqrt{m_e}\right)\right]^2 \]

With the measured electron and muon masses and the α above, it gives 1776.938 MeV. The measured tau mass is 1776.93 ± 0.09 MeV. The working paper carries the full computation, with live figures.

What a sharper tau mass decides

Koide's relation of 1981 says the three charged-lepton masses satisfy

\[ \frac{m_e+m_\mu+m_\tau}{\left(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\right)^2}=\frac23 . \]

Given the electron and the muon, it puts the tau at 1776.969 MeV. UMFT's relation puts it at 1776.938 MeV. Both sit inside today's measurement, 1776.93 ± 0.09 MeV, and they are 31 keV apart.

The electron and muon masses are known so well that their uncertainty moves either prediction by less than a twentieth of a keV, so the gap is the whole contest. A measurement with five keV of uncertainty would separate them. A proposed scan of the tau-pair threshold at a future high-luminosity collider projects about one keV. That scan would also produce tauonium, the bound state of a tau and its antiparticle. The measurement will then pick UMFT's number, Koide's, or neither.

How the numbers were reached

UMFT has been worked out since 2025 in conversation with general reasoning models, with the measured values in view. The ratio of ten-thirds was first read from the measured lepton masses. Its loading was matched to the tau's measured shortfall and derived afterwards. For the fine-structure constant, the strength of the final toroidal coupling was first solved from the gap left to the measured value, and the formula that produces it came after. The two numbers above are therefore a reconstruction: one structure that ties them together, built while looking at them. A formula with the same leading term and a single simple correction, 137 + (π²/2)/Λ − (2/9)/Λ², reaches the same digits, so the digits alone carry less weight than the structure does. A blind test, its rules committed before anything was computed, found that coupling both neighbouring boundary modes rather than one moves α⁻¹ about 74 measurement uncertainties from CODATA, and that the same 3 and 10, read as neutrino mixing, put the smallest mixing angle θ13 at 6.9° against a measured 8.6°. What remains is for the field action to force each term before anyone checks the answer, and to produce a stable electron with its charge, spin and mass scale. The tau measurement is the part of that test the next colliders can run.

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