If space can curve negatively, how small can an entire universe become? And what does that minimum know about the mass of the muon?
The muon is a fat electron. Same charge. Same spin. Same forces. The only difference is its mass: 206.77 times heavier. The Standard Model describes the muon with extraordinary precision — predicting its magnetic moment to eleven decimal places — and offers absolutely no explanation for that factor of 207.
That number is inserted by hand. Measured, plugged in, never derived. It is one of approximately nineteen free parameters in the Standard Model that have no theoretical origin whatsoever.
HFG reproduces the leptonic CP phase, the electron–muon mass ratio, the electron–tau mass ratio, and all three PMNS lepton mixing angles (θ₁₂, θ₂₃, θ₁₃) from the geometry of a single hyperbolic 3-manifold — with zero adjustable parameters.
What if the answer is geometric? What if the mass ratios, the mixing angles, the CP-violating phase are not arbitrary numbers but invariants of a specific mathematical space — fixed by geometry, not by accident?
This article is about the evidence that this is actually the case.
In the nineteenth century, mathematicians proved that Euclid's geometry is not the only consistent geometry. There exists a geometry of constant negative curvature — hyperbolic geometry — where triangles have angle sums less than 180° and space expands exponentially as you move outward.
A compact hyperbolic 3-manifold is a finite chunk of this curved space, closed on itself like a doughnut but with hyperbolic geometry inside. These objects obey a theorem that has no flat-space analogue:
This is why hyperbolic 3-manifolds can potentially encode physical information without any tuning. The encoding is exact, or it is not an encoding at all.
Parallel lines stay parallel. Area grows as r². Space feels familiar.
Parallel lines diverge. Area grows as cosh(r)−1 ≈ e^r/2. Space expands explosively.
Drag the slider to increase the radius. Watch how Euclidean and hyperbolic areas diverge. This single demonstration explains why hyperbolic manifolds are so rigid and information-rich.
At the bottom of the ordered census of compact hyperbolic 3-manifolds — the manifold with the smallest possible volume — sits the Meyerhoff manifold, volume 0.9813688289. It is constructed from m003 by Dehn surgery along the slope (−2,3).
A systematic scan of the SnapPy census, optimizing a fitness function measuring how well the holonomy group reproduces the observed PMNS lepton mixing matrix, found the global minimum at this manifold. Fitness: 0.005087. Free parameters: zero.
The lepton mass ratios appear as Eisenstein norms — exact algebraic integers in the number field generated by the manifold's cusp shape:
In the number field K = ℚ(w), w⁴ = w+1, the tetrahedral shapes of m019 are explicit units. They orbit under T(z) = 1/(1−z). The Bloch-Wigner value never changes.
Every cusped hyperbolic 3-manifold in the SnapPy census with cusp field discriminant −283 has volume that is an exact rational multiple of v₀. All six known examples. All verified to machine precision.
Click any bar to see details.
The minimum-volume hyperbolic universe encodes the PMNS mixing matrix. Its cusp-shape Galois group is the Weyl group of SU(2). Its tetrahedral shapes are units in the discriminant −283 number field, forming a period-3 orbit that forces their Bloch-Wigner values to equal its own volume. Every manifold in its arithmetic family has a volume that is a rational multiple of that volume.
Whether this is a deep connection between arithmetic geometry and particle physics, or an elaborate numerical coincidence, remains to be determined. The mathematics is real. The programme continues.
Marvin L. Gentry · Independent Researcher · Seattle WA
Preprints: SSRN ·
Code: GitHub ·
Website: hyperbolicflavorgeometry.org