Clifford Torus Particles

The Clifford Torus

In the CPT framework, the physical universe is modelled as a 3-sphere (S³). Within this space, there is a unique flat torus known as the Clifford torus. We define it as the set of points:

Torus Knots T(p,q)

Particles are not point masses; they are one-dimensional closed curves winding around this Clifford torus. These are mathematically described as torus knots . A curve winds times around one cycle and times around the other.

For the curve to be a true, single continuous knot rather than a set of unlinked loops (a link), the winding numbers must satisfy the condition:

Self-Linking Number

The self-linking number of a torus knot determines fundamental particle properties, such as its fermion or boson nature via the FR (Finkelstein-Rubinstein) theorem. For a torus knot , the self-linking number is .

• T(1,1): The electron class. Simplest knot, self-link=1, fermionic nature.
• T(2,3): Proton. Self-link=6, acts as a bosonic nucleus in this context.
• T(3,2): Neutron. Self-link=6, acts as a bosonic nucleus.

Why the Clifford Torus?

Why do particles live on the Clifford torus and not on any arbitrary submanifold? The Clifford torus is the unique flat torus in S³, meaning it has zero Gaussian curvature intrinsically. Furthermore, it sits exactly equidistant from the two poles of the Hopf fibration, providing a perfectly balanced symmetric "equator" for particle states. All knots with form the skeletal structure of the particle spectrum.