foundations

MatureSelection rules are established, but specific winding assignments beyond the proton/neutron remain mutable.

Torus Knot Spectrum

The allowed torus knots T(p,q) with gcd(p,q)=1 form the particle spectrum skeleton. This page maps known particles to winding numbers and explains the selection rules.

The Spectrum Generating Rule

In the CPT framework, the zoo of fundamental particles is elegantly reduced to topological taxonomy. The particle spectrum is generated by the allowed set of torus knots on the Clifford torus.

Selection rules for fundamental states

Low-Lying Knot Assignments

The simplest knots correspond to the most stable, fundamental particles. The properties of the particle are dictated heavily by the self-linking number, which determines the topological phase under exchange.

Self-linking number of a T(p,q) knot

  • T(1,1): The simplest non-trivial winding. Self-link is 1. This represents the electron class, exhibiting fundamental fermionic behaviour.
  • T(2,1) or T(1,2): Self-link is 2. This state is bosonic. It is currently a strong candidate for the pion.
  • T(2,3) and T(3,2): Different embeddings with the same self-link of 6. These represent the fundamental nucleon pair: the proton and neutron. They act as bosonic nuclei in low-energy limits.
  • T(3,4): Higher winding numbers correspond to heavier, more excited topological states.

Mass Hierarchy

The intrinsic mass of a state is not arbitrary but is a function of its topological complexity, specifically relating to its self-linking number and the curvature of the background universe (radius ).

The Open Problem in 8D

While the low-lying states (electron, nucleons) map cleanly to simple knots, the full mapping of all quark generations and heavy resonances remains an active area of development. It involves embedding these structures into higher dimensional spaces, specifically 8D octonionic geometries, to capture the full flavour symmetry.

[M]
Mutable: The fundamental selection rules ( and the FR phase rule based on self-linking) are firmly established [E]. However, the exact, exhaustive dictionary mapping every known high-energy particle to a specific winding pair beyond the core nucleons is still mutable and subject to refinement.

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

Documents open in the CPT Research Library — a formatted viewer for all source research files.