Particle Spectrum Skeleton

The allowed torus knots T(p,q) with gcd(p,q)=1 form the particle spectrum skeleton. The lattice of coprime pairs, ordered by self-linking number pq, gives a predictive hierarchy of particle types.

The coprime lattice defines the fundamental skeleton of the particle spectrum. By taking , we construct a Farey-type lattice of allowed particle labels. The constraint is critical because non-coprime pairs with form -component links rather than elementary knots.

Self-Linking Ordering

Ordering the lattice by yields the known particle families: maps to the electron, and to pion candidates, to the proton, and to the neutron.

Euler's totient function dictates the multiplicity of these states. It counts the number of knots with and .

Multiplicity Function

The precise mapping of pairs to all observed hadrons beyond the simple nucleons requires an 8D octonion extension. Finally, a knot is stable if no smaller knots sum to the same topology with lower self-link, providing a purely geometric stability criterion.

Integrity Status: Established [E]

The topological classification of particle states by coprime pairs and self-linking number is mathematically robust and forms the foundational skeleton of the CPT framework.

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