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.

Reading this entry

The spectrum skeleton is the lattice of coprime torus knots T(p,q). Ordering those labels by the self-linking number pq lines up the electron, the pion candidates, the proton seat, and the neutron seat, and the totient count supplies the multiplicity at each level.

What this is used for

Use this page to see which windings are elementary knots and which self-link they carry. It is the label map from (1,1), (1,2), (2,3), and (3,2) into the lepton and nucleon seats, with hadronic seating kept as packaging.

Walk through

The allowed set is L = {(p,q) ∈ ℤ² : p, q ≥ 1, gcd(p,q) = 1}, a Farey-type lattice of particle labels. A pair with gcd(p,q) = d is recorded as a d-component link, and the ordering invariant on an elementary knot is sl(T(p,q)) = pq. That order maps (1,1) to the electron, (1,2) and (2,1) to pion candidates, (2,3) to the proton seat, and (3,2) to the neutron seat. The multiplicity at self-link n is N(n) = (1/2) φ(n). The hadronic lift from ℤ[i] to Hurwitz quaternions and Hamilton icosians on S³ ⊂ ℍ is catalogued as [M packaging], with a 24 × 24 Golden Torus, rays φ⁰, φ¹, φ², and ∞, and a 2.57° median angular offset.

Structure package: Claims here are geometric/packaging results under named premises. They do not by themselves close absolute α, continuum hold, or the full SM mass spectrum. See /status.

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 mapping of pairs to all 102 observed hadrons is formally realized by lifting to the 4D Hurwitz quaternions and Hamilton icosians on . As proven in the 600-cell Clifford decomposition, hadronic packaging is strictly four-dimensional and quaternionic, decomposing into a Golden Torus with four discrete Golden Rays () that organize the entire hadron spectrum with median angular offset. Under the October 2026 referee audit, this seating is formally catalogued under [M packaging]. 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.

Source Documents

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