Investigation PPT01 — Topology & the Mass Spectrum
A particle is a self-sustaining vortex on a genus-1 torus. It circulates two ways at once — the long way around the ring and the short way around the tube. One rule governs both: the speed budget closes at c. Only a discrete set of aspect ratios survive.
A spation cannot relay faster than c in any combination of modes — and a vortex spends that one budget three ways at once.
For a vortex on a torus of major radius R and minor radius a, three velocity components share a single fixed allowance: toroidal (v_T, around the ring), poloidal (v_P, around the tube), and translational (v_C, the centre-of-mass drift). Law V binds them:
This is the whole engine of PPT01. Drag the dial below: as the particle accelerates through the lattice, its spin slows so the total stays pinned at exactly c².
The two spins are not free. Self-consistency forces the aspect ratio to be exactly √W.
A circulation completing W poloidal loops per toroidal revolution traces a helix whose geometry fixes the velocity ratio:
Then the torus is the flow: each radius equals the Compton wavelength of its own circulation, so R = ℏ/(m·v_T) and a = ℏ/(m·v_P), giving R/a = v_P/v_T. Substituting collapses the system to one clean result:
Stable particles are torus knots. A single topological test sorts the entire particle zoo.
Generalising to a full (p,q) torus knot — p toroidal wraps, q poloidal wraps — the mode equations stay exactly solvable:
A (p,q) curve is a true knot — it cannot be deformed to a point, so it cannot decay — only when gcd(p,q)=1 and both p≥2 and q≥2. Spin the dials and draw the knot.
| (p,q) | Name | v_T/c | v_P/c | R/a | Knotted? | Candidate |
|---|
The electron is the only stable unknot — held together not by topology but by the displacement well V_disp. Everything stable above it is a genuine knot.
Topology sets the shape; size sets the mass. The spectrum is discrete-in-topology, continuous-in-size.
Self-consistency fixes the mass in terms of the geometry:
The (2,3) trefoil's major radius comes out to exactly half the measured charge radius. The missing factor of 2 is the W+1 = 4 winding conjecture: R_charge = (W+1)·ℏ/(m_p·c).
The mass ratio m_p/m_e = 1836.15 does not emerge from topology alone. It needs the size ratio a_e/a_p ≈ 2011, which comes from the displacement volume V_disp — a separate derivation handed off to PPT03. Topology gives the velocity partition and stability; it does not hand you the numbers for free.
Every relation closes to machine precision — these are not fits, they are algebraic identities forced by the budget.
The solver lives in the engine alongside laws.hpp. The whole spectrum is three lines of geometry:
| Proof | Description | Result |
|---|---|---|
| PPT01-1a | W=1 budget = c² | PASS |
| PPT01-1b | W=3 budget = c² | PASS |
| PPT01-2a | a_e/a_p consistency | PASS |
| PPT01-3a | (2,3) trefoil budget = c² | PASS |
| PPT01-4a | W=2 lighter than W=1 | PASS |
The discrete mode spectrum follows exactly from the speed budget with helical coupling. Stable particles are torus knots: the electron = (1,1) unknot (equal partition, R/a = 1, held by V_disp) and the proton = (2,3) trefoil (R/a = √(3/2) = 1.225, topologically protected — it cannot decay). W=2 is a lighter unknot that decays, matching the absence of any such particle. The m_p/m_e ratio is deferred to PPT03.