The first rung of the chain that builds the IDWT sector set — drawn as linked circles in 3D.
Every circle you see is a fiber — a copy of S¹ sitting over one point of the base sphere CP¹ (which is S²). No two fibers touch, yet every pair is linked. Together the circles fill the 3-sphere S³, shown here by stereographic projection into ordinary 3D space.
The fiber S¹ is the U(1) circle — the gauge structure of electromagnetism. In IDWT this U(1) is the geometry of the d=2 sector, not something a particle carries between others. ⭐ identity
The base CP¹ is the d=2 sector geometry; the total space S³ is the d=3 sector geometry. The complex Hopf chain continues CP¹ → CP² → CP³ → CP⁵, generating the even sectors, while the odd spheres S³ and S⁵ sit at d=3 and d=5. This chain is what fixes the active sector set D = {2,3,4,5,6,10}. ✅ structural (Part 9 T3)
Drag to rotate · scroll to zoom. The fibers are circles in S³; the projection sends one point of S³ to infinity, which is why fibers near it open into the large sweeping arcs.