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 \(S^1\) sitting over one point of the base sphere \(\mathbb{CP}^1\) (which is \(S^2\)). No two fibers touch, yet every pair is linked. Together the circles fill the 3-sphere \(S^3\), shown here by stereographic projection into ordinary 3D space.
The fiber \(S^1\) is the \(\mathrm{U}(1)\) circle — the gauge structure of electromagnetism. In IDWT this \(\mathrm{U}(1)\) is the geometry of the \(d=2\) sector, not something a particle carries between others. ⭐ identity
The base \(\mathbb{CP}^1\) is the \(d=2\) sector geometry; the total space \(S^3\) is the \(d=3\) sector geometry. The complex Hopf chain continues \(\mathbb{CP}^1 \to \mathbb{CP}^2 \to \mathbb{CP}^3 \to \mathbb{CP}^5\), generating the even sectors, while the odd spheres \(S^3\) and \(S^5\) 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.