ℂℙ³ possibility — antipodal pairsτ projection — half-turn along the fiberℝℙ³ actuality
S³ → ℝℙ³ along Hopf fibers
Exact geometry — stereographic projection of S3 ⊂ ℂ² into 3-space. The circles are true Hopf fibers, and −p really is a half-turn along the fiber through p, so the identification p ∼ −p acts along each circle, and the Hopf base S² never moves — τ works purely along fibers. The amber dots are points of ℝℙ3: the point each pair becomes. Only the cadence is schematic; the identification itself is timeless.
Hopf base S² — the τ projection never moves the base
one fiber — the antipodal pair, the τ projection, actualization