Hopf Coils

Here is the most beautiful idea on the S^3 ladder: a point can have a visible position and a hidden circular phase at the same time. Move by the common phase, and the visible position does not budge. That hidden circle orbit sits inside a Hopf fiber.

Lesson Goal

See how the Hopf map pairs a visible base coordinate with a hidden phase, and check the key fact by hand: multiplying by a common unit phase leaves the visible base point unchanged.

The Idea

Take a unit pair of complex numbers (z₁, z₂) with |z₁|² + |z₂|² = 1 — a point of S^3. The Hopf map sends it to a visible point on an ordinary sphere S^2:

base = ( 2·Re(z₁ z̄₂),  2·Im(z₁ z̄₂),  |z₁|² − |z₂|² )

Now multiply both coordinates by the same unit phase λ = e^{iθ} (with |λ| = 1). The visible base point does not change at all. As λ turns, the source point traces a circle orbit in S^3 that lands on the same spot of S^2. That circle of choices is the hidden phase direction certified here.

This is the S1 idea returning in disguise: S1 gave you one visible circular coordinate; Hopf gives you a visible base plus an invisible circular coordinate riding along underneath.

Worked Example

Start at (z₁, z₂) = (1, 0). Then:

z₁ z̄₂ = 1·0 = 0
base = ( 0, 0, |1|² − |0|² ) = (0, 0, 1)   ← the north pole

Multiply both by λ = e^{iθ}: (z₁, z₂) → (e^{iθ}, 0).

z₁ z̄₂ = e^{iθ}·0 = 0   (still 0)
base = ( 0, 0, |e^{iθ}|² − 0 ) = (0, 0, 1)   ← still the north pole

The point on S^3 rotated all the way around as θ ran from 0 to 2π, yet its visible image stayed pinned at the north pole. That whole circle of S^3 points is a common-phase orbit inside one Hopf fiber.

What Is Proved On The Real Sphere

This is not only a picture. On explicit real coordinates (z₁ = a + bi, z₂ = c + di), the Hopf map is now Lean-proved to:

satisfy the unconditional norm identity |h(a,b,c,d)|² = (a²+b²+c²+d²)²;
therefore send any point of S³ to a point of the actual unit 2-sphere S²;
be fixed by the common-phase circle action in all three output coordinates, so that common-phase circle orbit stays inside one Hopf fiber.

That is the genuine coordinate-and-fiber spine of the fibration S³ → S². (What is not yet claimed, and is tracked as roadmap: classifying every point in each fiber, local triviality, and the full fiber-bundle structure.)

Common Mistake

The norm identity, the landing on S², and common-phase invariance are proved; the full fiber classification and bundle topology (local triviality) of S³ → S² are still future work, not claimed. The widget is a coordinate explorer, not a proof. See What “Proved” Means Here.

Checkpoint

What does the current theorem prove when you multiply both complex coordinates by the same unit phase? (Answer: the visible Hopf image stays fixed; the converse fiber classification is later work.)

Source Trail

Showcase evidence: SHOW-007 quaternion, Hopf, and phase coordinates.

Paper source: S3 Hopf Coils