Universal One-Dimensional Solenoid

Abstract

The universal one-dimensional solenoid carries its visible projection and dense real flow.

The carrier is the compatible family of circle phases indexed by positive integers under divisibility. Coordinate one defines a continuous, surjective additive projection to the visible circle.

A real parameter maps to the family represented in coordinate m by t/m. This is a continuous additive flow, its visible projection is t modulo one, and its image is dense. The density proof exactly matches every finite coordinate window by passing through a common multiple.

Theorem 1.1 (The real flow projects visibly and has dense range).

$$\pi (\mathrm{realFlow}(t))=t\mathrm{mod}1.$$

Proof. Machine-checked in Lean as D5/S1/Dynamics/UniversalSolenoid.projection_realFlow (✓ std3). ∎

Source. Repository-derived.

Commentary.

The projection formula is machine-checked directly. The same module proves dense range and derives connectedness from it.

References

• Truth anchor: D5/S1/Dynamics/UniversalSolenoid.projection_realFlow