Proven and open
Proven: the slide-rule globe is the Riemann sphere, where infinity is one point and x ↔ 1/x is a mirror. Layer m's ticks are the m-th roots of unity, the points where xᵐ = 1 on the unit circle. Its fresh ticks are the primitive ones, the ones no smaller layer already had. Every layer's ticks split into the fresh ticks of each layer that divides it (Gauss). The fresh ticks always balance to +1, −1 or 0 (Möbius, Ramanujan).
Checked: after layer 200, the running balance has stayed inside 0.57 × √n for every layer up to ten million.
Open: the Riemann Hypothesis is exactly the claim that the running balance never grows faster than √n times an arbitrarily small extra power of n. The stronger guess that it never passes √n is known to fail, somewhere astronomically far out.
Siblings: Prime Tread Wheel · Problem Wheels