Counterfactual gravity

Why is gravity inverse-square? In a possible world where F = −μ/rᵖ, do orbits still close — or do they precess?

How the lab tests it

Make the force exponent p a knob and orbit a planet about a fixed star (symplectic leapfrog); measure the apsidal angle Φ (the angle swept between successive closest/farthest points) and compare it to the near-circular law Φ = π/√(3−p).

What it looks for

Bertrand's theorem: a bound orbit closes (Φ a rational multiple of π) for ALL initial conditions only at p = 2 (Kepler) and p = −1 (Hooke); every other exponent precesses into a rosette