An Introduction to the Kepler-Heisenberg Problem

The paper’s Theorem 3 proves that a zero-energy Kepler–Heisenberg trajectory is determined by its restriction to a fundamental domain between two successive oriented z-crossings, via a rotation, a Heisenberg dilation, and a time reparametrization, with λ = exp(s(t2)−s(t0)) and ϕ = θ(t2)−θ(t0) (and an explicit τ) . The candidate solution reproduces this structure: it uses the symplectic cotangent lifts of rotations and dilations, the scaling H∘δ̂λ = λ^{-2}H, conservation of angular and dilational momenta on H=0, and the oriented section Σ+ to show S = R̂ϕ ∘ δ̂λ maps [t0,t2] to [t2,t4] with time rescaled by λ^2, hence the entire orbit is generated by iterates. There is, however, a minor algebraic slip in the model’s quadratic discriminant for pz on Σ+: the correct discriminant depends only on D (J), not on L, matching the paper’s z-axis threshold |J| = 1/(2√π) for admissible crossings . This slip does not affect the uniqueness/self-similarity conclusion.