Fold Formation and Branch Information in Vlasov–Poisson Flows with a Repulsive Harmonic Field

We study the three-dimensional gravitational Vlasov–Poisson equation with an external repulsive harmonic field. A time-dependent canonical dilation removes the explicit harmonic acceleration while preserving the Hamiltonian character of the phase flow and introducing different time-dependent weights in the transport and Poisson terms. The main result is a local fold-formation theorem for the exact self-consistent characteristic flow. Suitably embedded Lagrangian tracer graphs develop a temporally transversal regular fold along a selected reference characteristic, with fold time t_kappa = kappa⁻¹ + O(kappa⁻³), while the six-dimensional Hamiltonian flow remains a diffeomorphism. No claim is made that this event is the first fold of the entire open material layer. A three-parameter family of nearby velocity-translated Lagrangian layers gives an exact finite-thickness disintegration of a positive portion of phase space. After folding, material identity is retained through push-forward measures and branch-resolved configuration-space projections. For finite-to-one projections, we separate branchwise Jacobian compression from the conditional entropy of the inverse branch. At a regular fold, the leading branch uncertainty is log 2, with the first correction determined by the odd normal derivative of the label density. The logarithmic Jacobian and projected entropy density remain locally integrable. The canonical scaling shifts spatial and conditional-velocity differential entropies by opposite amounts while preserving their sum and the admissible Vlasov Casimirs.