Hartree-Refreshed Continuum Density-Selected Phase Drives in the Regular Mean-Field Regime

We formulate a regular-regime theorem for continuum density-selected phase drives

refreshed along a Hartree reference path. In contrast to a prescribed continuum drive, the

density path is generated by the reference Hartree solution itself, namely ρt = |φt|2

. The

microscopic comparison is nevertheless made against the corresponding time-dependent

Hartree-refreshed reference field, not against a fully microscopic feedback law depending

on the evolving reduced density of the same N-body state. In this regular finite-window

setting we isolate three mechanisms: construction of the Hartree-refreshed reference branch,

convergence of finite refreshed phase schedules to an abelian multiplication product integral,

and a Pickl-type propagation-of-condensation estimate for the associated many-body reference

dynamics.