Conditional KPZ reduction in a one-dimensional model of bosonic dark matter

Wave-like dark matter described by a high-occupancy self-gravitating bosonic field provides a microscopic setting in which both amplitude and phase are dynamical. We study a one-dimensional Gross–Pitaevskii–Poisson toy model and ask which coarse-grained variable, if any, can be meaningfully compared with the 1+1-dimensional Kardar–Parisi–Zhang (KPZ) fixed point. We show that the relevant field is not the raw microscopic phase but a branch-resolved coarse-grained phase built from the sound sector. Above the Jeans scale and below the microscopic cutoff, self-gravity acts as a weak deformation of local sound dynamics. In this window the exact linear modes admit a local sound form, and a weakly nonlinear projection yields a nonvanishing same-chirality Burgers self-coupling. Under one-branch dominance together with a local Markov closure, the dominant branch reduces conditionally to a KPZ-type equation. We also formulate a dictionary from microscopic initial data to the canonical curved, flat, and stationary KPZ benchmarks. Our results do not establish KPZ universality for self-gravitating bosonic dark matter, but they identify the proper comparison field and the controlled regime in which an exact fixed-point test can be posed.