Moving prime environments and arithmetic cluster limits

We study shrinking-target occurrences of completely multiplicative functions with arbitrarily moving phases on a fixed finite set of exceptional primes and independent Haar values at the remaining primes. After exact removal of the environment-only points, effective integer-subgroup profiles classify all subsequential marked cluster limits, while target-scale pair overlaps characterize Poisson convergence. The quantitative input is a direct Haar comparison with the moving germ kernel, uniform in the exceptional phases and with stretched-exponential error. For deterministic profiles and interval targets, the complete spatial covariance measure reconstructs the unmarked law through primitive multiplicative rays and finite interval geometry. A positive forest formula makes this reconstruction valid even with unbounded cluster sizes. For a fixed known bounded Borel target, retaining positions and angular marks yields the corresponding identification from the complete marked covariance, using signed overlap channels. Restricting the observation to an interior spatial window imposes both angular and logarithmic-spatial constraints. The maximal number of descendants of one germ becomes a rectangle-occupancy number for a subgroup of R², finite exactly when that subgroup is discrete. Explicit rank-one laws illustrate what counts forget, while environmental mixtures show why deterministic identification does not extend by averaging covariances.