Poisson laws and arithmetic clusters for rare multiplicative patterns

We determine when rare observations of completely multiplicative functions have a Poisson law and when arithmetic relations force clusters. For independent prime values on the circle with uniformly bounded L2 densities, we approximate the field of hits of site-dependent targets of Haar mass O(N−1) on deterministic masks in [2,CN]. The target retains the true prime-site intensities and uses Haar at composite sites, even when the total prime contribution diverges. The exponent 2 is sharp in the uniform Lq scale for this target and class of masks. Below this threshold, even correct prime intensities can fail at site resolution while the total count remains asymptotically Poisson. A fixed finite exceptional prime environment, possibly atomic and internally dependent, instead produces a conditional cluster law together with an exact environment-determined configuration. With Haar outside that environment, its integer relation lattice determines the compound-Poisson limit of the corrected prefix count and characterizes simple Poisson convergence. Further applications treat rational observations on sparse moving supports, permanent bounded prime biases, and—using a binary arithmetic relation profile—long Borel patterns in a finite abelian group times a torus. Classical marked Chen–Stein comparison provides a common interface. A separate deterministic-sequence interface gives correlation identities, transfer criteria and a resolution obstruction, without proving cancellation for Liouville.