Rare events in multiplicative sequences: Poisson laws, clusters, and observation

A multiplicative identity can persist at every scale while disappearing from a rare-event count. Another identity can survive as a cluster whose points lie far apart. Which distinction matters depends on what is observed. We explain this through one completely calculated model: a uniform germ can trigger two occurrences in disjoint spatial windows. Its two marginal counts become Poisson, but their sum need not do so. A coupling identifies the correct same-grid target and separates the mean number of points from the mean number of active germs. We then isolate the additional work required by multiplicative sources: private prime coordinates, weighted relations between windows, corrections to prime-site intensities, and relations visible at the target scale. A final passage from marked configurations to records explains the roles of ties, recognition and observation clocks. Selected results are stated with their hypotheses; a short concordance leads to their proofs. The elementary calculations are proved here, while the arithmetic and functional limit theorems are taken from the four cited research articles.