From lattice Poisson staircases to pure-death flows and Gaussian endpoint fields

This paper develops a functional limit theory for lattice Poisson fields with geometric marks. A canonical intra-cell randomization converts integer-valued geometric lifetimes into exponential lifetimes while preserving every integer threshold simultaneously. This yields a continuous pure-death flow, exact laws for its extreme gaps, scalar and spatial Ornstein–Uhlenbeck limits, discrete and continuous threshold white-noise limits, and an asymptotically independent Gaussian–Poisson endpoint decomposition. The results are formulated as consequences of a total-variation approximation of the underlying complete lattice field, making the probabilistic mechanism independent of the arithmetic construction that produces that field. Random completely multiplicative functions provide the principal application through the discrete-field results established in the companion work Critical Poisson fields for long constant stretches of a random completely multiplicative function.