A Monotone Diamond Covering Functional for Gaussian Primes in Z[i]

This manuscript introduces a monotone finite-window extremal functional for Gaus- sian primes in the ring Z[i] under the boxcar (Manhattan, L1) metric, and proposes a calibrated extreme-value constant and a dimensionless normalization (the Sakib Index) for nearest-prime distance fields. While boxcar-metric “prime gap” frameworks in Z[i] exist in the literature, the present work isolates a distinct monotone diamond (increasing-window) covering-radius functional, provides a dataset-based computational study (no schematic/simulated plots in the 10 principal figures), and packages a norm- generalized constant law that reduces to a candidate constant cSakib = π/2 for the L1-ball geometry. All principal illustrations are generated directly from computed Gaussian-prime data (sieve + exact distance transform), with additional conceptual diagrams provided for clarity.