Bounded-Step Walks on Gaussian Primes

We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.