The Gaussian moat problem

Theorem 1.1: for every real $D$ there is a finite $B_D$ such that every connected component of the graph on Gaussian primes with edges of length at most $D$ has at most $B_D$ vertices, independent of the starting prime and including primes on the axes; so every walk through distinct Gaussian primes with steps at most $D$ has at most $B_D$ terms. The proof finds, for each $D$, a finite set of split primes whose joint sieve leaves a periodic set with no infinite bounded-step walk, using geometric sampling and entropy estimates. $B_D$ is nonexplicit, so no numerical moat width is given for any particular $D$.

Theorem 1.1: for every real $D$ there is a finite $B_D$ such that every connected component of the graph on Gaussian primes with edges of length at most $D$ has at most $B_D$ vertices, independent of the starting prime and including primes on the axes; so every walk through distinct Gaussian primes with steps at most $D$ has at most $B_D$ terms. The proof finds, for each $D$, a finite set of split primes whose joint sieve leaves a periodic set with no infinite bounded-step walk, using geometric sampling and entropy estimates. $B_D$ is nonexplicit, so no numerical moat width is given for any particular $D$.