Step-Norm Thresholds and Prime-Walk Connectivity in the Eisenstein Integers

Source abstract

The moat problem asks whether one can move arbitrarily far from the origin using prime elements as stepping stones while keeping every Euclidean step below a fixed bound. We study finite-radius prime-walk graphs in the Gaussian and Eisenstein integers and introduce a discrete step-norm framework for the Eisenstein lattice. For radius $R$ and step bound k, prime elements are vertices, and two distinct vertices are adjacent when their Euclidean distance is at most k. A selected connected component is computed by breadth-first search from a deterministic minimum-norm prime. At k=2, the selected Gaussian component has 720 vertices for $R=50,100,150,200, whereas the selected Eisenstein component grows from 1{,}974$ to 4{,}186. The principal theoretical contribution is a threshold-invariance theorem: because squared Eisenstein displacements have the form a^2-ab+b^2, the finite graph can change only when k^2 crosses a represented norm. At R=200, the selected Eisenstein component grows from 23 vertices at k=1 to 1{,}402 at \sqrt3, 4{,}186 at 2, and 22{,}474 at \sqrt7; at k=4, all 22{,}810 enumerated prime vertices lie in one truncated component. These results identify a pronounced finite-radius transition without asserting an infinite walk or a global moat.