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.

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