When Does a Square Board Split?

We study graphs whose vertices are the points of an integer square and whose edges have Euclidean length 1/λ or √2/λ. These are the step graphs of five-in-a-row under a scaled Euclidean metric. On the infinite lattice, their components are cosets of a Gaussian ideal. If every prime divisor of the reduced denominator k > 1 of λ² is 1 (mod 4), the square of side N ≥ 2 is connected if and only if N ≥ 2r(k), where r(k) = min{max(|x|, |y|) : x² + y² = k}. For split rules we determine when the finite components coincide with the coset intersections, except for one family of scales. In that family we give a smaller upper bound, reduce the conjectured threshold to one board for each prime 1 (mod 4), and establish it for every such prime below 2000 and for 2017. The proof uses residue-class gluing and a two-color leaper theorem whose square case was stated by Jelliss in 1985. Every square containing a winning line splits only along the cosets. We also determine the limiting distribution of the gain over a single primitive leaper orbit, give finite error bounds using rational approximations to its angle, classify two-row rectangles by Pell equations, and provide a finite connectivity test with an explicit cutoff for every fixed width.