When Does a Square Board Split? Connectivity of Scaled Euclidean Step Graphs

We study graphs whose vertices are the points of an integer square andwhose edges have Euclidean length \(1/λ\) or \(√2/λ\).These are the step graphs of five-in-a-row under a scaled Euclideanmetric. On the infinite lattice, their components are cosets of aGaussian ideal. If every prime divisor of the reduced denominator\(k>1\) of \(λ^2\) is \(14\), the square of side \(N≥2\)is connected if and only if \(N≥2r(k)\), where\(r(k)=minx^2+y^2=kmax(|x|,|y|)\).For split rules we determine when the finite components coincide withthe coset intersections, except for one family of scales. In that familywe give a smaller upper bound, reduce the conjectured threshold to oneboard for each prime \(14\), and establish it for every such primebelow \(2000\) and for \(2017\).The proof uses residue-class gluing and a two-color leaper theorem whosesquare case was stated by Jelliss in 1985. Every square containing awinning line splits only along the cosets. We also determine the limitingdistribution of the gain over a single primitive leaper orbit, givefinite error bounds using rational approximations to its angle, classifytwo-row rectangles by Pell equations, and provide a finite connectivitytest with an explicit cutoff for every fixed width.