Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal Arrangements B(n,k)

We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making it a partial cube and a median graph, with distances given by the Hamming distance and geodesics described by symmetric differences. We also prove a Poisson limit and a sharp threshold for overlaps of random circuit families, revealing an underlying hypercube geometry.