B-Spline Laws and Nonreal Zeros for Weighted-Cycle Internal DLA

We give an exact finite-time distribution for single-source internal diffusion limited aggregation driven by an arbitrary positive nearest-neighbour chain on the integers. The probability of each occupied interval is a B-spline value on the harmonic-scale knots of the chain. Lifting a cycle until its penultimate occupation yields a last-site law and a divided-difference formula for arbitrary positive edge weights. For the four-cycle we characterize the full attainable probability region and every reversible realizing resistance vector up to scale. The shifted last-site polynomial has only negative real zeros for every positive nearest-neighbour chain on three or four vertices, but a five-cycle with resistances (10,10,1,1,20) has a nonreal conjugate pair. Thus five is the smallest possible cycle size for this obstruction.