Classify decidability for specified small-local-dimension and frustration-free one-dimensional chains. A concrete promise problem takes a finite rational/algebraic nearest-neighbour interaction 0≤h≤I and rational γ>0, with open-boundary HL=∑j=1L−1hj,j+1 frustration-free for every L, and asks whether the gap above its ground space is eventually at least γ or tends to zero. Identify which fixed on-site dimensions and structural restrictions admit a terminating algorithm.