A finite gap criterion for level rank-three lattice-path matroid polytopes

Let P be the base polytope of a connected rank-three lattice-path matroid on n elements. We give an explicit finite criterion, in terms of the gaps between the two bounding paths, for the Ehrhart ring of P to be level. Four prefix constraints partition the ground set into at most five blocks. Up to permutations within these blocks, all relevant vertices are generated by at most 14,112 templates, each with at most three coordinates at the upper bound and at most one free coordinate per block. Splitting a template is equivalent to the absence of a negative cycle in a directed multigraph with at most six vertices and thirty edges. Its cycle weights are affine in the dilation degree, so two endpoint evaluations eliminate all degrees. The same construction gives the codegree, the least degree in which levelness fails, explicit certificates, and the degree support of the minimal generators of the canonical module. The template count bounds a generating family.