Cover Relations and Lagrange Collisions in Rational Dyck Path Orders

Schiffler defined matching and Lagrange scores on rational Dyck paths, producing two strict partial orders, and asked for a path description of their cover relations and whether equal Lagrange values force isomorphic band graphs. We determine the cover relation in both orders. Exact matrix bounds for every rational-Dyck prefix cylinder yield a terminating best-first traversal of the path-prefix tree whose consecutive score batches are exactly the cover layers. Equivalently, a pair is a cover precisely when a finite prefix-antichain certificate excludes every intermediate score. The matching order has fixed endpoint parity, explicit initial levels in D(n,n-1), and covers at Hamming distance 2n-6; hence its cover geometry is not generated by uniformly bounded local changes. An exact four-case continuant formula describes every proposed local interchange. We also classify band-graph isomorphism classes at every coprime endpoint as orbits of an explicit reversal involution, giving closed fibre counts. Distinct fibres collide in D(17,9): two paths have the same Lagrange value but nonisomorphic band graphs, and independent exact enumerations show that total length 26 is minimal. All finite certificates use integer arithmetic or reduced rational squares.