The Coefficientwise Order on Markov Numerators

Classical Markov orderings compare values attached to rational slopes. We establish their coefficientwise polynomial refinement for homogeneous numerators of lower Christoffel paths. In the chart Q_{p,q} (t, w) = P_{p,q} (1 + t, 1, w), for pairs (p, q), (p′, q′) with 1 ≤ p ≤ q and 1 ≤ p′ ≤ q′, coefficientwise domination holds exactly when q ≤ q′ and p+q ≤ p′ +q′. Thus the three classicaldirections are part of a complete endpoint-to-endpoint coefficientwise order. The proof combines the directional inequalities with an endpoint-lattice decomposition; degree and constant-term statistics give the converse. Specialization at unit weights recovers the established numerical orderings while retaining the finer information in the full path polynomial.