  We predict neutron-proton scattering cross-sections and polarization
observables up to next-to-next-to-next-to leading order in a
renormalization-group invariant description of the strong nucleon-nucleon
interaction. Low-energy constants are calibrated to phase shifts, sub-leading
corrections are computed in distorted-wave perturbation theory, and we employ
momentum-cutoff values 500 and 2500 MeV. We find a steady order-by-order
convergence and realistic descriptions of scattering observables up to a
laboratory scattering energy of approximately 100 MeV. We also compare
perturbative and non-perturbative calculations for phase shifts and cross
sections and quantify how unitarity is gradually restored at higher orders. The
perturbative approach offers an important diagnostic tool for any power
counting and our results suggest that the breakdown scale in chiral effective
field theory might be significantly lower than estimates obtained in
non-perturbative calculations.
