The flux relation is a conservative lower bound to the measured spectra (Auger, HiRes) in the region 10^17 eV < E < E_GZK ~ 5 x 10^19 eV; the paper’s quantitative production-rate calculations (Appendix E) use the actual Auger spectrum up to 2 x 10^20 eV, interpolated linearly, with a nucleon-nucleon inelastic cross section of 100 mb. For nuclei only E/A per nucleon is available, so a pure-iron composition reduces the above-threshold rate by a large factor — which is why composition worst cases are carried through the paper’s bounds.