Energy Lost by a 10^20 eV Proton Crossing 50 Megaparsecs of Microwave Background

A proton launched at 1e20 eV toward an observer 50 megaparsecs away does not arrive at 1e20 eV. Combining photopion losses, Bethe-Heitler pair production, and adiabatic expansion in a continuous-loss approximation gives an effective loss length of 2.419e+01 Mpc at that energy and an arrival energy of 1.265e+19 eV, a retained fraction of only 1.265e-01. At 6e19 eV the retained fraction over the same distance rises to 4.086e-01, and at 4e19 eV the proton keeps most of its energy, which is the arithmetic behind the horizon: sources of the highest-energy events must be nearby by cosmological standards. A density bound at intergalactic densities changes none of these loss lengths at any relevant precision, for the same density-ratio reason established in the threshold calculation; the horizon is standard physics and the frame neither extends nor shortens it. This study is simulated and computed from stated assumptions; it is not a measurement.