Air-Shower Maximum Depth for a 10^20 eV Proton Under a Density Ceiling

When a 1e20 eV proton strikes the top of the atmosphere, the cascade it initiates reaches its largest particle count at a slant depth that fluorescence telescopes can photograph. A Heitler-model calculation anchored to published shower-model values gives a proton maximum depth of 8.100e+02 g per square centimetre at 1e20 eV, against 7.138e+02 g per square centimetre for iron at the same total energy, a separation of 9.615e+01 g per square centimetre, with an elongation rate of 5.500e+01 g per square centimetre per decade. The air density at the altitude of shower maximum, about 0.5 kg per cubic metre, stands in a ratio of 5.145e-59 to the density bound, so the ceiling on density plays no role in shower development: cascades are electromagnetic and hadronic physics in a medium forty-plus orders of magnitude too dilute for the bound to enter. The computed proton value sits modestly deeper than the mean depths Auger reports at the highest energies, which is the familiar composition hint, not a frame effect. This study is simulated and computed from stated assumptions; it is not a measurement.