The Ankle of the Cosmic-Ray Spectrum as a Pair-Production Feature: Threshold Arithmetic

The hardening of the cosmic-ray spectrum near 5e18 eV, the ankle, is tested against the pair-production interpretation by direct threshold arithmetic. For a proton colliding head-on with a microwave-background photon, the Bethe-Heitler threshold for producing an electron-positron pair is 7.567e+17 eV at the mean photon energy and 2.998e+17 eV for a Wien-tail photon, with 4.797e+17 eV at 1e-03 eV. The propagated spectrum of extragalactic protons develops its characteristic dip where pair losses peak in efficiency, and the recovery from that dip produces a hardening whose energy scale is set by these thresholds accumulated over cosmological distances, landing within a factor of 1.668e+01 of the observed ankle position for the tail-photon value. The arithmetic therefore supports the pair-production reading of the ankle for a proton-rich extragalactic flux, while the composition analysis shows the real flux is mixed, which keeps the competing galactic-to-extragalactic transition reading alive. A density bound contributes nothing at these densities. Simulated and computed; not a measurement.