  We explore particle physics beyond accelerator energies, motivated by
questions exposed in astroparticle physics observations: 1) Are there
reasonable modifications to the standard extrapolations of LHC-tuned hadronic
interaction models, so that ultrahigh energy cosmic ray (UHECR) showers are
well-described with a purely protonic primary composition rather than requiring
a tuned, energy-dependent composition mixture as needed in conventional models?
2) What modifications to standard models can solve the deficiency in the
predicted ground signal found in hybrid UHECR observations? We find that a pure
proton composition provides an excellent fit to shower observations, if the QCD
inelastic cross section increases more rapidly above $E_{\rm cm} \approx 60$
TeV than in conventional models, and speculate as to possible reasons this may
happen; the "muon deficiency" can be cured by relatively minor modifications to
particle ratios in unexplored kinematic regimes below and above LHC energies.
