The Muon Puzzle in Ultra-High-Energy Air Showers, Priced: The Per-Generation Neutral-Pion Shift the Auger Deficit Requires, and What Accelerators Allow

Air showers above about 10 EeV contain more muons than any current hadronic interaction model predicts for the composition inferred from the depth of shower maximum. Adopting a muon enhancement factor R_mu = 1.45 at 10 EeV (published range 1.30-1.60) and R_mu = 1.62 at 100 EeV, a calibrated Heitler-Matthews cascade (4.94 generations at 10 EeV, effective multiplicity 57.7, baseline beta = 0.90) is used to compute the per-generation change in the neutral-pion energy fraction f0 that would accumulate to the observed enhancement. Uniform placement: f0 must fall from 0.333 to 0.281 at 10 EeV (-15.6%) and to 0.272 at 100 EeV (-18.4%) -- inside an adopted +/-20% accelerator band in size, but forbidden in placement, since most generations occur at energies already measured. Restricted to the 1.14 generations above LHC proton-proton energy: f0 must fall to 0.076 at 10 EeV (-77.1%) and 0.115 at 100 EeV (-65.6%), roughly four times outside the band set by LHCf forward neutral-pion spectra, FASER forward neutrino fluxes, and the July 2025 LHC proton-oxygen run. Verdict: no placement of the required change is simultaneously sufficient and accelerator-allowed.