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$<X_{max}>$ Uncertainty from Extrapolation of Cosmic Ray Air Shower Parameters

Recent measurements at the LHC of the p-p total cross section have reduced the uncertainty in simulations of cosmic ray air showers. In particular of the depth of shower maximum, called $X_{max}$. However, uncertainties of other important parameters, in particular the multiplicity and elasticity of high energy interactions, have not improved, and there is a remaining uncertainty due to the total cross section. Uncertainties due to extrapolations from accelerator data, at a maximum energy of $\sim$ one TeV in the p-p center of mass, to 250 TeV ($3\times10^{19}$ eV in a cosmic ray proton's lab frame) introduce significant uncertainties in predictions of $<X_{max}>$. In this paper we estimate a lower limit on these uncertainties. The result is that the uncertainty in $<X_{max}>$ is larger than the difference among the modern models being used in the field. At the full energy of the LHC, which is equivalent to $\sim 1\times10^{17}$ eV in the cosmic ray lab frame, the extrapolation is not as extreme, and the uncertainty is approximately equal to the difference among the models.