  At $\sqrt s = 57\pm 7$ TeV, the Pierre Auger Observatory (PAO) collaboration
has recently measured the proton-air inelastic production cross section
$\sigma_{\rm p-air}$. Assuming a helium contamination of 25%, they subtracted
30 mb from their measured value, resulting in a p-air inelastic production
cross section, $\sigma_{\rm p-air}=475 \pm 22\ ({\rm stat.})\pm^{20}_{15} \
({\rm syst.})$ mb, exclusive of helium contamination. Using this result in a
Glauber calculation to obtain the $pp$ inelastic cross section, they found the
inelastic $pp$ cross section $\sigma_{\rm inel}= 90\pm 7\ ({\rm stat.})
\pm^9_{11} ({\rm syst.}) \pm 1.5 {\rm \ (Glaub.})$ mb. Parameterization of the
$\bar pp$ and $pp$ cross sections incorporating analyticity constraints and
unitarity has allowed us to make accurate extrapolations to ultra-high
energies, and using Glauber calculations, accurately predict cosmic ray results
for $\spai$. In this update for 57 TeV, we predict i) a $pp$ total cross
section, $\sigma_{\rm tot}=133.4\pm 1.6$ mb, using high energy predictions from
a saturated Froissart bound parameterization of accelerator data on forward
$\bar pp$ and $pp$ scattering amplitudes and ii) a p-air inelastic production
cross section, $\sigma_{\rm p-air}=483\pm 3 $ mb, by using $\sigma_{\rm tot}$
together with Glauber theory, allowing us to determine independently that the
helium contamination was 19%, in reasonable agreement with their estimate of
25%. Our predictions agree with all available cosmic ray extensive air shower
measurements, both in magnitude and in energy dependence. By using our value
for the $pp$ total cross section at 57 TeV, Block and Halzen \cite{blackdisk}
have predicted that the $pp$ inelastic cross section is $\sigma_{\rm inel}=
92.9\pm 1.6$ mb, in agreement with the measured POA value.
