  For the $\pbar p$ and $pp$ systems, we have used all of the extensive data of
the Particle Data Group[K. Hagiwara {\em et al.} (Particle Data Group), Phys.
Rev. D 66, 010001 (2002).]. We then subject these data to a screening process,
the ``Sieve'' algorithm[M. M. Block, physics/0506010.], in order to eliminate
``outliers'' that can skew a $\chi^2$ fit. With the ``Sieve'' algorithm, a
robust fit using a Lorentzian distribution is first made to all of the data to
sieve out abnormally high $\delchi$, the individual i$^{\rm th}$ point's
contribution to the total $\chi^2$. The $\chi^2$ fits are then made to the
sieved data. We demonstrate that we cleanly discriminate between asymptotic
$\ln s$ and $\ln^2s$ behavior of total hadronic cross sections when we require
that these amplitudes {\em also} describe, on average, low energy data
dominated by resonances. We simultaneously fit real analytic amplitudes to the
``sieved'' high energy measurements of $\bar p p$ and $pp$ total cross sections
and $\rho$-values for $\sqrt s\ge 6$ GeV, while requiring that their asymptotic
fits smoothly join the the $\sigma_{\bar p p}$ and $\sigma_{pp}$ total cross
sections at $\sqrt s=$4.0 GeV--again {\em both} in magnitude and slope. Our
results strongly favor a high energy $\ln^2s$ fit, basically excluding a $\ln
s$ fit. Finally, we make a screened Glauber fit for the p-air cross section,
using as input our precisely-determined $pp$ cross sections at cosmic ray
energies.

High Energy Physics - Phenomenology;Astrophysics