  High energy measurements of the inelastic proton-proton cross sections, at
the LHC at $\sqrt s$=7 TeV and by Auger at 57 TeV, have validated previous
evidence from data collected over a wide range of energies that the total and
inelastic cross sections for $pp$ and $\bar pp$ interactions saturate the
Froissart bound of $\ln^2 s$. Although the data themselves did not cover truly
asymptotic energies, our recent analysis of these data obtained the asymptotic
ratio $\sigin/\sigtot=0.509\pm 0.021$, consistent with the value of 1/2
required for scattering by a black disk; further, the forward scattering
amplitude became purely imaginary for $s\rightarrow \infty$, confirming the
black disk interpretation. In addition, the limiting black disk behavior has
been independently confirmed by an analysis of Schegelsky and Ryskin including
LHC data on the shrinkage of the slope of the forward elastic scattering cross
section. Unless one considers these results, emerging from an analytic
amplitude analysis of data over an energy range of $6\le \sqrt s\le 57000$ GeV,
a complete numerical accident, we rule out any new physics thresholds that
contribute higher powers of $\ln s$ or, worse, powers of $s$ to the energy
dependence of cross sections {\it at any energy}. This includes theories with
additional dimensions of space-time, whose existence is challenged.
