  It is well known that high energy data alone do not discriminate between
asymptotic $\ln s$ and $\ln^2s$ behavior of $pp$ and $\bar pp$ cross sections.
By exploiting high quality low energy data, analyticity resolves this ambiguity
in favor of cross sections that grow asymptotically as $\ln^2s$. We here show
that two methods for incorporating the low energy data into the high energy
fits give numerically identical results and yield essentially identical tightly
constrained values for the LHC cross section. The agreement can be understood
as a new analyticity constraint derived as an extension of a Finite Energy Sum
Rule.
