A new analysis of high-resolution data from the Atacama Large Millimeter/submillimeter Array (ALMA) for 5 luminous or ultra-luminous infrared galaxies gives a slope for the Kennicutt-Schmidt (KS) relation equal to $1.74^{+0.09}_{rm -0.07}$ for gas surface densities $Sigma_{rm mol}>10^3;M_odot$ pc$^{-2}$ and an assumed constant CO-to-H$_2$ conversion factor. The velocity dispersion of the CO line, $sigma_v$, scales approximately as the inverse square root of $Sigma_{rm mol}$, making the empirical gas scale height determined from $Hsim0.5sigma^2/(pi GSigma_{rm mol})$ nearly constant, 150-190 pc, over 1.5 orders of magnitude in $Sigma_{rm mol}$. This constancy of $H$ implies that the average midplane density, which is presumably dominated by CO-emitting gas for these extreme star-forming galaxies, scales linearly with the gas surface density, which, in turn, implies that the gas dynamical rate (the inverse of the free-fall time) varies with $Sigma_{rm mol}^{1/2}$, thereby explaining most of the super-linear slope in the KS relation. Consistent with these relations, we also find that the mean efficiency of star formation per free-fall time is roughly constant, 5%-7%, and the gas depletion time decreases at high $Sigma_{rm mol}$, reaching only $sim 16$ Myr at $Sigma_{rm mol}sim10^4;M_odot$ pc$^{-2}$. The variation of $sigma_v$ with $Sigma_{rm mol}$ and the constancy of $H$ are in tension with some feedback-driven models, which predict $sigma_v$ to be more constant and $H$ to be more variable. However, these results are consistent with simulations in which large-scale gravity drives turbulence through a feedback process that maintains an approximately constant Toomre $Q$ instability parameter.
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