  A well-known result of Carrillo, Choi, Tadmor, and Tan states that the 1D
Euler Alignment model with smooth interaction kernels possesses a 'critical
threshold' criterion for the global existence or finite-time blowup of
solutions, depending on the global nonnegativity (or lack thereof) of the
quantity $e_0 = \partial_x u_0 + \phi*\rho_0$. In this note, we rewrite the 1D
Euler Alignment model as a first-order system for the particle trajectories in
terms of a certain primitive $\psi_0$ of $e_0$; using the resulting structure,
we give a complete characterization of global-in-time existence versus
finite-time blowup of regular solutions that does not require a velocity to be
defined in the vacuum. We also prove certain upper and lower bounds on the
separation of particle trajectories, valid for smooth and weakly singular
kernels, and we use them to weaken the hypotheses of Tan sufficient for the
global-in-time existence of a solution in the weakly singular case, when the
order of the singularity lies in the range $s\in (0,\frac12)$.
