Spinors are lightlike. How do they combine to make massive particles? We visit the zoo of Lagrangian singularities, or caustics, in spacetime projections from spin space- the phase space of lightlike, 8- spinor flows. We find that the species living there are the elementary particles. Codimension J=(1,2,3,4) phase singularities - vortex lines, sheets, tubes, and knots, are classified by the Coxeter groups generated by multiplicity s reflections: "mass scatterings" off the vacuum spinors, that keep chiral pairs of matter envelopes confined to a timelike world tube, endowing a bispinor particle with mass. Using the volume in spin space as the action, the particle masses emerge in terms of the multiplicities, s: the number of null zigzags needed to close a cycle of mass scatterings. These mass values (calculated to lowest order in the vacuum intensity) are within a few percent of the observed masses for the leptons (J=1) and hadrons (J=3) ; but are up to 25 percent off for the mesons (J=2).