Kinetically-constrained models (KCM) are widely used in the study of jamming and glass transitions in granular and amorphous materials[1]. Most KCM become jammed only at the limit of zero temperature or equivalently full occupation or when considering finite-sized or confined systems. An interesting exception is the Spiral Model, which has been proven to undergo a directed percolation phase transition even in the thermodynamic limit[2]. Beyond the critical density unjammed particles become caged within structures formed by jammed clusters of particles.  We introduce a 3D variant of the Spiral Model[3] in which beyond the critical density for jamming there are (initially) no caging effects. We give an intuitive geometrical explanation for the phenomenon and use a culling algorithm to show that in order to get a caging regime density must be increased even further to a second critical density in which both jamming and caging occurs. By analyzing cluster scaling we characterize the second phase transition and show that between the two critical densities a new regime exists where the system behaves qualitatively different than behaviors observed in 2D.

[1] F. Ritort and P. Sollich , Adv. Phys. 52 219 (2003)

[2] G. Biroli and C. Toninelli, Eur. Phys. J. B 64, 567 (2008)

[3] A. Ghosh, E. Teomy and Y. Shokef, Europhys. Lett. 106, 16003 (2014)