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We study the evolution of the core (r<1 kpc) and effective (r<r_e)stellar-mass surface densities, in star-forming and quiescent galaxies. Sincez=3, both populations occupy distinct, linear relations in log(Sigma_e) andlog(Sigma_1) vs. log(M). These structural relations exhibit slopes and scatterthat remain almost constant with time while their normalizations decline. ForSFGs, the normalization declines by less than a factor of 2 from z=3, in bothSigma_e and Sigma_1. Such mild declines suggest that SFGs build dense cores bygrowing along these relations. We define this evolution as the structural mainsequence (Sigma-MS). Quiescent galaxies follow different relations (Sigma^Q_e,Sigma^Q_1) off the Sigma-MS by having higher densities than SFGs of the samemass and redshift. The normalization of Sigma^Q_e declines by a factor of 10since z=3, but only a factor of 2 in Sigma^Q_1. Thus, the common denominatorfor quiescent galaxies at all redshifts is the presence of a dense stellarcore, and the formation of such cores in SFGs is the main requirement forquenching. Expressed in 2D as deviations off the SFR-MS and off Sigma^Q_1 ateach redshift, the distribution of massive galaxies forms a universal, L-shapedsequence that relates two fundamental physical processes: compaction andquenching. Compaction is a process of substantial core-growth in SFGs relativeto that in the Sigma-MS. This process increases the core-to-total mass andSersic index, thereby, making compact SFGs. Quenching occurs once compact SFGsreach a maximum central density above Sigma^Q_1 > 9.5 M_sun/kpc^2. Thisthreshold provides the most effective selection criterion to identify thestar-forming progenitors of quiescent galaxies at all redshifts.

A universal structural and star-forming relation since $z\sim3$: connecting compact star-forming and quiescent galaxies