What causes the formation of discs and end of bursty star formation?

As they grow, galaxies can transition from irregular/spheroidal with 'bursty' star formation histories (SFHs), to discy with smooth SFHs. But even in simulations, the direct physical cause of such transitions remains unclear. We therefore explore this in a large suite of numerical experiments re-running portions of cosmological simulations with widely varied physics, further validated with existing FIRE simulations. We show that gas supply, cooling/thermodynamics, star formation model, Toomre scale, galaxy dynamical times, and feedback properties do not have a direct causal effect on these transitions. Rather, both the formation of discs and cessation of bursty star formation are driven by the gravitational potential, but in different ways. Disc formation is promoted when the mass profile becomes sufficiently centrally concentrated in shape (relative to circularization radii): we show that this provides a well-defined dynamical centre, ceases to support the global 'breathing modes' that can persist indefinitely in less-concentrated profiles and efficiently destroy discs, promotes orbit mixing to form a coherent angular momentum, and stabilizes the disc. Smooth SF is promoted by the potential or escape velocity Vesc (not circular velocity Vc) becoming sufficiently large at the radii of star formation that cool, mass-loaded (momentum-conserving) outflows are trapped/confined near the galaxy, as opposed to escaping after bursts. We discuss the detailed physics, how these conditions arise in cosmological contexts, their relation to other correlated phenomena (e.g. inner halo virialization, vertical disc 'settling'), and observations.