Abstract: The importance of star formation “feedback” to the energetics of the interstellar medium (ISM) has been appreciated throughout the modern history of astronomy. Star formation is inefficient in gas consumption because feedback efficiently maintains the pressure support against gravity, which is otherwise rapidly lost via cooling and turbulence dissipation. At the same time, collective actions of feedback drive galactic-scale outflows, controlling the baryonic cycle in galaxy halos. In this talk, I will introduce the TIGRESS framework and its non-equilibrium cooling and radiation (NCR) extension. We solve magneto-hydrodynamics equations in a local shearing box representing a patch of galactic disks to take advantage of limited outer dimensions (~kpc) to achieve uniformly high resolution (~pc). The TIGRESS-NCR framework synthesizes our current best knowledge on governing physics of the star-forming ISM, including supernova and UV radiation feedback as well as photochemical reactions associated with UV (and cosmic rays) to set radiative heating rates and abundances for major coolants self-consistently. I will present the first results from a suite of simulations using the TIGRESS-NCR framework and explain the co-regulation of SFRs and the ISM. Specifically, I will delineate the self-regulation of SFRs in the context of pressure-regulated, feedback-modulated star formation theory and ISM phase structure and energetics with detailed breakdowns into energy source/sink from different processes and in different phases. Finally, I emphasize that having such a numerical framework is a departure point for further numerical experiments, including models with spiral arms and at low metallicities.