Terrestrial gamma ray flashes (TGFs) are brief bursts of gamma rays observed by satellites, typically in coincidence with detectable lightning. We incorporate TGF observations and the key physics behind current TGF production theories with lightning physics to produce constraints on TGF production mechanisms. The combined constraints naturally suggest a mechanism for TGF production by current pulses in lightning leader channels. The mechanism involves local field enhancements due to charge redistribution in current pulses and draws seed energetic particles from cold runaway in breakdown processes. The mechanism can reproduce the observed TGF time scale including multipulse TGFs, is in agreement with existing production altitude estimates, and may help explain TGF spectra by naturally producing unbeamed emissions.