Why does the required distance margin for biological extinction from a GRB escalate non-linearly with distance?

The atmospheric chemistry response to $ ext{NO}_x$ production is non-linear; once saturation for $ ext{NO}_x$ is hit, damage escalates disproportionately

The relationship between distance and destructive outcome from a GRB is not simply governed by the inverse-square law applied to energy absorption alone, especially concerning atmospheric damage. Once the initial gamma pulse generates nitrogen oxides ($ ext{NO}_x$), the resulting ozone destruction escalates disproportionately. If a burst at one distance causes 50% depletion, a burst only twice as close might cause near-total stripping because the chemical process hits a saturation point where nearly all available ozone molecules in the beam path are rapidly consumed. This non-linear escalation means the safe distance margin is considerably tighter than expected from simple energy attenuation calculations.