Crossing atmospheric-recovery distributions under fixed cumulative CO2 constraints

Carbon budgets constrain global warming remarkably well, but they do not uniquely determine the timing of atmospheric CO2 recovery. Here we compare feasible emissions pathways with exactly equal cumulative 2026–2100 anthropogenic CO2 across 841 calibrated FaIR 2.1.3 configurations and treat sustained atmospheric-CO2 decline as a first-passage distribution rather than a single recovery date. At 695 GtCO2, stronger early abatement advances median recovery by 8 years and lowers median integrated warming exposure above 1.5 °C by 3.18 °C yr, yet delays the 95th-percentile recovery time by 27 years. The recovery distributions therefore cross. Across a complete 525–772 GtCO2 integer sweep, tail-sensitive criteria reverse pathway preference before central statistics: q95 reverses at 643–644 GtCO2, whereas the median reverses only at 771–772 GtCO2, with no re-entry across the tested range. The delayed-recovery tail is strongly predictable from calibrated carbon-cycle parameters and is progressively recruited as cumulative mass increases. These results show that cumulative carbon can remain an effective scalar constraint on warming while being insufficient to identify a unique recovery-optimal pathway when atmospheric-recovery risk is distributional. This record includes the manuscript, Supplementary Information, source data, figures, exact computational inputs, archived FaIR environment, reproducibility code, verification outputs and integrity manifests.