Equal carbon budgets produce opposite atmospheric-recovery and thermal-exposure rankings

Under the same cumulative carbon budget, earlier atmospheric recovery can coexist with greater thermal exposure. Across 841 calibrated FaIR configurations, two pathways with identical 2025–2100 cumulative fossil emissions (1532.094261 GtCO₂) reach sustained atmospheric CO₂ decline in 804 versus 78 configurations. Among the 78 in which both recover, the front-loaded pathway turns a median 20 years earlier, yet 69/78 accumulate greater exposure above 1.5 °C, with a median excess of +1.63826 °C·yr. Across all configurations, median extra exposure is +1.862893 °C·yr while the median temperature difference at 2100 is only +0.0003157 °C. An independent MPI-ESM1-2-LR exact-equal-budget experiment reproduces the recovery–exposure inversion in 3/3 realizations. Exact FaIR reruns across SSP1-2.6, SSP2-4.5 and SSP3-7.0 show that thermal-exposure separation is robust across backgrounds, whereas atmospheric recovery first passage is background-dependent. Source-convergence and common-tail controls isolate inherited carbon-cycle state, and AERA-MIP GFDL-ESM2M provides a full-ESM source-order counterexample accompanied by stronger ocean and net-land uptake. Equal carbon budgets therefore do not preserve atmospheric-recovery and thermal-exposure rankings. Version 3.0 substantially extends and supersedes Version 2.0 as the current version; Version 2.0 remains unchanged and citable.