= Solution
For a gas whose lifetime is much less than the selected horizon, the numerator of its <global warming potential> has nearly saturated: $\int_0^H a_i m_i(t)\,dt\simeq a_i m_i(0)\tau_i$. The carbon-dioxide denominator continues to accumulate because of its long-lived response. Consequently \b[a shorter horizon usually gives a larger GWP for this short-lived substitute; a longer horizon gives a smaller GWP]. If the horizon becomes comparable to, or shorter than, the substitute's lifetime, the numerator also depends strongly on the horizon and cannot be replaced by its infinite-time limit.
The horizon is a value choice, and integrated forcing is not the same as <temperature> at a chosen future date. Equal-GWP pulses can therefore have different near-term warming and long-term <temperature> paths. The result also depends on <greenhouse gas radiative efficiency>, background <concentrations>, lifetime estimates and which indirect chemical effects or feedbacks are included. A pulse-based metric does not by itself describe a continuing emissions trajectory, nor does it measure <ozone> depletion or local air-pollution impacts. Comparisons should state the horizon, the emission scenario and the chosen climate outcome.<The Earth's Energy Budget, Climate Feedbacks, and Climate Sensitivity by Piers Forster and others>[IPCC AR6 Working Group I, Chapter 7, section 7.6.1: physical emissions metrics and time-horizon dependence]{ref}
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