Abstract National emissions targets are collectively insufficient to align with the Paris Agreement. The fair-share literature assesses whether these targets are fair and ambitious in comparison to emissions trajectories based on equity principles. Such emissions trajectories commonly start at present-day emissions levels. Here we show that these continuous trajectories inherently reward past inaction and increasingly do so with their iterative updates. We provide an approach to allocating emissions trajectories based on equity principles applied with immediate effect. The resulting discontinuous national trajectories not starting at current emissions levels imply significant immediate international support to fund rapid mitigation globally. Modelling allocations with or without continuity has remarkable consequences for the relative implied contributions to international support among high-income countries. We find that emissions targets of G7 countries, Russia and China are responsible for most of the global 2030 ambition gap, while only some countries align with their 1.5 °C allocation.