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Super-eruptions that dwarf all historical volcanic episodes in erupted volume and environmental impact are abundant in the geological record. Such eruptions of silica-rich magmas form large calderas. The mechanisms that trigger these super-eruptions are elusive because the processes occurring in conventional volcanic systems cannot simply be scaled up to the much larger magma chambers beneath supervolcanoes. Over-pressurization of the magma reservoir, caused by magma recharge, is a common trigger for smaller eruptions, but is insufficient to generate eruptions from large supervolcano magma chambers. Magma buoyancy can potentially create sufficient overpressure, but the efficiency of this trigger mechanism has not been tested. Here we use synchrotron measurements of X-ray absorption to determine the density of silica-rich magmas at pressures and temperatures of up to 3.6 GPa and 1,950 K …