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In this work we generalize the notion of convective inhibition to apply in cases where there is an infinite reservoir of condensible species (i.e., an ocean). We propose a new model for the internal structure and thermal evolution of super-Earths with hydrogen envelopes. We derive the criterion for convective inhibition in a generalizes phase mixture from first principles thermodynamics. We then investigate the global ocean case using a water-hydrogen system, for which we have data, as an example. We then extend our arguments to apply to a system of hydrogen and silicate vapor. We then employ a simple atmospheric model to apply our findings to super-Earths and to make predictions about their internal structures and thermal evolution. For hydrogen envelope masses roughly in the range between 0.1-10% Earth's mass, convective contact between the envelope and core may shut down because of the compositional gradient that arises from silicate partial vaporization. For envelope hydrogen masses that cause the associated basal pressure to exceed the critical pressure of pure silicate (of order a couple kilobars), the base of that envelope and top of the core lie on the critical line of the two-phase hydrogen-silicate phase diagram. The corresponding temperature is much higher than convective models would suggest. The core then cools inefficiently, with intrinsic heat fluxes potentially comparable to the Earth's internal heat flux today. This low heat flux may allow the core to remain in a high entropy supercritical state for billions of years, but the details of this depend on the nature of the two-component phase diagram at high pressure, something that is currently unknown. A supercritical core thermodynamically permits the dissolution of large quantities of hydrogen into the core.