The "new core paradox" suggests that the persistence of the geomagnetic field over nearly all of Earth history is in conflict with the core being highly thermally conductive, which makes convection and dynamo action in the core much harder prior to the nucleation of the inner core. Here we revisit this issue by exploring the influence of six important parameters on core evolution: upper/lower mantle viscosity ratio, core thermal conductivity, core radiogenic heat rate, mantle radiogenic heating rate, central core melting temperature, and initial core-mantle boundary (CMB) temperature. Each parameter is systematically explored by the model, which couples mantle energy and core energy-entropy evolution. A model is successful if the correct present-day inner core size is achieved and the dynamo remains alive, as implied by the paleomagnetic record. In agreement with previous studies, we do not find successful thermal evolutions using nominal parameters, which includes a core thermal conductivity of 70 Wm(-1)K(-1), zero core radioactivity, and an initial CMB temperature of 5,000 K. The dynamo can be kept alive by assuming an unrealistically low thermal conductivity of 20 Wm(-1)K(-1) or an unrealistically high core radioactive heat flow of 3 TW at present-day, which are considered "unsuccessful" models. We identify a third scenario to keep the dynamo alive by assuming a hot initial CMB temperature of similar to 6,000 K and a central core liquidus of similar to 5,550 K. These temperatures are on the extreme end of typical estimates, but should not be ruled out and deserve further scrutiny. The cooling of Earth over geological time is of interest to many scientific disciplines and has been studied intensively. Maintaining the geomagnetic field requires continuous core cooling to drive fluid motion in Earth's liquid iron outer core. Recent studies have shown that the materials that make up Earth's core can conduct heat very efficiently, which makes fluid motion in the core more difficult. In this study we investigate the thermal and magnetic evolution of Earth using a core-mantle cooling model. In particular, we investigate how several important aspects of Earth's interior that control its thermal and magnetic evolution over 4.5 billion years. In agreement with previous studies, we confirm that typical models fail to reproduce the thermal and magnetic evolution found in the geological record. We identify a potentially new solution that invokes (a) that the core is initially super hot following the formation of the Earth, and (b) that the melting temperature of the core material is lower than most estimates. This potential solution should be investigated further by studying the initial core temperature following planet formation and careful measurement of the melting temperatures of iron alloys.
