Core Mass Bounds from Mean Density for Uranus and Neptune

This paper examines the mass of the dense central core in each ice giant, bounded by mean density alone. Using only published input values and transparent formulae, the calculation addresses what core mass follows from a two-component interior in which a rocky core of stated density is embedded in a mantle of stated density, with the mixture required to reproduce the measured mean density exactly. The headline results are: core_volume_fraction_uranus = 0.0248; core_volume_fraction_neptune = 0.0361; core_radius_fraction_uranus = 0.292. The method, inputs and limitations are stated in full so that any reader can reproduce or rescale every number from the deposited calculation script. The central finding is that mean density alone bounds the Uranian core only loosely: a modest change in the assumed mantle density moves the inferred core by Earth masses, which is why gravity harmonics and moment-of-inertia data, rather than density alone, carry the real discriminating power. Agreement and disagreement with independent published estimates are reported at their actual size, and the conclusions are limited to what the budget arithmetic supports.