 Solution

Distinguish the depth-dependent mantle temperature from mantle potential temperature $T_p$: the latter is the temperature a parcel would have after reversible decompression in an adiabatic process to a chosen reference pressure. A convecting mantle is approximately adiabatic through much of its interior, with steep thermal boundary layers near the surface and the core-mantle boundary. It is neither isothermal nor wholly molten.

Early accretion, large impacts and gravitational planetary differentiation supplied heat; metal segregation released gravitational energy and extensive melting could produce a magma ocean. Rapid surface heat loss then crystallized much of the silicate mantle. Subsequent evolution reflects competition between radiogenic heating, heat entering from the core and heat carried out through the upper boundary. In a schematic mantle energy balance,Here $C_m$ is mantle heat capacity, $H_m$ mantle radioactive power and $Q_m$ heat leaving the mantle; continental crustal radioactive production is not included in $H_m$. The convective Urey ratio $H_m/Q_m$ compares internal production with mantle heat loss. A value below one does not alone prove cooling, because core heat also enters; cooling requires $H_m+Q_{\mathrm{cmb}}<Q_m$. Radioactive decay reduces the heating supply, while mantle convection, plate tectonics, melt extraction and changing viscosity control the heat loss. A hotter mantle does not automatically imply proportionally faster plate motion: melting, dehydration and boundary strength can alter that relationship.

The robust long-term picture is an energetic early Earth followed by substantial cooling and a presently mostly solid, convecting mantle. The shape of the solid-state history is less certain. Petrological reconstructions commonly place late-Archean ambient $T_p$ above modern values. One reconstruction infers about $1500$–$1600{}^{\circ }\mathrm{C}$ at $2.5$–$3$ Ga versus about $1350{}^{\circ }\mathrm{C}$ today, and permits warming before that late-Archean maximum after the initial magma-ocean cooling.^[ref] This is a model-dependent ambient-mantle history, not proof that the accreting Earth was cooler than the Archean mantle or that every mantle reservoir followed one curve. An early molten transient, subsequent solid-state heating or cooling, and later secular cooling must be distinguished.