modelled using the SPIDER code (Bower et al. 2018, 2019), which simulates the thermal evolution of a magma ocean as it transitions from a fully molten state to partial or complete solidification.

The model solves the vertically resolved energy balance using an entropy-based formulation of the conservation equation.

The full numerical interior framework and implementation are described in detail in Bower et al. (2018), and have been adopted in previous PROTEUS studies.

Here, we briefly summarise the key aspects of the interior
model, as our work focuses on how interior thermodynamics and melting behaviour influence atmospheric evolution.

In this context, the thermal evolution of the mantle is governed by the integral form of the energy conservation equation expressed in terms of entropy:
ρ T ∂S
�
∂t dV = − �
F·n dA + �
ρ H dV (2)
V
A
V
with S specific entropy, ρ density, T temperature, F heat flux vector, n unit normal vector to the surface, H internal heat generation per unit mass, t time, and V volume.

The model takes into account conduction, mixing, gravitational separation and convection as energy transport mechanism, which contribute to the total flux F.

Convective heat flux is computed through mixing length theory, enabling this mechanism to be represented as a diffusive 1D process (Wagner et al. 2019).

In this work, we adopted a constant mixing length parametrization, following the framework explored in Bower et al. (2018).

In this way, the eddy diffusivity (κh) scales with a characteristic velocity depending on the local Reynolds number (Abe 1995; Stothers & Chin 1997), and affects the convective heat flux:
Fconv = −ρTκh ∂S
∂r . (3) Energy transport contribution due to gravitational separation only occurs in the mixed phase region, by permeable flow of melt from the solid background, and depends on the flow mechanism (crystal settling or flotation), Fgrav = a2 g g ρ � ρliq −ρsol � ζgrav(ϕ) T ∆S fus / ηm. (4) where ag is the grain size, ηm is the melt viscosity, ρliq and ρsol are the melt and solid densities at the liquidus and solidus, respectively, ∆S fus = S liq −S sol is the entropy of fusion defined as the difference between liquid and solid entropies and ζgrav(ϕ) which parametrizes the dependence of the flow law on melt fraction ϕ, and accounts for the transition from crystal-rich suspensions at high melt fractions to porous flow at low melt fractions (Abe & Matsui 1988; Bower et al. 2018).

Phase mixing quantifies the energy transport carried by the latent heat which is released/absorbed as crystals form/melt while they are being transported through the solidifying mantle, Fmix = −ρT ∆S fus κh ∂ϕ
∂r , (5) We considered a single-component system with two phases (solid and liquid), such that the melting region is represented by a finite interval between the solidus and liquidus.

In SPIDER, these melting curves are implemented through entropy–pressure lookup tables that define the boundaries of this two-phase region.

Because SPIDER solves the conservation of energy in terms of specific entropy, all thermodynamic quantities including temperature are obtained from tables T(S, P) and associated properties.

At each pressure, the liquidus and solidus entropies �S l(P), S s(P)�determine the local melt fraction, which is computed as a linear function of entropy between these bounds (Eq. 6).

This representation allows SPIDER to flexibly incorporate melting relations from experimental petrology (e.g., peridotite and chondritic solidus/liquidus data) and to smoothly transition material properties across phase boundaries using numerical smoothing routines (Bower et al. 2018).

Therefore, the lookup tables encode the full thermodynamic structure of the Article number, page 5 of 19 showed a strong dependence of the solidus on fOmelt under oxidizing conditions (IW+2.0)

at pressures of 16-26 GPa.