termination criteria is met: (i) whole-mantle melt fraction de-
creases below a critical threshold ϕ < 0.05, indicating solidifi-
cation; (ii) the system reaches global energy balance, where the
melt fraction becomes constant in time and the net radiative flux
(iii) a maximum integration time is reached (set to 200 Myr).
This upper limit was chosen because the key phases of magma
ocean evolution, including rapid crystallisation and convergence
toward long-lived equilibrium states, occur within the first 10-
100 Myr (Nicholls et al. 2024). Our focus is therefore on the
Article number, page 4 of 19 terior–atmosphere feedbacks throughout the planet’s evolution (Tsai et al.

2021; Pierrehumbert 2010).

In this case, we used the star with a 50th percentile of rotation speed for its mass and age (Gallet & Bouvier 2013).
2.3.

Atmospheric model The energy transport in the atmosphere is computed with the open-source code AGNI (Nicholls et al. 2024; Nicholls et al. 2025b,c), a radiative–convective model that calculates the net atmospheric energy flux as a function of volatile composition, instellation flux, and the temperature at the atmosphere–mantle boundary (Nicholls et al. 2025b).

We employed AGNI in a configuration such that the atmospheric temperature–pressure profiles are convective (adiabatic).

Based on this structure, AGNI calculates spectrally resolved radiative fluxes using the SOCRATES code under the two-stream approximation (Edwards & Slingo 1996; Amundsen et al. 2016; Sergeev et al. 2023), with gas opacities from the DACE (Grimm & Heng 2015; Grimm et al. 2021) database and collision-induced absorption parametrised using HITRAN (Gordon et al. 2022).

We assumed the atmosphere is isochemical at the outgassed composition (see below, Sect. 2.4), with eight main volatile species: H2O, H2, CO2, CO, SO2, S2, N2, and CH4.

We assumed that the total number of moles of each volatile element is conserved throughout the simulation and do not model atmospheric escape.

Consistent with our maximum integration time of 200 Myr, we focused on the early coupled interior–atmosphere evolution and do not attempt to model long-term atmospheric loss or the subsequent evolution toward present-day conditions.

The impact of this assumption is discussed in Sect. 4.
2.4.

Outgassing model The exchange of volatiles between the molten mantle and the atmosphere is handled by the CALLIOPE outgassing module, based on (Bower et al. 2022).

This module applies mass conservation to solve for the C-H-O-N-S volatile partial surface pressures at thermochemical-solubility equilibrium at the magma ocean surface (Chase 1996; Bower et al. 2019; Shorttle et al. 2024; Nicholls et al. 2024).

The corresponding abundances are computed based on their solubility and equilibrium between the melt phase and the atmosphere.

During the evolution and without considering atmospheric escape, we conserved C-H-N-S and partitioned them between melt and atmosphere using solubility laws that depend on surface temperature, melt fraction, and imposed fO2 (Bower et al. 2025).

Oxygen is not tracked as a finite reservoir; instead fO2 is prescribed (infinite-reservoir approximation), so the atmosphere may effectively draw on or return O to the mantle to satisfy stoichiometry (Bower et al. 2022).
2.5.

Interior model The planetary interior is 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.