Nonlinearity of the post-spinel transition and its expression in slabs and plumes worldwide

Domain: earth_science
Published: 2025-01-26
Editorial outcome: minor_revision

Abstract
Phase transitions in the mantle control its internal dynamics and structure. The post-spinel transition marks the upper–lower mantle boundary, where ringwoodite dissociates into bridgmanite plus ferropericlase, and its Clapeyron slope regulates mantle flow across it. This interaction has previously been assumed to have no lateral spatial variations, based on the assumption of a linear post-spinel boundary in pressure and temperature. Here we present laser-heated diamond anvil cell experiments with synchrotron X-ray diffraction to better constrain this boundary, especially at higher temperatures. Combining our data with results from the literature, and using a global analysis based on machine learning, we find a pronounced nonlinearity in the post-spinel boundary, with its slope ranging from –4 MPa/K at 2100 K, to –2 MPa/K at 1950 K, and to 0 MPa/K at 1600 K. Changes in temperature over time and space can therefore cause the post-spinel transition to have variable effects on mantle convection and the movement of subducting slabs and upwelling plumes.

Introduction
High-pressure phase transitions in mantle minerals cause seismic discontinuities and influence convection in the Earth’s interior. One such transition is the pressure-induced dissociation of ringwoodite (rw, (Mg,Fe)2SiO4) into bridgmanite plus ferropericlase (bm+fp, (Mg,Fe)SiO3+(Mg,Fe)O)1–4. Often referred to as the post-spinel transition, it corresponds to a global seismic discontinuity in the mantle at a depth of about 660 km5. At this depth, the mantle experiences significant changes in temperature, material flow, and buoyancy, the effects of which are largely controlled by the Clapeyron slope of the post-spinel transition (γ post-spinel = dP/dT = ΔS/ΔV post-spinel, where ΔS and ΔV are the entropy and volume changes across the transition, respectively)6–8. For example, if γ post-spinel is more negative, this transition can cause subducting slabs to stagnate at mid-mantle depths9,10, and it can also impede the upwelling of mantle plumes11,12.

The post-spinel transition between rw and bm + periclase (pe) in the iron-free system (Mg2SiO4 ↔ MgSiO3 + MgO) is useful for understanding this transition in the Earth’s pyrolitic mantle, since the post-spinel Clapeyron slopes in these two compositions are nearly identical1–4,7,13 (Figure S1). Numerous experimental studies have directly investigated the phase stability of rw and bm + pe using in situ X-ray diffraction14–19, but with large variations in their derived Clapeyron slopes, which range from about –3 MPa/K to +1 MPa/K. The previous studies in the type of data. This is equivalent to assuming that upwelling or downwelling mantle flow across this boundary will experience the same Clapeyron slope, regardless of temperature variations in space and time. In fact, thermodynamic models have suggested that the post-spinel boundary may be nonlinear, with a slope that varies with temperature6,20,21. If the boundary is nonlinear, temperature differences (for example, between the present-day and early Earth, or between slabs and plumes) could lead to different effects of the same post-spinel transition on mantle flow due to local differences in the Clapeyron slope. However, the empirical thermodynamic models rely on experimental data collected at low temperatures, such as heat capacity (<500 K)20,21, which, when extrapolated, produce inaccurate entropy and volume at the high temperatures relevant to the mantle transition zone (1600–2200 K), thus compromising their reliability in predicting the Clapeyron slope. Chanyshev et al. 19 present the first experimental evidence that the post-spinel boundary is nonlinear, but do not quantify the variation in its Clapeyron slope. To date, there is no robust estimate of the nonlinearity of the post-spinel transition.

To understand the effects of the nonlinear post-spinel transition on mantle convection, we have placed new experimental constraints on the Clapeyron slope of this transition and clarified how this slope changes with temperature. Previous efforts to determine this slope from direct observations of phase stability have faced challenges due to: 1) relying on visual inspection and freehand drawing of the boundary, which can introduce biases based on the researchers’ expectations; 2) using individual datasets rather than combining them, which prevents evaluation of the accuracy of each dataset; and 3) obtaining in situ observations on Mg2SiO4 primarily only at temperatures of 1500–2000 K, which leads to errors when extrapolating the phase boundary to higher temperatures.

To address these challenges, we have performed laser-heated diamond anvil cell (LH-DAC) experiments using synchrotron X-ray diffraction (XRD) on Mg2SiO4 across a range of pressures (P) and temperatures (T) relevant to the Earth’s mantle transition zone (16–28 GPa, 1573–2723 K). Our LH-DAC data span multiple phase boundaries in Mg2SiO4, with a particular focus on the previously-understudied triple point of wd–rw–bm+pe. Combining our dataset including the triple point with previous high-precision multi-anvil (MA) press datasets, we used logistic regression and supervised learning to identify the phase boundaries in Mg2SiO4 under mantle transition zone conditions and determine the nonlinearity of the post-spinel boundary. Finally, we modeled the lateral variations in the post-spinel Clapeyron slope at the base of the transition zone due to variations in temperature, and evaluated its impact on slabs, plumes, and the ambient mantle.

Results
In situ experimental observations on the Mg2SiO4 phase diagram at transition zone conditions
We present an experimental dataset showing high P–T phase stability observations of Mg2SiO4. These data were obtained using synchrotron XRD in nine LH-DAC experiments, which included 17 laser-heated spots (Supplementary Data 1; Figure S2). Figure 1 shows examples of high P–T XRD patterns that constrain phase boundaries in Mg2SiO4. In Fig. 1a, the wd ↔ rw transition is highlighted. Initially, the wd phase remained stable at 1616±162 K and 20.6±1.3 GPa. However, in the subsequent diffraction pattern collected at 1785±187 K and 21.0±1.5 GPa, a transition to the rw phase began. Figure 1b focuses on the post-spinel transition, where the rw peaks initially continue to grow and increasingly overlap with the relic wd peaks. In the following XRD pattern, bm peaks appear for the first time, with the rw phase beginning its transition to bm+pe at 2173±252 K and 22.2±1.9 GPa.

To combine our results with literature datasets and perform a global analysis of the Mg2SiO4 phase diagram, we have compiled all available in situ phase stability data on the Mg2SiO4 system from previous studies (Supplementary Data 2). Here we utilize a selected compilation of four specific datasets19,22–24 (Fig. 2). These data were all obtained from in situ high-precision MA experiments using an MgO pressure calibrant, and were calculated their pressures so that all data in the compilation are based on internally-consistent pressure calibrations25,26. We also corrected the thermocouple-based temperature measurements of these studies for the effect of pressure on electromotive force27,28 to maximize consistency with the spectroradiometric temperature measurements in this study.

Global inversion of the post-spinel phase boundary and its degree of nonlinearity
After correcting and combining the four MA datasets from the literature19,22–24, we globally inverted them into a P–T phase diagram of Mg2SiO4, first excluding our LH-DAC dataset (Fig. 2a, inset). These literature data were all obtained at temperatures below ~2100 K. This initial analysis produces linear boundaries for both wd ↔ rw and the post-spinel transition. However, the limited temperature range of this MA dataset creates challenges in accurately determining the P–T conditions of the triple point. For example, the initial global analysis suggested a triple point involving wd, rw, and bm+pe above 2600 K, which directly contradicts all existing experimental and theoretical evidence15,29–32. This discrepancy highlights the unreliability of extrapolated phase boundaries and emphasizes the need to combine MA and LH-DAC data for phase boundary determination over a wider temperature range.

Our LH-DAC dataset fills this experimental gap, providing abundant in situ phase stability observations for wd, rw, and bm+pe over a temperature range of 1573–2723 K at mantle transition zone pressures (Figure S3). In our second and final global inversion (Fig. 2b, main), incorporating our latest LH-DAC data with the four literature MA datasets19,22–24, we found a nonlinear Clapeyron slope for the post-spinel transition. Specifically, the slope changes from –4.1+2.1/-7.6 MPa/K at 2150±50 K to –1.7+1.1/-1.4 MPa/K at 1900±50 K and finally to 0.0+1.7/-2.2 MPa/K at 1600±50 K. The wd ↔ rw and wd ↔ bm+pe transitions were found to be nearly linear with slopes of 5.7+1.7/-1.0 MPa/K at 1800±200 K and 0.8+1.3/-1.2 MPa/K at 2300±200 K, respectively. We estimated the location of the triple point to be ~22.2 GPa and ~2190 K.

The post-spinel boundary is remarkably nonlinear, in contrast to the wd ↔ rw boundary over a similar temperature range. This clear difference is consistent with the thermodynamic properties of the Mg2SiO4 polymorphs. The Clapeyron slope of a phase transition depends on ΔS and ΔV across the transition, γ = ΔS/ΔV. Both ΔS and ΔV are influenced by the thermal expansivities (α) of the phases involved. For the wd ↔ rw transition, αrw and αwd show similar temperature dependencies. However, for the post-spinel transition, αrw increases more slowly than αbm at high temperatures32,33. This discrepancy in the temperature dependencies of these thermal expansivities means that ΔV wd$rw and thus also γ wd$rw varies less than ΔV post-spinel and γ post-spinel with changes in temperature. Consequently, the thermodynamic properties of these minerals are consistent with the findings of our global inversion, with a boundary that appears linear for the wd ↔ rw transition and clearly nonlinear for the post-spinel transition.

Our LH-DAC dataset covers a wider temperature range that extends beyond the triple point of wd–rw–bm+pe, which is critical for constraining the nonlinearity of the post-spinel boundary and is only possible in the LH-DAC. However, using the LH-DAC to achieve these higher temperatures comes with the trade-off of lower precision (both in pressure and temperature) than in a multi-anvil press19. Here, we combine our LH-DAC data with the selected MA data from the literature, to take advantage of the higher-precision constraints on the phase boundary at lower temperatures obtained in those MA studies while significantly improving constraints near the triple point, and thus the nonlinearity of the post-spinal boundary, through the addition of the higher temperature LH-DAC data. Future work should aim to probe closer to the post-spinel boundary with improved experimental setups that balance achieving high temperatures and measurement precision.

Mantle expression of the nonlinear post-spinel boundary in slabs and plumes worldwide
Solid–solid phase transitions shape mantle convection. While a small subset of cold slabs and hot plumes are likely influenced by the akimotoite (ak) ↔ bm and post-garnet transitions4,19,34, the dynamics of the Earth’s mantle today, especially for most slabs and plumes, are primarily controlled by the post-spinel transition1–4. When the post-spinel boundary has a more negative Clapeyron slope, it impedes mantle flow more strongly7. The nonlinear nature of this boundary indicates that the magnitude of its negative slope, and hence its impedance to mantle flow, varies with temperature. To understand how such nonlinearity manifests itself in the mantle, we constructed a temperature map at the “660 km” discontinuity (labeled T660; Fig. 3a, b). We then examined variations in its Clapeyron slope (γ post-spinel) in slabs, plumes, and ambient mantle at this depth (Fig. 3c).

For slabs undergoing the post-spinel transition (T660 ≥ 1400 K, 46 of the 50 slabs we studied35; Supplementary Data 4), the average T660 is 1600±130 K (standard deviations applied throughout), and the average γ post-spinel is –0.2±0.4 MPa/K. In contrast, plumes associated with the post-spinel transition (T660 < 2150 K, 18 of the 26 plumes36 we studied; Supplementary Data 5) have an average T660 of 2060±80 K and a γ post-spinel of –3.6±1.4 MPa/K. For plumes, the average slope is eighteen times more negative than its value for slabs. Such a strong contrast in their post-spinel Clapeyron slopes would result in a significantly different impedance to flow for subducting slabs and rising plumes depending on their locations (Fig. 3).

Half of the subducting slabs in Earth’s mantle today are relatively cold, with T660 = 1400–1600 K, most of which are clustered in the western Pacific region. These slabs have a Clapeyron slope of ~0 MPa/K. As a result, they would encounter minimal impedance to subduction at 660 km. In contrast, warmer plates near regions such as southern Chile (with T660 = 1760 K), Colombia/Ecuador (with T660 = 1830 K), and northern Cascadia (T660 = 1940 K) have more negative slopes ranging from –0.7 MPa/K to –1 MPa/K and –1.6 MPa/K, respectively. These warmer slabs are therefore expected to encounter greater resistance during subduction at the post-spinel transition.

Many mantle plumes associated with the post-spinel transition today (T660 < 2150 K) would exhibit different dynamic behaviors and morphologies11, consistent with its nonlinear slope. Some of these plume heads (e.g., St. Helena, Canary, and Reunion) tend to be trapped at around 660 km depth with γ post-spinel more negative than about –3.4 MPa/K, while others (e.g., Ascension, Cameroon, and Cape Verde) continue to rise through the transition zone with γ post-spinel more positive than about –2.8 MPa/K. In the range of γ post-spinel between –3.4 and –2.8 MPa/K (e.g., Azores and Kerguelen/Heard), ring-shaped secondary plumes can form. In addition, the dynamic topography of these plumes can also be influenced by the nonlinear post-spinel boundary, causing the surface subsidence to decrease with a more positive γ post-spinel37: from 100 m with γ post-spinel = –3.0 MPa/K at T660 = 2055 K, to 200 m with γ post-spinel = –3.2 MPa/K at T660 = 2065 K.

On one hand, this nonlinear post-spinel boundary has implications for the evolution of mantle convection from the early Earth to the present. During the Archean, slabs, plumes, and the ambient mantle may have been 150–200 K hotter than today38,39. As a result, these slabs, plumes, and ambient mantle would have had more negative post-spinel slopes compared to their present-day equivalents. These substantial differences could have led to a form of dynamic layering within the early mantle that differed from the whole-mantle convection of today7,40. On the other hand, this nonlinearity in the post-spinel boundary also affects the thermal evolution of individual stagnant slabs and plumes at any given time. The variable effects of rheology and buoyancy, driven by the temperature-dependent post-spinel Clapeyron slope, could significantly influence their dynamics and shapes. This influence could persist throughout their lifetimes until they reach thermal equilibrium with the ambient mantle.

Our global map of lateral variations in the post-spinel Clapeyron slope (Fig. 3f) provides a spatial guide for studying slabs and plumes worldwide, each with its specific Clapeyron slope. Furthermore, the nonlinear post-spinel boundary introduces varying dynamic behaviors of the mantle through time. This variability is intrinsically linked to mantle temperature differences across time and space, and is largely unexplored in existing geodynamical models. Mantle flow is influenced by several factors10, including viscosity and compositional differences in the mantle, and the configuration and history of the specific slab and plume, in addition to the Clapeyron slopes of major phase transitions, so it is not always straightforward to predict slab or plume behavior at 660 km depth from only the post-spinel Clapeyron slope. Therefore, it is imperative for future studies to accurately model the thermal and dynamical evolution of slabs, plumes, and the ambient mantle consistent with a temperature-dependent post-spinel Clapeyron slope to fully understand mantle dynamics and its temporal and spatial variability.

Methods
Synchrotron x-ray diffraction experiments
Short symmetric diamond anvil cells and a gas membrane cell were used to generate high pressure conditions. Double-sided laser heating and synchrotron XRD measurements were performed at beamline 13-ID-D (GeoSoilEnviro Center for Advanced Radiation Sources, GSECARS)41,42 and beamline 16-ID-B (High Pressure Collaborative Access Team, HPCAT)43 at the Advanced Photon Source (APS), Argonne National Laboratory (ANL). The starting materials were powdered synthetic forsterite (Mg2SiO4) mixed with tungsten (W), sandwiched between two layers of potassium chloride (KCl) and loaded into a sample chamber predrilled into a rhenium (Re) gasket. The KCl layers served as the pressure medium, thermal insulator, and primary pressure standard26,44, while the W was used as a laser absorber and secondary pressure standard25. Both W and KCl have melting temperatures higher than Mg2SiO4 45,46 and are chemically inert in this pressure and temperature range. The P–T path of each experiment is shown in Figure S2. Further details of our experimental procedures and results are given in the Supplementary Information.

Phase identification and phase boundary detection
We adopted two main criteria to detect phase transitions. First, we observed the emergence of the first peaks from any new phase between two consecutive diffraction patterns. Second, we monitored the peak intensity of existing phases in each heating cycle. However, the intensities of these peaks can change due to the preferred orientations of new crystal grains or peak sharpening during heating19. As an added measure, we used the “spottiness” of the Debye rings in 2D XRD images to track the stability of existing phases when the intensities of their diffraction peaks do not change uniformly. It is important to distinguish actual phase transitions from changes in peak intensity and d-spacing caused by temperature fluctuations. To avoid misinterpretation, we did not use any diffraction patterns in which the temperature fell by over 50 K during heating.

The main goal of these LH-DAC experiments was to investigate the location of the Mg2SiO4 triple point and obtain phase stability data in the high-temperature region between the triple point and the solidus. To achieve these goals, we began our experiments at pressures 3–4 GPa below our target range to accommodate thermal pressure upon heating. We initially compressed and heated our samples to 17–20 GPa and 1300–1600 K, then to our target range of 21–23 GPa and 1900–2300 K. Since these P–T conditions are so close to the wd–rw phase boundary19, we have consequently often observed the new rw phase and the metastable wd phase coexisting. Detection of the initial rw peaks was challenging due to the high diffraction peak density of the orthorhombic wd structure.

Although it would be ideal to reverse the P–T path and observe these phase transitions upon cooling, this proves challenging in LH-DAC experiments. Completely recombining bm+pe back into a single phase within the fast-paced experimental timeframe is difficult due to the sluggish kinetics of the back-transformation47,48. Results from multiple heating–cooling cycles can be controlled simultaneously and incrementally. However, the accuracy of the temperature measurements has limitations, which are affected by thermocouple calibrations and pressure effects on electromotive force at temperatures over ~1000 K. These can lead to significant systematic errors in temperature measurements, causing, for example, a >80 K error at 1773 K with a type D thermocouple, which impacts the determination of Clapeyron slopes27. Hence, we found LH-DAC experiments effective for collecting a larger dataset with smaller systematic errors, despite the potential for greater random errors in temperature measurements.

Selection and correction of phase stability observations from the literature
We performed a global analysis of the Mg2SiO4 phase diagram by incorporating our LH-DAC dataset with selected MA datasets from the literature. We used four key datasets from Katsura et al.22, Inoue et al.23, Katsura et al.24, and Chanyshev et al.19. All of these data are from in situ MA experiments with MgO as the primary pressure scale. The pressures from these datasets were recalculated using an MgO pressure scale that was cross-calibrated against KCl by Sokolova et al. and Tateno et al.25,26. The rest of the existing experimental datasets on Mg2SiO4 from the literature can be found in Supplementary Data 2–3, but they were omitted from our global analysis for one or more of the following reasons (Figure S4):

• Ex situ experiments: Many earlier experimental studies determined phase boundaries using ex situ methods. MA examples include works by Ito and Takahashi1 and Ishii et al.49, in which pressures were based on “fixed point” calibrations (typically with one calibration at room temperature and one at high temperature). These experiments are expected to have variable pressures with changing temperature, influenced by factors like thermal pressure and material relaxation50. DAC examples include Chudinovskikh and Boehler29, in which the pressures were based on estimated thermal pressures from past experiments. Comparing and correcting the pressures from these two type of ex situ datasets is impossible, and hence, they were excluded.

• In situ experiments with Au and/or Pt pressure scales: Notable differences in phase boundaries are found among literature datasets with different pressure scales, even after recalibrating them51. This phenomenon may be due to severe anharmonic effects in Au and a large deviation from Debye-like behavior in Pt at high temperatures. To maintain maximum internal consistency, datasets using Au or Pt pressure scales, including Irifune et al., Shim et al., Kastura et al., and Ghosh et al.16,18,50,52, were excluded.

• In situ experiments with uncorrectable pressures and/or temperatures: Some in situ experimental studies, such as Fei et al.17, used a type C thermocouple (W95Re5–W74Re26) that lacks an extrapolatable calibration for its pressure effects. As a result, these datasets were also excluded.

Multi-class logistic regression and supervised learning
Determining phase boundary locations can be considered a classification problem53–55: assigning observations to a specific stable mineral phase under certain P–T conditions. We used multi-class logistic regression here for determining the phase diagram of Mg2SiO4. This approach can accommodate multiple stable phases and gives probabilities of observing these phases at specific P–T conditions, p^ (Y=k|P,T), which can be expressed as a logistic function, ln p^ (Y=k|P,T) / p^ (Y=j|P,T) = Σ βk i,j PiTj = f(P,T) (1)

We can then convert probability estimates from three separate models (with k = wd, rw, or bm+pe; K=3) to one set of probability estimates using the “softmax” function53,55. The rescaled probability estimates add up to 1. We take the stable phase to be the class with the highest probability, and the tripe point occurs where p^ (Y=wd) = p^ (Y=rw) = p^ (Y=bm+pe) = 1/3. The coefficients βk i,j are estimated by minimizing a combined negative log-likelihood function, or total cross entropy, ΔL53:

ΔL = - 1/M Σ Σ [tm,k ln p^ m(Ym=k) + tm,l ln (1-p^ m(Ym≠k))] (2)

where tm,l(Ym=k) is 1 if and only if the observation m belongs to phase k, and p^ m(Ym=k) is the output probability that the observation m belongs to phase k.

We then used supervised learning (Figure S5) to constrain this multi-class logistic model using the experimental phase stability observations on Mg2SiO4 from this study and from the literature. We first divided the compiled data, with 70% being a train set and 30% being a test set. Evaluating the model on both of these sets was crucial to avoid overfitting because the model may overfit the train set to predict perfect responses but fail to perform well on the unseen test set. Our approach was to describe the log-odds using polynomials of P and T from degree 1 to 10 (n=1–10) in Eq. 1. However, overfitting can occur if a high degree polynomial for the log-odds is used, and thus we applied a regularization method, called Lasso or L1 regularization. This regularization method constrains or regularizes the coefficient estimates, βj, by modifying the least-squares loss function in Eq. 2, L(β), into a regularized loss function, LLasso(β) = L(β) + λ Σ |βi,j|, where λ is a scaler that assigns weights to the regularization term, or the regularization strength. When λ is sufficiently large, the regularized loss function LLasso(β) becomes increasingly sparse, with many of the βi,j = 0, thus avoiding overfitting and ensuring a successful convergence.

We optimized the hyperparameter C (the inverse of regularization strength, λ = 1/C), along with other parameters in the “scikit-learn” package such as “multi_class” and “solver”56. Parameter tuning was achieved through a grid search with k-fold cross-validation, which involved further splitting the train set into k folds (k=3–5). We then trained the model using k–1 folds, validating it on the remaining fold. The final evaluation of the parameters was based on the average over the k folds.

For model evaluation, we used a combination of two metrics: precision (true positive / (true positive + false positive)) and recall (true positive / (true positive + false negative)), which measure the proportion of accurate positive predictions and the ability to find all the positive samples, respectively. As precision and recall are inversely related, we used their harmonic mean, known as the F1 score (2 × precision × recall / (precision + recall)), as our evaluation metric to maximize both precision and recall simultaneously. This score helps maintain a balance between precision and recall, with a high score indicating a more accurate model (Figure S6). We then fitted the tuned model to the train and test sets. The log-odds polynomial degree with the highest F1 score on the test set was selected as the best degree. Lastly, we utilized the multi-class logistic regression with the chosen log-odds polynomial degree to fit the entire dataset (the recombined train and test set) and obtained an optimized phase diagram (Figure S7).

With the selected best model, we estimated the uncertainty in our predicted phase boundaries using bootstrapping. We resampled the compiled dataset with replacement (67%) and created 5×103 copies of the bootstrapped dataset. We then fit the best model to each resampled copy of the dataset, resulting in 5×103 sets of phase boundaries. We report 1–99% confidence intervals of the simulated distributions as the allowed P–T regions (or uncertainties) for the phase boundaries (Fig. 2).

A composite mantle temperature model for the “660-km” discontinuity
Our composite mantle temperature map for the “660-km” discontinuity provides an estimate of temperature variations at 660 km depth within the Earth’s mantle, labeled as T660. It is a combination of three existing thermal models: 1) 2D kinematic models for subducting slabs from Syracuse et al.35; 2) global shear velocity constraints for plume-fed hotspots from Bao et al.36; and 3) globally-compiled S660S observations for the ambient mantle from Waszek et al.57. The values of T660 are estimated as follows:

• Slabs: To estimate slab temperatures at 660 km depth, we extrapolated the slab surface P–T paths of all arc segments, using the W1300 case from [S10]. This process involved: 1) analyzing subduction temperature profiles, which become mostly constant after 150 km depth; 2) calculating the average slope of each temperature profile at 150–250 km depth; and 3) estimating T660 for each subduction segment by extrapolating their P–T paths to 660 km depth using their respective temperature gradients.

• Plumes: For plume-fed oceanic hotspots, T660 estimates were based on mantle potential temperature estimates from [B22]. This process involved: 1) calculating mantle adiabats for each plume-fed hotspot using the HeFESTo code30,58, assuming a depleted MORB mantle (DMM)59; and 2) extracting the mantle temperature at 660 km depth from each mantle adiabat.

• Ambient mantle: The same process was applied to the ambient mantle using the global mantle potential temperature map from [W21]. Here, we assumed a mechanically mixed (MM) mantle composed of 20% basalt and 80% harzburgite57,59.

We note that our estimates for T660 could have significant errors, especially for the slabs, due to the simple extrapolation of the slab surface temperature with a constant temperature gradient beyond 250 km depth. This could lead to deviations from the actual mantle temperatures at 660 km depth for some slab segments60. The estimates for the slabs serve here only as an upper bound and will not represent the minimum temperature within the slabs, which some regional models suggest could be a few hundred K lower, e.g., in the western Pacific61, and thus our estimates for γ post-spinel serve here as a lower bound. In addition, another bm-forming reaction, ak ↔ bm, may replace the post-spinel transition in a small number of extremely cold slabs where T660 is below 1400 K, such as the Tonga slab (T660 = ~1360 K). Such extremely cold conditions for the ak ↔ bm transition are unlikely to be prevalent in the ambient mantle today, but they may be more common than those identified in the slabs of Fig. 3, if the actual T660 is lower than our estimates.

Despite these potential errors, our composite T660 model provides a reasonable first-order approximation of the lateral variations in mantle temperature at the 660-km discontinuity. The mantle temperature distributions in these three independently-constrained thermal models are consistent with one another; for example, mantle temperature is lowest around the western Pacific subduction zones, while plume-fed hotspots coincide with positive T660 anomalies in the ambient mantle (Fig. 3e).

Data availability
The source data used to reproduce all of the figures in this study can be accessed in the Supplementary Data 1–5 and are available on Zenodo at https://doi.org/10.5281/zenodo.14188625.

Code availability
The code used to construct the Mg2SiO4 phase diagram in this study, including detailed documentation and a benchmark case, can be accessed in the Supplementary Data 6–7, and are available on Zenodo via https://doi.org/10.5281/zenodo.14188625.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Review of the manuscript \"Nonlinearity of the post-spinel transition and its expression in slabs and plumes worldwide\" by\nDong et al. to Nature Communications.\nThis is a revision of the manuscript that was previously submitted to Nature Geoscience. I appreciate the authors' efforts to\nimprove the manuscript. However, the main issue remains unresolved. Consequently, I do not believe it is suitable for high-\nlevel journals such as Nature Communications.\nI previously raised concerns about the motivation for the current study. The authors have added a paragraph to the\nintroduction addressing this point. They claim that the experimental temperatures in previous studies on the Clapeyron slope\nwere significantly lower than those in the mantle transition zone (1600 - 2200 K), thereby questioning the reliability of\npredicting the Clapeyron slope in the mantle. However, this assertion is quite misleading. As demonstrated in previous\nresearch, such as Chanyshev et al. (2022), the experimental temperature range actually spans from 1200 to 2100 K.\nRegarding the thermodynamic model, I fully agree that classification models can provide reasonable (or sometimes even\nbetter) fittings compared to thermodynamic models. I also concur that existing thermodynamic models in the literature may\nnot accurately predict the post-spinel transition slope. However, this is because some thermodynamic parameters have not\nbeen precisely determined in the literature. As the authors mentioned, the ΔS and ΔV values in the literature are unreliable,\nthus impairing the thermodynamic model's ability to describe the post-spinel phase transition.\nGiven this, a more reasonable strategy would be to obtain more precise thermodynamic parameters using the current data,\nrather than constructing a classification model, which lacks physical meaning. A classification model is appropriate only\nwhen the physical connections between the values are unknown. For the current case, we already know that the phase\nboundary is determined by thermodynamics. Therefore, the task for experimentalists should be to precisely determine the\nthermodynamic parameters, such as ΔS and ΔV.\nAs I mentioned in my previous review, the main problem with previous studies is the limited precision (or accuracy) in\ndetermining the P-T conditions of the phase boundaries. While the authors have added more data points, these additions do\nnot offer an improvement over the previous studies. In fact, some of their data points, such as the orange symbols in the wd\nand rw stability fields, contradict their results. Consequently, their findings do not provide a better constraint on the\nClapeyron slope compared to previous studies. The authors have discussed the difference between precision and accuracy,\narguing that multi-anvil experiments may have systematic errors in temperature readings. This might be true, but following\nthis logic, the authors should focus on obtaining more data points closer to the phase boundaries to better constrain them.\nWhile I agree that data points far from the phase boundaries can help to some extent, \"direct observations\" near the phase\nboundaries provide the most precise constraints for the phase boundaries. We can easily imagine that IF there are only data\npoints far from the phase boundaries, we cannot precisely determine the phase boundaries. In contrast, data points (if they\nare correct) near the phase boundaries can directly determine these boundaries with much higher accuracy. This is the\nfundamental principle of mathematics.", "reviewer_2": "The authors have addressed all my concerns. It may be accepted for the publication in nature communications.", "reviewer_3": "I am satisfied with the discussion of the limitations of the presented temperature model T660. I support the publication of the\nmanuscript.", "reviewer_4": "Dear Editor,\nThe Claperon slope at the 660 km discontinuity is an important question with significant geophysical implications. Ideally,\nthe problem can be resolved by covering the relevant pressure-temperature space with high-density experimental points\n(many high quality experiments precision and high accuracy). However, high-temperature and high-pressure experiments\nare expensive and time-consuming, leaving the problem still unresolved. Dong et al. provide new experimental results in this\nstudy and use a statistical/machine learning approach to investigate the problem. The approach is promising and advances\nour understanding of the Claperon slope near 660 km depth, which might be non-linear.\nThe study provides statistical and machine learning evidence by:\n1) Compiling and filtering the experimental results to construct a self-consistent database. This is important because\nalthough the accuracy of temperature is not good enough for diamond anvil cells, the dataset is self-consistent in revealing\nthe Claperon slope.\n2) A multi-class logistic regression is used, which is reasonable for the phase diagram's classification problem.\nI evaluate the data, method, and results, in particular of the statistical and machine learning parts. Given the data in the\nsupplementary Table, I organized the data and investigated the problem independently using Geochemistry π Python\nFramework version 0.6.1 (ZhangZhou and He et al. 2024 G Cubed). I tried several classification machine learning methods.\nThe results suggest non-linear phase boundaries are more consistent with the available experiment data.\nThe results will generate broad interest in the experimental, geodynamic, and seismological fields. I recommend publication\nin Nature Communications after minor revision.\nI have two suggestions:\na) The authors must provide the codes and training/test data in Zenodo for a final review before acceptance. The code\nshould open with the publication and be clearly annotated for the dataset and code. In my test exercise using Geochemistry\nπ Python Framework, I have to work on the supplementary Table to get the training data.\nb) Support Vector Machines (SVM) are the classic tool for classification problems (Cortes and Vapnik, 1995) and, therefore,\nis usually the first algorithm considered for classification. With kernel tricks, SVM can also achieve non-linear boundaries.\nThe authors can consider mentioning SVM in the manuscript to acknowledge people's work on classification problems using\nmachine learning. I think it is redundant to present SVM, XGBoost results in the manuscript. Logistic regression can justify\nthe non-linear phase boundary problem.\nDataset S5a: Plume-fed oceanic hotspots accompanied by the post-spinel transntion (transition, typo)\nFig.S1. I suggest putting the P-T phase boundary calculated by thermodynamic code HeFESTo (e.g., blue lines in Figure\nS1) as an equation here. The authors can compare the classification results of the equation (HeFESTo model) with\ncompiled experimental data (confusion matrix results). Furthermore, the authors can compare machine learning models\n(different degrees of polynomials) with experimental data. The above results can be put in Dataset S6. I think this can further\njustify the advancement brought by machine learning.\nReferences:\nCortes and Vapnik (1995) Support-vector networks. Machine learning, 20, 273–297. https://doi.org/10.1007/BF00994018\nJ ZhangZhou, Can He, Jianhao Sun, Jianming Zhao, Yang Lyu, Shengxin Wang, Wenyu Zhao, Anzhou Li, Xiaohui Ji, Anant\nAgarwal (2024) Geochemistry π: Automated machine learning python framework for tabular data. Geochemistry Geophysics\nGeosystems 25, e2023GC011324. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2023GC011324"}, "author_response": "REPLY TO REVIEWER #1\nREVIEWER’S COMMENTS IN BLACK\nAUTHORS’ RESPONSE IN RED\nMANUSCRIPT REVISION IN BLUE\nReviewer #1 (Remarks to the Author):\nReview of the manuscript \"Nonlinearity of the post-spinel transition and its expression in\nslabs and plumes worldwide\" by Dong et al. to Nature Communications.\nThis is a revision of the manuscript that was previously submitted to Nature Geoscience. I\nappreciate the authors' eXorts to improve the manuscript. However, the main issue\nremains unresolved. Consequently, I do not believe it is suitable for high-level journals\nsuch as Nature Communications.\nI previously raised concerns about the motivation for the current study. The authors have\nadded a paragraph to the introduction addressing this point. They claim that the\nexperimental temperatures in previous studies on the Clapeyron slope were significantly\nlower than those in the mantle transition zone (1600 - 2200 K), thereby questioning the\nreliability of predicting the Clapeyron slope in the mantle. However, this assertion is quite\nmisleading. As demonstrated in previous research, such as Chanyshev et al. (2022), the\nexperimental temperature range actually spans from 1200 to 2100 K.\nWe apologize for any confusion on this point; we have now clarified in the main text\n(L59) that:\n“Our new data span multiple phase boundaries in Mg SiO , with a particular\n2 4\nfocus on the previously-understudied triple point of wd-rw-bm+pe. Combining\nour LH-DAC dataset including the triple point with previous high-precision\nmulti-anvil (MA) press data, we used logistic regression and supervised learning\nmethods to identify the phase boundaries in Mg SiO under mantle transition\n2 4\nzone conditions and determine the nonlinearity of the post-spinel boundary.”\nThe particular utility of our new higher temperature dataset (to ~2600 K, instead of\n~2100 K in Chanyshev et al.) is that it significantly improves constraints on the location\nof the triple point at ~2200 K and thus the slopes and non-linearity of the phase\nboundaries that meet there, in addition to our new constraints on the post-spinel\nphase boundary directly. While the Chanyshev et al. dataset does span much of the\nMTZ temperature range, the phase boundary slopes at MTZ temperatures are much\nbetter constrained with the inclusion of our new data spanning a wider range of\ntemperatures.\nWe also added an additional SI figure to contrast the P-T coverage of our work (red)\nwith that of previous work (blue). The location of the triple point is indicated by a\nyellow star to emphasize its critical role in understanding the nonlinearity of the post-\nspinel transition:\nSI, adding Fig. S3:\nFig. S5. Comparison between the P–T ranges investigated in previous\nexperiments and in this work. The red arrow and symbols represent the\nexperimental P–T range covered in this study (1500 K to 2500 K), while the blue\narrow and symbols indicate the P–T range covered by previous studies,\nincluding Chanyshev et al. (2022) (19), which is limited to 1500–2100 K. The\nyellow star marks the triple point derived from our analysis, which plays a\ncritical role in constraining the shape and nonlinearity of the post-spinel\nboundary and is better determined with the addition of our new data to 2500 K.\nRegarding the thermodynamic model, I fully agree that classification models can provide\nreasonable (or sometimes even better) fittings compared to thermodynamic models. I also\nconcur that existing thermodynamic models in the literature may not accurately predict the\npost-spinel transition slope. However, this is because some thermodynamic parameters\nhave not been precisely determined in the literature. As the authors mentioned, the ΔS and\nΔV values in the literature are unreliable, thus impairing the thermodynamic model's ability\nto describe the post-spinel phase transition.\nGiven this, a more reasonable strategy would be to obtain more precise thermodynamic\nparameters using the current data, rather than constructing a classification model, which\nlacks physical meaning. A classification model is appropriate only when the physical\nconnections between the values are unknown. For the current case, we already know that\nthe phase boundary is determined by thermodynamics. Therefore, the task for\nexperimentalists should be to precisely determine the thermodynamic parameters, such\nas ΔS and ΔV.\nWe appreciate your comments regarding measurements of thermodynamic\nparameters such as ΔS and ΔV, and we agree that having more accurate values for\nthose parameters would be very helpful to determining the phase boundary and\nshould be a goal for our community. However, to our knowledge, these parameters\ncannot be reliably derived from our dataset or any others on the relevant phases at the\nrelevant conditions. For example, current experimental capabilities for calorimetric\nmeasurements of ΔS are not sucicient to resolve the nonlinearity of the post-spinel\nboundary. We have stated in the main text:\nMain text lines 43-46:\n“However, the empirical thermodynamic models rely on experimental data\ncollected at low temperatures, such as heat capacity (<500 K)20, 21, which, when\nextrapolated, produce inaccurate entropy and volume at the high temperatures\nrelevant to the mantle transition zone (1600–2200 K), thus compromising their\nreliability in predicting the Clapeyron slope.”\nFor this reason, we have chosen to rely instead on direct experimental observations of\nphase stability. The classification model serves as a complementary approach (to the\nthermodynamic model) to interpret phase relations over a broad temperature range,\nintegrating both new and existing data. While complete thermodynamic information\non all phases involved would indeed be ideal, our main goal is to estimate the\nnonlinearity of the post-spinel boundary and more importantly understand its\nimplications for mantle dynamics. We consider the quantification of the post-spinel\nnonlinearity from the classification model adequate for the purpose, and our use of\nthis model, combined with our larger dataset that extends to higher temperatures,\nprovides significantly better constraints on the nonlinearity of this phase boundary\nthan in any previous work\nAs I mentioned in my previous review, the main problem with previous studies is the limited\nprecision (or accuracy) in determining the P-T conditions of the phase boundaries. While\nthe authors have added more data points, these additions do not oXer an improvement\nover the previous studies. In fact, some of their data points, such as the orange symbols in\nthe wd and rw stability fields, contradict their results. Consequently, their findings do not\nprovide a better constraint on the Clapeyron slope compared to previous studies. The\nauthors have discussed the diXerence between precision and accuracy, arguing that multi-\nanvil experiments may have systematic errors in temperature readings. This might be true,\nbut following this logic, the authors should focus on obtaining more data points closer to\nthe phase boundaries to better constrain them. While I agree that data points far from the\nphase boundaries can help to some extent, \"direct observations\" near the phase\nboundaries provide the most precise constraints for the phase boundaries. We can easily\nimagine that IF there are only data points far from the phase boundaries, we cannot\nprecisely determine the phase boundaries. In contrast, data points (if they are correct) near\nthe phase boundaries can directly determine these boundaries with much higher accuracy. This is the\nfundamental principle of mathematics.\nWe agree that observations closer to the phase boundary provide the most direct\nconstraints, and fully recognize their importance, especially the contributions from\nthe in situ high-precision multi-anvil results. We also acknowledge that the laser-\nheated diamond anvil cell (DAC) technique used in our study has larger random errors\ncompared to large-volume press experiments under similar pressure-temperature\nconditions (but the relatively small number of data in Fig 2 in the wrong stability field\nare all within uncertainty (in P–T) of the phase boundaries, shown in Fig. S2). However,\nreaching the higher temperatures necessary to study the triple point is only possible\nwith the LH-DAC, and our study specifically aims to 1) provide experimental\nconstraints on the triple point as it provides critical information on the shape of the\npost-spinel boundary, and 2) combine the LH-DAC data including the triple point and\nthe high-precision MA data at lower temperature to determine the post-spinel\nboundary. We have added a paragraph to the main text discussing this issue and our\nexperimental limitations (L110–118):\n“Our new LH-DAC dataset covers a wider temperature range that extends beyond the\ntriple point of wd–rw–(bm+pe), which is critical for constraining the nonlinearity of the\npost-spinel boundary and is only possible in the LH-DAC. However, using the LH-DAC\nto achieve these higher temperatures comes with the trade-oc of lower precision (e.g.,\nin P and T) than in a multi-anvil press19. Here, we combine our new LH-DAC data with\nthe selected MA data from the literature, to take advantage of the higher-precision\nconstraints on the phase boundary at lower temperatures obtained in those studies\nwhile significantly improving constraints on the triple point, and thus the nonlinearity\nof the post-spinal boundary, through the addition of our higher temperature data.\nFuture work should aim to probe closer to the post-spinel boundary with improved\nexperimental setups that balance achieving high temperatures and measurement\nprecision.”\nDespite this limitation, we believe that our current methodology, which combines new\ntriple point data (between ~1500 and ~2700 K, by LH-DAC) with curated literature data\n(between ~1500 and ~2100 K, by MA) and advanced data analysis, provides a\ncomprehensive constraint on the nonlinearity of the post-spinel transition that was\nnot previously possible. We consider this work to be a significant step forward in\nunderstanding this complex but important transition in the mantle.\nREVIEWER’S COMMENTS IN BLACK\nAUTHORS’ RESPONSE IN RED\nMANUSCRIPT REVISION IN BLUE\nReviewer #2 (Remarks to the Author):\nThe authors have addressed all my concerns. It may be accepted for the publication in\nnature communications.\nThank you for your support in the publication of the revised manuscript.\nREVIEWER’S COMMENTS IN BLACK\nAUTHORS’ RESPONSE IN RED\nMANUSCRIPT REVISION IN BLUE\nReviewer #3 (Remarks to the Author):\nI am satisfied with the discussion of the limitations of the presented temperature model\nT660. I support the publication of the manuscript.\nThank you for acknowledging that we have addressed your concerns about the 660-km\ntemperature model. We appreciate your support in publishing the revised manuscript.\nREVIEWER’S COMMENTS IN BLACK\nAUTHORS’ RESPONSE IN RED\nMANUSCRIPT REVISION IN BLUE\nReviewer #4 (Remarks to the Author):\nDear Editor,\nThe Claperon slope at the 660 km discontinuity is an important question with significant\ngeophysical implications. Ideally, the problem can be resolved by covering the relevant\npressure-temperature space with high-density experimental points (many high quality\nexperiments precision and high accuracy). However, high-temperature and high-pressure\nexperiments are expensive and time-consuming, leaving the problem still unresolved.\nDong et al. provide new experimental results in this study and use a statistical/machine\nlearning approach to investigate the problem. The approach is promising and advances our\nunderstanding of the Claperon slope near 660 km depth, which might be non-linear.\nThe study provides statistical and machine learning evidence by:\n1) Compiling and filtering the experimental results to construct a self-consistent database.\nThis is important because although the accuracy of temperature is not good enough for\ndiamond anvil cells, the dataset is self-consistent in revealing the Claperon slope.\n2) A multi-class logistic regression is used, which is reasonable for the phase diagram's\nclassification problem.\nI evaluate the data, method, and results, in particular of the statistical and machine\nlearning parts. Given the data in the supplementary Table, I organized the data and\ninvestigated the problem independently using Geochemistry π Python Framework version\n0.6.1 (ZhangZhou and He et al. 2024 G Cubed). I tried several classification machine\nlearning methods. The results suggest non-linear phase boundaries are more consistent\nwith the available experiment data.\nThe results will generate broad interest in the experimental, geodynamic, and\nseismological fields. I recommend publication in Nature Communications after minor\nrevision.\nThank you for recognizing the novelty and importance of our work, and we are\nparticularly grateful for your e?orts to independently verify our conclusions. We\nappreciate the constructive feedback and address the suggested revisions as follows.\nI have two suggestions:\na) The authors must provide the codes and training/test data in Zenodo for a final review\nbefore acceptance. The code should open with the publication and be clearly annotated\nfor the dataset and code. In my test exercise using Geochemistry π Python Framework, I\nhave to work on the supplementary Table to get the training data.\nWe apologize for any inconvenience caused by the lack of access to the source code.\nWe have now organized and thoroughly annotated the code, which is included as a\nsupplementary file in this resubmission (‘dataset-s7.zip’). The code is designed for\ngeneral use in phase diagram construction, with a benchmark example of Mg SiO\n2 4\nprovided to reproduce the results presented in this manuscript. Auxiliary files,\nincluding a README.md with detailed instructions and the Python environment files\n(environment.yml and requirements.txt), are also provided to ensure ease of use and\nreproducibility. In addition, the code has been deposited in a GitHub repository,\n(MLPD: https://github.com/dong2j/MLPD.git) for wider access and archived on Zenodo\nfor version control (MLPD: https://doi.org/10.5281/zenodo.13988951).\nWe have added the description and link of the source code in the main text and in the\nsupplementary material of the manuscript:\nMain Text Line 312–318, adding:\n“Data and code availability statements\nAll data and codes used to construct the Mg SiO phase diagram in this study,\n2 4\nincluding detailed documentation, are available as Supplementary\nInformation. For long-term accessibility and version control, the code has also\nbeen deposited in a GitHub repository (https://github.com/dong2j/MLPD.git)\nand archived on Zenodo (https://doi.org/10.5281/zenodo.13988951).\nb) Support Vector Machines (SVM) are the classic tool for classification problems (Cortes\nand Vapnik, 1995) and, therefore, is usually the first algorithm considered for classification.\nWith kernel tricks, SVM can also achieve non-linear boundaries. The authors can consider\nmentioning SVM in the manuscript to acknowledge people's work on classification\nproblems using machine learning. I think it is redundant to present SVM, XGBoost results in\nthe manuscript. Logistic regression can justify the non-linear phase boundary problem.\nWe appreciate your additional context on classification algorithms. We have revised\nthe text to acknowledge methods such as SVM and XGBoost and to direct readers to\ngeneral ML resources such as scikit-learn as well as geochemistry-specific ML tools\nsuch as Geochemistry π:\nSI Lines 172–180, adding:\n“To construct the phase diagram, we used logistic regression, which is a well-\nsuited model for this study due to its simplicity and interpretability. While more\ncomplex classification algorithms such as Support Vector Machines (SVMs) and\nXGBoost are able to model nonlinear boundaries using kernel functions or\ndecision tree ensembles, they often produce unrealistic oscillations in the\nphase boundaries. This behavior is particularly problematic when working with\nlimited or noisy experimental data, as these classifiers often overfit the data to\nproduce frequent sign reversals in the phase boundary slope---such behavior\ncontradicts the thermodynamic expectation for solid-solid phase transitions. In\ncontrast, the result of the logistic classifier is more physically consistent with\nthe gradual, non-reversing changes in slope characteristic of solid-solid phase\ntransitions. Details on the basics and di?erences of several common\nclassification algorithms can be found in general machine learning resources\nsuch as ``scikit-learn'' (17) as well as geochemistry-specific machine learning\nresources such as ``Geochemistry π'' (18).”\nWe also agree that it is redundant to present results from SVMs and XGBoost for this\nstudy. We chose logistic regression for the following reasons:\n1. Interpretability: SVMs use kernel functions to transform data, while XGBoost\nrelies on ensembles of decision trees, both of which make interpretation\ndi?icult. In contrast, logistic regression is mathematically simple, and its\nfunctional form matches well the physical process expected in first-order\nphase transitions.\n2. Thermodynamic consistency: Our thermodynamic understanding suggests that\nsolid-solid phase boundaries should be nonlinear over a broad P-T space, with a\ngradual change in slope that does not exhibit abrupt sign reversals. However,\ncomplex models such as XGBoost and SVMs can produce boundaries with\nunrealistic swings, especially when data are limited or noisy (as in this high-\npressure experimental context). The simplicity of logistic regression minimizes\noverfitting, making it more suitable for this study.\nDataset S5a: Plume-fed oceanic hotspots accompanied by the post-spinel transntion\n(transition, typo)\nWe have fixed the typo in Dataset S5.\nFig.S1. I suggest putting the P-T phase boundary calculated by thermodynamic code\nHeFESTo (e.g., blue lines in Figure S1) as an equation here. The authors can compare the\nclassification results of the equation (HeFESTo model) with compiled experimental data\n(confusion matrix results). Furthermore, the authors can compare machine learning\nmodels (diherent degrees of polynomials) with experimental data. The above results can be\nput in Dataset S6. I think this can further justify the advancement brought by machine\nlearning.\nThank you for your suggestion to add the confusion matrices to demonstrate and\nfurther justify the progress made by machine learning. We have now included a\ncomparison of the confusion matrices between the logistic model and the\nthermodynamic code HeFESTo as SI Fig. S8 instead of adding the results to dataset S6.\nAs expected, the logistic inversion of the experimental data yields a higher\nclassification accuracy compared to the thermodynamic model of the post-spinel\ntransition of HeFESTo.\nSI Lines: 48–49, adding:\n“We also examined the confusion matrices and found that our logistic inversion\nof direct experimental observations provides higher classification accuracy\ncompared to the HeFESTo prediction at the post-spinel boundary. (Figure S8)”\nadding Figure S8 with a caption:\n“\nFig. S8. Comparison of normalized confusion matrices for phase diagram\nmodels using (a) the logistic classifier (this work) and (b) the thermodynamic\nmodel (HeFESTo), evaluated for the post-spinel boundary between rw and\nbm+pe. The diagonal elements represent correctly predicted phases, while the\no?-diagonal elements indicate misclassified phases. The logistic classifier\nachieves higher scores for both phases, with an accuracy of 79% for rw and 90%\nfor bm+pe, compared to 74% and 89% for HeFESTo, respectively.”\nReferences:\nCortes and Vapnik (1995) Support-vector networks. Machine learning, 20, 273–\n297. https://doi.org/10.1007/BF00994018\nJ ZhangZhou, Can He, Jianhao Sun, Jianming Zhao, Yang Lyu, Shengxin Wang, Wenyu Zhao,\nAnzhou Li, Xiaohui Ji, Anant Agarwal (2024) Geochemistry π: Automated machine learning\npython framework for tabular data. Geochemistry Geophysics Geosystems 25,\ne2023GC011324. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2023GC011324\n6"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Although I do not fully agree with the authors' points, I acknowledge that they have provided reasonable counterarguments.\nTherefore, I will not insist on rejecting it. I have no further comments.", "reviewer_4": "The authors have answered my questions. Data and codes are both available to readers. I think the manuscript is good to go.\nOne suggestion:\nDataset-S1: In the read me document, you can define t_err=temp-temp_ds. Otherwise, it is confusing at a first glance to see\nthe negative values of t_err in the worksheets in Dataset-S1.\n(Remarks on code availability)\nYes. There are README files for both code and data, which are enough for installation and run the code. See attached\nsupplementary material for the reproduction result."}, "author_response": "REPLY TO REVIEWER\nREVIEWER’S COMMENTS IN BLACK\nAUTHORS’ RESPONSE IN RED\nReviewer #1 (Remarks to the Author):\nAlthough I do not fully agree with the authors' points, I acknowledge that they have provided\nreasonable counterarguments. Therefore, I will not insist on rejecting it. I have no further\ncomments.\nWe appreciate your input and thank you for acknowledging our reasoning that\nsupports the conclusion!\nReviewer #4 (Remarks to the Author):\nThe authors have answered my questions. Data and codes are both available to readers. I\nthink the manuscript is good to go.\nOne suggestion:\nDataset-S1: In the read me document, you can define t_err=temp-temp_ds. Otherwise, it is\nconfusing at a first glance to see the negative values of t_err in the worksheets in Dataset-\nS1.\nWe fixed the missing definition in the README file.\nReviewer #4 (Remarks on code availability):\nYes. There are README files for both code and data, which are enough for installation and\nrun the code. See attached supplementary material for the reproduction result.\nWe appreciate your e@orts to replicate our work and your support for\npublication!\n1"}]

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