<table><caption>**Table 16** Model evaluation metrics calculated using uma with MP-20 as the reference set after relaxation: Distribution, Diversity, and HHI. All models provide 2500 structures except Crystalformer (1000 structures). Submissions are organized into four categories separated by horizontal lines: (1) models that incorporate relaxation as part of their generation pipeline (MatterGen, PLaID++, WyFormer, WyFormer-DFT), (2) other generative models, (3) samples from real-world datasets that contribute to LeMat-Bulk (Alexandria, OQMD) as baselines, and (4) samples from the AFLOW dataset and MP-Exp structures. Best values are shown in bold, second-best values are underlined. Arrows indicate whether higher (↑) or lower (↓) values are better.</caption><thead><tr><th rowspan="2">Model</th><th colspan="3">Distribution</th><th colspan="4">Diversity</th><th colspan="2">HHI</th></tr><tr><th>JS ↓</th><th>MMD ↓</th><th>FID ↓</th><th>ElemDiv ↑</th><th>SGDiv ↑</th><th>SizeDiv ↑</th><th>SiteDiv ↑</th><th>Prod ↓</th><th>Res ↓</th></tr></thead><tbody><tr><td>MatterGen</td><td>0.43</td><td><strong>0.000</strong></td><td>0.10</td><td>0.64</td><td>0.30</td><td>0.24</td><td>12.80</td><td>3.51</td><td>2.72</td></tr><tr><td>PLaID++</td><td>0.48</td><td>0.049</td><td>0.31</td><td>0.68</td><td>0.28</td><td>0.21</td><td>6.01</td><td>5.23</td><td>3.36</td></tr><tr><td>WyFormer</td><td>0.42</td><td>0.002</td><td>0.35</td><td><strong>0.74</strong></td><td><strong>0.50</strong></td><td>0.27</td><td>24.27</td><td>3.52</td><td>2.71</td></tr><tr><td>WyFormer-DFT</td><td>0.42</td><td>0.002</td><td>0.30</td><td>0.73</td><td>0.49</td><td>0.27</td><td>23.57</td><td>3.49</td><td>2.74</td></tr><tr><td>ADI T</td><td>0.43</td><td><strong>0.000</strong></td><td><strong>0.02</strong></td><td>0.71</td><td>0.43</td><td>0.24</td><td>14.51</td><td><strong>3.41</strong></td><td>2.70</td></tr><tr><td>Crystal-GFN</td><td>0.56</td><td>0.066</td><td>1.32</td><td><strong>0.74</strong></td><td>0.12</td><td><strong>0.32</strong></td><td><strong>31.50</strong></td><td>3.45</td><td><strong>2.33</strong></td></tr><tr><td>Crystalformer</td><td>0.42</td><td>0.002</td><td>0.13</td><td>0.71</td><td>0.34</td><td><strong>0.31</strong></td><td>21.62</td><td>3.54</td><td>2.68</td></tr><tr><td>DiffCSP</td><td>0.44</td><td>0.003</td><td>0.26</td><td>0.73</td><td>0.34</td><td>0.25</td><td>14.42</td><td><strong>3.27</strong></td><td><strong>2.59</strong></td></tr><tr><td>DiffCSP++</td><td>0.42</td><td>0.001</td><td>0.09</td><td>0.72</td><td>0.50</td><td>0.27</td><td>20.90</td><td>3.55</td><td>2.77</td></tr><tr><td>LLaMat2</td><td><strong>0.42</strong></td><td>0.001</td><td><strong>0.05</strong></td><td>0.71</td><td>0.38</td><td>0.24</td><td>11.91</td><td>3.84</td><td>2.79</td></tr><tr><td>LLaMat3</td><td>0.44</td><td>0.007</td><td>0.68</td><td>0.72</td><td>0.17</td><td>0.27</td><td><strong>28.26</strong></td><td>3.48</td><td>2.93</td></tr><tr><td>SymmCD</td><td>0.42</td><td>0.002</td><td>0.27</td><td>0.73</td><td>0.48</td><td>0.27</td><td>20.84</td><td>3.51</td><td>2.68</td></tr><tr><td>Alexandria</td><td><strong>0.38</strong></td><td>0.004</td><td>0.91</td><td>0.71</td><td>0.50</td><td>0.26</td><td>13.50</td><td>4.02</td><td>3.33</td></tr><tr><td>OQMD</td><td>0.43</td><td>0.012</td><td>0.51</td><td>0.63</td><td>0.46</td><td>0.24</td><td>12.57</td><td>4.22</td><td>3.29</td></tr><tr><td>AFLOW</td><td>0.44</td><td>0.003</td><td>0.53</td><td>0.69</td><td><strong>0.50</strong></td><td>0.25</td><td>10.07</td><td>4.12</td><td>3.22</td></tr><tr><td>MP-Exp</td><td>0.49</td><td>0.013</td><td>0.12</td><td>0.72</td><td>0.68</td><td>0.28</td><td>54.79</td><td>2.78</td><td>2.27</td></tr></tbody></table>

## D Environmental and Sustainability Considerations

The application of generative models to materials discovery presents significant opportunities for advancing environmental sustainability goals. As global challenges related to climate change, resource depletion, and environmental degradation intensify, the need for novel materials with reduced environmental footprints becomes increasingly urgent. Generative approaches can accelerate the discovery of sustainable alternatives by explicitly incorporating environmental criteria into the design process.

One promising direction involves the targeted generation of materials with reduced reliance on critical or environmentally problematic elements. By conditioning generative models on compositional constraints that exclude toxic, rare, or environmentally harmful elements, researchers can guide exploration toward more sustainable regions of chemical space. Similarly, models can be trained to prioritize earth-abundant elements and avoid those associated with problematic extraction practices or geopolitical supply risks.

Energy-related applications represent another frontier where generative models could significantly impact sustainability outcomes. The discovery of more efficient catalysts for renewable energy production, improved battery materials for energy storage, and novel photovoltaic materials could accelerate the transition away from fossil fuels. By specifically targeting properties relevant to these applications, generative models can focus computational and experimental resources on high-impact sustainability domains.

Life-cycle considerations present a more complex but equally important target for integration with generative approaches. Ideally, materials should be designed not only for performance but also for recyclability, biodegradability, or other end-of-life scenarios that minimize environmental impact. Incorporating such considerations into generative frameworks remains challenging due to the complex, multi-faceted nature of life-cycle assessment, but represents a crucial direction for future research.

The computational efficiency of generative processes themselves also warrants consideration from a sustainability perspective. As models grow in complexity and scale, their energy consumption and carbon footprint increase accordingly. Developing more efficient architectures, training procedures, and sampling approaches could reduce the environmental impact of the discovery process itself, aligning computational means with environmental ends. This consideration becomes particularly important as generative approaches scale to industrial applications and high-throughput discovery platforms.

The ultimate success of generative approaches in advancing sustainability will depend not only on technical
capabilities but also on intentional alignment with environmental objectives. By explicitly incorporating sustainability metrics into reward functions, objective functions, and evaluation criteria, the materials community can ensure that generative models contribute to addressing environmental challenges rather than merely accelerating traditional discovery paradigms without regard for sustainability implications.