Atom probe informed persistent homology analysis of precipitation topology and hardening

• Application of persistent homology enables quantitative analysis of precipitate structure evolution during ageing of maraging stainless steel, revealing a multi-stage three-dimensional topological transformation. • Topological descriptors correlate with mechanical property trends, with H2 topology (void-like 3D features) showing a strong correlation to precipitation hardening (R 0.9). • Wasserstein and bottleneck distances quantify topological changes in precipitate structures, linking them to observed hardening behaviour: the VR bottleneck distance shows a strong negative correlation with hardness difference, while the Alpha Wasserstein distance exhibits a strong positive correlation. • Topological descriptors reveal a non-monotonic relationship between spatial heterogeneity and peak hardening. • A unified topology-microstructure-strength framework is established, demonstrating the superior predictive potential of topological descriptors over conventional structural metrics.