ALMAGAL. VI. The spatial distribution of dense cores during the evolution of cluster-forming massive clumps

High-mass stars and star clusters form from the fragmentation of massive dense clumps driven by gravity, turbulence, and magnetic fields. The extent to which each of these agents impacts the fragmentation depending on the clump mass, density, and evolutionary stage is still largely unknown. The ALMA evolutionary study of high-mass protocluster formation in the GALaxy (ALMAGAL) project, with sim1000 clumps observed at ∼1000,au resolution, allows a statistically significant characterization of the fragmentation process over a large range of clump physical parameters and evolutionary stages. Our goal is to characterize where and how the dense cores revealed by ALMA are distributed in massive potentially cluster-forming clumps to trace how fragmentation is initially set and how it proceeds before gas dispersal due to stellar feedback. We characterized the spatial distribution of dense cores in the 514 ALMAGAL clumps that host at least four cores, using a set of quantitative descriptors that we evaluated against the clump bolometric luminosity-to-mass ratio, which we adopted as an indicator of the evolution of the system. We measured the separations between cores with the minimum spanning tree (MST) method, which we compared with the predictions of gravitational fragmentation from Jeans theory. We investigated whether cores have specific arrangements using the Q parameter or variations due to their masses with the mass segregation ratio, Λ_MSR. ALMAGAL cores are distributed throughout the entire area of the clump, usually arranged in elliptical groups with an axis ratio e _ although high values with e,≥,5 are also observed. We found a single characteristic core separation per clump in sim76% of cases, suggesting that multiple fragmentation lengths may be frequently present. Typical core separations are compatible with the clump-averaged thermal Jeans length, łambda^ th J . However, we found an additional population of cores, typical of low-fragmented and young clumps, which are on average more widely separated with l,≈,3 _ th J . By stacking the distributions of the core separations in clumps of similar evolutionary stage, we also found that the separation decreases on average from l in younger systems to l in more evolved ones. The ALMAGAL cores are typically distributed in fractal-type subclusters, while centrally concentrated patterns appear only at later stages, but we do not observe a progressive transition between these configurations with evolution. Finally, we also found 110 ALMAGAL systems with a signature of mass segregation, with an occurrence that increases with evolution.