Rebuilding microbiome diversity theory on the closed simplex

Ecological diversity theory links diversity within local communities to diversity of higher-level ensembles, but this scale structure is largely absent from microbiome analysis. Alpha diversity is usually treated as a within-sample summary, whereas "beta diversity" often denotes pairwise dissimilarity and gamma diversity is rarely explicit. We restore the local-regional architecture for environmental, host-associated and longitudinal microbiomes and distinguish regional beta diversity from pairwise dissimilarity and predictor-associated compositional variation. To quantify these objects for sparse compositions, we introduce Hellinger-Riemann intrinsic coordinates (HRIC), a one-to-one, bounded normal-coordinate representation of the closed simplex that retains exact zeros. The same coordinates yield Simplex Hellinger alpha and gamma diversity, additive regional beta diversity, taxon contributions, pairwise dissimilarity and model-explained dispersion. Simulations established the correspondence between HRIC dispersion and the between-condition component of PERMANOVA. Across Arctic and North Atlantic communities, local diversity relative to each regional benchmark covaried similarly with vertical environmental gradients despite partly different taxon-level associations. In a randomized autologous faecal microbiota transplantation trial, recipients returned earlier towards their personal pre-transplant compositions, whereas the alpha-diversity difference was smaller and less precise. Explicit local and regional referents therefore connect diversity partitioning with compositional analysis across microbial systems.