Arithmetic Power Geometry Descriptors for Interpretable Structural Analysis of Single-Cell PBMC Transcriptomes

Single-cell RNA sequencing produces high-dimensional molecular profiles, whereas compact cell-level summaries often reduce each transcriptome to one diversity or concentration statistic. We developed an Arithmetic Power Geometry (APG) representation that maps each nonnegative expression vector to normalized squared weights and then measures entropy, concentration, effective dimension, a bounded exponent-deformation curve, and integrated deformation. The method was evaluated on the official 10x Genomics Fresh 68k peripheral blood mononuclear cell dataset from Donor A, comprising 68,579 cells. APG descriptors were compared directly with Shannon entropy, Rényi-2 entropy, Simpson diversity, Gini inequality, Herfindahl concentration, participation ratio, and detected-gene count. Marker-supported scoring yielded confident provisional annotations for 56,903 cells across eight immune or blood-associated classes, while 11,676 cells were retained as uncertain. Programme-level APG was computed across nine curated immune and cellular programmes. In a balanced 6,400-cell internal benchmark with identical three-fold stratified splits, classical diversity descriptors achieved macro-F1 0.3027 ± 0.0058; adding gene-level APG increased macro-F1 to 0.3792 ± 0.0104. The combined classical, gene-level APG, and programme-level APG representation achieved 0.5299 ± 0.0037 macro-F1 and 0.5400 ± 0.0020 balanced accuracy. Programme APG maximum concentration, integrated deformation, and D(3) showed the largest between-class effect sizes (ϵ² = 0.563, 0.541, and 0.526, respectively). Under binomial count thinning to 20% of the original depth, normalized APG entropy, D(3), and integrated deformation retained Spearman correlations of 0.917, 0.878, and 0.893 with their full-depth values. All tested compact spaces nevertheless had negative silhouette coefficients, and the classification labels were derived computationally from marker modules. The results therefore establish a reproducible one-donor methodological benchmark rather than external biological or clinical validation. APG contributes a multi-exponent structural profile that complements, but does not replace, established single-cell representations and diversity measures.