Background. Sleep-stage EEG analysis conventionally relies on spectral and complexity measures. Whether the geometric organisation of neural dynamics provides complementary information remains untested. Here I test whether a topological recurrence descriptor derived from persistent homology indexes a dimension of sleep-stage EEG organisation that is statistically distinct from spectral and complexity measures. New method. K ₀ , a within-subject z-scored summary of persistent homology of time-delay embeddings (   ,   ) of single-channel frontal-midline (Fpz–Cz) EEG, was computed on 197 nights from 100 subjects (Sleep-EDF Expanded, a public polysomnography corpus). Mixed-effects models, binomial generalised linear models (GLMs), and leave-one-subject-out classification benchmarked K ₀ against band power, spectral entropy, permutation entropy, and Lempel–Ziv complexity. Results. K ₀ , varying systematically across sleep stages, is statistically distinct from spectral content (Pearson r   to   with log band powers; opposite rapid eye movement (REM)–Wake direction), with construct validity supported by independent-cohort replication, twofold lifespan attenuation, and within-subject benzodiazepine invariance. The REM–Wake separation is large (   standard deviation (SD);   ) and preserved across embeddings. Comparison with existing methods. A binomial GLM confirms K ₀ carries information statistically independent of band power, especially for REM vs active-ocular wake (ΔAkaike Information Criterion (AIC)   , odds ratio (OR)   ). Leave-one-subject-out area under the receiver operating characteristic curve (AUC) values are 0.79 ( K ₀ ) and 0.91 (band power); combining adds a small gain (ΔAUC   , paired-bootstrap   ). Conclusions. K ₀ is a topologically distinct, spectrally non-redundant descriptor of sleep-stage EEG organisation.