Sleep as Geometric Flow on a Compact Manifold: Closed-Form Torus Features Reveal a Continuous Pentagon in 280,000 Epochs

A geometric reframing of sleep staging. We propose that sleep is usefully described not as a classification problem over discrete stages but as geometric flow on a compact manifold of neural states, with the Clifford torus (T² ⊂ S³ ⊂ ℝ⁴) serving as a natural closed-form projection onto which this flow becomes observable. Using Takens delay embedding (d = 4, τ = 25 at 100 Hz) followed by a parameter-free projection onto the Clifford torus, we extract 8 interpretable geometric features per 30-second epoch across 8 PSG and wearable datasets totalling 280,000 epochs from 299 subjects. Pre-registered core findings (Appendix A). (1) A universal sleep gradient N3 < N2 < REM < N1 < W on the winding-number axis ω₁, replicated in 3/3 PSG datasets and classifier-independent. (2) ω₁ tracks homeostatic sleep pressure (Process S), with epoch-level anti-correlation r = −0.427 confirmed in 142/146 HMC subjects. (3) REM emerges as a geometric superposition REM = 0.38·Wake + 0.62·N3 (cos = 0.96), with the coefficient β pre-registered to be stable under noise — confirmed in feature space (β within 0.515, 0.518 up to 5× σ noise) and in raw-signal noise (β = 0.72 → 0.65 across SNR = ∞ → 0 dB). Exploratory findings. β is a probe-dependent observable (values ≈ 0.55–0.66 under multi-modal / multi-scale measurement, collapsing to ≈ 0.22 with single-channel EEG). Candidate biomarker for REM behaviour disorder (CAP cohort, Cohen's d ≈ 2 vs matched healthy). Three interpretable PCA coordinates of sleep (depth, exploration, external-vs-internal consciousness; 99% cumulative variance). Classification κ = 0.652 (combined 325 features; below deep-learning SOTA but with full interpretability and linear-time complexity). Honest limits. Persistent homology on aggregated EEG does not yield β₁ = 2 (expected, given coarse-graining over ~10⁸ neurons; Gardner 2022 Nature confirms toroidal topology at single-neuron resolution; Bressloff 2002 shows V1 orientation space = T²). The torus is therefore a natural lens, not the intrinsic topology of sleep-EEG-as-measured. A pharmacological validation (Prediction 3, Appendix B) is pre-registered but pending dataset access. Ethical considerations including diagnostic equity across measurement modalities are discussed in §7. Reproducibility. Source markdown, pre-registration records, pandoc compile header, individual high-resolution figure files, and an alternative "clinical-softened" abstract variant are bundled in the supplementary ZIP. The full analysis code will be released at a GitHub companion repository (URL to be announced).